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Sores os : ; ~~ : . ; = =e <<. eymrases. eae g oo seis istetentes =S= =: Sima size: rt > em itt? th tol De He gt if , ite fp as ' an be t ; 240d Re : Mai ae ; ; peek hy 4 ie ‘ia ud 2 se ue F bP tity My Sint sea Ps thie HER abides Hu A ey thik . ie ae ii ia Hen ph eater Bet eS ge ae Fe 7 aed a in 5 matinee ae a” 2. a. n o : did — aS = - ee — WACOM ILI NENENED CIDA OLIN OIONIOIOLNOLIOKS SIAN 3 We Lic DS, ° eae é> e | nf ans (vs D, . eee] Q rt ws ei Labuare & mt = ; 5, m0 i ya % = Dae 9 2) mad vs a w eds o% iy) Lal Cro ue 8 oe = Ss r’ o A femme (% & 0 ae 1% SL E> CG & (a) rewt Votaed > lo fomurs va é 6 8 O fame 9) 6 2 wae | CX g Sg NM MMMMMAMMAAMAaMBMwABMAMMMs = a eee ee denen — = = LETS —— a ye GO iret my ay ee ee tees SST ES an I a te Sere vo > - a a a ne ae . ee me cr RS — = = comes : = . : ; . - i ee + Ba — > aoe oan Si - * “= “ ~ . “ o . . . m . = . . - : fs om ’ ‘ea : «Ss . J - 2 ee Po en ee ‘. ry > nan 4 . , fi 2i* eh . Pal Sn : ‘ : . _- + . . vs . s- * ei 7} a LONDON, EDINBURGH, anp DUBLIN PHILOSOPHICAL MAGAZINE AND * JOURNAL OF SCIENCE. ' GONDUCTED BY SIR OLIVER JOSEPH LODGE, D.Sc., LL.D., F.R.S. SIR JOSEPH JOHN THOMSON, M.A., Sc:D., Tk IDs Thabo JOHN JOLY, M.A., D.Sc., F.B.S., EGS. GEORGE CAREY FOSTER, B.A., LL.D., F.R.S. AND WILLIAM FRANCIS, F.1.8. ‘Nec aranearum sane textus ideo melior quia ex se fila gignunt, nec noste? vilior quia ex alienis libamus ut apes.” Just. Lips. Polit. lib.i. cap. 1. Not. VOL. SKE SUNTH: SEP ES: wy alge a LONDON: TAYLOR AND FRANCIS, RED LION COURT, FLEET STREET. SOLD BY SIMPKIN, MARSHALL, HAMILTON, KENT, AND CO., LD. SMITH AND SON, GLASGOW ;— HODGES, FIGGIS, AND CO., DUBLIN; VEUVE J, BOYVEAU, PARIS ;——-AND ASHER AND CO,, BNRLIN. Ui 3 OQ “Meditationis est perscrutari occulta; contemplationis est admirari perspicua .... Admiratio generat queestionem, questio investigationem, inyestigatio inventionem.”— Hugo de S. Victore. “ Cur spirent venti, cur terra dehiscat, Cur mare turgescat, pelago cur tantus amaror, Cur caput obscura Phoebus ferrugine condat, Quid toties diros cogat flagrare cometas, Quid pariat nubes, veniant cur fulmina ccelo, Quo micet igne Iris, superos quis conciat orbes Tam yario motu.” J. B. Pinelli ad Mazonium. ALERE FLAMMAM, CONTENTS OF VOL. XXI. (SIXTH SERIES). NUMBER CXXI.—JANUARY 1911. Dr. James G. Gray and Mr. Alexander D. Ross on Magnetic 2. SS TUITION SI AGEs © RMDP ie aCe ore eon RUPEES SN SP Prof. M. 8. Smoluchowski: Some Remarks on Conduction of iene nrouph, aretied Gases. oa s/.)3, 2: sa; biiele 2). a este oe Mr. 8. B. Mclaren on Hamilton’s Equations and the Parti- tion of Energy between Matter and Radiation .......... Dr. A. 8. Eve on the Ionization of the Atmosphere due to Peuorenive Mather) Wee yh) eel Btls, iat ah Lyles Mr. Hugh Mitchell on the Ratios which the Amounts of Substances in Radioactive Equilibrium bear to one another. Mr. Norman Campbell on a Method of Determining Capacities fipveasmmements of Tomization’), .....:. 723 Mr. C. VY. Raman on Photographs of Vibration Curves (Plate VT.) 2050 5 SUS a See See ee A er Mr. C. V. Raman on the Photometric Measurement of the Obliguity Factor of Diffraction. (Plate VIT.).......... Mr. Norman Campbell on Relativity and the Conservation of Pome tum: +. a ie teh eee he eta tao ate te "chor Prof. W. M. Thornton on Thunderbolts’). 02>.... 23s Dr. William Wilson on the Discharge of Positive Electricity prom tLot, Bodies -.sc/) sis. 72 ee. ye ee eee ace ior Mr. G. H. Livens on the Initial Accelerated Motion of a Perfectly Conducting Electrified Sphere .............. Prof. Charles G. Barkla: Note on the Energy of Scattered REP AMIALTON. L). 0). ca ee tee ee het en eee Miss Ruth Pirret and Mr. Frederick Soddy on the Ratio between Uranium and Radium in Minerals ............ Dr. Charles A. Sadler and Mr. Alfred J. Steven cn an Apparent Softening of Réntgen Rays in Transmission phron et WPaGher toi oles eo I oe ee CONTENTS OF VOL. XXI.——-SIXTH SERIES. Prof. E. Rutherford on the Seattering of a and 6 Particles by Mattemand the Structure) of the! Atom .3).4 0). )60.-.- Profs. H. Rubens and O. von Baeyer on Extremely Long Waves emitted by the Quartz Mercury Lamp . Proceedings of the Geological Society :— Mr. T. O. Bosworth on the Keuper Marls around Charn- wood Forest aitaxhet (wire) el ofa) Well eiive) beth eis) 1B) c@ nate oii) OPM repre Hew @)n Shia deh) © 16) 18/18, NUMBER CXXVI.—JUNE. Lord Rayleigh on the Motion of Solid Bodies through Viscous RIOR R ae Sa scape) Src ses a(S Micra, er cy alate nN reba ay chee oiceka ahah enema NN Prof. H. A. Wilson on the Velocity of the Ions of Alkali SomaPOUrS TO. WITNESS Sc ye orca k yoe alatanas acces) elteatate Prof. H. A. Wilson on the Number of Electrons in the Atom Dr. R. W. Boyle on the Behaviour of Radium Emanation at omg Pemiperavurest ss 0.45 titers «oye tee fey sith eat gees Dr. W. F. G. Swann on the Longitudinal and Transverse iscsnoie ai WLCCEOM i aie caper Medes tn) aye ara pebie «mi el yey allel. Mr. John R. Airey on the Oscillations of Chains and their Relation to Bessel and Neumann Functions............ Prof. James E. Ives: An Approximate Theory of an Elastic String vibrating, in its fundamental mode, in a Viscous Medium eee eo Fo coe ee se oe ew eee wr ese er eee eee ere eo ew ew wm woe ow Mr. H. Bateman on some Problems in the Theory of MOAI 4.35 sins cy OM sel NEEM Nu. foe LEAT L Prof. R. A. Millikan and Mr. Harvey Fletcher on the Question ot Valency in Gaseous; lomiadiieny 7 clio sehen el an Mr. Arnold L. Fletcher on the Radioactivity of some Igneous Rocks trom} Antarctic Resions) 02.5) 20 oe... eee ie Mr. Andrew Stephenson on Water Waves as Asymmetric Oscillations, and on the Stability of Free Wave-Trains .. Index @) ©) 6) 0) ce) 0) .e) uly 0) eo! 0) ere) O40 1 )4) 00 2) 8) cei ieia)) Belle, e) O1@ elie a) em e@) & © ee @ else @ «@ « Vil Page 669 689 6V5 VIL. PLATES. . Illustrative of Sir J. J. Thomsun’s Paper on Rays of Positive Electricity. . Illustrative of Prof. R. W. Wood and Mr. J. Franck’s Paper on the Transformation of a Resonance Spectrum into a Band Spectrum by the Presence of Helium. . Illustrative of Prof. Wood’s Paper on the Destruction of the Fluorescence of Iodine and Bromine Vapour by other Gases, and of Mr. J. Franck and Prof. Wood’s on the Influence upon the #luorescence of Iodine and Mercury of Gases with different Affinities for Electrons. . Illustrative of Prof. A. A. Michelson’s Paper on Metallic Colouring in Birds and Insects. . Illustrative of Mr. A. M. Tyndall’s Paper on the NSE from an Electrified Point. . Illustrative of Mr. C. V. Raman’s Paper on Photographs of Vibration Curves. Illustrative of Mr. C. V. Raman’s Paper on the Photometric Measurement of the Obliquity Factor of Diffraction. THE LONDON, EDINBURGH, ann DUBLIN PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCIENCE: [SIXTH SERIES.] JANUARY 1911. I. On Magnetic Testing. By James G. Gray, D.Se., FR S.E., Lecturer on Physics in the University of Glasgow, and ALEXANDER D. Ross, M.A., B.Sc., FR S.E., Lecturer on Natural Philosophy in the Ur miner siti y of Glasgow 28 a methods in use for the experimental determination of the magnetic constants of materials have been brought to a high state of perfection, and are thoroughly understood by the majority of the investigators engaged in carrying out research work in magnetism. But the pre- cautions which must be taken in order that the results yielded by application of these methods may be accurate, are, however, not so well understood. It is proposed in the present paper to bring forward some facts bearing on this point which have been brought to light by recent work earried out in the Physical Institute of the University of Glasgow. It is well known, and is pointed out in all the text-books, that the magnetic properties of a test specimen depend on its previous magnetic history. In this connexion Ewing { has drawn attention to the very interesting fact that it is possible to put a specimen—say, a long rod cf iron having uniformity of physical structure throughout its length—through a series of magnetic operations such that it is left devoid of residual * Communicated by the Authors. + J.A. Ewing, “ Experimental Researches in Magnetism,” Trans. Roy. Soc. clxxvi. p. 5338. Phil. Mag. 8. 6. Vol. 21. No. 121. Jan. 1911. B 2 Dr. J. G. Gray and Mr. A. D. Ross magnetism and ina field of zero intensity, but with vastly different magnetic properties for the two ways in which it may be magnetized in the direction of its length. If we denote the ends of the specimen by A and B. then the per- meability of the material in the direction AB differs from that in the direction BA for magnetizing forces which are less than the maximum force employed in the operations. In other words, the chains of molecular magnets, in the closedness of which the external neutrality of the bar consists, are left by the process referred to in an arrangement which is unequally affected by a magnetic field according as it is applied from A to B or from Bto A. This difference of response to magnetic force in the two directions is due to the magnetic treatment which the specimen has sustained, and is ~ not removed by annealing unless the temperature is raised above the so-called critical temperature of the material. For this reason it is usual to divest a specimen of its previous magnetic history before submitting it to test. This is accom- plished by exposing the specimen to the action ofan alternating magnetic field which diminishes gradually from a high value to z2ro- But it is not generally known that purely thermal treat- ment, such as that involved in the process of annealing a specimen, no matter what temperature is reached in the process, develops in the specimen a peculiar state which renders additional precautions necessary. This fact has been disclosed in experiments carried out by the authors, which are described in papers * read before the Royal Society of Edinburgh. It is there shown that an alteration of the tem- perature produces what we have called a “ Sensitive State” of the material. A magnetization curve yielded by the specimen following upon any change, or cycle of changes, in the temperature, is not what we may here call the Hate magnetization curve of the material. This will be clear from fig. 1, which shows the results obtained on testing y,at room temperature by the magnetometric method, a specimen of hard steel which had been heated to 900° C. and allowed to cool slowly. Previous to being heated the specimen had been rendered neutral, and during the process of heating and cooling had been exposed to no magnetizing forces. On submitting the specimen, in the condition brought about by the thermal treatment described, to the action of a mag~ netizing field which increased gradually from 0 to +9 C&s. * J. G. Gray and A. D. Ross, “On a “Sensitive State induced in Magnetic Materials by Thermal Treatment, ” Proc. Roy. Soc, Edin. xxviii. pp. "239 & 615 (1908). on Magnetic Testing. 3 units, then diminished gradually from +9 c¢.a.s. units to —9 ¢.G.s. units, and finally increased in like manner from —9 c.a.s. units to +9 c.4.s. units, the intensity of magneti- zation Followed the curve OA BCDE. 5 Ale ie e° — I in c.GS.uNITS. It will be seen that the point E does not coincide with the point A, and that the vertical distance of A above O is much greater than the vertical distance of C below O. The specimen was now thoroughly demagnetized, and the process repeated, when the curve O A’B’C'D’H’ was obtained. It is important to notice that the point HE’ coincides with A’; the curve is now closed. Further, an inspection of the figure shows that the hysteresis loop is now symmetrical. Cyclic alteration of the magnetizing force between limits H=+9 o.@.s. units resulted in the curve being repeated over and over again. O A’ is the true magnetization curve for the specimen in the annealed condition; it is this curve, and not the curve O A, that is characteristic of the magnetic properties of the material. Table I., which is taken trom the figure, shows how serious are the errors involved in neglecting to render the specimen neutral before proceeding to carry out the tests. It will be B2 a 4 Dr. J. G. Gray and Mr. A. D. Ross seen that the necessity for rendering the specimen neutral is brought about by the thermal treatment, and not by any magnetic treatment to which the specimen has been exposed. (ess or ne by Value of I in c.a.s. units. Value of H in : C.G.8. units. | Given by specimen Given by specimen | after thermal treatment.| in neutral condition. | ho he) iw — Ov 4 67 47 6 136 100 8 | 220 173 9 | 275 ; 206 So far, in describing the phenomena of this “ Sensitive State,’ attention has been confined to the case of steel annealed at 900° C. It is not necessary, however, for the production of this state that the temperature change should be great : even a moderate alteration in the temperature is sufficient to bring about the effect to a marked extent. Moreover, it is not necessary that the alteration should consist of a step up in temperature followed by a step down of the same amount. Experiment has shown that if the temperature of a specimen, which is initially neutral, altered from any one temperature to a second, the specimen assumes the ‘ Sensitive State” at the second temperature. The further fact was established that once the specimen was rendered neutral it remained neutral provided that the temperature was maintained constant. In other words, the “ Sensitive State ” cannot be induced by prolonged exposure of the specimen to either a high or a low temperature ; it is brought on in the process of heating or cooling the specimen. peo eace may here be made fon the interesting fact that once the “ Sensitive State” has been induced ina specimen, it cannot be got rid of except by submitting the specimen to the demagnetizing process. Thus a specimen of aunealed steel was tested after having been laid aside for a period of several weeks, and gave results precisely similar to those yielded by it when in the freshly annealed condition. Further experi- ments revealed the fact, which is of great importance in connexion with what immediately follows, that once a spe- cimen of steel in the sensitive condition has been exposed to a magnetizing force in one direction, the ‘ Sensitive State ” on Magnetic Testing. 3 cannot be got rid of, or even affected, by any magnetic operation or process of operations whatever, unless the sign of the fieldis changed. Thusif onapplying a field of intensity + H the corresponding intensity of magnetization is J, eyclic variation of the field strength between the limits + H and H—h, where h may have any positive value ranging from 0 to H, does not result in the value of I corresponding to the field + H being diminished. If, however, h becomes greater than H, in other words if the field is reversed, then the value of I corresponding to +H is reduced. The effect of applying even an extremely small negative field is to bring about a perceptible diminution in the value of I, that is to reduce, more or less, the ‘‘ Sensitive State.” A further series of experiments showed that the “‘ Sensitive State ” induced by equal augmentations or diminutions of the temperature are of widely different amounts depending on the position of the temperature ranges on the temperature scale. ven a small increment of only 25° C. in the neigh- bourhood of 180° C. produced, in the case of the steel tested, a percentage ‘* Sensitive State” of 10 for a field of 10 c.a.s. units. Such facts have a very important bearing on magnetic testing. For example, suppose that it is required to test the magnetic properties of a specimen at 200° C. We shall consider the specimen to be initially neutral and at room temperature. It may now be heated to 200° C., and tested at that temperature. Or again it might be heated to say 250° C., then cooled to 200° C., and tested in like manner at that latter temperature. From what has been said it will be evident that if the specimen has not been rendered neutral at 200° C. in each of the two cases previous to exami- nation, the results of the two tests will not in general be in agreement. Or suppose that it is desired to test the magnetic properties of a specimen at temperatures lying between room tempe- rature and the critical temperature of the material. The specimen might be heated up above the critical temperature and allowed to cool very slowly, tests being carried out at different stages of the process. This method has been adopted by many experimenters. If the specimen were com- posed of steel, it would be heated up to the neighbourhood of 900° C. and then cooled. A test carried out at 800° C., without the specimen having been rendered neutral, would give a certain magnetization curve. The specimen might now be submitted to the process of reversals, allowed to cool to 700° C., and then tested in like manner. This procedure, 6 Dr. J. G. Gray and Mr. A. D. Ross if carried out at intervals during the cooling process, would result in a series of magnetization curves being obtained, from which it might be supposed that the permeability of the material for all temperatures lying between room temperature and 800° C. might be deduced. The properties so obtained, however, depend not only on the temperature, but on the particular procedure adopted. The fact that the specimen was tested at 800° C. determines to some extent the mag- netization curve which is yielded at 700° C.; in other words, if the specimen had been heated to 900° C., then cooled to 700° C. and tested at that temperature, the magnetization curve obtained would be distinctly different to that yielded by the specimen at 700° C. following upon the test at 800° C. Again, if the specimen were tested at 850° C., 750° C., and so on, a new set of curves would result. From the foregoing discussion it will now be evident that the permeability of the specimen at any one temperature for any one particular field as determined from the second set of curves would not in general agree with that obtained from the former Set If, however, the specimen were submitted to a process ‘of reversals just previous to the tests being carried out at any one temperature, the results would be in strict agreement. For example, if this precaution is observed a magnetization curve obtained say at 400° C. following upon tests carried out at 800° C., 700° C., and so on, would coincide with one obtained at 400° C. without intermediate testing. The magnitude of the errorsinvolved in testing a specimen of annealed steel by the magnetometric method without pre- viously subjecting it to the process of reversals has already been pointed out. It is now proposed to investigate the nature of the errors which would be introduced into the results yielded by a like specimen when tested by Rowland’s ‘¢ Method of Reversals.”’ In this method the magnetizing coil is wound with a secondary coil which is connected up in series with a ballistic galvanometer. Experiment has shown that if the specimen has been divested of its magnetic history in the © manner previously described, then, on reversing in turn in the magnetizing coil each of a series of currents of increasing magnitude, the inductions calculated from the corresponding kicks of the galvanometer give, when plotted against the magnetizing forces in the coil, a curve which coincides with the true magnetization curve for the specimen; that is, the eurve which would be obtained if the specimen were subjected to the process of reversals previous to the establishment of on Magnetic Testing. — 7 each ofthe currents in the coil. This will be seen from fig. 2, which exhibits some results obtained on testing a specimen of steel. OABCD is the Se Eee curve for the material in the neutral condition. If now, starting with the specimen in this condition, the strength of the field is changed from 0 to Hy, the indoctien becomes B, ; that is te say, the point A on the curve is arrived at. Now let the current in the ere coil be reversed. The induction changes from +}, to —B,; the kick of a ballistic galvano- meter connected to a secondary coil wound on the specimen would be proportional to 2B,. If the field is once more reversed the induction becomes + B,; that is, after the double reversal the point A is once more reached. Let now the field be gradually increased from +H, to + Hy, and the induction in the specimen will follow the curve from A to B. If after arriving at the point B the field is changed from +H, to —H,, the induction changes from +B, to —B, ; the kick of the galvanometer is therefore proportion: al to 2By. Following this procedure the complete curve may be traced out. 8 Dr. J. G. Gray and Mr. A. D. Ross It seems necessary to add that this only holds for a series of fields which are in ascending magnitude, and it is essential that the specimen be initially neutral. For the method to yield accurate results the hysteresis loop must be not only closed but symmetrical. If, however, the neutral condition of the specimen has been interfered with by magnetic or thermal operations, the method is not applicable. This will be apparent from fig. 3, which is intended to show the results Fig. 3. InpuctTion. edel +H / D Maenetisine Force. 2 obtained on applying the method to the examination of a specimen which had been exposed to thermal treatment. On establishing the field +H, the induction became +B, ; on reversing the field the induction became — By, and B, differs widely from B,. The change of induction is B,+B,, and the kick of the galvanometer is proportional to this change, and hence proportional neither to 2B, nor to 2B,. The method of reversals is therefore not applicable unless the specimen is in the strictly cyclic condition. On once more — reversing the field and increasing it to + Hy, the induction followed the curve A’D’C. If the field +H, had been applied in the first place, a widely different value of the induction would have been obtained. Thus the magneti- zation curve yielded by this method, when applied to a specimen in the condition brought about by thermal treatment, depends largely upon the procedure. Table Il. shows the results obtained on starting and. on Magnetic Testing. 9 reversing a number of times, a,magnetizing field of 11 c.a.s. units in a solenoid containing a specimen of steel in the freshly annealed condition. It will be seen that the change of induction brought about by the first reversal 1s 7572 C.«.s. units, whereas that brought about by the tenth is 6248 c.a.s. units. Hence the induction calculated from the first throw of a ballistic galvanometer connected to a secondary coil wound on the helix would be over 20 per cent. in excess of that which is characteristic of the material when in the cyclic condition. Tasue II. Value of Hin Induction in eee C.G.S. units. c.G.s. units. oa ey +11 4186 579 — ll 3386 6859 | +11 3473 6746 —l1 3273 6559 +11 3286 6484 —ll 3198 6396 +11 3198 6359 —ll 3161 6285 +11 3126 6250 —ll1 3124 6248 +11 3124 The table shows that the results of tests which have been carried out on annealed steel are very approximately correct, even though the specimen had not been submitted to the process of reversals, provided that the magnetizing current has been reversed nine or ten times previous to the galvano- meter-throw being observed. A magnetization curve so obtained would, however, lie slightly above the curve which is characteristic of the material when in the truly neutral condition. Fig. 4 shows the magnitude of the “Sensitive State” induced in aspecimen of steel by cooling from room tempe- rature to that of liquid air boiling under normal conditions (about —190° C.). Curve IJ. is the magnetization curve for the material at room temperature when in the neutral condition. This curve having been obtained, the specimen was subjected to the neutralizing process and brought to the 10 On Magnetic Testing. temperature of liquid air. On being tested at this temper- ature it gave results which are shown in curve II. Finally Fie. 4. te) 300 Po fo} te) 3 [in c.¢.s. UNITS. o 0 4 8 fz H in c.6.5. units. the specimen was submitted once more to a process of re- versals and retested, care being taken to keep the temperature at —190° C. throughout. The results shown in curve III. were then obtained. Curves I. and IIT. are the true magnetization curves for the material at room temperature and at —190° C. respectively; it is these curves, and not curves I. and II., which must be employed in contrasting the magnetic pr -operties of the materials at the two temperatures. If, for any reason, it is desired to obtain curve IL., it is evident that a method of experimenting must be employ ed which does not involve reversal of the magnetizing force. Tests carried out by the method of reversals upon a specimen in the condition brought about by the cooling alone would yield neither curve II. nor curve IIT. In carrying out the tests which furnish the magnetization curve for the specimen at the temperature of liquid air, it is necessary that not only the tests, but also the neutralization process, should be carried out at that temperature. This is Conduction of Heat through Rarefied Gases. iE very important, as even a small temperature change is sufficient to produce quite an appreciable “‘ Sensitive State.” Research work in magnetism at temperatures other than ordinary room temperature is somewhat difficult to carry out; low temperature research is moreover very costly; and it seems to the authors of great importance that the special precautions necessary in such work should be widely known. Il. Some Lemarks on Conduction of Heat through Rarefied Gases. By Prof. M.S. Smotucuowskx1, Ph.D., LL.D* T has been shown, by researches published more than te n years agoT, that the apparent decrease of thermic con- ductivity of gases, with progressing rarefaction, is due not to a decrease of the coefficient of conductibility, but to a surface phenomenon, analogous to the sliding of gases discovered by Kundt and Warburg. There also the kinetic explication of this phenomenon has been given, which leads us to expect the existence of a very simple law of conduction for extreme rarefactions, the quantity of transmitted heat then being proportional to the gas pressure and independent of the thickness of the layer of gas. xperimental evidence of such relations has been given first in some experiments of C. F. Brush{, and recently in a careful investigation by F. Soddy and A.J. Berry$. The form of the law for extreme rarefactions being estab- lished, the question arises as to the value of the constant of proportionality, or as these authors put it: the quantity of heat, referred to unit of hot surface, one degree of difference of temperature and 0:01 mm. of mercury pressure, which they designate by Q. I had found by a roughly approximative reasoning|| (assuming the molecules to be divided in three parts, moving parallel to the axes) that the flux of heat carried for one degree of difference of temperature ought to be of the * Part of a paper which will be published in the Bulletin de ?Acad., Cracovie, 1910. Communicated by the Author. + Smoluchowski, Ann. d. Phys. Ixiv. p. 101 (1898); Wien. Akad. Sitzgsber. cvii. p. 8304 (1898), evili. p. 5 (1899); Phil. Mag. xlvi. p. 199 (1898) ; Gehrcke, Ann. d. Phys. ii. p. 102 (1900). { Phil. Mag. xlv. p. 51 (1898). § Proc. Roy Soc. lxxxii. A. p. 254 (1910). i| Wien. Ahad. Stézgsber. evil. p. 828 (1898). 12 Prof. M. 8. Smoluchowski on Conduction of order of magnitude =. where p=density, s= specific heat, c=mean molecular velocity. This result applies to the case when every molecule assumes, by its impact on the solid wall, the vis viva corresponding to the temperature of the latter ; but it ought to be multiplied by ize if only a partial equalization of temperatures is taking place, according to the formula J 65=B(Ou— 0), &, @m, 3 denoting the temperatures of the wall, of the impinging and the emitted molecules. Messrs. Soddy and Berry use the same formula, with slight difference of notation, putting n HG apie ae where n=number of molecules per cm.? at 0°01 mm. pressure, N=number contained in one gram, H=molecular heat at constant volume, G=mean molecular velocity. Their experiments enabled them to determine the ratio of the observed transport of heat K to the calculated value Q for 11 gases, and from these numbers, ranging between 1°09 and 0°25, they attempt to draw conclusions about the factor which in the above has been accounted for by introduction of the coefficient @. These results, however, seemed rather strange, since only values inferior to unity could be expected. But when exact numbers are in question, the rough estimate referred to above is evidently insufficient, and an exact calculation ought to be substituted instead. Consider the gas contained between two parallel horizontal plates, the upper one of temperature 6, the lower one of temperature @, (supposed to be one degree lower). It is convenient then, instead. of making the above supposition about 8, to follow Maxwell’s supposition* that the surface of a solid acts as a partial reflector, by reversing the normal velocity component for the fraction (1—/) of the incident molecules, while the rest are “‘absorbed”? and emitted with the velocity distribution corresponding to the temperature of the wall. We suppose the gas to be so rarefied that the mean free path of the molecules is much greater than the distance of * Maxwell, Phil. Trans. clii. p. 231 (1879). Heat through Rarefied Gases. 13 the plates, and consequently the influence of the mutual encounters of the molecules to be negligibly small. Then the whole number of molecules in unit volume n mal be composed of four parts: Mate eg tals te YS ll hoy) Shey iL) V1Z.: m, molecules moving upwards with mean velocity ci, corresponding to temperature 6), 1 d9 ” downwards 9 a” rh) 9 ) 5 4, upwards with mean velocity ¢,, corresponding to temperature 0), » downwards ” ” ” ” ” These four kinds, each with velocities distributed according to Maxwell’s law, do not undergo any mutual influence, they are subjected only to the impacts on the walls. The number of impacts, for unit of time and surface, is given by if n denotes the number of molecules for unit of volume, moving only upwards or only downwards. Now considering the process at the lower plate, we see that the molecules » my are made up of the fraction (1—/) of the incident molecules n;' and of the fraction f of the whole number of molecules which are impinging on the lower plate, whence nye (Lf) m/e. +7 Gu 614-15 ¢3), » . « (2) and similarly Ro Gail faa Coty ils A Acne nica 4 (O:) By adding these two equations we vet an equation expressing the fact it no one-sided current will originate : Ny Cy + Nglg= Ny Cy + No’ Co. This equation and equation (8) and a similar one for the molecules moving in reverse direction take the form (14 — 24! )oy = (ng! — ng) cy M2= (1 —f) ng’ ny =(1—f)n, whence follows Ng Cg = Nyy é e ° e e (4) ny Ov tg=(1—f) nye, 14 Conduction of Heat through Rarefied Gases. The quantity of heat lost by the lower plate is ?ms 2ms Q= a/ ba [ Azn'Co + 61n,'¢)— Onto — O17, ¢; | = Sez (0, —6,)fnyey. Now equations (1) and (4) give Denon nO repeat so we have __ 2fmns C105 N= Jor) e402" : ae 2€1C5 iB z If we put c= ee and f=1—£, we get finally ae eee ole) Je) ee 1) This is the exact value for the conduction of heat in highly rarefied gases ; we see that it is greater than the value ealcu- lated before, and all the numbers given by Messrs. Soddy ca T and _Berry Oey ok ought to be multiplied by the factor vis 5 ° . G = 9° 7236, which gives the series : A. Ne N, 0, CO, NO. C,H, CO, CHgamaamn G0 0:75 068 062 039 O56 052 052 049 037 O18 It proves that the coefficient 8 is never to be neglected, that is to say, that the heat interchange of the molecules on impact is always imperfect. ‘Tle order of gases suggests the rule that the interchange of energy is worse for the lighter molecules and for the diatomic and polyatomic ones in com- parison with the heavier and monatomic ones. The first part of this rule is to be explained by a simple mechanical reason- ing, showing the interchange of energy between colliding spheres to be the more imperfect, the greater the difference of their masses (here we have the molecules of the gas colliding with the heavy platinum molecules). Also the second part of this rule seems to be in accordance with other phenomena of conduction of heat, showing intramolecular energy being comparatively less disposed to equalization by impacts than energy of progressive motion. Lemberg University. [eto] Ill. Hamilton’s Equations and the Partition of Energy between Matter and Radiation. By 8S. B. McLaren, Assistant Lecturer in Mathematics in the University of « Birmingham. § 1. INTRODUCTION. N this paper two things are done. Maxwell’s theory of A partition is extended to forms of energy not quadratic in the velocities or momenta, and it is applied to the inter- action of matter and radiation. As to the form, it is enough if we are assured that for finite energy all momenta are finite. That condition is satisfied by the expressions for energy which arise in the electromagnetic theory of mass, by the formula c(p?+ ’c’)’, for example, which gives the energy of Lorentz’s deformable electron. There p is the momentum, mp the mass for an infinitesimal value of p, and c the velocity of light. For the rest (§ 4), I have tried to fill the gap which Larmor (Bakerian lecture) has remarked in the work of Jeans and Lorentz on radiation. With Jeans the radiation is confined to a finite space bounded by reflecting walls. Since within these there is no ordinary matter, all radiation falls at once into the normal vibrations proper to the space enclosed ; no redistribution of energy is possible, and the amount of energy to be assigned to each normal mode is fixed once for all by applying Fourier’s analysis to the original field of force arbitrarily given. : it Jeans can nevertheless draw conclusions as to the partition of energy, it is because he assumes that there is one dynamical system of which matter and ether are parts, that Maxwell’s statistical method can be applied to it, and that in any complete formula for the energy his expression for the radiation will form a part. It is such a formula I give here; but I have not reached it except by assuming the atomic structure of matter. My electron is an invariable distribution of charge free only to move as a whole. It then appears that the whole electromagnetic energy must be regarded as belonging to the radiation, excepting only a term which depends on the position of the electrons at any instant and would be the electrostatic energy if they were at rest. In this division nothing is left tor kinetic energy of the electrons, the system of equations cannot be brought to Hamilton’s form, and it does not seem that Maxwell’s statistical method can be * Communicated by the Author, 16 Mr. 8. B. Mclaren on Hamilton’s Equations and the applied. It may be, however, that there is true material energy as well as electromagnetic; and then Maxwell’s. reasoning can be resumed, with its old paradoxical con- clusion of equipartition. To many it has always seemed that the method of sta- tistics builds much, and to little purpose, on a very unsure foundation. I recall its postulates before deducing from them the results just described. § 2. THE GENERAL DynamicaL MEtTHov. (1) The laws of heat are dynamical. This is fundamental and at once raises the question of reversibility. Fire may freeze a kettle instead of boiling it, only the law of chances favours the more familiar process (Jeans). Let there be a dynamical system with a very large number of coordinates. The equations of motion are in Hamilton’s form : dq, ivak dp,, dH = == ‘Spree s SS = Bay (= Te eee n). dt ap) nds dq, The p’s represent momenta; the q’s are coordinates of position. Suppose the initial values are regarded as mere matter of chance. Then it may be shown (Jeans’ ‘ Dynamical Theory of Gases’) that in the vast majority of cases the distribution of energy and momentum is near what may be called the temperature distribution. It is therefore very long odds that, if we start with any arbitrary heat distribution, the end will be equality of temperature. With this we are brought naturally to the second assumption. (2) Any set of values of the coordinates and momenta is possible provided it is consistent with the constancy of energy. (ay All these configurations are of equal probability. For what is possible by (2) must have attached to it a definite measure of probability. ‘lhe successive configura- tions assumed by any one system are equally probable, and so far as we know they have only one thing in common— the fact that their energy is the same. Hence (3), and hence without further assumption the law of equipartition. The only condition to be satisfied by H has already been stated. ; The truth of (2) is certainly not at all obvious. When the number n is finite it may indeed be allowed very Partition of Energy between Matter and Radiation. 17 plausible. Although we cannot’ deal with single molecules, it seems impossible short of that to fix any limit for the initial distribution. This can hardly be interpreted in terms of dynamics, save by assuming that any values of the co- ordinates may occur. But (2) is certainly not always true when n is infinite. Thus the vortex theory of matter allows only such values of the variables as satisfy the condition of constant circulation; for that is the essence of the distinction between matter and what is not material. Iffor any reason (2) be abandoned, it need not be concluded that there is no state of temperature equilibrium. We infer rather that the theory: of heat depends upon properties of matter more special than the abstractly dynamical, an inference which is in itself made certain by the observed laws of radiation. These involve an absolute constant, a length in dimensions, which cannot enter into the purely dynamical scheme (Jeans). The conclusion thus forced upon us ought to be as welcome to the physicist as it is distasteful to the mathematician. A deduction of Wien’s law from Hamilton’s equation adds nothing to our knowledge of the nature of matter; an explanation of it may yet add much § 3. Tau Partition or ENERGY IN A HAMILTONIAN SYSTEM. The equations of motion are Gadi dp, Rae mie ae dt do ae a Ride (Geek Za e A c (1) The kinetic energy belonging to the rth degree of freedom I define as equal -to dH tn, =. e . . ° ° 2 ° ° 2 ee. (2) This agrees with the ordinary definition when /7 is quadratic. Notice also that if Z is the Lagrangian function, we have Cis Oe L= > ra— Hi, ° Py ° ° ° ° 3 ice ' (L+H) = ERD 2 ° e e * ° e (4) L is in the simple case equal to the difference of the kinetic and potential energy; H is equal to their sum. Thus the kinetic energy again appears in (4) as a sum of terms of the form (2). We write MING ==) Ay dois a On AG AGs\ ons AQnes cares CO) Bhi Mages. Gs Voli21. Nov 12k, fan. 1911. C 18 Mr. 8. B. McLaren on Hamilton’s Equations and the The product on the right-hand side of (5) is known to be a differential invariant. Let there be systems all satisfying (1) but starting from all initial conditions. Let the number originally having their coordinates within the limits dp,, dg,, and so on be dN. This distribution is invariant, in consequence of the invariant character of dN. Such of the systems as have originally identical values of H will always have the same value for it. Hach of the systems is on the average in a state of temperature equilibrium, and the distribution of energy proper to the temperature exists in it. This distribution of energy can be inferred by averaging over all the systems, since it exists in each. The theorem of equipartition requires \>.Z ax = (>. ao . , 62 (6) is to hoid for all values ol pans It will be enough to show ea oy Ws = \\og Ps ap = dp. dps. H>H,, and the difference H,— H, can be made infinitesimal. Y Suppose the curves =H, and [= H, are represented in the plane of coordinates ; p, may be measured along OX und ps along OY. Partition of Energy between Matter and Radiation. 19 The condition as to 7 ensures that these curves have no branches at infinity and are therefore closed. (7) becomes dH al ; {fe dudy ={{y age aoe a SS CS) The integration is to be taken all over the areas enclosed between H=H, and H=H,. In the figure given the curve H= H, has a double point. Now pee he tae. ( EN a Sih ier du dy =| vGeae dy ~{) Tay birdy 1) In (9) the right-hand side represents the difference of the integrals taken over the area enclosed by H= H, and over the area enclosed by H=H,. It is conveivable that the curves may consist of two or more separated portions and that H=H, may enclose H= H, in one portion and be enclosed by it in the other. aH a ene Ales du dy -\\z, (yf)dy dx -{f Hl dy dx = ( vq yds -|{ Hdy de 2 = H,A,— ( HI dy dx. . ° ° ° ° ° (10) yq and yp are the ordinates where PQ cuts H=H,. In (9) A, represents the area enclosed by the curve i H,. In exactly the same way di | {\ oT. dx dy = ths. {V JElOniGhe: he MOLL From (10) and (11) the truth of (8) follows. There is equal partition of the kinetic energy. Further important results can be proved. In (8) we may substitute for « and y any two of the 2n coordinates. Thus (at ax = |r Ban, bi agente dq, and (12) is true for all values of r and s. For the proof it is only necessary that H considered as a function of the g’s C 2 20 Mr. 8. B. McLaren on Hamilton's Equations and the should satisfy the same general conditions as before. Fora finite value of A all the g’s must he finite. (12) 1s evidently connected with the virial of Clausius, which is, in fact, equal to dH 5 q+ ——— , dg; Thus the virial is equally distributed and it is equal to the kinetic energy. In particular, if His a quadratic function the virial is identical with the potential energy and the potential and kinetic energies are equal. One more point may be noticed, SS a bore Ch ASS tN {\ S2ae ay =o. + — aS In (13) the integration is over the same area as in (8). For f 2 ut dy = || eee dy ~\| dae i on ide Jeg uy : (14) (aie My aemerian i Wed 2 iy | (Vey dee =f og Heide = | since Hy = dele Similarly the second term on the right of (14) vanishes and (13) 1s proved. For x and y in (14) we may substitute any two of the p's, or if the necessary conditions hold also for the q’s, any two of the 2n coordinates. So ?> een ie NO eo ae Oe fo “as - 0,2 a (pines. {a Goan ere. (18) (15) and (16) hold so long as 7 and s differ ; (17) and (18) hold for a! values. | Partition of Eneray between Matter and Radiation. 2 § 4. Disrrrpurion oF Enercy iy Marrer AND RADIATION. For the ether 1 & , we oe ap) e+ sme SMa ice DOs) Lede N u g@ and F are the scalar and vector potentials. wis the vector velocity of Hoa oly of density p. For any electron =| Roden. dss Dalinvalnedty fade) In (21) p is the true material momentum of the electron, supposed rigid we must remember. E’ is the whole electric . . e Je ° d ° . 1 intensity at any point of this moving electron. al indicates differentiation at a point moving with the electron. In (21) the integration is all over the volume of the electron H/ = —~— Fa [i Cun rs 022) _At the perfectly reflecting boundary there is the condition that the tangential component of 1 d¥ rhe ae is zero, and everywhere we have ° if dd MS Div F + c dt — 0. 2 e e ° ° (23) Now F and @¢ are clearly indeterminate to the extent that we may add to KF a term of the form Vo and to ¢ the term 1 dw me a The electric and magnetic forces remain unaltered, an but (23) requires wm can be chosen to ay this equation and to have any assigned value at the surface : we can therefore so choose w as to make the value of ¢@ always zero at the surface. Then the condition that E is normal at the surface requires also that F is normal. 41390 22 Mr. 8. B. McLaren on Hamilton’s Equations and the Suppose F=ehtVY-. - « 2. ees and DivF,=0, . . 3° then Div F = Vp ld or aT —o = Vb, < Sic (ete (26) or by (19) ~ Ld V7(¢ Pe = +4mp = 0 (27) (27) shows that (6+ 5 4) is the potential due toa | distribution of density p within the space. If we choose ay zero at the surface, then since ¢ is also zero, it follows that d+ 2 ae is the potential due to charge of density p and the charge it induces on the surface, also at the surface V7 and ~f are both zero, the former by (26), Again at the surface Fis normal. (24) shows that F, is also normal since yr=0. In view of (24) and the surface condition, F, can be expanded in a series of normal functions (Poincaré, Am. Journal of Maths,), Reser... iL where (=i 0) Div F, = 0 | (29) EF normal at surface | ( F2dV= 4or | dV is an element of the volume enclosed, the numbers x, are arranged in an ascending series. The last of (29) is required to determine F, absolutely. When ¢ and s differ we have the normal property {E,F,dV=0.... . , 2 It may be in certain cases that x, = #s, but (30) still holds, We can easily see from (28), (29), (30) that {FF dV = 412,, Og wee ee Partition of Energy between Matter and Radiation. De also {ver av= fav RF, av+ (4 oF i Ss (32) d Ta) denoting differentiation along the outward drawn normal ri at the element of surface dS. At the surface F, and F’, are normal, /’, and F, are their normal components. From (25) and (29) it can be shown easily that 1 oa way Here p, and pz are the principal radii of curvature at the dF di’, haat. ea ava Using this and (31) we can reduce (82) to WENZEL, aN = Age a) (3d) We now: use the value of I by (24) in the equations of ether (20) and the equations of motion (21). First in (20) and we have iL @ ; IL ad (vi~s oe) Pit V(Vi= sap) et ame, = = Multiply this equation throughout by F,dV and integrate through the volume. By (81) and (33) 1 dar a? Unces 0 —Ar nf 5 3 op ie “z.)+ | F.V(v?- a EY aN + tp: F7v=0. The term involving W is easily seen to vanish on partial integr ation when we recall the surface conditions pa0, Vp=0. We have finally surface. These give FI, dee. 7 u = Oe +e tay, =e; Fide, leap bella ty and the integral on the right hand of (84) is taken wherever there is electric charge. 24 Mr. 8. B. McLaren on Hamilton’s Equations and the Use (24) next to transform (22) dy ehhh ee 1d 1 : B=— > S2-Vv($+5 G)—5[s Col Bi], Cat ce dt or 1 dak ] ARI ANN Gripe a gm ' ake B= —- ice Curl F, | von ldwy here a Pies C dt , Sida, Substitute this value of E’ in the equations of motion of the nth electron dp, U,, ‘ ea dt’ ff es C aa a|- Curl Fy] + Vy} pden=0: this ean be written dp, iPyait) Uy Pee: “ae! +| {3 ar + V7 do- “VE, bpde.=0. The electron has a movement of translation only, eaci point of it has the same velocity u,, — | V dopdvn is the electrostatic force due to the field of potential ,. Let Vy be the electrostatic energy of this field. It is a function of the positions of the electrons. If qx be the vector position of a point of the nth electron. Then dVx Ww d OR So aes { Pop ae Also \ 7 Fypdin= + ai 1 \2 Un F pdv, e on the right the differentiation is partial with respect to q. the function being expressed in terms of the velocity Un and the coordinate q,. The eae of motion becomes Aes Un dt! 4 Pat: “{¥ spades | ~ ta, —{ 18 F pdv,, —V a =0. (35) In (35) insert the value of F, from (28), and write : { Ee pdu =i Ff, 1s a function merely of the position of the nth electron, | Partition of Energy between Matter and Radiation. 2d (34) and (35) can now be written d/l da n=N 3 —— ss gece — = ° ry e 7 4 az dt & talrn ener Ge) d ae dad 0) = : dt (p.+ Ed Fn) — aa i 0,2, Fn — Vn =0. (35) Suppose p, is derived from a Lagrangian function ZL, for the purely material energy. Then the whole system of equations (34)' and (35)’ has the Lagrangian function ZL. Ne rao Pee eae VAAN qilh L=ly—Vxyt DS > Unc Lyon + > A = P= secant 2 ree rol 2¢ dt remembering that F,, is a function only of q,. The corresponding Hamiltonian function 1s n=N H= > tee Gore line dicot eel Shiga | (yaw a e. Int Vat > {aalae) +36}. GN) The “ gyrostatic terms ” linear in u, and a, disappear. The need for a function Ly appears at once if we are to transform (34)/ and (35)/ to Hamilton’s form. . dl, ; If f,, is zero the momentum fa Joes not contain the ne velocities at all. The 2N material momenta are functions of the coordinates «,and q, only. Therefore the velocities u, cannot be expressed as functions involving the corresponding momenta. By (85)' we can derive relations between the various velocities so that these are not independent. The statistical method cannot be applied. If, on the other hand, fy 1s not zero we may apply Maxwell’s method in the ordinary way. [By the first part of this paper the equi- partition of kinetic energy is expressed by the equation 1 dH_1 dH Pr dp, = BP dp, the bars denoting average values. Or enh Ban, eT Many Wiig du,’ where u; and us denote different velocities. —_——< 26 Dr. A. 8. Eve on the Jonization of the da Here put —— for uw, and u, for us, so that u, denotes i dt the component of Up 1n the direction of the wz axis. 1 sda, 2 il ad if por | a Un ae =e Dy Un > a Eons 1 2 Oey eg dun or Te ee Gh eee al “) = os ra —" on re Sapa “ Fa, denoting the « component of F;. Hach degree of ether has the same kinetic energy equal to the kinetic energy of each material degree of freedom. In the latter the gyro- static term reappears as the second part of (38). Thus the kinetic energy of the electron so far as it is electromagnetic in origin is not any part of the total energy H. It will be objected that what appears as radiation in this paper is not the real radiation. The field immediately sur- rounding an electron ought to be reckoned as part of the electron’s energy. If, however, weattempt thus to distinguish radiation from what is not radiation, the result is a set of equations which do not belong to a dynamical system at all. With that I am not here concerned, my object has been to consider the results of the general dynamical method wherever it is adopted. ————— iV. On the Ionization of the Atmosphere due to Radioactive Matier. By A. 8. Eve, WA., D.Sc., McGill University, Montreal*. f ieee has been quite recently a well-marked advance in the accuracy of the determination of important radioactive constants. This progress is due to improvement in method and technique, and also to the cumulative evidence derived from different types of experiments. Three notable instances may be given: the determinations of the electronic charge, of the rate of discharge of a particles from a gramme of radium, and of the number of ions produced in air by the various types of « particle. It is therefore now possible to estimate, with a reasonable degree of accuracy, the relation in atmospheric electricity between the radioactive agents and the ionizing effects attributable to them. A balance-sheet may be attempted between cause and effect; and although it will not be possible here to review all the valuable work done by many observers * Communicated by the Author. Atmosphere due to Radioactive Matter. a4 in the past few years, yet we may concentrate the best results together, ascertain the measure of success attained by the radioactive theory of the ionization of the atmosphere, and indicate those outstanding difficulties which require further explanation and investigation. At the outset it may be stated that the radioactive theory of atmospheric ionization holds the field, and that in most respects it appears sufficient and satisfactory. Yet there are some particulars in which this theory is detective or lacking, so that either there may be causes still undetermined, or our knowledge of the ascertained causes must be incomplete. Also we will consider only normal atmospheric conditions, and not such abnormal ones as thunder-storms—recently explained by Simpson as sometimes due to vertical air currents causing the breaking of water-drops with the well known consequent ionization. In this paper the value of the electronic charge e will be taken as 4°6x107" electrostatic unit, and all numerical results will be given on that understanding. Also the value of a, the coefficient of recombination of small ions, will be taken as 3420e or 1°57 x 1078, at standard temperature and pressure. When the ionization is steady, there will be the well-known relation g=an” between q the rate of production of ions and zm the number of small ions actually present, in each case per cubic centimetre. So that we have the following table for the corresponding values of g and n. qg n. | q n 1 800 5) 1784 1‘57 1000 | 6 1960 2 1131 6°28 2000 3 1386 9 2400 4 1600 14:13 3000 Unfortunately these simple relations rarely if ever exist in the atmosphere, because many small ions become transformed into large ions owing to the presence of water vapour, dust, mist, smoke, or other physical impurities in the atmosphere. Recombination is thus delayed, and the number of ions present is exceedingly variable and hard to determine. Other difficulties arise when we seek to determine gq, the rate of 28 Dr. A. S. Eve on the Ionization of the production of ions per cubic centimetre, for then an electro- scope must be employed, and at once there is an uncertain. factor, namely the ionization due to the intrinsic radiation from the sides of the employed vessel, probably attributable to radioactive matter in the walls of the electroscope. Moreover we are seeking for the value of g in free air, while in the electroscope the ionization is in part due to the secondary rays (@ rays) from the electroscope, and this sécondary radiation from the walls is in excess of that from free air. Recollecting all these pitfalls, we will commence with a careful estimate of the rate of production of ions due to the radium products known to be present in the atmosphere. Alpha Rays in the Atmosphere. The most simple and accurate method of measuring the amount of radium emanation in the atmosphere is that of collecting the emanation either by adsorption, using coconut charcoal, or by condensation with liquid air. The results obtained are in fairly good agreement. Eisele Nemece ee ho ACE callie Sara 60 Dacverley hen wee ice 100 ANGhimam (ig epee ee 689 Miami wast 83 We will select 80 as the mean value, and state that one cubic metre of the atmosphere near the earth’s surface con- tains the amount of radium emanation in equilibrium with 80 x 10>" cramme of radiumy;: 9... . . . 2) There are of course considerable fluctuations with time and position, due to barometric pressure, wind, rain, and to the varying radium-contents of the underlying soils, so that it is only an approximation to the mean value which is here quoted. It is marvellous that the properties of this gas are such as to enable us to detect its presence and measure its amount, when it constitutes less than one thousand million © billionth (1/10") part of the atmosphere. From the results of Rutherford and Geiger § we can now calculate the number of ions produced per cm.*, due to radium * Phil. Mag. Oct. 1908. + Amer Journ. Sci., Aug. 1908. t z.e. 80x10—!2 Curies. § Proc. Roy. Soc. July 1909. Atmosphere due to Radioactive Matter. 29 emanation, A and (, in the atmosphere, namely, 80 x 10-22 x 8-4 x 10!(1-74 + 1°87 + 2°37)10° per m.%, Bee Ox O49) 98 x 10> per m.*, mae ies tons per’ cme per SeC,. iia pe is a ss | (2) Thorium and Alpha Rays. We are not in a position to make any such accurate calcu- Jation for thorium. It is known, however, that the active deposit due to thorium on a negatively charged wire is, on an average, about 60 per cent. of that due to radium ; and we may perhaps guess, for it is little better than a guess, that the « rays from the thorium products in the atmosphere will contribute about, and not more than, HOV aOMy pera cha Me TASe Coty io i/o) Minar) Gamma Rays from RaC in the Earth. Next the reasonable assumption will be made that the penetrating radiation, discovered by Rutherford, Cooke, and McLennan, is y radiation due to the presence of uranium, radium, and thorium. It is noteworthy that the amount of the radioactive products in the soil is of the right order of magnitude to account for the ionization effects due to the penetrating radiation, at least near the earth’s surface; also that the amount of radium emanation is of the right order to account for the active deposit which may be coilected on a negatively charged wire. ‘hese relations encourage the belief that we are dealing with atrue cause. It has also been shown repeatedly that the natural leak of an electroscope may be decreased in rate by surrounding it with screens ot lead or water. Thus McLennan and Wright at Toronto have found that the water and ice of Lake Ontario form an efficient screen from the rays from the radioactive matter in the land beneath. KF. W. Bates has verified this result over the ice on the River St. Lawrence near Macdonald College, and Gockel also has repeated the observation in a boat overa Swiss lake. There is therefore abundant and certain evidence that at least a part of the ionization of the air in an electro- scope is due to y rays from the radioactive matter in the earth. When we come to examine this question in more detail there are several points of considerable importance and interest. We will first quote C. S. Wright’s figures (Phil. Mag. Feb. 30 Dr. A.S. Eve on the Lonization of the 1909), obtained at Toronto over the land, and over the ice on Lake Ontario, correcting them for the new value, Lead. Zine. Alumininm. Over ice, Lake Ontario ...... 6-4 as) ir 4°8 7 Over land, Toronto............ 9°8 82 (0 Dntlerencern ny. ssas sues ey 38 i 2°9 Le om This indicates a difference of about 3 ions per cm.? per sec. within an aluminium electroscope due to the penetrating rays from the earth near Toronto. We have now to consider what would be the effect in free air, where the secondary radiation is less than in a metal electroscope. If radium is placed outside electroscopes of different metals the ionization within is dependent on the nature of the metal employed. Thus four observers have found, to an arbitrary scale :— Wright. Soddy. Bragg. Hye. ede ee aco e000 |) | a TDS OT ae AP ee | 81:5 7d 1315) 63 Aluminium ...... | TEND) Dh 49 54 wherein the variations between observers are no doubt largely due to different dispositions of apparatus. McLennan found that in lead the secondary radiation was twice that due to the primary, but in aluminium one-half. On the other hand, W. Wilson found that in aluminium the secondary radiation gave rise to six times as much ionization as the primary; but his apparatus was not altogether satisfactory for this determi- nation. Some experiments which I am making, not yet completed, seem to indicate that the ionization in an aluminium electroscope is only ten to twenty per cent. more than in free air; and Bragy finds that the ionization in a cardboard electroscope is about the same as in free air. He advocates the view that the primary y rays do not ionize It at all, but give rise to secondary 8 rays, and that these Hh alone produce ionization. From Wright’s figures we may | therefore conclude that in free air the penetrating rays from i the radioactive matter in the earth generate about | | 2: 1ons per em.® per secs” [s+ 0) a Atmosphere due to Radioactive Matter. 31 This value is smaller than that found by several observers for the total ionization due to the penetrating radiation as determined by complete screening with lead or water. This fact suggests that a considerable part of the penetrating radiation might come from the radioactive matter in the air. Subsequently it will be shown, however, that such a con- clusion appears altogether untenable, according to the present state of our knowledge and information. Theoretical Consideration. If there be a quantity Q grammes of radium at a point source, then the ionizing effect at a distance r in free air will be proportional to Q/rre™, where 2 is the coetficient of absorption of the y rays in air. If N isthe rate of production of ions in a cubic centimetre, we may write N=K(Q/7e’. This constant K was measured by me* with a fair degree of accuracy, and its value inside a thin aluminium electroscope is 3°-4x10%. Its value in free air at standard pressure and temperature is therefore about 3°1x 10%. I am hoping to evaluate shortly this somewhat ake constant with more AeeUACY. ).-\.. sy tee ike” aestahescierus( Gi) Integrating from 0. to <0 we can ‘calculate the total number of ions which one gramme of radium can produce in virtue of the y rays from the equilibrium amount of RaC. We have {e Kemtdn on™ Orel, OK) Nears wy eccen(e) 0 Now A/density is equal to ‘034 according to McClelland and to ‘04 according to Soddy. Selecting, in the present case, the former value, we have A for air equal to -OO00044. Hence the total number of ions due to the y rays from a gramme of radium in equilibrium is 8°5 x 10™ per sec. . (8) But trom Geiger’s} experiment on the ions produced by a rays, we know that Ra Em, A and C in equilibrium with one gramme of radium produce 2 x 10% ions per second, or about 23 times as many as the ¥y rays, so far as the radium products in the atmosphere are concerned. Near the earth the eftect of the y rays must be halved, so that we get a ratio of 1 to 46 or ey Teios sd atalal DyUae, Ja\eie Oni cae are ae 1°63 GyjsleaN Guy uch 2 einer Nae RN 0 do Cl vi SOD iva: (a) The addition for the ionization of the primary @ rays is at * Phil. Mae. Sept. 1906. 7) eroe, Roy. Soe. July 1909. az Dr. A. S. Eve on the Ionization of the present unknown, but it appears to be about equal to that of the y rays in effect. Perhaps the distinction is an erroneous: one, but in any case we obtain an estimate of this sort— | Rays. Ra. | Sub Total. | | = eee: | SR ee ih: 163 LOO | 263 | | | Bea yee 035 025 | 06 | rete eines 035 025 06 Ole p Aeon | ee | | | (10) Thus 2 7 ions per cm.” per sec. is our value for the mean total ionization, due to all the radioactive matter in the atmosphere, near the earth’s surface. It will be seen that the a rays are mainly effective. We can, however, calculate the ionization due to the y rays from radium Cina more direct way than above. If N is the number of ions produced per cm.? per sec., X the coefficient of absorption of the y rays by air, Q the amount of radium C per cm. of the atmosphere, expressed in terms of the number of grammes of radium with which it is in equilibrium, then N= | QrdrK Ye“ =P QK). . eget “0 Here Ko =3:1 x10", X= 000044, O—8 x10 whence N=2ax3:1x10®x8~x 107/44 x 107° ty : (12) =°035 lon per em. per sec. J This agrees with the former result (9). It is possible to use a formula exactly similar to (11) in order to calculate the rate of production of ions near the earth’s surface, due to the radium C in the earth. In this case K has the same value as before, but Q’/=1:4 x 10-! x 2°7, the mean value found for sedimentary 1ocks by Strutt, and r’ = "034 x 2°7 (McClelland). Hence N'=27rQ’K/a! Rea 8 sc OP G4 Sn) soley » 034 2°7 20x31 x 1A We isa = 0°80 ion per cm? per sec. . - (ia) es Vie N’/N=Q'ei Qo’ = Atmosphere due to Radioactive Matter. 33 This is the calculated value of the mean ionization at a given locality, near the earth, due to the penetrating rays from the radium © in the earth. It will be nearly constant at any given place, but it will vary from place to place according to the amount of radium present in the surface soils and rocks of the more immediate neighbourhood. It may be noted in particular that the ratio of N to N’ is equal to that of Q/rX to Q'/r’, whose accented letters refer to the earth. But X and 2X’ are proportional to p and p’, the densities of the earth and soil. Hence ex 10n 2x O01 en Lacs 7A Tie ROY > Gol Oa 3 The penetrating radiation from the earth is therefore normally about 23 times as great as the penetrating radiation from the atmosphere, at least so far as radium C is concerned. I see no escape from this conclusion*, and it appears to x00 2332 0 * We may, however, consider an extreme case. Suppose that the active deposit of radium and thorium were distributed uniformly, as near the earth’s surface, for a height of 5 kilometres; and suppose, further, that the whole of this active deposit were carried suddenly to the earth by « fall of rain. There would then be on or near the earth’s surface 5X10°X8x 10-1!" or 410-1! gramme of RaC (expressed in terms of radium). ‘This is equal to the amount of radium C in 80 or 4U centimetres of the soil, in each case per square centimetre. The thin layer of RaC derived from the atmosphere would, however, be by far the more efficient ionizer, because the y rays would not have to penetrate through, and be absorbed by, the surface lavers of the earth. ‘The increase of the penetrating radiation during rain has been observed by Mache, and by Gockel. Such variations, however, were not found by McLennan at Toronto, The motion of RaC earthwards, not carried by a water drop, would be only about 50 metres an hour, due to the potential gradient. I have calculated the ionization at various altitudes due to this very hypothetical surface layer of radium C, and obtain, 1 Lee Na2nok | aes cat 0 whence Altitude. Tons/em.? sec. 1m. : 10 1:05 100 32 1000 negligible On comparing this table with result (18) it will be seen that, in extreme cases, near the earth’s surface the ionization due to the RaC carried down by rain or snow migat equal or exceed that due to the itaC contained in the soil and rocks. In general, the active deposit would doubtless enter the soil with the rain. Phil. Mag. 8. 6. Vol. 21. No. 121. Jan. 1911. D (14) ot Dr. A. 8. Eve on the lonization of the negative the view that the penetrating radiation comes equally from all directions. So far as our knowledge of the distri- bution of thorium at present goes, we may also predict that a similar statement will be found to hoid good for the y rays of thorium C. Returning to the result previously stated (13), we will next assume that the work of Blanc, Joly, W. Wilson (Phil. Mag. Feb. 1909), and others indicates that the penetrating radia- tions from the radium and from the thorium in the earth are about equal, and we can now set forth a complete table. Type. Whence. Kas) | Ry: Total. eae —— — —— hatanvat cee: sel Air 1:63 1:00 2-63 | 1 Teoh 1d ae hd Vo AG “035 025 | 06 | Povavias a fekz nega” Air ‘O35 025 | 06am eee Wes ta, isharetes os Earth “30 ‘80 1-60 | 250. | 135 , |e | | (15) This rate of production of ions per cubic centimetre, namely 4°35, corresponds to the presence of 1660 ions per cm.’, supposing that they were all small ions. The maximum value of ions measured in clear weather by an Ebert ion- counter is usually somewhat less than this. It must be remembered that the numbers quoted in the preceding table are subject to considerable variations both with time and place; and are only intended as a guide to the mean values, which will no doubt in due course of time be determined with an accuracy to which my estimates can of necessity make no claim. It will be seen that the theoretical value found, 1°6 ions per cm.® per sec. for the penetrating radiation from the earth, is less than that obtained by Wright at Toronto, namely 2-5. It is also considerably less than the number of ions per em.® per sec. lost when an electroscope is well screened by lead, amounting to about 6. It is true that my calculations do not take account of the feeble and negligible y radiation from actinium, the distribution of which is unknown. Further, it has been shown by Soddy (Phil. Mag. Oct. 1909) that the y rays from uranium X are more penetrating than was at first supposed. The addition which should on this account Atmosphere due to Radioactive Matter. 35 be made is of an uncertain character*. An increase in the value of q will affect the value of n, which is already in excess of that observed. Yet with every allowance for the value of g, due to penetrating radiation, the theoretical value appears to be smaller than the observed. We may deduce either that the radioactive matter in the earth has been under- estimated, which is improbable, or that the value found tor K is too small, or that the observed values of q are too great, or that radiation exists of a type not yet expected or discovered proceeding from matter rather than from radioactive matter. Ionization over the Ocean. It would be interesting to make a table corresponding to that given above (15), but with values for mid-ocean. Un- fortunately we have no certain measurements of the mean quantity of radium or thorium emanations over the ocean. From experiments by many observers, it appears that sea breezes on the western side of a continent contain less radium C than do land breezes. It is, however, certain that radium C does exist over the ocean in measurable quantities, and there is little doubt that radium emanation escapes from the ocean with greater facility than from the land, and this fact compensates to some extent for the minuteness of the radium contents of the ocean as contrasted with the land. There is also some uncertainty as to the mean value of the amount of radium in sea water, for Joly (Phil. Mag, Sept. 1909) has found 1:'1x10-4% gramme of radium per cm.’ for sea-water from various localities, and I have found 10- for sea-water from the middle of the North Atlantic. For the present we will take the higher value given by Joly. The ratios of the penetrating radiations from the RaC in land, sea, and air are then, necessarily, near the earth’s surface, as Q/p . Q'/p': Oe where the Q’s denote quantities of radium © per ecm.?, ex- pressed in equivalent grammes of radium, and the p’s denote the respective densities. * Tf equal to radium we get 3xX0'8=2°4, agreeing with Wright, see (15). D 2 36 Dr. A. 8S. Eve on the lonization of the Thus we have:— Penetrating Radiation, RaC. Land Sea Air Qe Valiak Se 43x10 77 §| 11x10") |) Se Ape Se IN ae nie 27 10 0013 2) Per ae oat a Ole 14x10 7 | 11x10 | Gascon Ration, eee Mean 130 1 5D GES pt G(R 23 18 1 Tons/em.? see. ...... “80 006 035 It will at once be seen that the penetrating radiation over the ocean from the radium in the sea is negligible, being less than one-fifth of that from the radium C in the atmosphere over land. Since the radium emanation in the atmosphere over the sea is probably less than over the land, it is altogether quite extraordinary that those observers who have used Kbert’s ion-counter, and those who have measured the con- ductivity of the atmosphere over the sea, have found values in each case not much, if any, less than over the land. It is possible that the greater purity of the air over the ocean— the absence of dust and smoke and physical impurities of that kind—may cause the small ions over the ocean to remain small ions, except during time of actual fog, while over the land they tend to become large ions. Yet such an explana- tion is very doubtful, and in this discrepancy les at present our chief objection to a purely radioactive theory of atmo- spheric ionization. Itis a question on which further investi- gation is needed, before any weighty opinion can possibly be hazarded. i We get a sidelight on this interesting and difficult point in the action of Hertzian waves in wireless telegraphy, which appear to have « larger effective radius in the dark, or during mist or fog, and are to a considerable extent absorbed in the presence of sunlight. It is possible that the small ions both render the air a conductor and tend to absorb or disperse the Hertzian waves, and that the sunlight may itself produce small ions. ‘There appears to be no direct evidence of the influence of sunlight increasing the ionization near the earth’s surface, but if the electrons were freed a little from the Atmosphere due to Radioactive Matter. 37 atoms by sunlight and then returned to the parent atoms, these might account for the greater absorption of the Hertzian waves during the daytime. The ultra-violet light from the sun seems to be mainly absorbed in the upper layers of the atmosphere, so that if 1s not an important lonizing agent near the earth’s surface. Further observations are needed on this point. Liffect of Altitude. In conclusion we have to discuss the effect of elevation on the radiation from the radium C in the earth. The y rays are absorbed rapidly in the earth, slowly in the air, and the ratio of the coefficients of absorption is nearly that of the densities, or as 2°7 is to ‘0013, about 2000 to 1. So that 1 cm. of the svil is as effective as 20 metres of air. In the case of a point source of radium C the law of inverse square of the distance also obtains, so that the radiation at a distance r from the source varies inversely as r?e". Hence if with such a point source the radiations were represented by 100 at 1 metre, there would be a rapid decrease with distance thus At 1 metre, radiation 100. 10 oe) ” "95 00) i 0064 WOO i “000012 But in the case of the earth we have a sheet of radioactive matter, distributed with some approach to uniformity through the surface layers, which alone is effective in the atmosphere. The calculation has not, I think, yet been given, and therefore it is set forth here with some fulness. Let P (fig. 1) be a point at elevation h above the earth’s Biow, A ai Nair rc) Ty surface, assamed plane. Let Q grammes of radium be the contents per cm.’ in the earth. Consider an elementary ring dx by dY, radius x, at a depth Y below the surface. Then. 38 Dr. A. 8. Eve on the Ionization of the the ionization at P will be { (?zs2ei0 8 oa XY gm (r+y) o4U iB where y is small compared with 7, = ‘ih 27rQK BSG COs alg ~— X\h cosec oy —X'Y cosec OqY 4 h? cosec? 6 e e — Ah cosec 8 HOOK bee cot 6dé Jo » cosec @ In QK (7/2 tes cos Ge Dee iis oX6 27QK (1 K = Sas arg ( A —Ah/z dz. A At i) r 0 In order to evaiuate this troublesome integral I have drawn the curve y=e-?, and determined the area from z=() to z=1, for various values of h, and substituted the values of A, A’, Q, and K given in this paper. When A is zero we have 27QK/N’, agreeing with our ne work. Thns the numbers of ions produced per cm.° per sec. at various altitudes by the pevetrating radiation due to radiuin © in the earth is given in the following table :— | Penetrating Radiation. Tons per cm.? Height in metres Ratio per sec. ieee ye, 100). 6 SOU TERE ed Bl 2 ‘98 | 78 {Hera Oke events ie ‘83 | 67 LOO see elon eee 36 | “29 NSIOGO>. £2 dA le ayer 008 (18) The values in the right column should perhaps be doubled if we add the effect of thorium to that of radium. In this calculation I have not made any correction for the decrease of density of the air with increase of altitude. This would tend to augment the lower figures, because the rays would be less readily absorbed ; but, on the other hand, the number of ions produced would He diminished with a decrenan of pressure. Almosphere due to Radioactive Matter. 39 The results shown in the table (18) are very remarkable. They indicate a decrease of ionization with altitude which is within the limits of detection for an elevation of 100 metres. They show, moreover, that at 1000 metres altitude the pene- trating rays from the earth are ineffective asan ionizer. ‘This result cannot well be put to the proof by ascending a mountain, because the radium contents of the ground beneath the observer are changing as he ascends. The experiment may be commended to those who are able to fly kites or captive balloons for 24 hours at a few hundred metres. It would be necessary to have an electroscope with a low natural leak and insulation of a high order fortified with a guard ring. Gockel has made observations in a_ balloon (not captive) at an altitude of 4000 metres on the penetrating radiation and found but a moderate decrease in the saturation current of an electroscope, carried upwards in the balloon. There are, as he points out, difficulties and uncertainties in making such measurements. lt will be seen from this paper that the radioactive theory of the ionization of the atmosphere is gradually approaching what may almost be termed an exact science, but that there are three outstanding problems of difficulty and importance. ‘These are: (1) the high value of the ionization over the ocean is not yet explained ( is lar ger than expected. q is unknown): (2) the rapid decrease of ionization with altitude has not been detected ; (3) the theoretical value of the ionization due to the penetrating radiation is dess than that found when observers completely screen an electroscope with lead. Until these three questions have been satisfactorily ex- plored we cannot rest wholly content with the theory, but have a vague feeling that all matter may give rise to feeble rays of the Rontgen type, of which the more penetrating may integrate into an effective total. ‘This last view is by no means advocated by the writer. Last May I attempted some measurements on the North Atlantic, and found that in a cabin of the iron s.s. ‘ Dominion,’ in mid-Atlantic, the leaf of an electroscope closed a little more slowly than on land, indicating a ‘slight decrease of penetrating radiation over the ocean. But I regret to add that the inherent dithculties of observations on “shipboard prevented me from reaching any exact or convincing results. The writer will be grateful for any information or sug- gestions tending to correct the somewhat tentative values given in this paper. July 1910. 40 Mr. H. Mitchell on Ratios which Amounts of Note added Oct. 26, 1910. Since this paper was written there have appeared two communications by Wulf (“e Radium, June 1910; Phys. — Zeit. 15 Sept. 1910). These throw much light on the subject of penetrating radiation and confirm some of the conclusions set forth above. Wulf, with the vessels he used, finds for the penetrating radiation in Holland g=10, in Paris g=6, and at a height of 300 m. on the Eiffel Tower g=3'5. The reduction in g over a lake was about 4:9, and a similar reduction was determined at a depth of 12 m. beneath the surface of the water. His interesting experiments seem to establish the terrestrial origin of the penetrating radiation and the diminution of intensity with altitude. The figures appear to require reduction for values in free air, perhaps on the basis of lead 100, zine 60, aluminium 52, free air 42. It will be seen that Wulf’s value for g, even at Paris, is larger than that calculated in this paper, and that a loss of 40 per cent. for an altitude of 300 m. is less than the loss of 64 per cent. which I calculate for an altitude of 100 m. Jt will no doubt take time to reconcile these points. At present it would seem that the penetrating radiation both passes through the air more readily, and also produces more ions than laboratory experiments, with Ra C and metal screens, lead us to infer. It will, however, be remembered that no direct measurements of the coefficient of absorption of the y rays by gases have yet been made. A. iS. aa: V. Note on the Ratios which the Amounts of Substances wn Radioactive Equilibrium bear to one another. By Hug MITCHELL, 14.A.™. - (1) HE law of radioactive change is that the rate of disintegration of an active substance is proportional to the amount of it present. A series of radioactive sub- stances is said to be in equilibrium when the ratio of the amount of one substance to that of any other substance in the series remains unchanged as time goes on. Itis known that assuming the disintegration rate of the parent substance to be small in comparison with that of any other in the series, the ratio of the amounts of any two substances in the series is approximately equal to the ratio of their average lives. The present note is intended to show that without * Communicated by F. Soddy, M.A., F.R.S. Substances in Radioactive Equilibrium bear to one another. 41 this assumption the theoretically correct deduction from the above law still remains very simple. (2) Let &, be the number of atoms initially and w, the Humber at time ¢ of a radioactive substance S,, which changes into a radioactive substance S,,,,) at the rate of A, «, atoms per unit time, where n=1, 2, 3...n. Then the average life Soy Mae a: of an atom of §, is — units of time and z, satisfies the differ- vv ential equation wa a Mo Lin Nin XU 9 in which when n=1, Az 1s to be taken as zero. (3) The general solution of this equation giving the quantity w, of the nth substance after time ¢ when the parent substance is initially free from products is en Ait en Ant e-ant @ = EMyAg.. Neola eee) Shah a arrete aa ey ae . ee where the expression II(A,—2,) is to be taken to mean the product of all terms of the type (A,—2,) when ), takes all possible values from ), to A,, except X,,. For example, ee NG a= Ee, and ea~ Ait @~Axt m= Es) ORR =M)OQU=M) ” Ca=Ag)Oe—= Ay) ea Ast edt + I (A1—A3)(Az— Az) (Ag—As3) fi (Ar Ag) As— Ag) (As— Aa) (4) When the parent substance is the longest lived member : : wv of the series, as ¢ increases, —“ approaches the value v7 AyXo...An—1 (Ag—Ay) (Ag—Ayz)...(An— Ay) Now At Aj4A9 1 + ——— —— ——. +.,, Ag—A1 i (Az— Aj) (Ag — Ax) 4 ey ane Noe (es ae (Mm —=y) (Ag Aa) (Np) n= 42 Mr. Norman Campbell on a Hence | SA MOE Na I EC “, Ay ay Ce NG ay i That is H Xn ne SSP BUA AP WE a> ab odli Ave | Hence in a series of substances in radioactive equilibrium i the ratio of the number of atoms of the xth substance to q those of it and all the preceding substances in coexistence {i with it is equal to the ratio of the average life of the nth substance to that of the first substance present in the series. ; | High School, Dundee. \ rs - — VI. A Note on a Method of Determining Capacities in Measurements of Lonization. by NormAN CAMPBELL*. | | | | fl aes ordinary method of determining the capacity of the ionization vessels and the measuring instrument used in observations on ionization consists in comparing the rate | of rise of potential of the electrode system for a constant | current (1) when a known capacity is put in parallel with the electrode, and (2) when that capacity is absent. The il advantages of the method lie in the simplicity of the appa- | ratus required ; its disadvantages are connected with the choice of the standard capacity. Standard condensers of not Jess than ‘Ol m.f. are easily procured, but their capacity is i very great compared with that of the usual measuring | systems, which is usually less than -0005 m.f. If they are | used, either the time for which the rise of potential is observed in (1) must be so long that difficulties are likely to arise from defective insulation, or the time for which the rise is observed in (2) must be so short that it cannot be measured with accuracy. On the other hand, accurate and | well-insulated standard condensers, the capacity of which is of the same order as that of the measuring instrument, are not easily procured, and the (unknown) capacity of the q conductors used to connect them to the measuring system may be too large a fraction of their capacity to be sately : neglected. q The following method, which appears not to be known generally, is free from these disadvantages. Though there i is nothing new in its principle, which is sufficiently obvious, | it may be worth while to draw attention to it. ) * Communicated by the Author, Method of Determining Capactties. 43 Let ¢,, ¢,’ be the inner and outer coatings of the condenser forming the measuring system, of which the capacity C, is to be determined. Let cg, ¢' be the coatings of the standard condenser of known capacity Cy, and cg, ¢’ those of an “auxiliary ” condenser of capacity Cy. The “inner” coating is that which is usually connected to the measuring system, the “outer” coating that which is usually connected to earth. Perform the following observations:—(1) Connect and earth c¢,, ¢ 3; also earth c,', c'. Insulate ¢,+c¢., and then raise c)’ to the potential V,, the potential of earth being 0. In virtue of the charge induced the electrometer (or other measuring instrument) indicates a potential v,. The charge induced on ¢c; must be equal and opposite to that induced on Cy: hence OR a Cy (v4 —V),) = 0. 5 G O 5 (1) (2) Connect and earth ¢,, ¢:, co; earth ¢4’, cy’, co. Insulate ¢,+Co+¢o, and then raise c,/ to the potential V.. Then, as before, if v2 is the potential indicated by the measuring instrument, (Cy + Co) Vg + Cy (v9 — V.)=0. 4 ° Py e (2) We have thus two equations to determine C,/Cy and C,/Cy in terms of the known quantities V,, Vs, vj, v,3; and the problem is solved. Iiet us now examine the accuracy of the method. For the determination of V, and V,, and also of the sensitiveness of the electrometer (which gives 7; and v, in terms of the deflexion), a resistance-box arranged as a potentiometer will doubtless be used. The potential (V) of the battery in the circuit need not be known, so long as it is constant through- out the observations. The error in V, or V, will be due to the deviation of the ratio of the resistances from their nominal value, and need not exceed 1 part in 1000 over a range from V to 1/1000 V. The advantage of this method lies in the fact that potentials, unlike times, can be measured with the same instrument over a wide range with great accuracy. On the other hand, the error in determining v7, or v, from the deflexion is likely to be about 1 part in 100 for a single observation. Accordingly we shall neglect the error in V, and V, compared with that in , or v. In practical measurements V, and V, will be so chosen that v; and v, are nearly equal. Jf e is the probable error in the observed value v, €;, €, €12 those in the deduced values of C,, Cy, Cy; +C,, the ordinary formule for the relation of AL A. Method of Deternuning Capacities. WEe pits ecruae give, when we put cj =t,=1, QC VVO-Wil eg Uy (Vi—v)(Ve— V3) ' “yD 9 e ° TUN ON see Nee ao = fa lw) a SORTA V/2V~_ OC, +0, 7 (V2= Vi) © So far as C, is concerned, equation (4) shows that the Wi A | e e . 2 € e accuracy increases with the ratio =~, 2. e. with ‘ ve Ci <5 Cs i But for C, the relation is more complex and involves the Seah enews ie) A Cot dae Sl ies ° 19 ° ° cho ratio G: However, we can avoid using C, explicitly, by 1 taking as the “auxiliary” condenser an ionization vessel which forms part of the measuring system. Cy, is then the capacity of the rest of the measuring system, and we are concerned only to know the value of C,+C,. The accuracy of C+C, increases also steadily with the ratio ——— ite, Hence, by the method of measurement here proposed, the greatest accuracy is attained, when the standard condenser used has as great a capacity as possible. The use of such large standards is convenient on other grounds. A limit is set to the possible magnitude of Cy) by the condition that both V, and V. must fall within the limits of measurement by the potentiometer ; but a capacity as large as 0-1 m4. would be convenient for determining the capacity of a system including an ionization vessel of reasonable size and an electrometer. If the insulation of the electrode is not perfect, a correction will have to be made for the time whieh elapses between raising c,’ to V; or V.and reading the electrometer. The correction is easily determined by watching the decay of the deflexion of the electrometer for a short time; and such observations will give the value of the insulation resistance, a quantity of which the knowledge is important apart from measurements of capacity. It will be observed that the method is useful for deter- mining the variation of the capacity of the electrometer with its deflexion—a troublesome relation to measure by any of the ordinary methods. For this purpose it is only necessary Relation between Viscosity and Atomic Weight for Gases. 45 to repeat the observations with different values of V, and V,, and, consequently, different values of v. The method may also be applied to the determination of the capacity of con- densers designed to serve as secondary standards. These condensers can be used as “auxiliary” condensers, and the determination of their capacities will have the same accuracy whatever the ratio of those capacities to that of the measuring system. It is found in practice that it is quite easy to attain such accuracy that the probable error of the value deduced from a single observation is not greater than 1 per cent. Leeds, September 1910. _ VIL. On the Relation between Viscosity and Atomic Weight for the Inert Gases; with its Application to the case of Radium Emanation. By A. O. Rankine, DSc., Assistant in the Department of Physics, University of London, University College *. | FURTHER examination of the data obtained in my measurements of the viscosities of the gases of the argon group f has brought to light a simple relation between viscosity and atomic weight. ‘This relation has some im- portant applications. It may be used, as will be shown, to form an estimate of the critical temperature of neon. Further, in conjunction with another relation previously published, it renders possible the estimation of the viscosity and molecular dimensions of radium emanation, upon the assumption that this gas belongs to the argon group. The data upon which the present paper is based are given in Table I. TABLE I. Gas, Niyo< Os C. | 1G eee amt Secs GR 1:879 70 BN AM ap aR A 2°981 56 CAI Ag ER TIA Ut 2°102 142 GIs Deas cent e: 2°3384 188 ENG Ree ite 2107 252 | * Cominunicated by Prof. F. T. Trouton, F.R.S8. 7 Proce. Roy. Soc. A, vol. Ixxxiii, p. 516, and Proc. Roy. Soc. A, vol. Ixxxiv. p. 181. 46 Dr. A. O. Rankine on the Relation between The numbers in the second column are the viscosities in absolute units at 0° C. multiplied by 10°. Those in the third column are the values of the constant C in Sutherland’s * equation =n) (Gon) n=mn( 9. C+T/ where 7 is the viscosity at the absolute temperature T. ; The first five points in fig. 1 show the viscosities at Fig. 1. viscosity A’ O°C. x 10” 0 To OCS. 200 ATOMIC WEIGHT 0° C. x 10* plotted against the atomic weight. The undula- tory nature of the curve is apparent, and it is at once recognized that no simple algebraic relationship obtains. Indeed, such a connexion is hardly to be expected in view of the fact that comparison is made of the viscosities of the gases at the same temperature. It seems much more reasonable to perform the comparison at corresponding temperatures—for example, the boiling points or critical temperatures of the respective gases. With the data available this procedure is only possible if the truth of Sutherland’s equation be assumed, and the values of the viscosity found by extrapolation. In this manner the viscosities of the gases have been calculated, with the striking * Phil. Mag. 1898, vol. xxxvi. p, 07. Viscosity and Atomic Weight for the Inert Gases. AT result that the viscosity-atomic weight curve becomes smooth, and the square of the viscosity at the critical point is found to be proportional to the atomic weight. This relation has been deduced from the values for argon, krypton, and xenon only, because the critical temperature of neon has not been determined, and that of helium is not known with certainty. The curve in fig. 2 is the parabola whose equation is n ay Me 3-93 x 10-2 A ee =) « = 3 = i A Em 2 = <= = E Cz YQ = 2 = ” Ke an 2 La) a > oN H j A Ne t He : | | 0 100 200 ATOMIC WEICHT. where 7, is the viscosity at the critical temperature and A is the atomic weight. The calculated value of », for helium (assuming the probable value 5° absolute for the critical temperature) is also shown on the curve. It will be seen that it is considerably removed from the theoretic curve, and it should be pointed out that the uncertainty as to the critical temperature is whoily insufficient to account for the departure. 48 Dr. A. O. Rankine on the Relation between Table II. shows the numbers relating to the curve. TABLE II. Gas. ies A. 104 7-X10* | Difference <3 y me (calculated), | per cent. ile anaes | @)s 5:96 0-021 0°32 Note. | 20-03 AR a | 1556 | 39:92 | 1-961 1-253 06 SSRI eee enoOss 83:0 1-828 1-806 1-2 Dh cate | 288 1380°7 2°220 2:266 | —2:1 The numbers in the fourth column are calculated from Sutherland’s equation using 7), C, and T-; those in the fifth column are deduced from the equation 7 = 3-93 x 10° a 3 With the exception of helium this equation is a good fit, and even the exception seems capable of reasonable ex- planation. In the cases of helium and argon we may avoid the necessity of such extensive extrapolation by using the experimental results recorded by Schmitt *. These measure- ments extend from —193°2 C. to +183°7 C., and the value of the viscosity of argon at the critical tempenanene can be deduced by interpolation. The value thus obtained is 1:25 x 107*, which is remarkably near the value calculated here, and suggests that in this case the extrapolation is valid. When we turn to helium, however, the results are very different. Ata temperature of 80° absolute the actual value of the viscosity exceeds that calculated by using Sutherland’s equation by 26 per cent., and the divergence | increases with fall of temperature. The temperature at which the viscosity is required is 5° absolute, or 75° lower, where we should expect the divergence to be more serious still. In fact, an es over this range of 75 degrees indicates that Oro2 105s much more probable value of the actual viscosity at ‘5° *) than 0:021x107*. This would bring helium into line with the other gases. It is quite possible, theretore, that helium also conforms with the law here pre- sented, and that the apparent divergence should be attributed to the failure of Sutherland’s equation at temperatures so near to absolute zero. * Ann. der Physik, Bd. xxx. Heft 2, p. 899 (1909). Viscosity and Atomic Weight for the Inert Gases. 49 The Critical Temperature of Neon. It is evident that, working backwards, this relation may be used to estimate the critical temperature of neon, upon the assumption that it applies for this gas. In that case the value of 7, for neon (shown by a cross in the diagram) would be 0°887 x 10-*. This, according to Sutherland’s equation, would be the viscosity of neon at a temperature of 61° 1 absolute. That is to say, the critical temperature of neon is 61°-1 absolute. The author * has already made an estimate of this tempera- ture based upon another relation, namely, that the critical temperature is proportional to Sutherland’s constant. This is expressed by the equation Le: Taking the value C=56 obtained from the experiments, this gives T,=62°7 absolute. ‘The agreement between these two figures 61:1 and 62°7 is remarkable, and constitutes weighty evidence of the probable accuracy of the estimate. Application to Radium Emanation. It is generally accepted that the emanation from radium belongs to the same group in the periodic table as the gases previously referred to. There is considerable justification, therefore, for applying the two above-mentioned relations to this case. The fact that the atomic weight and critical temperature of the emanation are now known renders it possible to estimate the viscosity, not only at the critical point, but also at any other temperature, together with the dependent molecular properties. The values of the atomic weight and critical temperature used are those obtained by Ramsay and Gray f, viz. A=222 and T.=377° absolute. In the first place, using the relation Wes OM Os we obtain C=337. Further, using the second relation 3°93 1. = 10910 4 A, n, is found to be 2°954x10-*. [This is shown by the cross marked Hm on figure 2.] * Proc. Roy. Soc. A, vol. Ixxxiv. p. 190. + Trans. Chem. Soc. 1909, p. 1073. Brit. Assoc. Report, Shettield, 1910. Plas Magus. 6s Vol. 20: Noi 125) Jan. L911. K 50 Dr. A. O. Rankine on the Relation hetween Now, making use of Sutherland’s equation in cay oe) ae ian ii lp Us ral : as applied to the emanation we obtain ny = 2-954( 203)" wn 10-4 3777 \610 = 2°130x 10™. This is the estimated value of the viscosity of radium emanation at O° C. It is recorded by means of the cross on fig. 1. It les between the values for xenon and krypton, and its position strongly suggests that the up and down. distribution of the points is not without significance. The alternations were already regular but of decreasing ampli- tude before the addition of the point representing the emanation ; its addition carries the rule a step further in both respects. Of course, it is possible that another un- identified member of this group of gases lies between xenon and radium emanation ; butit will be seen that a place could be found for it on the diagram. Indeed, a point higher than X or Em and lower than Kr, placed about midway between X and Em, would be more in keeping with the general trend of distribution than a single step from X to Em. It is now possible to compare the molecular dimensions of emanation and helium, since the viscosities at the same temperature are known. The connexion according to the kinetic theory is SHe Nim Pe where s is the radius of the molecule and p the density of the gas under constant conditions. ‘This gives 2 SHe and consequently v ~Em — 16:97, UHe where v denotes the molecular volume. Thus the molecule of emanation is larger than that of any other known gas in the group, as will be seen from the following table. The other figures are copied from a * ik Neel | Viscosity and Atomie Weight for the Inert Gases. previous paper for purposes of comparison, helium in all eases being taken as unity. The last three columns refer to the molecular dimensions corrected according to Sutherland’s theory. This is that owing to the increased frequency of collision caused by molecular attractions, the molecules behave as though their dimensions were greater than their reil values. If we wish to find the true radius we must diminish that calculated by the simple theory in the proportion 1:(14 7). Tha 4 numbers in the last three columns are obtained from columns 3 to 5 by the application of this process. Aa UOT. With Sutherland correction. eee atue’ | vainus lo ceectows| ade’ | Voline bftaenc. 1-00 1-00 1-00 1-00 1:00 1-00 1-00 5-06 1-19 1-69 299 1 120) L798 2°83 10:08 1°68 4°74 2128 ie 153 3°59 2°81 20 96 1-91 6-98 3°00 1-65 453 | 463 ...| 3301 2°25 11°37 2°90 1:83 6-11 5-40 Em..., 555 | 257 | 16°97 3°28 1°93 | 716 C75 Finally, using the value N = 2°8 x 10° for the number of atoms per c.c. at N.T.P., we obtain for the absolute molecular radius of radium emanation Spo Ome emit This is the uncorrected value, 2.e. the value obtained without applying the Sutherland diminution, it being still customary to record molecular dimensions in this way. In order to avoid confusion with regard to these two separate measures of the molecular radius, the following consideration may be found useful. Rayleigh * has pointed out that if the molecules mutually attract one another, mutual repulsions must also exist, and that the laws of variation of force with distance are different * Phil. Mag. xxx. pp. 285 & 456 (1890). K 2 92 Relation between Viscosity and Atomic Weight for Gases. in the two cases, the variation being necessarily more rapid for the repulsive force. Perhaps from this point.of view. we - should regard the uncorrected radius as referring more particularly to distances from the centre of the atom at which the values of the attractive force become serious, and the corrected or Sutherland radius as the distance at which the forces of repulsion and attraction balance one another. Connexion between the Critical Temperature and Molecular Radius. It will now be shown that a simple relation exists between the critical temperatures of these gases and their molecular dimensions. Starting from the equation 7 = inmVxr obtained from the kinetic theory; 7 being the viscosity, nm the number of molecules per c.c., m the molecular mass, V the root man square velocity, and » the effective mean free path. Hence ! 0? = tn'm Vr. Now, mV? is proportional to T, the absolute temperature. Therefore ; se T= Kina, . 3. 208, nr where K is a constant not depending on the particular gas. Further, of sal Sota / 2n77s* where s is the effective radius. According to Sutherland, Y = 4(14 a where Sp is the real radius. Whence MS : aSs (2) 2aPntsi( 1 + T) Combining with (1) we obtain | : T a7 Top hc g being a constant, . Now- the. author has shown that at the Vibrations of a Circular Membrane. 53 2 critical temperature aaa te L are constant. Therefore, from (3) m t ney = constant. S0 That is, the critical temperature, and therefore C also, is proportional to the fourth power of the true atomic radius. The truth of this law depends on the accuracy of the two laws previously given, and the figures given in column 4 of Table IV. are expected to be constant. The numbers in column 2 are relative to helium. ' TABLE IV; § Gas. Soe Te ret Eh eee | 100 5? 0-669 Ne 2 ee | 1:21 62? 0-433 i 1:53 1556 0433 i ae | 1-65 210°5 0-433 Sl ee | 1:83 288 0-444 ee Poo | 1:93? 377 0-438 The values to which doubt attaches or which are aeonaed from previous considerations in this paper are marked with a query. Helium, as one would expect, is an exception. This and the constancy of the remaining ratios are direct Soma eg prety of the relations previously given. ’ VIII. Note on Bessel’s Tone as applied to the Vibrations - of a Circular Membrane. By Lord RAYLEIGH, ote ziieby 21S." T often happens that physical considerations: point to analytical conclusions not yet formulated. The pure mathematician will admit that arguments of ‘this kind are suggestive, while the physicist may regard them as con- clusive. The first Cre -here to be touched upon relaree to the dependence of the roots of the function J» (¢) upon the order * Communicated by the Author. o4 Lord Rayleigh on Bessel’s Functions as applied n, regarded as susceptible of continuous variation’ It will be shown that each root increases continually with 2. 3 Let us contemplate the transverse vibrations of a membrane ~ fixed along the radii 0=0 and @=8 and also along the circular are r=1. A typical simple vibration is expressed by” : w=dn Gr) ssin' nO. cos\(c@t), where 2% is a finite root of J, (z)=0, and n=7/P. OF these finite roots the lowest 2) gives the principal vibration, 2. e. the one without interna] circular nodes. For the vibra~ tion corresponding to 2@ the number of internal nedal circles is s—1. As prescribed, the vibration (1) has no internal nodal diameter. It might be generalized by taking n=v7/8, where v is an integer ; but for our purpose nothing would be gained, since § is at disposal], and a suitable reduction of 8 comes to the same as the introduction of v. In tracing the effect of a diminishing @ it may suffice to commence at B=a,orn=1. The frequencies of vibration are then proportional to the roots of the function J,. The reduction of @ is supposed to be effected by increasing without limit the potential energy of the displacement (w) at every point of the small sector to be cut off. We may imagine suitable springs to be introduced whose stiffness is gradually increased, and that without limit. During this process every frequency originally finite must increasef, finally by an amount proportional to d@ ; and, as we know, no zero root can become finite. Thus before and after the change the finite roots correspond each to each, and every member of the latter series exceeds the corresponding member of the former. As 8 continues to diminish this process goes on until when 8 reaches 37r, n aguin becomes integral and equal to 2. We infer that every finite root of J. exceeds the correspond- ing finite root of J,;. In like manner every finite root of Js exceeds the corresponding root of J., and so on. I was led to consider this question hy a remark of Gray and Mathews {—‘ It seems probable that between every pair of successive real roots of J, there is exactly one real root of Jnii. It does not appear that this has been strictly proved ; there must in any case be an odd number of roots * ‘Theory of Sound,’ §§ 205, 207, + ZL. c. §§ 88, 92 a. t Bescel’s Functions, 1895, p. 50. to the Vibrations of a Circular Membrane. D0 in the interval.” The property just established seems to allow the proof to be completed. As regards-the latter part of the statement, it may be considered to be a consequence of the well known relation Init (2)= = In (2) Ie! (2). iuceiee LCD When J, vanishes, J,4; has the opposite sign to J,’, both these quantities being finite *, But at consecutive roots of Jn, J,/ must assume opposite sions, and so therefore must Jy41. Accordingly the number of roots of Jni1 in the interval must be odd. The theorem required then follows readily. or the first root of J,,; must lie between the first and second root of Jp. We have proved that it exceeds the first root. If it also exceeded the second root, the interval would be destitute of roots, contrary to what we have just seen. In like manner the second root of J,4; lies between the second and third roots of Jz, and soon. The roots of Jn4i separate those of ont The physical argument may easily be extended to show in like manner that all the finite roots of J,’ (z) increase con-~ tinually with n. lor this purpose it is only necessary to alter the boundary condition at »=1 so as to make dw/dr=0 instead of w=0. The only difference in (1) is that 2 now denotes a root of J,!(z)=0. Mechanically the membrane is fixed as before along 0=0, 0= 8, but all points on the circular boundary are free to slide transversely. The required con- clusion follows by the same argument as was appled to Jn. It is also true that there must be at least one root of J'n41 between any two consecutive roots of J,/, but this is not so easily proved as for the original functions. If we differentiate (2) with respect to z and then eliminate Ja between the equation so obtained and the general differential equation, viz. 2 Jn. 25,/+(1— 5) Jn=0, ye ee ea * If Jn, Jn41 could vanish together, the sequence formula, (8) below, would require that every succeeding order vanish also. This of course is impossible, if only because when 2 is great the lowest root of Jp is of order of magnitude n. + I have since found in Whitaker’s ‘ Modern Analysis,’ $ 152, another proof of this proposition, attributed to Gegenbauer (1897). re 56 Lord Rayleigh on Bessel’s Functions as applied we find (1-4) It 5 Eas 1=2)5/+(1-" 3") 3," =0. (a In (4) we suppose that z is a root of J,’, so that J,’=0. The argument then proceeds as before if we can assume that 2?—n? and z?—n(n+1) are both positive. Passing over this question for the moment, we notice that J,'’ and J',,1 have opposite signs, and that both functions are finite. In fact if J, and J,’ could vanish together, so also by (8) would Ja, and ayain by (2) Ja41; and this we have already seen to be impossible. At consecutive roots of Jn’, J,’ must have opposite sions, and therefore also J’,4;. Accordingly there must be at least one root of J’,1; between consecutive roots of Jz’. It follows as before that the roots of J’,4; separate those of Jn’. It remains to Pee that z? necessarily exceeds n(n+1). That 2? exceeds n? is well known”, but this does not suffice. We can obtain what we require from a formula given in ‘Theory of Sound,’ 2nd ed. § 339. If the finite roots taken in order be 2, 2,... Z;... , We may write log ie (z) = const. + tl aes log z+ log (1—27/z.*), the summation including all finite values of 2:3; or on dif- ferentiation with respect to z ne) n—l We" FT This holds for all values of z. Jf we put z=n, we get 2, ee =, since by (3) J, (n) +J,' a)=— In (5) all the denominators are positive. We deduce zy —n? 2n and therefore —n* 2,°—n’? pA) =1+ =, 2 De Rape Py >1; . (6) “9 —nN Zz >n?+2n>n(n+]1). Our theorems are therefore proved. ; * Riemann’s Partielle Differential Gleichungen; ‘Theory of Sound,” 210. i to the Vibrations of a Circular Membrane. aa ~ Tf a closer approximation to 2,” is desired, it may be obtained by substituting on the right of (6) 2” for 2?—n? in the numerators and neglecting n? in the denominators. Thus ' ao 2 Ua LN SOR eo eS Dasa ee Sa 2n 3 3 ) 1 ‘ a—2 ~—2 —2 — —— + >1+2n = aes + 25 + ooo arn} Now, as is easily proved from the ascending series for J,’, ee Gates | n+2 pen Re ae ae Beli ~ 4n(n+1) so that finally 3 zy 74 2, a nif ticters esumnaries ey aia (n+1)(n+2) 7) When 7 is very great, it will follow from (7) that 2° > n?+3n. However, the approximation is not close, for the ultimate form is * 2), =n? +1:02684 n*?. As has been mentioned, the sequence formula 2n Cy di, (2) ay (z) tJnay (z) oft hse (8) prohibits the simultaneous evanescence of J,_; and Jz, or of Jn-1andJ,41. The question arises—can Bessel’s functions whose orders (supposed integral) differ by more than 2 vanish simultaneously ? If we change n into n+1 in (8) and then eliminate J,, we get An (n+1 sa hale { nae —] hoan=d" , ar Int Tae as (9) from which it appears that if J,-1 and J,4. vanish simul- taneously, then either Jnii=0, which is impossible, or 2=4n(n+1). Any common root of J,_1 and Jz. must therefore be such that its square is an integer. Pursuing the process, we find that if J»_1, Jnz3 have a common root z, then (2n4+1)2?=4n (n+1)(n+ 2), so that 2? is rational. And however far we go, we find that * Phil. Mag. vol. xx. p. 1003, 1910, equation (8). 58 Mr. H. H. Poole on the Rate of the simultaneous evanescence of two Bessel’s functions requires that the common root be such that z* satisfies an algebraic equation whose coefficients are integers, the degree of the equation rising with the difference in order of the functions. If, as seems probable, a root of a Bessel’s function cannot satisfy an integral algebraic equation, it would follow that no two Bessel’s functions have a common root. The question seems worthy of the attention of mathematicians. IX. On the Rate of Evolution of Hzat by Pitchblende. By Horace H. Poorer *. A DESCRIPTION was given in a former paper (Phil. Mag. Feb. 1910) of a determination of the rate of evolution of heat by Joachimsthal pitchblende. As the results obtained were considerably greater than was to be expected, further experiments seemed to be desirable. The same method and apparatus were again employed, and for a description of them reference must be made to the previous paper. Before insertion in the calorimeter the pitchblende was gently heated for several hours on a metal plate. It was then placed while still hot in the calorimeter in which the flexible thermo-junction had already been inserted, and molten parafhu-wax was run in to completely fill uj all the interstices. The calorimeter was gently tapped till air-bubbles ceased to rise to the surface. The neck of the calorimeter was then closed as before and the whole put aside to cool. The calorimeter contained 525 grams of pitchblende. A week later the calorimeter was buried in ice in the usual way. Another couple was placed in the ice vessel with its junctions at different points in the ice, to indicate what temperature differences might occur in the ice itself. The readings of the couples are shown on the chart (fig. 1), on which the upper line indicates the temperature of the pitchblende and the lower the differences of temperature between the terminals of the second couple. As the latter was slightly more sensitive than that employed with the pitchblende, the readings have been reduced to the same scale, 2. e. scale-divisions indicated by the pitchblende couple. This and the galvanometer were the same as those previously employed, but the distance from the galvanometer mirror to the scale having been increased from 102 cms. to 107 ems., * Conimunicated by the Author. Evolution of Heat by Pitchblende. 59 the arrangement was about 5 per cent. more sensitive, so that 1° C. corresponded to about 1200 scale-divisions. The chart begins two days after the ice was packed. It will be seen that for a week after packing the temperature difference Bigs I Ae PaaS 0 . ——— ae naraom ieee ae N ~ Q wu z SESS _ = S \ 4 indicated by the couple in the ice never exceeded 0°:0004 C., and was sometimes in one direction sometimes in the other ; but as time went on the differences and irregularities became greater and greater, the maximum difference of temperature recorded being nearly 0° 0020 C. There can be little doubt that the irregularities observed in the apparent temperature of the pitchblende are to be ascribed to fluctuations in the temperature of the outer junction. These fluctuations almost invariably increase with the time that has elapsed since packing. These variations of temperature may be due to the action of the water on the zine of which the vessel is composed, or to regelation effects ; these points will be further considered later. The mean temperature between the pitchblende and the ice for the last 30 days was 8°3 scale-divis‘ons. As the heat escaping from the calorimeter in calories per hour is 6*2 times the temperature difference between the interior and the ice, 8:3 x 6°2 the leat evolution per gram of pitchblende is i900 x 598 60 Mr. B, H. Poole on the Rate of i.e. 8'15X10-> calorie per hour. This result being even higher than the previous ones, the calorimeter and pitchblende were put aside unopened for some months so that any chemical aciion which might be occurring might cease. As a short time previous to this experiment the wax had been melted, it is possible that crystailographic changes may have been occurring in the wax. During the interval, in the course of some experiments on orangite which will be described in a later paper, a new Inner ice vessel was made with comes walls and lid, the space between the walls, about + inch, being filled with cotton-wool, This vessel is watertight, and arranged so that water from the outer ice cannot enter round the lid. In these experiments the ice employed was specially frozen by an ice-making firm out of water distilledin the laboratory. As, however, the temperature variations in this ice seemed quite as large as those in the ordinary commercial ice, the latter was again employed in the subsequent experiments. The distilled-water ice attacked the zine vessel much more than the ordinary ice, as the sides were found to be coated with a thin layer of oxide. This was scrubbed off and the vessel thoroughly dried and coated while warm with vaseline. Nine months after the insertion of the pitchblende in the calorimeter it was again placed in the ice. The second couple was not employed on this occasion, as there seems to be no doubt that variations in the ice temperature will be detected by the couple used with the pitchblende, and it was thought best to have as few leads entering the ice as possible. The temperature of calorimeter is shown on the chart (A, fig. 2). As the distance from galvanometer-mirror to ( tS) —_ g NX ~ a) scale had teen reduced to 161 cms.,.1°.C.. corresponds. to about 1138 scale-divisions. As usual, the irregularities get Evolution of Heat by Pitchblende. 61 larger as time goes on, finally becoming so large that the experiment was discontinued. The mean for the last 20 days, which, however, is not very reliable, is 6°25 scale-divisions, corresponding toa heat evolution of 6°5 x 10~° calorie per hour per gram. va It is evident that in all previous experiments the first fortnivht during which the ice temperature is fairly steady is expended in the cooling of the pitchblende. So after the conclusion of the last experiment, before the pitchblende had warmed up more than 0°2 C., the ice vessel was repacked. As will be seen (B, fig. 2) the temperature became steady four days after packing, and remained so for seven days, after which irregularities again made their appearance. The mean 5°9 scale-divisions corresponds to a heat evolution of 6:2 x 10-° calorie per hour. This figure agrees very well with the mean of the previous results, 6°1x10-°. In this experiment the pitchblende had been in the wax nearly eleven months and had been approxi- mately at 0° C. for about two months. The steadiness of the final temperature attained also lends weight to the result. : The variations in the temperature of the ice may be due to traces of zinc oxide which, however, is almost insoluble in water, but, as coating the zinc with vaseline, though it stopped all apparent action. did not effect any improvement, it seems more probable that the variations are in some way caused by regelation. The ice when put in is in very small pieces and almost resembles snow in appearance; but when the vessel is opened the ice is always found to be frozen into a glassy cake evidently pierced by numerous small air spaces. -Itis hard to say whether these are isolated cavities or whether they may form connected channels through which air might circulate. No connexion could be traced between the tempe- rature variations and the occasions on which the outer ice vessel was repacked. The three experiments recorded give for the heat evolution— (1) 8:15 x 107° calorie per hour per gram. (2D DP Ss aay a ss Caoee oc Oia | Mean 6:95 x 1075 > Doe bP) 39 The mean. obtained in the previous paper was 61 x 107°; so if we take the mean of all the results we obtain 6°5 x 107°. The discrepancies between the various results are so large 62 Dr. J. W. Nicholson on the Bending of that the exact value of their mean is not very important, but the later experiments strongly support the general result before obtained, that the hext evolution is at least 6x 1075 calorie per hour per gram, instead of 4:4 x 10-5, which is the figure obtained on the assumption that each gram of radium generates 110 calories per hour. Physical Laboratory, Trinity College, Dublin. October 1910. X- On the Bending of Electric Waves round a Large Sphere. Iif.* By J. W. Nicuotson, I7.A., 1.Se.F Determination of the constant B. 9) eee present section of the paper takes up the problem of the determination of 8, the numerical coefficient which is necessary to a final tabulation of the intensity at any point of the surface of the obstacle. This coefficient is defined by the fact that the first root of Qfdz.2tKn(z)=0, «2 . . « (102) where the Bessel function involved is regarded as a function of m, has an imaginary portion —ves@, and a real portion whose most significant term is z. It was shown before that no root of this equation ovcurs whose real part has not an order < at least (if the real part is to be preponderant as supposed), and therefore that the first possible root should be sought within such a region of variation of m that | m—z| is of order 23. Now for values of m and z corresponding in this way, a development of K,,(z), suitable for the present purpose, can be deduced at once from the results of a previous papert | to which reference has already been made. For we may write, if p=(m—z)(6/c)3, and if the terms of the series converge, 1/6 (_(1\_ 7 2\ Sor p?/ 3\0 Mae Jim 2)= — |] — a es Bos @ ai rd (=) ae } Q =e) { 1(5 )eos? + Pg eee Hg 3) ooo eae the error involved being of order z~! at most. * Part I., Phil. Mag. April 1910. Part II., Phil. Mag. July 1910. * Communicated by the Author. { Phil. Mag. August 1908. Electric Waves round a Large Sphere. 63 In the paper in question, a portion of the proof assumes that m is real, but the extension to complex values whose real part is large and positive may be shown to be lawful. By reference to the original definition of K,,(z), namely, a I __ ylmr —limr ie Tea mar | J—ml2 ee In(2) be ee it may be shown that ene) Tet rs) +e") rite ® Jad. . (103) 0/02 - 2? Kn(z)=0 0/d<. zw (p) =0, where v(p) = r( 0 +e el ir )(1+eH")+ He 0/02 . z6h(p) =2*h'(p) dp[dz— anes = 2-5 {p'(p 0328 + 4H (p) since for a value of p of order unity, dp/d-= ae as in a previous section. ‘lhe first term of the last expression is therefore preponderant, and the equation for the required zero may be treated as v'(p)=0 to a sufficient order for the present purpose. We must therefore solve the equation (ae yn(s)a (FOZ) Ss He) (5)S i. Thus the equation becomes But gu 7 pP {to e jy eae: foe =0, certain of the terms vanishing. This may be written in the form pet Pe ma dpe) fr1i 4s 9 p° 5 ea lS AR S ecaneaeee ° e (104) and —78 is the imaginary part of the first root of this equation. 64 Dr. J. W. Nicholson on the Bending of In determining the first approximation to this root, we may shorten the equation to po ae yn De la ne {145 331 provided that the other terms are really negligible for the ensuing value of p. This is found to be the case. If p>=w (substituting the values of the Gamma functions), w is a root of w(1+ 2) = 27-895 (142) Bei iis. and this is a quintic with real coefficients, and therefore with at least one real root. It is found by the usual method of trial that there is only one real root, given by w= —3°115. With this value, higher approximations to the solution of the actual equation may be deduced by the method of continued approximation, but this is not necessary for the present purpose, as an examination of the neglected terms indicates at once. Thus (m—z)3° =—3'115, or ; m—Zz=—23 (1, w, w?)(°5192)3, where » and »? are cube roots of unity, given by 4(—1 +4 V3). But two of these voles may be rejected, for, p® being real, (104) indicates that p? must be proportional to —e7, or p proportional to e—s™, and therefore to 1—z v3 with a positive real part. The only root available is therefore given by m—z==12t (1—z /3) (5192), . . (4106) the others being introduced in cubing (104). Thus for the first root of 0/ z.2tK,,(¢)=0, the imaginary part is on reduction 696 7a, ee This imaginary part is negative, as stated in the previors section, and pes 696, roughly equal to 2/3*. With this value, the ratio of disturbed to undisturbed amplitude is given by (101) as (8m sin @)3 (ka)e tan $0 B- e— (ha) BO, * This was inadvertently quoted as 1/3 in the previous section, but no deduction was made from it. Electric Waves round a Large Sphere. 65 Application to ENE Boks teleyraphy. As a typical case, suitable for the tabulation of the formula (101), we may take that of electric waves and the earth, the height of the antennz being about 260 feet, and the corresponding wave-length therefore a quarter of a mile. In this case, ka is 1:01.10". When the orientation of the receiver from the oscillator is not greater than about ten degrees, or seven hundred miles, a convenient practical formula for the ratio of the diffracted amplitude to the amplitude corresponding to an undisturbed oscillator is found to be OD SIG Goma ee Maan GLO) where @ is in degrees. The value of ka corresponding to average practice has been inserted. A change of the wave- length, say from a quarter to a fifth of a mile, does not change the order of magnitude, or in general, by more than unity, the most significant figure of this formula. The variation of this function is exhibited in the following table. TABLE I. Q. Amp. ratio. Terrestrial | 9, Amp. ratio. Terrestrial tiles. miles. 19 033 Re es GSO ate 22 "054 138 Me "022 483 ae 057 Zor ne O15 552 4° ‘050 276 9° ‘O11 621 ise) 040 345 OS ‘007 690 Beyond 10°, the amplitude ratio in the same case may be computed from the formula 50:2 tan 40 v/ sin) O'574)89, 3). (109) where @ is again measured in degrees. This is tabulated below for every degree from 0° to 30°, and for every five degrees from 30° to 90°. The values given by the two formule for 2=10° are in agreement, and between 10° and 30°, an interpolation method may be used for intermediate cases. The convenient notation ‘0 m has been used for *m10-”. Piul. Mag. 8. 6. Vol. 21. No. 121. Jan. 1911. iy 66 Dr. J. W. Nicholson on the Bending of TABLE LI. | __ | Terrestrial || terteeeeeaae | 0. Amp. ratio. yh ‘ | 0. Amp. ratio. Beale | 11° 0047 ease ea 21° 048 1449 Pb fie -0030 828 92° .0'30 1518 ones -0020 897 93° ‘018 1587 ide 0013 966 24° Ol 1656 15° 0°81 1035 aes? ‘0568 1725 16° “0352 1104 | 96° 0°42 1794 17° 0332 1173 [ie as 0°25 1863 18° ‘0320 1242 | 28° 0715 1932 19° 0713 1311 - 9ge 0°93 2001 20° ‘0°78 1380 30° ‘0°56 2070 TABLE Til. Al 2) | bs 0. Amp. ratio, CEN. | 0. Amp. ratio. Terrestrial miles miles. 35° 0744 AU ness W466 4485 40° “0334 2760 | 70° 0546 4830 45° 0°25 S105) || 775° “01639 5175 50° ‘0°18 3450 || 80° ‘0 722 5520 55° 04113 Spoon) | ih) eebe O15 5865 | 60° 01394 4140 | 90° ‘010 6210 These tables are in the form which indicates most readily the amount of shadowing due to the earth at any point of its surface, without confusion with the ordinary fall of in- tensity due to distance from the oscillator. ‘The shadowing soon becomes very complete, for the ratio of the mean energies per unit volume in the two cases is given by the square of the amplitude ratio. But in estimating the capacity of diffraction for giving an explanation of the great success of experimenters, other tables are required, for the degree of sensitiveness possible to the receiver plays a determining part. Let us suppose that a given receiver will function at a distance of about seventy miles or one degree from the oscillator. Then the further degree of sensitiveness necessary, in order that it shall detect radiation at an orientation @, may be found by comparing the two cases 0=@° and @=1°, without reference Electric Waves round a Large Sphere. 67 to the undisturbed oscillator. In the following tables this comparison is made both for the amplitudes and the energies :— TaBLE IV. Q Amp. at 9°) Energy at 0° Terr. al Amp. at 6°| Energy at 0° Terr. "| Amp. at 1°} Energy at 1°] miles. Amp. at 1°| Energy at 1°| miles. pee 1 1 69 || 16°) -0996 0°93 1104 2° "812 659 LSS WT ‘0°57 0°33 1173 3° D7 1 326 ZOT || 18° 0334 sOcIst 1242 4° 378 143 Geto rs "0°20 0739 1311 a 243 059 345 || 20° 0712 ‘O"14 1380 6° 153 023 414 || 21° 0'69 0°47 1449 fa 2) ) O95 ‘0°89 483 || 22° °0*40 0716 1518 oP 058 "0°34 552 | 23° 0424 "0°56 1587 a 035 ‘0712 621 || 24° “O14 0°19 1656 10° 021 “0°46 690 || 25° "0581 0'°66 1725 ae 013 0716 W592 26°) 0747 ‘0'922-—«| «1794 12° 0°77 (U5) 828 || 27° 0°27 “Ou74 1863 13° "0746 0121 Soi 28. O21 01126 1932 | 14° 20227 0575 966 || 29° 0°94 01289 2001 ie 0716 0°26 1035 || 30° "0°55 01739 2070 TABLE V. Amp. at 6°} Energy at 9° | Terr. Amp. at 6°} Energy at 6°) Terr. 0. Amp. at 1°| Energy at 1° | miles. & Amp. at 1°| Energ y at 1 | miles. 35°} 037. | 013 «| 2415 | 65°) 0127 074 | 4485 40° ‘0°24 O59 2760 || 70°|) = -01517 03130 4830 45° ‘0°16 025 3105 | oxen SOL 5033112 5175 A0° sORLO 0711 3450 || 80°) -0'869 03°47 5520 55°| -01266 or44 | 3795 | 85°] -043 019 | 5865 | 60° 01343 07°18 4140 | SOF 27 0°73 6210 | , The orientation taken as a standard of comparison must be such as to make the distance from oscillator to receiver large in comparison with the height of antenna. ‘This condition is satisfied sufficiently by taking 1° as the standard. It is difficult to draw any conclusions from these tables for small orientations, ous the lack of a reliable series 1 2 68. Bending of Electric Waves round a Large Sphere. of quantitative measurements of the effect at the receiver for different distances from the sending apparatus. But when — such measurements are available, it will be possible to decide at once what proportion of the observed effect is due to diffraction round the surface. That diffraction may be a very important factor for small orientations is not con- tradicted by the above figures, although the investigation by Sommerfeld* of the effect of conduction through the earth must be borne in mind. But the smallness of the numbers corresponding to larger orientations shows very clearly that, as already pointed out, diffraction must be a relatively insignificant agency in the success of experiments such as those of Marconi. For example, the ratio of the energy falling in unit time ona receiver at a distance of 2000 miles, to that falling of the same receiver at a distance of 70 miles, if diffraction alone were the agency, would be of order 10”. It is improbable that any receiver could be sensitive enough to record so small an effect, and we are compelled to seek an explanation of the experiments elsewhere. Two alternatives have been proposed. The first is that of Sommerfeld, who, by taking an infinite plane to represent the surface of the earth for convenience of mathematical analysis, concludes that tha finite conductivity of the earth may be sufficient to account for experimental success. But the simplification thus intro- duced into the analysis may render the results, at the same time, inapplicable to large orientations, and for these great distances a more rigorous investigation is desirable. Un- fortunately the method used by Sommerfeld does not : appear to be applicable to the case of an obstacle with a spherical boundary. The other alternative is the hypothesis that upper layers of the terrestrial atmosphere may, by being rendered conducting, reflect back to the earth the radiation which they receive from the sending apparatus. This seems to furnish a hopeful line of attack on the problem, and the view is supported by several known experi- mental results, in particular by the difference experienced in signalling by night and by day. But no investigation of the matter from a theoretical standpoint has yet been published. The next section of the paper is concerned with the deter- mination of a higher approximation to the effect 1 in the region previously called the “ region of brightness.” * Ann. der Phys. 1909, March 16. LG i XI. New Determinations of some Constants of the Inert Gases. By Cuive Curapertson, fellow of University College, London*. ie a recent paper in the Philosophical Magazine, Dr. G. Rudorf f calculated some of the molecular constants of the inactive gases with the object of seeing how far the values obtained by various methods agreed among themselves. The results were disappointing. Since that time three papers { on the physical properties of these gases have provided additional data for their study, and a comparison of these figures, obtained by different methods, points to some interesting conclusions. RELATION BETWEEN DETERMINATIONS OF VOLUME OF MoLECULES FROM VISCOSITY AND REFRACTIVITY. On certain assumptions it is possible to calculate the volume occupied by the atoms of a gas both from the vis- cosity and from the refractivity. We find in Rankine’s first paper the first complete set of measurements of the viscosity of the five inert gases. The experiments were made at atmospheric temperature, with the same apparatus under similar conditions. The results are therefore of high value. Rankine gives the figures shown in the first columns of the table below for the relative viscosity, mean free path, and molecular radius of the five gases, taking the figures for helinm as unity in each case. He adds that, in order to obtain the absolute values of the radii, each number in the third column must be multiplied by 1°68x10°. The number of molecules per unit volume * Communicated by the Author. + “The Molecular and some other Constants of the Inactive Gases,” by G. Rudorf, Ph.D., B.Sc. Phil. Mag. June 1909, p. 795. t “On the Viscosities of the Gases of the Argon Group,” by A. O. Rankine, B.Sc. Proc. Roy. Soc. vol. Ixxxiii. p. 516 (1910). “On the Variation with Temperature of the Viscosities of the Gases of the Argon Group,” by A. O, Rankine, B.Sc. Proc. Roy. Soc. vol. Ixxxiv. p. 181 (1910). ‘On the Refraction and Dispersion of Argon and Redeterminations of the Dispersion of Helium, Neon, Krypton and Xenon,” by © -* M. Cuth bertson. Proc. Roy. Soc. vol, Ixxxiv. p. 18 (1910), 70 Mr. Clive Cuthbertson on New Determinations of TABLE I. if 2) le eat 4, Bes: 6. a. Relative | Relative Ce ce A 22 ie THe Relative | mean free | molecular teat volume (Ha —1)5. , a viscosity. path. | radius. x 108. Pee | Ree 6. a | Helium ...... 1-000 1000 1000 4 0000695 0000231 | 30 Neon 2.2 1-585 0704 | 1:19 ‘9996 | -0001171 | 0000444 | 2:53 Argon ...... 1-124 0°354 1-68 14112 | -0003296 | 0001848 | 1-785 Krypton ...) 1-253 0274 | 1°91 16044 | -0004844 | 0002791 | 1733 | i Xenon. ..... 1136 0198 | 2:25 1:890 | -0007918 0004545 | 1-742 | } is taken as 2°8x10%. The calculation is based on the relation L= Aygo . Niro”, where L is the mean free path and o is taken to be the molecular radius. It is, however, more usual to take o as : the radius of the sphere of action of the molecule and equal | to the molecular diameter. Hence the absolute value of the molecular radius is given by multiplying Rankine’s figures by | 168 x 10° 2 - | The sum of the volumes of the molecules per unit volume of gas is ~ Atra* N—. 3 These figures are given in the fourth and fifth columns of the table. Turning now to the refractivities, we start from Clausius’s well-known equation ee tot 7 ave? where g denotes the fraction of the volume containing a gas | which its molecules actually occupy. This becomes some Constants of the Inert Gases. (il In the case of gases, where »—1 is small, 9 — 5 (4 —1) approximately. The refractivities of the inert gases given in the paper quoted above are shown in the sixth column of the table, D) ; multiplied by 5 and in the last column are shown the ratios of the molecular volumes as determined from the viscosities to those derived from the refractivities. The constancy of the ratio in the cases of argon, krypton, and xenon is re- markable, and, considering the dissimilarity of the two methods and the number of assumptions upon which they rest, the concordance between the two sets of values in absolute measure is not less surprising. RELATION BETWEEN THE NUMBER OF HLECTRONS IN THE ATOM AND THE RADIUS OF THE SPHERE OF ACTION. Till recently the refraction and dispersion of gases have usually been expressed in terms of Cauchy’s formula AB sie 9 fp—l=A+ 5+ 57.-.-- But in view of the success which had attended the use of a formula of Sellmeier’s type N Ny? — 2? ae in explaining the observed facts of dispersion in other cases, it was thought better to express the refractivities of the inert gases in this form in the paper quoted above. Here N is a constant, my is the frequency of the free vibration of the parts of the atom (assuming there to be only one) and n that of the light whose refractivity is measured. When this equation is transformed into the shape it assumes on the electronic theory, as developed by Drude, N becomes proportional to the number of electrons in the molecule which ure effective in influencing disperston. Table II. shows the values of the constants N and n,? obtained experimentally *. * In the paper quoted above the values for N were doubled for com- parison with similar figures for diatomic gases. The numbers here given are those calculated from the actual observations by the method of least squares, 72 Mr. Clive Cuthbertson on New Determinations of Papin i. il 2. Element. Nx 10-27. m2 10727, lela 2eoeenccececee ce 1:21238 34992 INCOR ee tecee een 2:59826 38916 AT CONG en cneeee net se 4°71632 17009 Kary piOms erect rere ee 53446 12768 EXC M Gly. Secrescee see 6°1209 8978 If the figures in the first column, which are proportional to the number of “dispersion electrons”? in the atom, are plotted against the square roots of Rankine’s values for the relative mean free path at 0° and 100° C., we obtain two Fig. 1.-—Inert Gases. Relative number of dispersion electrons plotted against square roots of relative mean free paths at 0° and 100°. | ELECTRONS ii ‘| 4 “5 -6 7 8 “9 1-0 II 12 : J MF. PATH. eurves which are shown in fig. 1. The radius of curvature is directed away from the origin, but the lines are so nearly some Constants of the Inert Gases. 73 straight that for present purposes the relation may be con- sidered linear. The numbers used for the mean free paths are given below. Jue 1UOE Square roots of mean free paths (Rankine). Element. a At 0° C, AE LOOC: LEIGINUDT SO esas ek eet rae 1 Heya COM ee oxo ke ‘839 927 HMR OM asin nae osiaeeciserces HVS 674 HOEY AOEOM ais: seltiuirecielsie vec 023 598 MEMORY resco. ce ccen cc ve “445 515 On the kinetic theory in its simplest form the root of the mean free path is inversely proportional to the radius of the sphere of action. To a first approximation, then, we find in these five gases a linear relation between the number of electrons influencing dispersion and the reciprocal of the radius of the sphere of action as determined from measure- ments of the viscosity. Another interesting result is obtained by plotting N? against o, the reciprocals of the numbers given in Table III. These numbers are shown in Table LY. TasLEe 1V.—Relative radii of the spheres of action of the inert gases at 0° and 100° C. At 0° O. At 100° C. JEIGINHN IT Ry onenae enc rene 1:0 | “902 INGOT css carat aus a, 1:19 1082 ANTE OM Rael dee sinus nse bets apl 168 1-482 | [cesraten Ne ae eee | Lvl 168 | Co a | 2:25 | Lod 74 *, Clive Cuthbertson on New Determinations of tg 2 shows that this relation also is approximately linear *. It denotes that the squares of the number of Fig. 2.—Inert Gases. Relative number of electrons squared (N*) plotted against radii of spheres of action (7) at 0° and 100° C. . an 1-5 2. 25 pe “dispersion” electrons are proportional to the radii of the spheres of action of the atoms diminished by a constant. This constant is about 95/100 of the radius of the sphere of action of helium at 0° C. RELATION BETWEEN THE NUMBER OF ELECTRONS AND THE CRITICAL TEMPERATURE. If the squares of the relative numbers of “ dispersion ” electrons are plotted against the critical temperatures of the gases, the four known points fall nearly on a straight line * For obvious mathematical reasons these relations cannot both be exactly linear. But they exhibit the broad relationships of the constants sufficiently well for the present. some Constants of the Inert Gases. io which passes near the origin (fig.3). The numbers employed are given in Table V. Fig. 3.—Inert Gases. Relative number of electrons squared (N’) plotted against critical temperatures. Ne £0 30 20 cine 0 . 100 200 S005): ABSOLUTE TEMPERATURE TABLE V. Relative number of | Critical Temperature Electrons squared. (absolute). | ie raw | | ELS Luu en eae eracccin | 1:46 5 DINE ee 6:7 — Moonie. ke sec) | 20°97 155'6 PaKasypton ee eceesieet ao: | 28:57 210°6 RPNeNONS (Pan cars ascruetes | 375 287°C | | | The critical temperature of neon is unknown: but caleu- lated from the ordinate it should be in the neighbourhood ot 46° A. 76 Mr. Clive Cuthbertson on New Determinations of Since it has been shown above that a linear relation also exists between the squares of the number of electrons and the radius of the sphere of action, it follows that a similar relation exists between the radius and the critical temper- ature. This is shown in fig, 4. The agreement is even better than in figs. 2 and 3. Fig. 4.—Inert Gases. Critical temperatures plotted against radii of spheres of action at 0° and 100° C. ABSOLUTE TEMPERATURE POL Ys eae = A | | | | 200 aaa RELATION BETWEEN THE NUMBER OF ELECTRONS AND THE CONSTANT C in SUTHERLAND’S FORMULA. In his second paper Rankine shows that the temperature coefficient of the viscosity may be of great importance. Using Sutherland’s formula n=KT/(1+ 9), where T is the absolute temperature and K and C constants for the gas, he obtains the following values for C from two observations of the viscosity :-— C. Relies ania. Ais ae weet oe 70 INV CIC aT | Pee peak meer 56 EASON ITC Rat caer ose 142 BSI pOM ent 188 Wen Om ee Oo be ete: 252 some Constants of the Inert Gases. 17 These numbers he plots against the critical temperature, and obtains a straight line running through argon, krypton, and xenon, which passes very near the origin, but helium is far off the line. } Since the critical temperatures are now shown to be proportional to the squares of the number of electrons, it follows that for argon, krypton, and xenon, C in Sutherland’s equation is nearly proportional to the square of the number of electrons. Remarks. These results throw additional light on Rankine’s discovery that the constant in Sutherland’s formula for the temperature coefficient of viscosity is intimately connected with the critical temperature ; and they suggest that the forces which, by acting between the atoms, cause both these phenomena, are to be found in the electric charges of the electrons which influence dispersion. Further they show that there is an intimate relation between the number of electrons in the atoms of these five gases and the radii of their spheres of action at the same temperature. One indirect inference may further be noted. Doubts have been felt whether it is permissible to express the refractivity of a gas by a single term of a formula of Sellmeier’s type, since this is equivalent to the assumption that the electrons influencing dispersion are all of the same type and have the same free frequency ; and it was recognized that such a simplification was only tentative. The facts that figures derived from this formula can be correlated with others based upon the kinetic theory of gases is evidence that, at least in the case of these gases, the hypothesis is substantially correct. Applacation to other Gases. The relations exhibited above cannot be extended to hydrogen, oxygen, or nitrogen. The best agreement is found in the case of the relation between the volumes, as determined from measurements of viscosity and refractivity respectively. The ratio in the case of oxygen is the same as in that of argon, krypton, and xenon, while in that of nitrogen it is not far different. But hydrogen departs widely from this value. It is probable that in the diatomic gases the conditions are more complicated than in the case of the argon group. [ea XII. On Magnetostriction. By R. A. Houstoun, M.A., DSc., Ph.D., Lecturer on Physical Opties in the University of Glasgow™. | ee object of this paper is to derive a relation connecting magnetostriction with the change of magnetization produced by stress and to test it by the results of experi- ment. In substance the relation is not new but in method of statement it is, and the derivation presented here is shorter and simpler than other methods. ‘The paper also proves in a new and simple way a theorem due to Lord Kelvin. Consider a ferro-magnetic wire hanging vertically inside a vertical solenoid with heating jacket, a pan being attached to the lower end of the wire for hoiding weights. Then, if hysteresis be neglected, the state of the wire may be regarded at any time as a function of the three independent vari:bles T, F, and H,—temperature, stretching force, and magnetic field intensity. If T, F,and H suffer small changes, then the heat received by the whole wire is given by dqg=cdT + bd¥ + adH. Let B denote the induction in the wire, v its volume, and x the vertical displacement of its lower end. Then the work done on the wire when F and H are increased is Fda+vHdB/47. Let U be the intrinsic energy of the wire and § its entropy. Then— ‘HdaB dU =dq+Fde+ ae: ei Ot ©) elem \ ox vHdB — (c+F Sr + ie SE Et (0+ F 5+ = sr )ae Oz . vHOB and dS= fal + ndl + Gall Since these are perfect differentials, we have the following six independent relations :— oc. oz Wirele 1 oF + or wok . . . . e ° ( ) ce, OB oe Oe dt ae (OLE L 5 Maire ie Beatie * Communicated by Prof. A. Gray, F.R.S. Part of the matter of this paper has appeared in “On Two Relations in Magnetism,” Proc. Roy. Soc. Edin. vol. xxx. p. 457 (1910). Dr. R. A. Houstoun on Mugnetostriction. 79 Oe CE ec) Sean Velanoel, |. 13e_ 9(b)_1db_b : Tor at\h)=T3e7 TL “) 1 d¢ = 2 (t)=aor- @ (5) TEL Cw) mW aL and 0b _ 0a (6) Saar ae The differential coefficients of a, b, and ¢ cannot be easily determined experimentally ; hence we eliminate them by combining (1) with (4), (2) with (5), and (3) with (6), and obtain or _ b aI Oba e Amo Tl and v OB oe dn gk OH The first of these is the well-known relation between the coefficient of linear expansion of a wire and the cooling effect produced on stretching it. The second relation states that if the induction in a wire increases with temperature, a is positive, and consequently the temperature will fall if H is increased; if the induction diminishes with temperature, the temperature of the wire will rise if H is increased. This theorem has been derived before by Lord Kelvin,* and in quite another way. He states the result differently,—that a substance in which the magnetism diminishes with temperature when drawn gently away from a magnet experiences a cooling effect ; a sub- stance in which the magnetism increases with temperature when drawn gently away from a magnet experiences a heating effect. This cooling and heating effect is probably always masked by the irreversible heating due to Foucault currents in the wire, and to the viscous resistance to the motion of the molecular magnets. The third relation is the one that the paper is mainly con- cerned with. It states, that if the induction increases when the wire is stretched, the length of the wire increases when it is magnetized, and vice versa. The connexion between * “On the Thermo-elastic, Thermo-magnetic, and Pyro-electric Properties of Matter,” Phil. Mag. [5] y. (1878) p. 4. 80 Dr. R. A. Houstoun on Aagnetostriction. magneto-striction and the effect of stress on magnetism has been worked out several times, amongst others by J. J. Thomson in his ‘ Applications of Mathematics to Physics and Chemistry.’ The result nearest mine is given by A. Heyd- weiller *. His equation is Ole) plo Biel i dieteyen: E being Young’s modulus for the wire and p the stress per unit area of cross-section. My equation can be pat in this form. Heydweiller starts with another term in the expres- sion for the external work done on the wire, namely, HBdv/47, but he makes approximations afterwards, which are equivalent to neglecting this term. Heydweiller’s formula has been proved experimentally by H. Rensing}f; and has been attacked by R. Gans f. On the suggestion of Dr. C. G. Knott I have tested the third relation with the data of Nagaoka and Hondag. Before stating the results of the test I shall, however, prove the relations in another way. ¢ a 4 i) t] ' ’ f 5 1 ‘ 1 U i Od Suppose that the magnetic field intensity remains con- stant and that the wire is carried through the cycle depicted in the adjoining diagram, AB and CD being isothermals and * “Zur Theorie der magneto-elastischen Wechselbeziehungen,”’ Ann. d. Phys. {4} xii. (1908) p. 602. + “Ueber magneto-elastische Wechselbeziehungen in para-magnetischen Substanzen,”’ lan. d. Phys. [4] xiv. (1904) p. 363. t “* Magneto-striktion ferromagnetischer Korper,” Ann. d. Phys. [4] xiii. (1904) p. 634. ‘‘ Zur Heydweillerschen Kritik meiner Formeln betreffend Magneto-striktion ferro-magnetischer Korper,” Ann. d. Phys. [4] xiv. (1904) p. 688. § ‘*On Maepneto-striction.” HH. Nagaoka and K. Honda. Phil. Mag. (5; xlvi. (1898) p. 260. Dr. R. A. Houstoun on Magnetostriction. 81 BC and DA adiabatics. On AB a quantity of heat qg is taken in; on CD a quantity g+dq is given out. The area of the figure is equal to the work done on the wire in the cycle and this is equal to dg. Hence dg—AC—AK—=AG x BJ. Also q/T=dq/adT. Therefore, eliminating dq, AG x BJ =qaT/T. But g, the heat received in the isothermal AB, is equal to bdF, i. e. bBJ. Substituting, 6 AG Mh ae But AG is the alteration of x, F constant in going from the one isothermal to the other ; hence b _ de eR Ole The second relation can be proved in an analogous manner by using the B, H diagram. If hysteresis be not disregarded, the cycle becomes irreversible, and in place of g/T=dq/dl we must use q_ 9t49 <9, a ey, Also the figure is no longer a parallelogram as the shape of the isothermals and adiabatics is different according to the direction in which the wire is being put through the change. In order to prove the third relation, assume that the wire is put through the cycle represented in the two following diagrams. hee NG The temperature is kept constant. The two diagrams represent the same cycle, the points ABCD in the one figure corresponding to the points A’B’C’D’ in the other. Pht Wags. 6. Vol. 21. No. 121, Jan. 1911; G 82 Dr. R. A. Houstoun on Magnetostriction. The area ABCD represents the work done against the stretching force, and v/4a multiplied by the area A/B/C'D' | the work done by the magnetic field intensity during the cycle. Therefore, by the principle of energy AROD= —— A'B'O D. 4c Now ABCD = dF 22) gH: land A’BC'D' aan ee oH OF Hence substituting and dividing out by dF dH, heal 4n OF OH’ Thus the third relation follows solely from the principle of energy ; the second law of thermodynamics has not been used at all, This is not apparent from the former method of proof. In the paper by Nagaoka and Honda measurements are given on the change in length of an ovoid of iron and a nickel rod of square cross-section when magnetized in the direction of the axis. ‘The major axis of the ovoid was 20 ems. long and its minor axis 0°986 cm.; the length of the nickel rod was 26 ems. and its breadth was 0°514 ecm. Measurements are also given on the effect of longitudinal pull on the magnetization of the same nickel rod and on the magnetization of a rod made from the same iron as the ovoid. Since we are dealing with nickel and iron we may write B=47I. ‘The relation to be tested then becomes » ol _ oe OF OH PMB Od ae OL Op 7 OE if a be the elongation per unit length and p the stretching force per unit area of cross-section. The results are given by the following curves (fig. 3). The curves marked by circles give OI/dp and those marked by crosses Qa/OH. They shou!d of course coincide. The differential coetticients were formed by taking the differences of the successive figures in the tables in the article. ‘The smaller value of Sp, 0°19 kg./sq. mm. was taken. The agreement, such as it is, is as good as that given by Kirchhoff’s theory which Nagaoka and Honda tested numerically, Law of Chemical Attraction between Atoms. 83 Similar relations can of course easily be derived for other forms of stress, torsion or hydrostatic pressure. We have only Fig. 3. Iron. 3 2 anion ays to substitute for F the twisting couple or hydrostatic pressure and for dx change of angle or of volume. But the change of magnetization produced by hydrostatic pressure is only of the same order as the change of volume produced. Hence in this case no agreement can be expected; the order of magnitude is, however, given correctly. XI. An Investigation of the Determinations of the Law of Chenucal Attraction between Atoms from Physical Data. By R. D. Kureman, D).Sc., B.A., Mackinnon Student of the Royai Society *. ‘JN this paper the writer proposes to investigate to what extent the law of chemical attraction can be derived from latent heat, surface tension, and other physical pro- perties of substances ; and an endeavour will be made to place the whole subject on a sound mathematical basis. ‘This is highly desirable as different investigators have obtained ditterent laws for the attraction. These laws will be briefly discussed and compared with the results obtained in this paper. ~ mere aaa py the Author, 84 Dr. R. D. Kleeman: Determinations of the Law of It can be shown strictly mathematically that it is impos- sible to determine completely the law of attraction between — atoms from latent heat or surface tension data ; in other words, the law deduced must contain an arbitrary function of the distance between the attracting molecules and their tempe- rature. Thus, let yw, (z, T) denote the internal heat of evaporation of a liquid into a vacuum, where 2 is the distance of separation of the molecules in the liquid and T is the temperature, which is supposed to be deduced from the true law of molecular attraction on the supposition that the in- ternal latent heat is the work done against the attraction of the molecules on their getting separated *. Since z may be expressed in terms of the density of the liquid this expression may be written Wr (p, T). The above statement can most Fig. 1. conveniently be proved graphically. Let the curve A, A; o fig. 1 denote the graph of the equation L=, (p, T) fora * We are assuming that the attraction between two molecules depends on their temperature as well as on their distance of separation. Chemical Attraction between Atoms from Physical Data. 85 given substance at the temperature T,, where L denotes the internal latent heat, and let B, B, denote the graph for the temperature T,, &c. Let the abscisse of the points a, bj, ¢, ..., denote the densities of the substance in the liquid state in contact with its saturated vapour at the temperatures i ks... aud the abscissee of the points ao, 05, ¢,..- :, denote the corresponding densities of the saturated vapour. The internal latent heat of evaporation of the liquid at the temperature T, is then the difference between the ordinates of the points a, and ag, and so on for the other temperatures. Now the finding of a formula for the latent heat consists in finding an equation for a series of curves Xj, Xg,.., which pass through the points a, dy. ), by, ..., as shown in the figure, but which need not coincide with any other points or parts of the curves A, A», B, B,.... It is obvious, then, that an infinite number of sets of such curves ean be found to each of which corresponds a formula for the latent heat. And since each formula corresponds to some law of attraction between the molecules (for given a law of attraction and we can at once deduce a formula for the latent heat), an infinite number of laws can be obtained in this way, but we cannot be sure that any one of them represents the true law of attraction without further evidence. It follows, therefore, that the law deduced from latent heat data should contain an arbitrary function. We have considered the most general case in latent heat in assuming that it is a function of the temperature as well as of z orp. But the same conclusions hold if we could prove that the attraction between two molecules a given distance apart is independent of the temperature. Let the equation of the latent heat of evaporation of a substance into a vacuum in this case be L=.(p), where p denotes its density, and the curve A,’ A,’ in fig. 2 its graph. Let the difference between the ordinates ¢ a2, b,. b,..., denote the latent heat of a liquid corresponding to the temperatures Tj, T,,... Let the curve N,, Ns, possess the property that the difference between the ordinates of ¢, and ay is equal to the difference of the ordinates of ¢, and a, in the curve A,', Ay’, and the difference between the ordinates of b, b, in the curve N, N, is equal to the difference of the corresponding ordinates in curve A,’ A,’, and so on. It is obvious that the curve N, N. subject to these conditions may have an infinite variety of shapes. Now the equation of the curve N, Nz gives the Jatent heat in terms of p, and as the curve may have an infinite variety of shapes there are an infinite number of sach equations. If the curve is expressed by a single equation it must contain an arbitrary function ; and the law of attraction 86 Dr. R. D. Kleeman: Determinations of the Law of to which it corresponds must therefore also contain an arbitrary function. } t f f j ? { 1 Next let us consider the determination of the law of molecular attraction from surface tension data. The surface tension of a liquid may be defined as the work done against the molecular attraction per cm.? of new surface formed in cutting a thick slab of liquid into two parts and separating © them by an infinite distance. If the liquid is surrounded by vapour of density p. we may suppose that an amount of matter of density p, remains stationary in space when the slabs are separated, which is equivalent to separating two slabs (not surrounded by vapour) each of density (p;—p,)*. Jiet the equation of the surface tension of a substance for any value of (py—pz) or ps, deduced from a knowledge of the law of attraction between the molecules, be A=wW; (T, ps). * This is strictly admissible only if matter consists of molecules in- finitely small in size, but devia icns due to this not being the case will occur only when the liquid and vapour have nearly the same density as occurs near the critical point. Chemical Attraction between Atoms from Physical Data. 87 sppese the vcurves As" As By! By)......5 im fig. 3 represent the graphs of this equation corresponding to the famperatures I), T,,.... het the abscissse of the points dz, b3,..-, denote the values of ps or (p;—psz) of a liquid in contact with its saturated vapour. ‘The ordinates of the points then give the surface tension at different temperatures. ‘The equation of a set of curves which pass through the points dz, bs,..-, is a formula for the surface tension. An infinite number of such sets of curves can be obtained. It follows, therefore, in the same way as before that the law of molecular attraction deduced from surface tension data should contain an arbitrary function. Next let us suppose that the attraction between two mole- cules a given distance apart is independent of the tempera- ture. Let the graph of the equation for the surface tension in this case be represented by the curve Ay’ A,'’’, in fig. 4, the points a3, 3, ..., having the same meaning as before. The equation of the surface tension of a liquid in contact with its saturated vapour is the equation of the curve through the points a3, 63,.... Now this equation can be approxi- mately found by trial or by the help of the Calculus of Finite 88 Dr. R. D. Kleeman: Determinations of the Law of Differences. It will represent the whole curve Aj!’ A,’”’ with a degree of closeness depending on the extent of the Fig. 4. part of the curve represented by experiment. On the above supposition we could thus approximately determine the law of molecular attraction. But if we arenot able to prova that the attraction is independent of the temperature the law deduced should involve an arbitrary function. The above result is, however, of importance, as it will enable us later to determine whether the attraction is a function of the temperature or not. We have then that an infinite number of formule for the surface tension or latent heat of a liquid can be obtained, each of which corresponds to a Jaw of molecular attraction, and that we cannot be sure without further evidence whether any one of the laws happens to be the true Jaw. All these laws are included in the general law that can be deduced which contains an arbitrary function. It is owing to this fact that different investigators of molecular attraction have obtained different results, and in some cases results have beet: deduced concerning the potential energy of a molecule and the origin and nature of its attractive forces which are quite absurd. Chemical Attraction between Atoms from Physical Data. 89 We will now iliustrate the result obtained in this paper by some examples. Mills* has put forward the theory that the molecular attraction which accounts for latent heat &c. is independent of the temperature and varies inversely as the square of the distance between the attracting molecules. This gives for the latent heat the equation MeO (one sab eed Whey ines) Ck) where C is a constant, which was found to agree well with the facts. According to what has gone before it should be possible to discover any number of laws of attraction which will give latent heat formule agreeing with the facts. Thus if we assume that the attraction varies inversely as the seventh power of the distance of separation of the molecules we obtain the equation L= W(p}—p3) for the latent heat, where W isa constant. This equation will at once be obtained bv substituting = for $(2)(ZV m,)? in the eeneral formula for the latent heat given by the writer +. This equation agrees } with the facts as well as that of Mills, and we ean therefore with equal justice assume that the attraction varies inversely as the seventh power of the distance of separation of the molecules. It can be very simply shown that the law of Mills cannot possibly account for the magnitude of the latent heat of evaporation and cannot therefore be true. The attraction between two molecules according to the law of Mills is Ke 2 where K, is a constant which is the attraction between the molecules at unit distance apart. Substituting this expression for the attraction for $(z)(=“m,)? in the general formula for tke latent heat quoted above we obtain K aH 4 ( 1/3 7 296, SS RT —— » Aue Pi P where L is the latent heat per unit mass in ergs and m is the mass of a molecule. By means of this equation the value of K, can be calculated. In the case of ether at 273° C. we have ipo Lox 4:2)x 10" eros, m—74 x 161 10-> om. pr= 1362, and p.,='0,827, which on substituting in the equation gives K,=89 x 1077! dyne. Since the gravitational attraction also varies in- versely as the square of the distance of separation of the * Journal of Phys. Chemistry, vol. vi. p. 209; vol. viii. p. $83 and p- 593; vol. ix. p. 402: vol. xi. p. 594 and p. 182. t Phil. Mag. May 1910, p. 801. { Phil. Mag. Oct. 1910, p. 678. 90 Dr. R. D. Kleeman: Determinations of the Law of molecules, it should be identical with the attraction we are considering. Now the gravitational attraction of two ether molecules one cm. apart is 1:84x10-*! dyne, whieh is a much smaller quantity than K,. Thus the gravitational attraction would have to be about 102! times its actual value to account for the latent heat of evaporation. It follows therefore that the molecular attraction must decrease at a much greater rate with the distance from the molecule than that given by the inverse square law. That Mills’ law cannot be true follows also from surface- tension considerations. If = is substituted for f(z)(& Vm)? in the general formula for the surface-tension given by the writer * we obtain X= D(p,—p,), where D isa constant which varies only with the nature of the liquid. In the case of chlorobenzene we have D=1°84 at 423° C. and D=-89 at 533° C. D is thus by no means constant, and the law there- fore not true. Several investigators of the law of attraction between molecules have assumed that it must be such as will make the internal heat of evaporation per gram of liquid inde- pendent of the mass of liquid allowed to evaporate completely. But it is not at all necessary that this condition need be fulfilled. Consider a spherical mass of liquid to evaporate till none is left. It is evident then that the energy expended to remove a molecule to infinity depends on the mass of the liquid only when the diameter of the sphere of liquid is equal to the radius of the sphere of action of a molecule, which is taken to be the distance of separation between two molecules for which their potential energy is small in com- parison with that corresponding to the distance the molecules are separated in the liquid state. Now the writer f has shown that the radius of the sphere of action of a molecule is of the same order of magnitude as the distance of separation of the molecules in the liquid state. It follows, therefore, that the latent heat will depend on the mass of liquid allowed to evaporate only when it is equal to about 10-4 gram, and the latent heat of a gram of liquid therefore independent of external conditions, as has been found to be the case. The law deduced from surface-tension or latent heat we have seen should contain an arbitrary function. But still the law may contain a good deal of useful information. Thus it might happen that the arbitrary function is an arbitrary function only of certain quantities or of definite functions * Phil. Mag. May 1910, pp. 789-792. 7 Phil. Mag. June 1910, p. 840. Chemical Attraction between Atoms from Physical Data. 91 of certain quantities. Further, the known part of the law may contain quantities which were excluded from the arbitrary function. Now the law of attraction between two molecules of the same kind deduced by the writer * is 2 T\(3V%m,) Were where z is the distance between the molecules, > 7; is the sum of the square roots of the atomic weights of the atoms oe Ne : BRE of a mo'ecule, and ¢, ie T) is a quantity which is the same sana for all molecules at corresponding temperatures. The “3 a A quantity b,( =, r) must therefore be a function of the € Cc ratio of the distance between the molecules to the distance of separation in the liquid.state at the critical temperature, and the ratio of the temperature of the molecules to the critical temperature. The function 1s arbitrary as its form cannot be determined from the data from which the expression for the law was obtained. We see that it does not contain = e heat quoted, we obtain L=H (p;"" —p””), where H is a con- stunt which should depend only on the nature of the liquid. But EH or itor Sn” is not constant for the same liquid, as is shown by Table J. The attraction between two mole- cules a given distant apart cannot therefore be independent of their temperature. TaseE I. | i Carbon tetra- Methyl Benzene. Chlorobenzene. allocides fotmete SER R Nee 853 | 4383 | 503 | 423 | 473 | 533 | 363 | 433 | 503 | 803 | 363 | 423 — 46-7) 46-7) 485) 21:3) 21:1| 21-4] 40:1] 40:3] 41:5 | 27-5] 27-4] 27-9 (0) Po) | L | | | pi/5—piv/a'-'| 198 | 210 | 267 | 75-4] 843) 102 | 11°6| 139] 18-0) 123 | 189 | 174 1 2 * The values of A contained in the tables in this paper are taken from a paper by Ramsay and Shields, Phil. Trans. of the Koyal Society, vol. clxxxiv. p. 647 (1893), and the values of L, e,, and p,, are taken from the papers by Mills quoted previously, who calculated values of L by Clapeyron’s equation using the density and pressure data of Ramsay and Young. Chemical Attraction between Atoms from Physical Data. 93 We have already obtained évidence of this before. The writer * has deduced a formula for the radius of the sphere of action of a molecule on the supposition that matter does not consist of molecules but is evenly distributed in space. The radius of the sphere of action obtained on this supposi- tion decreases with rise of temperature. The decrease of the attraction with rise of temperature may be partly direct and partly indirect. Indirectly a change could be, and probably is, produced in the following way. We have obtained some evidence Tf that the deviation from the additive law for the attraction of a molecule is due to an interaction between the atoms which decreases their attrac- tion. Now the atoms in a molecule are probably in rotation round the centre of gravity of the molecule, equilibrium being maintained between the centrifugal forces and the forces of attraction. The speed of rotation very likely increases with an increase of temperature as this produces an increase of the internal energy, and in order to maintain equilibrium between the forces the molecule must contract. This brings the atoms still further under each other’s influence, which decreases the force of attraction of the molecule at an external point. 2 =) boo, If we assume that is a function of the temperature only so that it may be written bo( = the equation for the latent heat becomes i L= Aid (7) Gi [os 9) and that for the surface-tension (an ce i Ashe (ple =O es where A, and A, arenumerical constants. If the assumption is true, the value of the expression 4/3__ 4/3 r(pi! P2 ) No L(p1— pz) should be independent of the temperature. This is, however, * Phil. Mag. June 1910, pp. 840-846. + Phil. Mag. May 1910, pp. 798-801. J4 Dr. R. D. Kleeman: Determinations of the Law of not the case, as is shown by Table II. The expression (er) is therefore a function of the distance between the molecules as well as of the temperature. TABLE IT. Carbon Methyl B ¢ : Saree Chlorobenzene. tetrachloride. formate. seeceeeeeees 353 | 433 | 503 | 423 | 473 | 583 | 363 | 483] 503 | 303 | 363 | 423 se Li... .eeeeeeeseeees 85°6 | 69°7 | 50°5| 65-8| 58°83} 49-1] 40-6] 33:3| 23-7] 107 | 85-1} 64-0 4/3 4/8 (pj —p9") | pes Lopap,) | 248 | 210 | 140 | +279) +243) 184) “341 | 270} -171) -228) 190] -135 On W125: If it could be proved that the above function consists of two factors, one a function of the temperature and the other of the distance between the molecules, say (9 i )=a(S =) el) then the form of both. of) mt 0(F) could be completely determined. The formule for the latent heat and surface-tension then become Le bie lbs = bi 7 7) be where d; and d, are known functions of Je ACP) (=) is 6 Eliminating bo) we obtain the equation a= S which 6 can be used to determine approximately the form of the function bo(= ) by taking a sufficiently large number of equations corresponding to different values of p. The form Chemical Attraction between Atoms from Physical Duta. 99 of the function dy 7") can then in the same way be deter- i mined by means of the first or second of the above three equations. If we further suppose that (=) may be ex- pressed in the form (=) , we have n=l n+l at mn : T : L=$(—)P, (o, ee ), A= di( oe Palo. pe) ns c and therefore L (ae ar) x == le3 n+1 ? where P, is a function of n. Applying this equation to the facts, we find that n lies between 7 and 11. This makes the value of b.(r) increase with the temperature, in other words, the molecular attraction increases with the tempera- ture. But this is highly improbable ; moreover, we will show in a subsequent paper that n cannot be as large as he above value, and it follows therefore that strictly the arbitrary function cannot be expressed as the product of two factors as we have suppesed. One of the methods used to obtain some information on the law of molecular attraction is based on the change of the coefficient ot viscosity of a gas with change of temperature, making the assumptions that the attraction is independent of the temperature and that the molecules behave simply as centres ot force. But this cannot give accurate results, as the attraction depends on the temperature, and moreover the results must be influenced by the fact that a molecule has u certain definite volume. If the attraction between two : a molecules is supposed to be given by the expression —, the values of x vary between the limits 5 and 12 according to the nature of the substance *. According to the investiga- tions of the writer, the law of attraction is the same for all substynces. If the attraction is a function of the tempera- ture, then according to the law obtained by the writer the value of n found from viscosity data should depend on the extent the temperatures at which the viscosity measurements have been carried out are removed from the critical tem- peratures. This is probably the explanation why the values of n increase in the order as the critical temperatures of the * Jeans’ ‘ Dynamical Theory of Gases,’ p. 287. 96 Dr. R. D. Kleeman: Determinations of the Law of substances decrease. On the whole, it appears that the method is of little value to determine the law of molecular attraction. The arbitrary function in the law of attraction given by the writer consists possibly of the sum of a number of positive and negative terms. In that case the force between two molecules will change in sign a number of times as they are brought nearer to one another. This could be experi- mentally investigated by allowing a monatomic gas to expand without doing work. and measuring the cooling or heating effect produced (Joule-Thomson effect). If a neat- ing effect is produced there is repulsion for distances of separation of the molecules lying between their distances of separation in the two stages ; if a cooling effect is produced the force is one of attraction. It seems best to use a mon- atomic gas, as we cannot be sure that the frequency of collision of a complex molecule—which changes in the above process—aiters the configuration of its atoms and thus causes the effect observed. It is interesting to observe that the distances between two molecules of the same kind for which the force changes sign depends, according to the form of the arbitrary function in the law of attraction, on the nature of the molecule. If two different pairs of molecules are at corresponding temperatures and the molecules of one pair are separated by a distance 2’, and those of the other pair by a distance 2’, the forces will have the same sign when pl yet ah Te x, x, ! that is, when the distances are to one another as the distances of separation of the molecules at the critical state, for this makes the values of the arbitrary functions assume the same sign in the two cases. It follows, therefore, that the distances between two molecules of a different kind for which the force changes sign are not the same as those when the molecules are of the same kind. The Joule- Thomson effect of a mixture of gases should therefore not be an additive property of the constituent gases, and this has been found to be the case*. It will also be easily seen that the sign of the Joule-Thomson effect should depend on the density of the gas before and after expansion, and keeping the conditions of expansion constant it should depend on the temperature. When a mass of gas expands * Preston’s ‘ Theory of Heat,’ p. 811, second edition. Chemical Attraction between nso Physical Data. 97 from the density p, to ps, the energy I, expended in over- coming the molecular forces is Qlv en * by mens Nit a ki po\* Wii i (3) Gad (@) a(S a) f where «, and «x, are the distances of separation of the molecules before and after expansion. The inversion of the Joule-Thomson effect occurs when Lao or (BY (2 )= mC a.) Therefore if T’ and T” are the temperatures of inversion of two different gases, when the experiments are carried out so that the conditions / U I ! P. my Pa Pe Bxan P2 i eae 7 ee are satisfied, then us ies ny = at it i from the above equation, or the temperatures of inversion are proportional to the eritical temperatures. These conditions have probably been approximately satished in the experi- ments on the subject, since gases with a low critical temperature were found to have a low temperature of in- version. A systematic investigation of the Joule-Thomson effect of a gas at different initial densities would furnish some useful information by means of which the exact form of the arbitrary function in the law of attraction might possibly be determined. Particular Cases of the General Law of Attraction a (2 ) (Se/mi)? oat dD ee We have seen that it is possible to find an infinite number of formulee for the latent heat and surface tension, each of which corresponds to a definite law of molecular attraction, but none of these laws can be taken to represent the law actually existing without further evidence. All these laws must be particular cases of the general law given above, obtained by giving the arbitrary function in the law detinite * Phil. Mag. May 1910, p. 794. Phil. Mag. 8. 6. Vol. 21. No. 121. Jan. 1911. H 98 Dr. R. D. Kleeman: Determinations of the Law of forms. This furnishes the means for further tests of the truth of the above law. We have seen that Mills, assuming that the attraction between two molecules separated by a distance z is given by a where K, is a constant, obtained the formula L= D(p®—p} 3) for the latent heat, which was found to agree well with the facts. To reduce the general law of attraction to this form we must put - Ak ) 215) eos = — =) Sa 9 $2 ele aie m where p, is the critical density and 5 is a numerical constant. Ss a The value of the constant K, is thus a (2,/m,)?, and the value of D therefore Bap: \/m,)?.. These values of D should agree with those obtained by Mills from latent heat data. This is tested in Table III. for a number of liquids, TaseE III. 2 mn. \2 i Name of liquid. a8 pa Name of liquid. Ra = HEED cee neeeceente: 104:4 127 IBEMZene tae ee eee ee 109°5 1076 Di isopropyl ...... 98:1 TT57 Hexametbylene......... 103°6 186°9 Di isobutyl ...... 86:3 100°3 Hilworbenzene) ee... 85°6 97°5 Tsopentane......... 105°4 119°3 Chlorobenzene ......... 81:2 77:3 Normal pentane .| 109°9 118-4 Bromobenzene ......... 56:1 56:4 » bexane...| 102°8 98:7 Todobenzene... ......... 44-4 41:8 » heptane.| 987 85°31 Carbon tetrachloride .| 44'1 52°8 ye) foctane).--|)) -Ia50 90°24 | Stannic chloride ...... 26.0 26:4 which contains the mean values of A =e obtained by S Mills, and the corresponding values of Bie 5( /m,)*, putting S=12,830. The agreement is fairly good. If we assume that the law of attraction between two molecules is sh, we have that K, must be equal to s(™ mm) Chemical Attraction between Atoms from Physical Data. 99 where } is a numerical constant. The equation for the Intent heat then becomes L=E(p;—>p3), where E is equal to z./m,)? : sey This equation we have already deduced pre- viously and found to agree well with the facts*. 6 If we give the arbitrary function the form u ( =) , where wu is a numerical constant, we obtain a very convenient expression for the surface tension. The attraction between : K; two molecules is then given by —;, where K,=ux(S Vm)?= u(t) (S Vimi)?, an expression for the attraction we have already used in this paper. This gives for the surface-tension the formula 2 X= F(p,:—p.2)*, where F= reeaay : Mpc In Table LV.(p. 100) the values of F are given for temperature intervals of 10° for anumber of liquids. It will be seen that they are approximately independent of the temperature over the ranges of temperature given in the table. If the general law of attraction is true, the value of F should be equal to Tap Awe i) . This is found to be the case, as is shown by Mpc Table V. (p. 101), the value given to u being determined by the method of least squares to be 32°96. If the attraction between two molecules is taken to be rr given by = where K; depends only on the nature of the liquid, then according to the general law of attraction K.=8,(2) (e /m)2, where S, is a numerical. If this constant expression for the attraction is substituted for (<)(& “m,)? in the equation f for the intrinsic pressure p, of a liquid, we obtain eels (S /m,)%2, where F is a numerical constant. Now van der Waals has proposed the * Phil. Mag. Oct. 1910, pp. 686-687. + Phil. Mag. Oct. 1910, pp. 666-667. H 2 f the Law of Dr. R. D. Kleeman: Determinations o 100 68: LF Go-4 FOLC. SPL-P 92.8 0296. 83-LF 06-9 TL09- CLF LP L810-1 TT-8z 90-4 E199: 61-49 G61 Q8LE- 86-UF LT-L 0929. FELP OF-G L901 98-13 08-9 1689: OF-€9 80-6 OGGF- G8: OF 91-8 O9F9: COL FE-9 OFIL-T 18-12 Pa-L QGL- GL-G9 88-6 POP: 8¢-0F C16 1099. £90-F 86-4 LOGL-T ¥9-12 98-8 9GGL- 2-69 LLG 986F- 91-9F 03-01 PES9- £90-F 96-8 CFGL-T PLL SL-OL OBL: 68-89 69-F 9TZo: TL-9F 63-11 CIOL: Ca0-F ¥6-6 666-1 GF-1G 6G-TT 6F08. 0-89 G9.g T LPS. 98-9F 98-21 QQ FL: LZ0-F CE-OT LIOG-L || 8-26 06-21 1828: 0929 69-9 LOLS: LL-9F Ch SI EEL: 810-F I@-TE G16G-T |, 88-12 62-F1 66F8: ZB-29 69-1 S16: L-OF L241 GLEL- 610-F CE-F1 COZE-T FFL OL-GT 8698: 6229 69-8 ZOT9: LL-9F TL-GT OTOL: 610-F L281 RLbE:T G18 eG-LT 8888: 98-19 19-6 1829. 8L-9F 98-91 FLL: L10-F CE-F1 6816-1 PP-LG 89-81 $106: 6-69 SLOT SPF9- 9L-OF 60-81 O88 || LFG-E $9-G1 FOOF-T | 66-24 G0-06 0926: 96-29 08-11 089. €9-9F 91-61 coos: || —L00-F SPOT OFZF-T | 19-96 94-16 OTF6: PL-E9 66-61 SLL9- BLOF 8z-0G STB: F10-F 09-LT PLFF-T OF-L3 60-83 916. 96-89 CO-FT LG89- eet) Ng Z%_ ld 5 ee) "Y qd hae) \ %g— Id dio): \ tq —ld Y x Nv x ‘o&GP-oSeR ‘Away, “o809-of 9G “Auta, ‘oSSP-o808 “Amey, ‘oSPP-OFTR “Amoay, ‘QuaZUOg ‘APlIO/Youljo} WOGludg ‘aqeurtoy [Aq] ‘apixo [AY "Al @1eVy, Chemical Attraction between Atoms from Physical Data. 101 TABLE V. APR (2A m,)? 32-96 (A= ey): m?p? IeMietlayl formate) ....ca2sece..0- 27°53 26°93 | | Carbon tetrachloride ......... 3°99 o'00 Benzene eee een eaten 46°5 41°76 | LD ae a eee AROS Nee ate eee 62:92 68°77 expression ap” for the intrinsic pressure of a liquid, where a is a constant which depends only on the nature of the liquid. From the above equation we see that van der Waals’ constant a—which he determines by means of his equation 1/3 i of state—is equal to Se (X,/m,)?. This is roughly the 7/3 case. But the above law of attraction cannot be exactly true since the attraction must be a function of the tem- perature ; moreover it gives a formula for the latent heat which does not agree with the facts*. However, it is probably a nearer approximation to the actual facts than most of the other definite laws that have been proposed. At the critical state of a liquid the average kinetic energy of a molecule must be approximately equal to its latent heat of evaporation into a vacuum. If this were not so, then only the molecules having a velocity above a certain limit could escape from the liquid, and it could be in equilibrium with vapour of a different density than the liquid. The internal heat of evaporation of a molecule from a liquid in the critical state into a vacuum is, according to the general law of molecular attraction, equal to Pec a3 —\. A,(2)" (Vim) where A, is a constant. This is proportional to the critical temperature, and we therefore have MA 2 iy Tee H3(2*) WERE where H, isa constant. This equation the writer has already * Ibid. p. 677. 102 Mr. A. L. Fletcher on the obtained previously in a different way *, and was fonnd to agree well with the facts. Similarly we can equate the expression for the latent heat and the temperature when the furmer quantity is derived from the general law of molecular attraction giving the arbitrary function a definite form. But in every case the equation should reduce to the above form. For example, Mills obtained in the above way from miko his law of molecular attraction that ee = constant, which was found to agree well with the facts. Now we have seen that according to the general Jaw of molecular attraction Kees Cease)’, m which reduces the equation to the same form as the preceding. London, Oct. 27, 1910. XIV. The Radioactivity of the Leinster Granite. By ARNOLD L. FLetcHER, B.A.J.T dl iene granite of Leinster extends for a distance of 70 miles. along the Hast Coast of Ireland, having an average width of 8 or 10 miles, and covering a land area of some 600 square miles. It extends in some places, near its northern extremity, beneath the Irish Channel, and is by far the largest granitic exposure in the United Kingdom. Its petrographical and mineralogical characters have formed the subject of many memoirs which deal for the greater number with special matters. The most important treatise dealing generally with the petrology and mineralogy of the granite is the paper by Professor Solias (Trans. R.LA. xxix. part xiv. 1891). The Leinster granite is throughout the main chain a trne granite, containing both biotite and muscovite micas. This character applies also to most of the outlying exposures of the rock. In two cases, Nos. 10 and 11 in the table on p. 106, biotite granite or granitite is found in such exposures; while in the cases of Nos. 1, 12, and 13 in the table, the mica is mainly of the white variety. In all the granites dealt with, with these exceptions, both micas are found. * Phil. Mag. Oct. 1910, p. 688. + Communicated by the Author. Radioactivity of the Leinster Granite. 103 Throughout the biotite of the Leinster granite pleochroic halos are abundant, and they are occasionally seen in the white mica. In the latter case they are less conspicuous. Primary muscovite often includes biotite flakes in which halos are plentifully developed, while they are absent or only faintly visible in the surrouuding muscovite. Professor Sollas considers the corroded edges of these included plates as evidence for the derivation of the mus- covite from originally large crystals of biotite, by a process of magmatic corrosion. ‘ 1n the Leinster granite, primary muscovite may be regarded as to a great extent the product of the action of the magma on the primarily formed biotite.” This view appears to involve an equal distribution of radio- active primary minerals, e. g. zircon, allanite, uraninite (7), in both primary muscovite and biotite. It is interesting to note that the recent advance in our knowledge of the origin of halos bears upon this question of the genesis of the muscovite from the biotite. Thus the fact that a zircon may he found in the muscovite and the segment of a corona, having the zircon for centre, in a closely adjoining flake of biotite, is no proof that biotite once extended to and surrounded the zircon, which would, in fact, have affected the biotite by « rays sent across the intervening muscovite. It has been found impossible in the experiments which are cited below to connect definitely the radioactivity of the rock directly with the visible abundance of dark mica _ present. Indeed, the highest result obtained was upon a somewhat gneissic granite from Glendalough which contained much silvery white mica, Prof, the Hon. R. J, Strutt has shown (Proc. R. 8. Ixxvill. A. p. 150) that in a granite, the heavier minerals include most of the uranium-radium elements, and that zircon was the most radioactive of the constituents in the granite examined by him. This result has been recently confirmed by J. W. Waters (Phil. Mag. June 1910) working in Prof. Stratt’s laboratory. Biotite being a mineral of early consolidation, and forming around the still earlier nuclei of the zircon and apatite, comes to possess in this way a greater radioactivity than minerals of later consolidation. From this point of view the radio- activity of the biotite is dependent upon its place in the order of consolidation, and its presence or absence does not affect the radioactivity of the rock. The inferiority of some of the muscovites is readily accounted for in the view advocated by Prof. Sollas, that much of it is derived from the alteration 104 Mr. A. L. Fletcher on the of felspathic substances, and is thus of quite secondary origiu. The main object of the determinations given below was to ascertain if any considerable variations in radioactivity were present in the mass of the Leinster granite. The experi- ments, as may be seen, do indeed show a fairly wide range in the quantities detectsd: from 0-41 gr. per gr. in a specimen of granite taken from the north side of Glenmalure, Co. Wicklow, to 4°36x10-" gr. per gr. in a specimen of granite also from Glenmalure. The fact that the two specimens of highest and lowest radioactivity measured come from the same loc: ality serves te illustrate the sporadic distribution of the radioactive matter. This is borne out by other results. For exanple, the deter- minations 3°62x10-" and 1°76x10-”% gr. per gr. were made upon specimens taken from points on Killiney Hill within a distance of half a mile of one ancther. Much difficulty has been experienced throughout in pre- paring solutions which are perfectly clear, or such as will retain their limpiaity during their period of storage. In some cases the solutions, while showing no visible precipitate, lacked the sparkling appearance, characteristic of perfect solution. It did not appear that those determinations made upon solutions containing precipitate fell short of the general mean obtained. Inthe case of the granite from Aughrim, the acid solution, after a preliminary estimation, was filtered, the precipitate re-fused, and added to the original acid solution, the whole being then reclosed fora further period of storage. A second determination, however, corroborated the first by yielding for the radioactivity nearly the same figure as before. The preparation and estimation of the solutions was carried out inthe same manner as before described (Phil. Mag. July 1910, p. 36). The same electroscopes were used and the same constants employed. In the majority of cases there was no appreciable quantity of radium in the alkaline solutions. Asa precaution, however, the alkaline solutions of No. 24—a mica showing a radio- activity of 4°48 gr. per gr. from its acid solution alone—and of Nos. 1 and 5 in the table, having been treated in the electroscope used exclusively for alkaline solutions, were acidified with hydrochloric acid, reelosed and retested as acid solutions. This confirmed the original result by failing to show any increase in the rate of collapse of the gold-leaf. For purposes of accuracy, and to determine whether there were any tendency toa general falling off in the quantity Radioactivity of the Leinster Granite. 105 of emanation extracted on subsequent ebullition of a solution, a series of thirteen redeterminations were made after the solutions had been stored during a further period of four months. The results were quite satisfactory, no tendency either toa lowering or a raising of the original determinations being apparent. Tn six of these cases the subsequent experi- -ments yielded figures identical with those originally arrived at. In the case of the granite from Aughrim four determinations vielded the same fioure. lor the remainder the subsequent results vary but slightly from the original ones with the single exception of No. 17. very effort was made to expel all the emanation by a vigorous ebullition, which was forced until the steam condensed in the receiver globe. Finely powdered tale spread over the inner surface of the condenser tube served as a perfect indication of the maximum height of condensation. The following table (p. 106) has been arranged with a view to bringing together results obtained upon specimens from the same locality. Some attempt was made to determine whether the radio- activity could be assigned to any particular constituent, or whether the distribution was uniform throughout the mass. A coarse granite from Ballyellin—No. 17 in the accom- panying table ,—hearing the typically porphyritic muscovite with included biotite, waschosen. The flakes of biotite when they appeared in the muscovite showed pleochroic halos and much irregular radioactive darkening. The granite was coarsely broken and about three grams of mica crystals extracted. An examination of this yielded for the radium present per gram the figure 448x107" gr. The original granite from the same hand specimen yielded a quantity 2:08 x 10-2 or. per gr., and assuming the mica to be about 20 per cent. of the whole mass (Haughton), it follows that the mica was HES Has for about 1x 105” or. per gr. of the granite, or about $ of the who'e quantity contained. Professor the Hon. ae J. Strutt found (Proc. R. 8. Ixxviii. A. p 150) that in one gram of Cornish granite over one half (approximately 3) of the radium present, was confined to the brown mica and the heavy minerals contained in it. Generally speaking, this relation therefore seems to hold good in the case of this granite. An examination of massive quartz, felspar, and muscovite mica, tree from biotite inclusions, all from large crystals ead; in the neighbourhood of the Three inoae Mountain, yielded respectively O00 0; 52, and 0°72 gram per gram for their contained radium. The mica having the radium content 106 Mr. A. L. Fletcher on the Radium Thorium Radium 7 Locality. Sea gr. per gr.| X 10-° gr. per gr. | Thorium x10". D. Klliney pitalitees sa 52 ene 3°62 0-81 43 Di) A aA RECT one 1°76 1°36 ia 3. Three Rock Mountain ...... 0-90 O71 12 AS a ie ee stati hss BoM 1-41 0:60 2:3 >. HI Chm ockges css. AeceeA. cia 1:00 0°33 30 0 80 \ 6. Glencullen (3) .........-+-+--- 1-20 l 0-31 26 I-14 : S15) 7. Dundrum * (2) vesseseseeeee: 1159 | o86 3-0 Sh Gilencrer Wa.kc ete cutee cee 2:05 1057 3-4 OG lendaloueh teeeeenasseeee. 1-04 O-d1 20 Hol@mGilenmaltre: <-coscccscssecces: 0-41 0-66 0-6 ele Dont Satan aA 0°65 0°30 2°2 12: PP WEB, uote s Tencton Reno 4°36 None detected. hs 13) Ballyknockan (2) <...2..5-..: ae 0-81 2:0 1-45 1-02 | 14. Aug (3) eos sssceese 6 1-02 | 0-71 1-4 1-02 | 1 02 Ie eBalliellin’ osc ccc seacnose sees 1:20 0-51 a5 16. RS ARR eI raed 1-44 0°68 21 ee a Oa eh NO Har } 089; 2°3 1S5 Banal SiON coc sct ac eteas t= 2 1:76 1:50 1:2 LOSING wkOWal) Lectoss tease tes ok 4-0 1:25 3:2 9). 20. Blackstairs Mountain (2)...|_ { 3.4) }o-so 30 | 2°57 | PA eBourish(S) ecteessccgpcasecess 2-00 j 143 1-4 { 2°19 DOMRGIGeAlyg |, beck -cwsenacstecses 1°66 0-67 2-4 DerGeretnileei) ot ate ae | oss 2:3 24. Muscovite with Biotite 4-48 None detected. from granite No. 16 (2°5 grs.) 25. Muscovite from Three 0°72 0-91 0-8 Rock (5°38 grs.). 26. Felspar ditto (9°3 grs.)...... 0:52 None detected. 27) Quartz ditto (3 27s.) ..-.-.. 0:00 None detected. 28. Halo -bearing Biotite from; 11:87 None detected. Ballyellin (1 gr.). Mean Sorc rceececses (of granite only)) 1:68 x107!? gr. per gr 0'70X10~° gr. per gr. * From a boring 190 feet below surface level. Radioactivity of the Leinster Granite. 107 of 11:87x10-” gr. per gr. was from a large hexagonal crystal of biotite, with a maximum width of 2°5 cms., enclosed in a crystal of muscovite. It revealed countless pleochroic halos under the microscope, and showed considerable radio- active darkening along cracks, and in the neighbourhood of the junction. Professor Joly obtained for a specimen of the Ballyknockan granite, using a very limpid solution, a radioactivity of 5°5 x 107! gr. per gr. on two closely agreeing experiments on the same hand-specimen (‘ Radivactivity and Geology,’ p- 43). In another experiment on granite from this Jocality, but using a different method of investigation, the result obtained by Prof. Joly was 2°6x10-” gr. per gr. In this case the melt from the crucible was broken upand closed with some distilled water in a flask for 21 days, when hydrochloric acid was carefully run in, the evolved gases passed over soda-lime, and finally the contents of the flask briskly boiled for fifteen minutes. The steam was returned by acondenser attached to the flask. The gases unabsorbed were collected over a potash solution and finally admitted into the electro- scope. In this procedure, undissolved or precipitated matter remained in the flask (see fig. 1). | i | ran is! i | 1 A third method was tried upon the same rock. The gases evolved during the decomposition of the rock in presence of the fusion mixture were, after absorption by soda-lime «c., admitted into the electroscope. In this case the charge, consisting of the powdered rock and the fusicn mixture of carbonates, is melted in an air-tight platinum still which can be heated to a high temperature over the blowpipe. A platinum tube leading from the top of the still conveys the 108 Mr. A. L. Fletcher on the evolved gases to the soda-lime tubes (see fig. 2). This method gave 2°3 x 10-” gr. per gr. for the contained radium. My own result on a Ballyknockan granite by the solution method used by Professor Strutt yielded a result of 1:6 x 10-!? gr. per gr. The variations in these results, together with other differ- ences in the quantities of contained radium, seem to indicate thata moderately large divergence may exist between closely adjoining specimens, but that the general average fairly approximates to the mean result in any particular district. In no case was the comparatively high result of 4x 10~” gr. per gr. maintained in a series of specimens from the same locality. These results have been included therefore in estimating the mean radium content of the whole chain. The thorium was estimated from the same solutions, by the method introduced and described by Prof. Joly (Phil. Mag. July 1909), using his apparatus, and the constants determined Py him. I adda description of this apparatus in its latest orm. Apparatus for Detection and Measurement of Thorium. The electrescope in which the thorium was estimated was made and calibrated by Prof. Joly (Phil. Mag. July 1909) from solutions containing a known quantity of thorite. The vacuum-pump P, w hich is fed from an overhead tank main- tained at a constant Jevel, is first started, the partial vacuum caused in the apparatus being observed by the rise of the oil in the tube T, reading on to an arbitrary seale S behind, which thus serves as an indication ot the velocity of the air- current. The vessels B, B serve to steady the influx of air, and thus keep the indicator steady. The actual rate of flow of the air is about 4 c.c. persec., and is measured by means of the 100c.c. flask F. ‘The stopcock © Radioactivity of the Leinster Granite. 109 is first closed and C, opened, so that the air is drawn entirely through the measuring flask. The end of the tube T, is ——— then dipped below the surface of the water in the vessel V placed in pesition for the purpose, and the time of filling to the fixed mark noted. ‘The friction opposing the entry of the water into the measuring flask is compensated for by lengthening the inner limb of the tube at M’, so that this takes the character of a siphon and a slight static head is given to the entering water. The additional length—about 3 cms.—was found by trial and error, the exact length being hit when no motion of the oil in the gauge was observed to attend the flow of water into the flask. The air flow was regulated by means of the stopcock X, which was then allowed to remain untouched throughout the remainder of the work. The standard height of the oil in the gauge was adhered to as far as possible throughout the successive experiments, by regulation of the vacuum-pump. The stopcock C is of course allowed to remain permanently open, and C, closed until itis thought necessary to again test the velocity of influx. The height of the thoria solution in the flask is marked, and is as far as possible maintained in successive estimations. Flasks of various sizes may thus be used, and standardized to a particular velocity of air-current. It was found possible to maintain a violent ebullition. The current of air entering the electroscope from the boiling flask is first dried by passage through the tube D, containing coarsely granulated calcium chloride, and finally through the tube U, filled with phos- phorus pentoxide. An asbestos card A was used to prevent the admission of gases from the flame along with the air-current. A small piece of the bulb of the electroscope was removed and a FBC eae Mr. A. L. Fletcher on the window w of thin cover glass let in for the better observance of the gold leaf. It was not found necessary, when observing the natural collapse of the gold-leaf, to remove the flask, but merely to discontinue the ebullition. The thorium content of this granite is very low. In some cases careful and repeated experiment failed to detect with certainty any trace. EHfforts were made, but failed to reveal any concentration of thorium existing in any particular con- stituent, as seems to be the case with radium. Thus one gram of the biotite showing nearly 12 x 10-¥ gr. radium per gr., showed no detectable trace of thorium ; while 2°5 grams of muscovite with contained biotite picked from dio. 17 in the table, also failed to reveal any thorium. In the case of those solutions containing any quantity of precipitate (either original or subsequently developed during the estimation of the radium), a second experiment was made on the clear solution after the precipitate had been removed by filtration. The effect generally was io slightly increase the collapse of the gold leaf. In the case of No. 1 in the table, an increase in nae rate of gain of 2 scale-divisions per hour, and of No. 2 in the table an increase of :5 seale- division per hour was inom The mean quantity of thorium found was 0°70 x 10-° gr per gr. of the rock. Professor Joly (Phil. Mag. July 1909) ob- {ained on nineteen various primary and secondary rocks a mean thorium content of 1:07 x 10~-° gr. per gr., and on fifty-one rocks, chiefly gneisses of exceptionally high radioactivity from the St. Gothard Tunnel, a mean of 1°12 x 10—° er. per gr. The approximately constant proportions obtaining between the thorium and the radium contained in the granites is a striking feature. The mean ratio borne by the radium to the thorium present is 2-4 x 10-7. Inno less than ten separate specimens, the proportion borne by radium to thorium varies between 2°0 x 10-7 and 2°6 x 10—, the three specimens from Ballyellin showing this very closely. Three remarkable examples, however, are those of Nos. 12, 24, and 28 in the accompanying table, these three specimens showing the highest radium content but no detectable thorium. On comparing “these latter results with the others, it would appear that an exceptional rise in the quantity of radium > sO far from being attended by any corresponding rise in the quantity of thorium, is actually attended by its absence. It cannot be said, however, that previous work supports so marked an inverse ratio; although it is apparent from the Radioactivity of the Leinster Granite. 111 examination of the published results that exceptional richness in the uranium-radium elements is not attended by a corre- sponding richness in the thorium series. This is more especially evident in the radium-thorium ratio of Vesuvian lavas, when contrasted with the ratio for other lavas (Joly, Phil. Mag. Oct. 1909). It seems desirable that the investigation of the radium content of rocks should be accompanied by an estimate of the thorium content also; not only to elucidate the question of a possible relationship between the radium and the thorium content, but in the interests of the geological applications of radioactivity. The same solution may be used in both deter- minations, and the experimental observation is actually more readily made and confirmed by repetition of the test in tie ease of thorium than in the case of radium. The results, it may be seen, are obtained from materials taken at points ranging from the most northerly exposure of the granite, and along the western and eastern side of the chain to nearly the southern limit. The specimens from Glendalough and Glenmalure may be regarded as from the central axis. The specimen from Dundrum—No. 7 in the table—comes from arecent boring at that place, and was taken from a point 190 feet below the surface level. The experiments therefore probably yield an approximation of say 1:7 x 10-" gr. per gr. to the mean radioactivity of the mass, so far asit is capable of being determined by the method adopted, and as carefully applied as was found possible. That there is radium throughout the entire great mass seems quite certain ; and it may be safely inferred that in a material so homogeneous a somewhat similar distribution of radioactive materials prevails at all points near the surface ; and indeed it is probably allowable to assume that the vertical distribution of radioactive elements throughout the mass is not very different. In conclusion I have to thank Prof. Joly both for his initiation of the work, and for his continual assistance throughout its performance. Geological Laboratory, Trinity College, Dublin. Nov. 24, 1910. a pe XV. On the Uniform Motion of a Sphere through a Viscous Fluid. By Prof. Horace Lams, £.1t.S.* A important contribution to this subject has recently a been made by Prof. C. W. Oseent, of Upsala, who calls attention to a certain limitation affecting the validity of the accepted solution {, however small the velocity may be, as regards points at a sufficient distance from the sphere. He proceeds to give an amended solution which appears to be free from this defect, whilst it gives the same distribution of velocity in the iminediate neighbourhood of the sphere, and consequently the same value of the resistance, as the older theory. Prof. Oseen’s analysis appears to have more than the immediate object in view, and is for the present purpose somewhat long and intricate. Considering the great interest of the question, it may be permissible to indicate a shorter way of arriving at his results, and to add a few comments dealing somewhat more explicitly than he has done with their interpretation, and with the estimation of the degree of approximation which is attained in different parts of the field. The problem is most conveniently treated as one of “ steady ”? motion, the fluid being supposed to flow with the general velocity U, say, parallel to x, past a fixed spherical obstacle whose centre is at the origin. The distribution of velocity is then given, on Stokes’s theory, by the formulee 1 aut u=U(1— v= Calera) ) fh san v= UGG peat os er oh Are (i) a ate ey ee i — {Uae Dee where a is the radius of the sphere, and » denotes distance from the centre. These formule are based on the assumption that the inertia terms in the hydrodynamical equations, which are of the second order in u, v, w, may be neglected. They were, in fact, only propounded as a limiting form to which * Communicated by the Author. + ‘‘Ueber die Stokes’sche Formel, und iiber eine verwandte Aufzabe in der Hydrodynamik,” Arkiv for matematik, astronomi og fysik, Bd. 6, no. 29 (1910). { Stokes, Camb. Trans. vol. ix. (1851) ; Sci. Papers, vol. iii. p. 55. Uniform Motion of a Sphere through a Viscous Fluid. 113 the distribution of velocity imay be supposed to tend as U is diminished indefinitely. It is known*, however, that results obtained in this way will accurately satisfy the equations, provided these be modified by the introduction of constraining forces X=wun—ve, Y=ut—wi, Z=vE—un, . . (2) - where Oey LOE) OM iiiy el Cui Ow om Bye: ‘ide & Oat S By een ee. These forces have a resultant R which is normal both to the stream-line and to the vortex-line, and whose magnitude is in the present case SOCOM al titerraea ay at CL) where q=V/(W+v?+w?), on /(E +n? +o). . « (5) The magnitude of these hypothetical forces, as compared with the viscous forces YG ea ee VT Ome LOE 8 a) Via vn Ph CO) where v is the kinematic viscosity, gives an indication of the degree of approximation which is attained in formule such as (1). Now from (1) we find 3 Uaz 3 Uay E=0, 1D Lass oF oe oy ai AU and) so that for large values of r : 3 U*ay 3 U%az : = (), ves re 5 1=—5 ee (8) For the viscous forces (6) we find 3 Ore Oe Org Dia he 8 ort gana gaa tn: 9” *Jr0z27 ° (9) The ratio of the former to the latter is ultimately of the order Ur/v, which increases indefinitely with +, however small U may be. This is, under a slightly different form, the objection which Prof. Oseen raises to the validity of (1) * Rayleigh, Phil. Mag. [4] vol. xxxvi. p. 354 (1893); Sci. Papers vol. iv. p. 78. The pressure must be supposed altered by a term, 3p(w?-+v?+ w*), which vanishes however at the surface of the sphere, on the hypothesis of no slipping. Phil. Mag. 8. 6. Vol. 21. No. 121. Jan. 1911. I 2 ipl Prof. H. Lamb on the Uniform at points distant from the sphere. Since, however, both the constraining forces and the viscous forces are in these regions relatively small, it does not necessarily follow that the character of the motion in the immediate neighbourhood of the sphere will he seriously affected. At points near the sphere the constraining forces tend to vanish, whilst the viscous forces are of the order vUa/r’. The innovation made by Oseen in the treatment of the question consists in writing U + for u, and neglecting terms of the second order in u, v, w only. These symbols now denote the components of the velocity which would remain if a uniform velocity —U were superposed on the whole © system. The hydrodynamical equations accordingly take ihe forms | ~Ov 1 Op Ler 5, +HvV'r, (10) ~ Ow 1 Op 2 a ed with 3 Ot. OU Ow. A oy MIE Gs . ° e ’ ° (11) The inertia terms are thus to some extent taken into account, but it is to be remarked that although the approxi- mation is undoubtedly improved at infinity, where u, v, w=0, it is in some degree impaired near the surface of the sphere where we now have u=—U. This will be a matter for subsequent examination. _ The solution of the equations (10) and (11) for the purpose in hand can be effected very simply. In the first place we have vp 205) 0g) «*.o i e and a particular solution is therefore obtained if we write . p=pU <> ° ° a leas ° (13) w= 88, en 8b. wa SP... fe Motion of a Sphere through a Viscous Fluid. 115 The solution is completed if we write 0d OR o¢ | ain ae +e’, ee Cena ot (15) where wu’, v’, w’ are solutions of the equations : oy, ‘] ad} oo ») fa, = (v 2b Yu Q, | (v-2%2)y'=0, | (16) Ox ; Miao | Bnei Vet kd (v a . ») and Chae Bn Ow! _» A A EEO DAN th) 4 2 “oy | 82 We have here written, for shortness, k=U/2v. PLE ARTCC UMTS OMI RCT MC) (18) Since the vortex-lines must be circles having the axis of x as a common axis, we may assume PODER A EO Uhatg) where y is a function of w and @ (the distance from the axis of z) only. It follows from (16) that we must have (vi-2% 2) y=0, . MGs es 0) an additive function of x being obviously irrelevant. Hence Du! _ a 99 BE__ (Bx, BY an ae POZO -(53 ii 52) ENO GS Ch Oe" as (et a Gee) ra Ties Cece ~ Ocay’ Si OLN allt On LOnN Qk? =V2w' = C5 Qe Oy Oe = ade 4 LV? 116 Prof. H. Lamb on the Uniform We thus obtain the solution pas NG): aaa Y= 9b 3% | aOR Be iy alone | TRE Bek y) which is easily verified. The equation (20) may be written (Ware =7=0,..2.<) eee (23) the solution of which is well-known, the simplest type being eit v=Ce-*Ir, Adopting this, we have, finally, 7 OO was Oe 5 Sl tar OV NOP Cen 5 = Oy T Oh By ’ r (24) OD 2 OX. | OT ge) Oe a where Ce-*r-2) s .. ete Os (25) Since @ must obviously involve only zonal harmonics of negative degree, we write mies Ove (=) v6 a te pheno z +. ea For small values of kr we have v=e(? kt 4.) be A which leads to WiOV ee Ce (EAB folte 1 ye al or are -H415.-8; Be oer o }, 1 Ox ae C ed 1) nee 22 X 2 ; 1 ox =-54 sing ee mec a 2k 02 7 Der) © Ouos7: Motion of a Sphere through a Viscous Fluid. IG, Hence the relations w= — U, v=0, w=0 which are to hold for r=e@ will be satisfied, provided +8 3 i C= 5 Ua, Ag= 5 v4; ea Ua, A e (29) approximately; and it will be noted that the condition for the success of the approximation is that ka, or Ua/v, should be small. The equations (24), (25), (26), (29) agree with Prof. Oseen’s solution of the problem, obtained by a different process. To find the distribution of velocity in the neighbourhood of the sphere we may use the formule (26) and (28), with the values of the constants given in (29). The result is identical with (1) if regard be had to the altered meaning of u. The resistance experienced by the sphere has the same value (64aU) as on Stokes’s theory *. In other respects the motion differs widely from that represented by Stokes’s formule; and the further inter- pretation is very interesting. In the first place, as pointed out by Prof. Oseen, the stream-lines are no longer symmetrical with respect to the plane z=0, the motion being in fact no longer reversible. Again, the (doubled) angular velocity of the fluid elements is ox . Oa and is therefore insensible, on account of the exponential factor alone, except within a region bounded, more or less vaguely, by a paraboloidal surface, having its focus at O, for which ‘k(r—az) has a moderate constant value. This region may be referred to as the “ wake,” although it includes a certain space on the upstream side of the sphere. If we superpose a general velocity —U parallel to w, the residual velocity tends, for large values of 7 and for points outside the wake, to become purely radial, as if due to a simple source of strength 4aAo, or 67va, at the origin. This is compensated by an inward flow in the wake ; thus for points. along the axis of the wake, to the right, where e=7, we find =SUaltir) Se, . . (30) o= aeWlow wus Was Cae aes ° ° ° ° ° (31) This indicates a velocity following the sphere (when the * The addition to the pressure (see the footnote on p. 113) is now 2pU* at the surface of the sphere. This being uniforni, does not affect the resultant. WWeu= VEO a yi 118 Prof. H. Lamb on the Uniform latter is regarded as moving through a liquid at rest at in- finity) which ultimately varies inversely as the distance, instead of as the square of the distance. It remains to estimate the degree of approximation which the preceding results afford in various parts of the field. For this we have recourse, again, to a comparison of the — ‘constraining’ forces, which would be necessary to make the solution exact, with the viscous forces. The former are still given by the formule (2), provided wu has its altered meaning. At distant points, well outside the wake, the terms in (24) which depend on y may be neglected, and we haye, ultimately, is Sn ig as Deng) i 3 The (32) CT Mee Magara anre 2 AS dain: s/n aZ t fr }: Zz s Also, from (30), 3 + < — ate g 3 = ?/ Be a ) % E=0, n=, Ukaye RS a €=— ~Uhkase Re), aed Aust gna Hence | 4 @ ki; g ra s = J ee) y 7 rt2,94% 4 —k(r—-2) x U 2 42 : Y a - U te : d WH 2 vee pe co 34° Z i aes U%a yee (< ) the resultant of which is Ce E ae SE i os 5 Ua? ae AG ae s e@ e © q (35) in a direction at right angles to the radius vector. The viscous forces may be found from (21) and (25). If we retain only the terms which are most important when 7 is large, we find y(r— ee iy Thee ae WWW = Mae (36) It pollens from i that the ratio of the forces is of the order (1/kr).(a/r). The approximation in this part of the field i is therefore amply sufficient. Motion of a Sphere through a Viscous Fluid. 119 At points well within the wake, where k(r—.) is small, we have eet ss OT) and em H mcaaey | £=0, n= 5 Uha%, C= — 5 Uka ss, ae (38) approximately. These make X=0, Y= 5 Ura, La Uke, Pape Go) The viscous forces are found to be , yyru= WhO, v= WhO, VVy2w= 2vkC > 2 0) approximately. The ratio of the magnitudes is of the order. ka. Near the surface of the sphere we have u=—U, v=0, w=0, approximately, and therefore, from (2) and (19), ON Ox , = (SSS Ve SU ee OR EN m= (0, U 5" U 5 (41) or, by (27) and (29), , i Bye Piya dth, 02 x0; Y=5U'a; De Wa or et (42) The resultant is therefore of the order U?/a. The viscous forces are obtained from (21) and (28) ; thus vy 9 , y? 3 3 Vy US 5 vUa iy oS 4 Ve Ww = a Ua, 3) giving a resultant of the order vU/a?. The ratio of the magnitudes is therefore Ua/v, which has already been assumed to be small. The approximation, although less perfect here than on Stokes’s theory, is seen to be adequate. I may repeat that the object of this note has been merely to givea simpler demonstration of Prof. Oseen’s results, and to elucidate a little more fully their scope and significance. I have not touched upon another aspect of the question which is referred to in his paper, and which is apparently to form the subject of a further investigation. 120 Prof. H. Lamb on the Uniform It is of some interest to apply the same method to the two-dimensional problem of the flow past a circular cylinder where, as is well-known, Stokes was led to the conclusion that a steady motion is impossible. It will appear that when the inertia terms are partially taken into account, in the manner above explained, this conclusion is modified, and that a definite value for the resistance is obtained. The hydrodynamical equations are now satisfied by Vo ROO eX | Ox 2k Ox } Ad po 2 Be, | oe 1 Oye oy p=puse, (45) provided Vi ¢=0, ,. . ... 2 een and (vr-22\y= 0. ea where Weoley. . aes The appropriate solution of (47) is i: KH Cer). :- 6.2 eee where * 1 outers ean es = tre gpl its) + =/(= Jew sig oF j 50} GE Sieg wan ae) For small values of kr we have X=—C(l+khe)(y+logtkr), . . . (51) whence OMe eee ee low alte Dia mee oie SA re ae ") 0 leis ion + an OE d fal g hh sn 8 Parks | ) (52) Loy ue se O° Bt by ~~ HEL Sy!98T Eh gape tf * The notation is that of Gray and Mathews, Treatise on Bessel Functions, p. 68, Motion of a Sphere through a Viscous Fluid. 121 _ Hence if we put ; | p=Aplogr +A 2 tog r+. Bas | \SGus 5S) we find that the conditions w=—U, v=0, w=0, will be satisfied for r=a, provided C=2U/(k-y— log tka), Ay=—C/2k, Ay=4Ca?, (54) a Hence near the sphere we have 2 a 5 + log; jkr 5 (8 @) Slog rh. ) | . (55) Or ye i — Of? a”) Ouvdy log ? e 2 e e es ° e es i vorticity is given by pa - St =e 2K (kr), ll 0565 which for large sick of kr takes the form gang [( sre... A MINER _ The general interpretation would follow the same lines as in the case of the sphere. To calculate the force exerted by the fluid on the cylinder we have to integrate the expression (—r+2nSi)eta(Se +S") x = pes wre + u(a oe yO"), natay | (OS) with respect to the angular coordinate (@) from 0 to 27. The products of plane harmonics of different orders will of course disappear in this process. The first term of (58) gives, when r is put equal to a, 21 : —pUAdl eos 0 dd = = mpl N= mu, . (Oo) 0 or by (45), (53), (54). The second term contributes, on substitution from (55), mwC. The third term vanishes identically, to our order of approximation. ‘The final value for the resistance per unit length is therefore ; oh sue! The investigation is Aeon as RB iore to the condition that ka, or Ua 2, is to be small. (60) [eal AVI. The Change of Resistance of Nickel and Iron Wires placed longitudinally in Strong Magnetic Fields. By Epwin A. Owen, 6.Sc., University Student, University College of North WV ‘ales, Bangor *. UMEROUS experiments have been carried out on the effect of magnetization on the resistance of ferro- magnetic metals; the most recent being those of Gray and Jones f, Barlow t{, and W. HE. Williams §. The experiments of Gray and Jones were carried out with the object of determining simultaneous values in a specimen of soft iron wire of the magnetizing force, the magnetization, and the change of resistance due to magnetization. They studied in particular the longitudinal. effect, when the direction of the electric current in the specimen is parallel to the lines of magnetic force, and found that Af=al* represented approximately, in soft iron for fields ranging from 30 c.G.s. to 250 c.G.s. units, the relation between the magnetization I and the fractional increase of resistance Ad, z. e., the increase of resistance divided by the resistance in zero field. Barlow studied the same effect in nickel and found that the change of resistance showed a decided maximum :— Ad = 0:0156, H = 2000 c.a.s. units, and in higher fields decreased continuously to the value Ad = 0°0100, H = 18,000 c.c.s. units. He suggested that this decrease was due to the end elements a his coil, which were transversely magnetized. The electrical Pictanes of nickel diminishes when trans- versely magnetized in fields stronger than 2000 C.G.s. ae and this may therefore explain the diminution in the val ue of Ad observed by Barlow in his experiments. The object of the experiments described in the present paper was to examine the effect in still stronger fields. and in doing so to employ specimens which were placed entirely longitudinally i in the field. For this purpose the specimens examined took the form of very thin straight wires, only about a millimetre long. These, while having deuheies! resistance to measure accurately, could be placed longitu- dinally in the narrow gap between the pole-faces of an electromagnet. * Communicated by Prof. KE. Taylor Jones, D.Sc. 7, RGye Soc lento 1900,. vol. Ixvii. p. 208. - t Roy. Soe. Proc. 1902, vol. Ixxi. p: 80. . § Phil. Mag. October 1902, Bee 1903, Saneary: 1905. Change of Resistance of Nickel and Iron Wires. 123 (1) Apparatus. A portion of the electromagnet by means of which the fields were produced is shown in section in fig. 1. The Q2DeeOoeeeD ro BPoOD ood oP OG @BSoO® currents used in the magnet ranged from ‘1 ampere to about 22 umperes. The current was allowed to flow through the inagnet for as short a time as possible, only during the few seconds required for obtaining the balancing-point on the resistance bridge. To prevent rise of temperature of the pole-pieces, water was arranged to flow along the hollow cores to within a few millimetres of the surfaces of the pole- pieces ; the water was previously passed through a spiral tube which was slightly heated to bring the temperature of the water up to that of the room. The strengths of the fields were measured directly after each set of readings for the change of resistance. This was done by the ballistic method,—the exploring coil being connected to a ballistic galvanometer provided with a tele- scope and scale. An arrangement was set up by which the exploring coil could be placed in the same position each time between the poles, and by which it could be suddenly removed from the field. The galvanometer was standardized before and after each set of readings by means of a standard solenoid and a secondary coil. No attempt was made to measure the magnetization or the field inside the specimen. No form of “Isthmus Method ” would in fact be suitable for such a thin specimen. The change of resistance was measured by means of a slide wire bridge, the metal parts of which were all mounted on ebonite. In order to obtain a large step on the bridge, a thick german-silver wire was used as the bridge-wire, and also two auxiliary coils of german-silver wire were placed in 124 Mr. E. A. Owen on Change of Resistance of Nickel the gaps R and 8. These two coils were immersed in the same oil-bath. Brass springs, attached to two ebonite riders R, and R, (fig. 2), were always in contact with the bridge- wire. These springs were connected to mercury keys which could be worked from a distance so as not to bring the hands Fig. 2. T: Terminals attached to bridge-table to receive cable. G: Com- pensating coil. R,S: Auxiliary coils. B: Broca Galvanometer. M: Mercury keys. near the bridge connexions. Jn the inner gaps, E and F, of the bridge were placed respectively the specimen examined anda very thin platinum wire used as a comparison resistance. These were mounted on two carriers, and placed in the same trough which had pure paraffin oil flowing slowly through it (see fig. 3a). They were placed close together so as to be subjected, as nearly as possible, to the same external changes of temperature. In series with the specimen examined was placed a compensating coil of platinoid wire, whose resistance was adjusted so that the temperature coefficient was the same for the two branches H and F. If G is the resistance of the compensating coil, then its ae where N is the value is given by the formula G=N. resistance of the specimen; @ and # the temperature coefficient of platinum and the specimen respectively. A number of different suspended coil galvanometers were tried, but not one was found sensitive enough for the experiment. The galvanometer ultimately used was a Broca galvanometer with coils whose combined resistance was about 90 ohms. This instrument was found to be quite sensitive enough and it worked exceedingly well. It was used with a telescope and scale, the lamps used to light up the scale being immersed in a glass trough through which a and Iron Wires placed in Strong Magnetic Fields. 125 Stream of cold water was passing. This precaution was taken to prevent any heat radiation from the lamps to the bridge connexions. 3 The battery circuit consisted of one small storage-cell in series with a resistance of 150 ohms, and a mercury key. The current was allowed to flow through the bridge for about an hour before any readings were taken, and it was not stopped until the set of readings was over. This was done in order that the rise of temperature due to the passage of the current through the specimen might have attained a steady value when the readings were being taken. The electromagnet was situated broadside on to the galvanometer, and at a distance of about 45 feet from it. In this position and at this distance, it was found that the field due to the magnet did not affect the galvanometer needle. The specimen examined and the platinum wire were connected to the bridge table by means of three lengths of thick cable twisted together and passing from the table over the beams of the laboratory to three terminals on a wooden upright fixed to the frame of the magnet and insulated from it. The cables were twisted together to prevent inductive effects when the wires moved in the earth’s field. The specimen was held in position in the field as shown in fig. 3a. A thin piece of ebonite, thickness about 1 mm., &F = Sue ID % A Se A Fig. 5a. Fig. 3. T: Terminals on wooden upright attached to frame of maenet. p g was cut into a rectangle 6 cm. by 1 em. Two strips of copper were fixed to each side of the ebonite, and three pieces of the specimen examined were soldered across the ends of these as shown in fig. 36. Great care was taken to solder them straight across and parallel to one another. The Tisto 126 Mr. E. A. Owen on Change of Resistance of Nickel comparison wire of thin platinum was mounted in a similar way on another ebonite plate. The whole was fixed in a trough of mica just wide enough to contain the specimens. The trough was held horizontally in a wooden stand resting on a slab of plate glass which was separated from the fixed table on which it stood, by a thick layer of paraffin wax. The trough, and with it the thin wires, could be withdrawn from between the poles of the magnet to a distance of about 80 cm. by lowering the top part of the stand and turning it in its socket through 180°. The whole of the apparatus including the cable, bridge- table, galvanometer, specimen, and all connecting-wires were thoroughly insulated from the earth. (2) Method of taking Readings. The specimen was demagnetized by reversals, and then removed to a distance of about 80 em. from the pole-pieces. When in this position, that is, when the specimen was in zero magnetic field, the balancing point was obtained on the bridge wire with the rider R, (see fig. 2). The specimen was then turned into the field. Preliminary experiments had shown that the magnetic change of resistance in both nickel and iron reached a maximum value in a certain field, and the current required to produce this field had been carefully determined. This current was now passed through the magnet coils and the balancing point obtained on the bridge with the rider Rg. The specimen was again demagnetized and the same opera- tion repeated until about a dozen readings were taken. The mean of these readings gave the maximum step on the bridge- wire. If x9 is the balancing point in zero field, and « that in the field, then (2#—.2)) gives the maximum step. The specimen was then permanently placed between the poles. Different currents were sent through the magnet alternately with the current which gave the maximum step, and the balancing points on the bridge wire found in each case,—R, for the fixed current, and R, for the other currents. The varying currents were not turned on regularly in ascending or descending order of magnitude, but quite irregularly. The mean of two consecutive readings with the fixed current was taken as the balancing point for this current. Letit be # If x is the balancing point for any other current, then (v—.)’) is the amount to be subtracted from the maximum step to give the step for that current. The riders were moved about by means of a long ebonite rod, so as to avoid bringing the hands near the bridge-wire. Changes ‘of resistance were examined for fields ranging from 500 c.a.s. units to 20,000 c.c.s. units with the pole- and Iron Wires placed in Strong Magnetic Fields. 127 ‘pieces at about 7 mm. apart, but for fields reaching up’ to - 30,000 c.c:s. units, the pole-pieces were about 2 mm. apart. (3) Results. The fractional change of resistance Ad in the specimen examined corresponding to the step Aw on the bridge-wire is calculated from the formula AND RS 7) @ ues AG= Thao ao Ag; where o =resistance of bridge-wire per unit length ; N=resistance of the specimen examined. Q=total resistance in the bridge-gap containing the specimen. This includes the resistance of the compensating coil and the connecting wires and cables ; Rand S=resistances of the auxiliary coils. Three experiments were carried out :— Experiment I. with three pieces. of nickel wire in series, diam. ‘0206 mm.* Experiment IT. with two pieces of nickel wire in series, diam. ‘0159 mm. Experiment III. with two pieces of iron wire in series, diam. ‘0208 mm. TasiE I.—Particulars of coils, &c. used in the experiments. : i us ie Pcs NA VR ea | Expt. 1. | Expt. II. | Expt. ILL. | (USED) ON UPS Oh) OO) Resistance of specimen in ohms. ...... 0-761 0835 0-916 Resistance of auxiliary coil R- ......... 0-615 0-615 1527 Resistance of auxiliary coil S............ 0-506 0:492 1:231 Resistance of compensating coil......... 0524 0-443 0-444 | Total resistance in gap containing} 1:240 1506 1530 | SPO CHMLCMMM AS ots eaGewsiec elle cme tees : Resistance of 95°5.em. of bridge wire .|. 00152 OO1S2 erly Cole) 1 Maximum step in cm. (%@—2p)......ee vee 18:01 16-49 6:28 * The nickel and iron wires were supplied by Messrs. Hartmann & Braun, Frankfurt. | Maximum | Hinces. | | value of Ag. units. | | | | Experiment I.—-Nickel wire; diam. -0206 mm. 001462 2800 | | | - | Experiment II.—Nickel wire; diam.0159mm., 001418 | 2800 | 7 | | | Experiment ITI—Iron wire; diam. °0208 mm. 0002225) 1900 | 1500 128 Mr. E. A. Owen on Change of Resistance of Nickel The results obtained are given in figs. 4, 5,and 6. It is observed that there isa decided maximum change of resistance e ! . e e s in each case (see Table II.), followed by a diminution in stronger fields *. . TasLE I].—Maximum values of Ad. \ Fig. 4.—Nickel (0296 mm. diam.). Temperature=18°'5 C. ve) 14-00 x S- J 1.300 1200 1100 $00 * It has recently been brought to my notice that a maximum value in the magnetic change of resistance of nickel was also observed by R. Dongier ( Betbliitter z. d. Ann. d. Physik, Bd. 27, p. 677, 1903), but I have not seen any details of his measurements. ant Iran Wires placed in Strong Magnetic Fields. 129 Fig. 5.—Nickel (0159 mm. diam.). Temperature=17°'5 C. 1.500 Ax 105 1400 1,300 1.200 1,100 0 10,000 20,000 30,000 HIN CGS. UNITS Fig. 6.—Iron (0208 mm, diam.). Temperature=17%5 C. 0) 10,000 20,000 30,000 HIN CGS. UNTS. Phil, Mag. Ser. 6. Vol. 21, No. 121, Jan. 1911, °° K 130 Messrs. A. S. Russell and F. Soddy on the- In the case of nickel, the change of resistance shows a tendency towards a constant minimum which is reached in a field of about 24,000 o.a.s. units. For the first specimen the total decrease in the value of Ad after the maximum is reached is 00030, and for the second specimen 0-0042. There is no tendency towards a constant minimum shown in iron in the fields examined, the fractional change of resistance steadily diminishing as the magnetic field increases. Comparing these results with those of Barlow, we find that the diminution in the value of Ad for nickel between the field strengths 2000 c.¢.s. and 18,000 c.a.s. units, is about one half that observed by him. It is possible, therefore, that the diminution of A@ observed by him in strong fields is not entirely to be accounted for by the end elements of his coil, which were magnetized transversely, but was made up of the decrease due to both longitudinai and transverse effects acting together. According to the electron theory of metallic conduction as given by Drude *, the change of resistance due to a longi- tudinal magnetic force is proportional to the square of the strength of the field in which the electrons are moving. In a magnetic metal this is presumably to be identified with the magnetic induction, and the change of resistance should therefore be proportional to the square of the magnetic induction. The results of the present experiments are thus not in accordance with this form of the electron theory. All the above experiments were carried out in the Physical Laboratory of the University College of North Wales. In conclusion, I desire to express my thanks to Professor EH. Taylor Jones for the interest he has taken in the work, and for much valuable help and advice. = XVII. The y-Rays of Thorium and Actinum. By ALEX- ANDER 8. RussELu, J.4., B.Sc., and FREDERICK SopDDY, YY Ee NT Sp a investigations to those described in two previous papers f on the y-rays of uranium X and radium C have been carried out with the y-rays of actinium C (or possibly it may prove to be actinium Dj and with the two types of powerful y-radiation in the thorium series, namely that given * Drude, Ann. der Phys. iii. p. 878 (1900). + Communicated by the Authors. t Phil. Mag. 1909 [6] xviii. p. 620: 1910, xix. p. 725. For brevity these two papers will be referred to throughout as I. and II. respectively. y-Rays of Thorium and Actinium. 13H by the short-lived @-ray product between mesothorium 1 and radiothorium, which we shall refer to as mesothorium 2 (Hahn, Phys. Zeit. 1908, ix. pp. 245 & 246), and that given by the last known product of the disintegration series, thorium D (Hahn and Meitner, Phys. Zeit. 1908, ix. p. 649). The paper conveniently divides itself into three sections. The first deals with the relative intensity of the y- and B-rays of these substances. In the second section a number of so far unexplained effects in the measurement of the absorption coefficients of the y-rays are described in detail. In the last section the penetrating power of the actinium and thorium types of y-rays arecompared with that of radium U. A brief indication of the general character of the results may con- veniently precede their detailed consideration. The two thorium products resemble radium C remarkably closely both in their y/ ratio, and in the penetrating power of their y-rays, and, although interesting differences exist, these are comparatively small. The most penetrating y-ray known is that given by thorium D, that of mesothorium 2, speaking in a general sense, being about as much less pene- trating than that of radium C as that of thorium D is more penetrating *. These three bodies are sharply distinguished from all the other @- and y-ray products by their high and similar y/@ ratio. At the other extreme are radium B and radium 1, which we have not examined, the y-rays from which are, either not at all, or only barely detectablef. It is perhaps still natural to leave radium B out of the compa- rison, as its 8-rays are excessively feebly penetrating; but the established very low value of the y-rays of radium H, taken in conjunction with the character of its @-rays, which, although feebly penetrating, are of the same order as those of actinium C and mesothorium 2, furnishes another example of the lack of connexion between the two types of rays. We have to consider actinium C and uranium X, each of which differs in this respect from all the other types. For uranium X, as we have previously found, both @- and * Apparently nothing has been previously published with reference to the y-rays of mesothorium 2; but it should be mentioned that [ve (Phys. Zeit. 1907, viii. p. 185), in comparing the y-rays of a preparation of radiothorium (thorium D) with those of radium, found that they were almost identical in penetrating power, although the measurements indi- cated that the radiothorium y-rays were a little the more penetrating. The difference, however, Eve considered to be within the error of the experiments. t H. W. Schmidt, Phys. Zeié. 1906, vii. p. 764; Ann. Physik, 1906 [4] xxi. p. 609. Meyer and von Schweidler, Wen. Ber. 1906, exy. ILa. p- 697. HL. W. Schmidt, Phys. Zeit. 1907, viii. p. 361. KK. 2 132 Messrs. A. S. Russell and F. Soddy on the y-rays are similar in penetrating power to those of thorium and radium C, but the y/8 ratio is of an entirely dif- ferent and lower order of magnitude. For actinium C, both 8- and Y-rays are less penetrating, and the y/f ratio, although of the same order, is distinctly greater than far uranium X. Indeed there is no rule about the matter. From the fact that hard y-rays usually accompany hard B-rays, it might be supposed that thorium D, which gives the most penetrating y-ray known, would have also the highest y/8 ratio. Whereas it is distinctly lower, both than that of radium C and that of mesothorium 2. Owing to the importance of the y-ray method asa standard means of comparison of radiouctive substances, we have thought it advisable in the second section to collect a number of observations showing how the most curious changes of penetrating power of the y-rays are brought about by the slightest change in the experimental disposition, although one lars he general explanation of these effects to offer. Section I.—The Ratio of the y- to the B-rays of Mesothorium 2, Thorium D, Actinium C, and Radium C. In a previous paper (I. p. 629) the ratio of the y- to the 8-rays of uranium X has been accurately compared under the same conditions with the same ratio for radium C. It was found, assuming the rays to be homogeneous, by mea- suring the y-rays through 1 cm. of lead and calculating the initial intensities from the known absorption coefhicients, that the y/8 ratio tor uranium X was about 50 times smaller than that for radium ©. When the y-rays were measured through 0°6 cm. of aluminium, the ratio, uncorrected for absorption, was 18 times less. Similar measurements have now been extended to actinium and thorium. It may be pointed out that we are measuring a highly complicated effect in this y/@ ratio, and that before any absolute com- parison is possible, it would be necessary to have a good deal more data than are at present available. We have been concerned only with the relative order of magnitude of the ratio sought, and have attempted to get at least a rough idea of this order by carrying out the y- and 6-ray measurements respectively for both substances under exactly the same conditions. Considerable differences will be found in the ratio according to the method of measurement employed, as is only to be expected ; but the general order of the effects measured 1s sufficiently correctly indicated by the mea- surements here given. y=Rays of Thorium and Actinium. 13: For measurement of 8-rays a cylindrical brass electroscope of the ordinary type, 13 em. high and 10°8 cm. diameter, was used. The thickness of the walls was 0°32 em. The base consisted of a layer of aluminium foil, 0'095 mm. thick. In the first series of measurements the preparation was placed at a distance of 60 cm. below the base, but later various shorter distances were also used. The apparatus was set up so as to reduce secondary 8-radiation to a minimum, The electroscope was flush with the table, which had a large hole cut in it, and was supported by light steel brackets from the wall. The preparation was supported centrally below the electroscope on a light framework of brass rods, made in sections screwed together, and atlached to the under side of the table, so that any distance between the preparation and base of the electroscope could be employed. For mea- surements of y-rays three different dispositions were used, as follows :— (y,) The active preparation was placed at a distance of 8°6 cm. below an electroscope of lead of the usual type having wall and base thickness 0°3 em. of lead ; (y2) at a distance of 3°3 cm. below a lead electroscope whose wall was 0°65 em. and whose base 0°975 em. thick. (y3) Ata distance of 8-7 cm. below the same electroscope as In 7. The radium © was prepared by exposing one side of a negatively charged brass disk to a quantity of radium emanation for 20 hours. After the exposure the disk was placed in a brass cell and a piece of 0°095 mm. thick aluminium foil was cemented over the cell to prevent any possible escape of adhering emanation. One hour after the preparation of the fiim of radium C, alternative measurements of the 8- and y-rays, in the four different dispositions detailed above, were taken over a period of three hours, and from the four decay curves obtained from these measurements the 6- and the three different y-activities could be readily deduced. The four decay curves obtained were found to be very nearly exponential with values of X very approximately the same, (1°39 (hour)—1). The relative values therefore of the intensity of the preparations measured under the four different dispo- sitions could be accurately obtained. For thorium D the procedure of preparation of the disk was exacily the same, the disk being exposed negatively charged to the thorium emanation from a preparation of radiothorium spread out ona shallow dish for 20 hours under conditions, such that only the one side of the disk was coated. ‘The disk was then: mounted and covered with the same EE 134 Messrs. A. S. Russell and F. Noddy on the thickness of aluminium as the radium © disk. Fifteen minutes after the disk had been removed from the emanation the first measurements were taken. Experiments in each of the four different dispositions were made with thorium D, just as they had been made with radium C, and the relative intensities of the rays at each disposition accurately determined. The mesothorium 2 used in the experiments was separated chemically from a preparation of mesothorium 1 previously prepared from several kilograms of thorianite. The preci- pitate, which weighed only a few milligrams, was evaporated down ona small watch-glass of diameter exactly the same as that of the disks used for the other two active bodies. ‘The quantity of matter was so small that self-absorption can be neglected. The watch-glass was covered with a piece ef aluminium foil, 0095 mm. thick, and several measurements were made for each disposition over a period of 33 hours. The four decay curves were exponential (\=0-12 (hour)—?). The residual activity remaining two days later was negligible. In the following table the actual results found for each disposition for each source are given :— 7/8 72/83. 73/8 Redium ©... i0ely, | lies |. ona Mescthorium 2... 1299 2099 | 0-s13K0 Thormimn Dias. | 0795 Ay iioon ae Pees a : | lB 4/9. lB | Deg RAO Mee PNR TEGoM ny Phen TaD 100g ae | Mesothorium 2... 143 | 1138 | 14 ieee | Spae Lreedls | 6. oTae min For these dispositions therefore mesothorium 2 gave 13 per cent. more y-rays per #-ray than radium C, and thorium D about 25 per cent. less. Also, the variation of the ratios thorium D to radium C, or mesothorium 2 to radium ©, with the thickness of the lead base is slight, if any. This is to be expected because, as will be shown in y-Rays of Thorium and Actinium. 13D another part of the paper, the penetrating powers of all three types of y-radiation are very similar. A second series of experiments, which embraced in addition the actinium active deposit, was started in order to investigate various influences which are likely to affect the values of the y/8 ratio. These are the difference in penetrability of the hard @-rays of each element and the effect of scattering of such rays by the air between the preparation and the electroscope. Some metal must necessarily intervene between the preparation and the electroscope, but the amount used was the minimum, so that the absorption of @-rays by it was small. When the ratio of y- and 8-rays for uranium X was measured in a former experiment it had been necessary to place the preparation at a great distance from the electroscope in order not to give too large a 6-ray effect. This was not essential in the case of active deposits having a quick rate of decay, so that the values of the S-activity of the preparations, placed at five different distances from the electroscope, has been measured in order to see how the distance affects the values of the y/@ ratio of thorium D, mesothorium 2 and actinium active deposit relatively to that of radium C. The distances used were 51°‘1 em., 35°9 cm., 20°6 cm, 13:0 cm., and 5°37 cm. These positions are denoted as 8, 82, 83, Gy, and 85 respectively. The dispo- sition y,, used before, was used again. The mesothorium 2, separated chemically as before, was this time mounted on a brass disk, exactly the same as those on which the active deposits of the other bodies were collected, placed in a brass ceil and covered with the usual aluminium foil. The sub- stitution of the brass for the glass had only a small effect on the value of the @-rays. The active deposits were prepared and mounted as before, the actinium active deposit being obtained in the same manner as that of thorium. The very fine actinium preparation, lent by Messrs. Buchler & Co., and described in Section III., was employed. Measurements were started in the y,and £; positions, and, when the pre- paration had become too weak to be easily measurable in the latter, it was placed inthe 8, position, and so on till the 8s position. The residual activity was also measured in the 85 position ; but since the @-rays of radium C and meso- thorium 2 reach a minimum and then again increase, owing to the formation of later @-ray products, the correction for the residual activity cannot be directly applied, but must be extrapolated. From the decay curves of all the bodies experimented with in the five different dispositions, corrected, where necessary, for residual activity, the y/@ ratios in the 136 Messrs. A. 8. Russell and F. Soddy on the different positions are readily obtainable. The final results are given in the following table, in terms of the ratio for radium C which is taken in each position ag unity. Positions : B,- Be. B3- Bs | Bs 7/8 Mesothorium 2 ...| 1-036 | 0-035 | 0-676 | 0-800 || 0-624 | | y,/3 Thorium D......... 0:690 | 0637 | 0:568 | 0510 | 0-336 | ] | H | | j | | | | For actinium the y-rays were measured also through 0°95 em. zine (4) as well as through 0°3 em. lead (¥y,). This was done because, as was shown previously by Godlewski and confirmed later in this paper, the y-rays of actinium are abnormally highly absorbed by lead. Radium (©, the ratio for which is again taken throughout as unity, was also measured under the same conditions for comparison. The final results are:— | Positions : | | | | | nb Wig aed Ge a Be | | | | ¥,/3 Actinium™@ -..0.22: | O-O77 | O-C65 0072 0 065 | 0 C67 | Le epSeeetalic vaueeLT fan | y,/ Actinium C......... 0-128 | 0-103 | 0-116 | 0-105 | 0-109 | It will be seen that the general effect of decreasing the distance of the pr eparation in the -ray measurements ; is to decrease the ¥/8 ratio, to about one half, over the range of distance examined, for mesothorium 2 rate thorium D ; but for actinium C ae difference is less marked. No deals the causes of this are very complex. In the first place it is to be expected that the scattering of the 6-rays by the air between the preparation and the electroscope will be the greater the less penetrating the @-rays, which, in descending order of penetrating power, Ae ian C, orien D, meso- thorium 2, actinium C. Then, with diminishing distance and greater angle of the cone of rays entering the’ electroscope, the effect of “reflexion” of the rays froin the inner sides -Rays of Thorium and Actinium. 137 yo Luady d of the electroscope will come in. According to the expe- riments of Kovarik and Wilson (Phil. Mag. 1910, xx. p. 849 & p. 866), this reflexion increases with the penetrating he of the rays up to a maximum, at A(cm.) ~) aluminium 20, and then again diminishes. This eftect for most of the rays ‘therefore would oppose that of scattering. An attempt to evaluate the absor ption in the thin aluminium foil covering the preparation led to the unexpected result that the ionization was slightly greater with the foil than without. This was at the time ascribed to a possible generation in the foil of secondary §-rays, an effect which had frequently been looked for previously but never actually obtained. The same effect is recorded by Kovarik (doc. c2t.), who made a closer exami- nation of it than we have done, and ascribed it to an effect of scattering. None of the effects considered affect the order of the y/B 1 ratio to a serious extent, and they are relatively unimportant. Probably from the theoretical point of view the original ratios obtained with the preparation at a great distance from the electroscope are to be preferred. For practical purposes the 4 position may be selected, as here the distance, 13 cm., isa very usualone. The ratios, relatively to that of radium G, are for these positions :— 1/8). ome © Wb Oeuf Ene 74/84. Meeeoriamn 26) | 1-036... |) o-800 ie “Sheetal 0-690 ESTO SU phe Re aH ie eo.) 60077 0.065 0128 | 0-105 The y/@ ratios are thus of the same order for radium C. mesothorium 2, and thorium D; while for actinium C it is only about one-eighth to one-sixteenth of radium ( and therefore more nearly approaches the order previously found for uranium X. Haperiments with Thorium Minerals.—Before any meso- thorium or radiothorium preparations were ready, Mrs. Soddy had carried out a comparison of the y/@ ratio for Ceylon thorite (containing about 66 per. cent. of thoria) and Joachimsthal pitchblende (containing about 40 per cent. of uranium oxide). ‘The former mineral contains practically no uranium, and the latter no thorium, so that they represent respectively the thorium and uranium disintegration series in equilibrium. The result may be briefly referred to here. The 7/8 ratio of these two minerals was practically the same i | 138 Messrs. A. S. Russell and F. Soddy un the | Bearing in mind the fact that the uranium mineral contains two 1 | products, uranium X and radium E, which contribute B-rays iq with little or no y-radiation, this result is in agreement with ii) | what is to be expected from the determinations just described Wh of the ratio for the three single products, radium ©, meso- thorium 2,andthorium D. The thickness of the lead through which the y-rays were measured also made no_ practical | difference in the ratio. Eve has already shown that the y-rays from thorium nitrate and uraninite have practically I the same penetrating power. The similarity of penetrating | power of the y-rays from thorium minerals is to be expected from the result, detailed in Section IIJ., that the radium C y-rays are intermediate in penetrating power between the two i types of y-rays given by the thorium series. | | There remains one practically important question with q reference to the y-rays of the thorium series. What is the relative intensity of the radiation contributed by the two products mesothorium 2 and thorium D?- Mr. Alexander | Fleck has done some experiments with Ceylon thorite with a | view to obtaining information on this point, and although, so far, only preliminary results are available, these may be given here. The thorite was dissolved in hydrochloric acid, preci- i pitated with excess of ammonia, filtered, and the precipitated | hydrates subjected to the same treatment five times, in | all, without unnecessary lapse of time. The filtrates were collected, evaporated and ignited together, and sealed up in a box with a thin aluminium lid for B-ray measurement. The filtrate from a sixth precipitation, carried out immediately atter the fifth, treated separately in the same way, was inactive, showing that by this treatment the whole of the mesothorium and thorium X had been separated. The 6-ray decay curve of the preparation was taken for some weeks. The intensity of the rays rose toa maximum after 2 days (due to thorium D) and then fell, until a constant minimum, due to mesothorium only, was attained. The decay curve for the Ht maximum onward was extrapolated back to an origin corre- it sponding to the time of the last precipitation with ammonia, i and from the initial and final value of the intensity of the §-rays, the proportion due to each product could be obtained. 4 The measurements indicate that the thorium D contributes i distinctly more §-radiation than the mesothorium 2. The | difference is not great,and may be estimated provisionally as from 25to 50 per cent. Since thorium D is richer in #-rays, relatively to the y-rays, than mesothorium 2, it follows that the y-radiation from the two types must be very similar in intensity. This result, although only approximate, will prove y-Rays of Thorium and Actinium. 139 useful in calculating the rise in intensity of the y-radiation of a mesothorium preparation with time, due to the generation of radiothorium and thorium D, provided that the fraction of the radiation due to radium is known. Section I].— Variations in the Values of the Absorption Coefficients of y-rays *. Effect of distance of the preparation from the electroscope.— In all this work a lead electroscope with detachable base, similar to that previously described (II. p. 752, fig. 17), has been used. It has been shown to exclude external secondary radiations effectually. Moreover it vives of all metals easily the maximum ionization for a given intensity of y-radiation (I. p. 642), which Bragg has since explained on the view that the A-ray, produced by the y-ray, possesses really : greater penetrating power in lead than in equal weights of other metals, though its trajectory is more entangled. Asa result of experiments on the effect of the distance of the preparation from the electruscope, it was found that at a distance above 14 em. the values of A became constant, the beam being now practically parallel (II. p. 736). In most experiments the distance employed was therefore 14 cm. The following table gives the value of A at various distances for radium y-rays, a thick lead base being used with the electroscope, the absorbing lead being laid directly on the preparation. | | | | | | Sisal Semi | 25 | 115 Distance (em.)...| 5°U | 6: | O17 0:500/ 0:500' 0:498| 0-498. Similarly, for zinc, the value of X at 13 cm. was 0°278, aud at 115em. 0°274, which agree within the experimental error. The “Hardening” of y-rays by passage through Lead.— Some further experiments were done on this effect. By passage through lead the value of > for lighter metals is reduced, though other bodies possess the same power as lead in lesser degree. In no case have we observed any “softening.” Hardening is more pronounced when the rays pass first through the lead and then through the lighter metal, than vice versa. Thus, with a brass base to the electroscope, 4 Meng k 0-574 | 0°553 | | c | | { | } * This section may be regarded as a continuation of Section IIT. of the previous paper (II, p. 744). 140 Messrs. A. S. Russell and F. Soddy on the | the values of \ for zinc were (1) 0°23, (2) 0°21 when 1°24 em. of 1 lead were placed (1) between the zine and the electroscope, (2) between the zinc and the radium. Similarly the values i for iron were (1) 0°304, (2) 0280, 1 cm. thickness of lead Hh being used. These results indicate at once that the values | found for > are to some extent influenced by the nature and Vi thickness of the base employed. Thus the results given, for | M | example, in Table II. (I. p. 644), in which the mean value of | r/d for Class II. bodies (y-rays of radium) was 0°040, refer 1 to a lead base (0°975 cm.). ‘Lhe mean values for other bases \ | were as follows :— BARE catenins Lead Lead Brass Brass 0-98 em. | 2°86 em. 0-6 cm. {73 cm i Mea nie Nyaa Aaa see eres 399 3°81 392 3°82 Hh An experiment was conducted with zine to see to what | degree the process of hardening could be pushed. ‘The dis- position used in this case was different to that of the last, the radium being placed 25 em. below the usual lead electroscope | and the absorbing zine clamped up to form the base. Seven i thicknesses of lead were placed in turn over the preparation | and the absorption by the zine for each thickness of lead ain measured. ‘The results are given in the following table™*. | 2°50 | 3°75 Aicneke oe lead . 0 | 0-124 0249 | 03731 0-64 | 1-24 | ni See | | 0265 0-258 | | | Com aes ie | 0-325 0822 0809 6-300) 0291) 0272 Thus the final value is 0°79 of the initial, and there does not seem to be a limit to the degree to which the hardening | process may be carried. The remarkable point is that the Ae curves are in every case, as nearly as can be seen, exponential with the new value of X, after from 1 to 0°35 cm. equivalent Mh | thickness of zinc has been penetrated, according to the amount Mh | of lead covering the preparation. For lead the value of A is, il as we have shown, independent of thickness up to 22 em. Cp. To2))- Provided there is a base of lead J cm. thick | * Incidentally it may be noticed that the value of for the base prepa- tt ration (0°325) is 7 per cent. greater than the value found in the standard iit series of measurements given later (Table B, p. 148). The cause of this NG difference is not yet clear, but may be due to the fact that in this experi- | ment the preparation rested in a groove in a lead disk, which somewhat confined the beam of y-rays. y-Rays of Thorium and Actinium. 141 and 0°5 cm. of lead over the preparation, the absorbing plates of lead may be placed anywhere between the pr eparation and the base, the value of ® is constant at 0°50 (cm.)—.. These two facts are extraordinarily difficult to reconcile. The first indicates that each successive thickness of lead traversed modifies the nature of the radiation remaining, continuously without limit, while the second indicates that the effect of each successive thickness is to absorb a definite fraction of the radiation without modification of the nature of the remainder. This difficulty suggested the possibility that a heterogeneous beam, consisting of two types of about equal energy but different penetrating power, might be transformed by a layer of the absorbent so as to act subsequently as a homogeneous beam. A heterogeneous beam of y-rays was made by placing two point sources, one of radium and the other of meso- thorium, side by side. The activity of the mesothorium was AQ per cent. of the total, measured through 1 cm. of lead. Lead laid over the preparations was used as absorbent. X for lead for radium for this disposition is 0°50, and for mesothorium 0°62. The differences are thus of the same order as that produced in the value of X (zinc) by hardening the y-rays of radium with lead. The absorption curve for the heterogeneous beam, over the 5 cm. of lead investigated, was not exponential and by no allowance for experimental errors could it be made so, the curve being plainly convex to the origin*. The theoretical values of the ionization of the beam for the different thicknesses of lead were calculated from the data known, and they agreed perfectly with the observed values. The only way of escape from the difficulty seems to be to deny that an exponential absorption curve necessarily means the stopping by the metal of a definite proportion of the incident radiation without modification of the fraction emerging. A similar state of things exists with regard to the @-rays. It has been fairly conclusively shown that an exponential absorption occurs with a heterogeneous beam, and that the part unabsorbed suffers reduction in velocity and penetrating power. But a true exponential curve for the @-rays has never been obtained by anyone over any considerable range, whereas with the y-rays under proper conditions the absorption curve never departs measurably from the scpeuenie| form. Initial Part of the Absorption Curves.—The deviations * This experiment suggests a possible means of detecting the adul- teration of radium by Ene cophoniunt without opening the tube, 142 Messrs. A. S. Russell and F. Soddy on the from the exponential curve over the first part of the range (0 to 1 em. thickness of lead or its equivalent) now call for remark. If we place a source of radium at a distance of 13 cm. below a lead electroscope and clamp up as the base varying thicknesses of absorbing material, we invariably get a similar result. Between a thickness sufficient to absorb all §-rays and a thickness equivalent to 1 cm. of lead the absorption curve is convex to the origin, 2. e. the value of X decreases continuously. From about 1 cm. onwards JX is constant. Experiments under identical conditions were carried out with lead, tin, zinc, and aluminium as absorbers. The absorption curves for each consist of two parts, an upper steep curved part and a lower straight part. The greater the density of the body the steeper is this upper part and the longer it takes to join the straight part of the range. If now 1:3 em. of lead be placed on the radium and the results repeated, the general character of the curves is the same as before. The slopes of the lower part are less, since the rays are hardened by the lead, and the upper part joins with the lower part at a smaller equivalent thickness (0°5 instead of 1:0), thus smoothing out somewhat, but not eliminating, the convexity of the curve. With lead as absorber the placing of 1 cm. of lead over the radium has little effect, the convexity over the first centimetre being only slightly diminished. This shows that the effect has nothing to do with a soft type of y-rays initially present with the hard type, as has previously been supposed. If we now carry out similar experiments with another disposition (permanent base of lead or brass, absorbing material laid directly on the radium) the results are very different. For lead the upper part of the curve is again convex, whether the base be brass or lead, the convexity being the greater for a brass base: with zinc and aluminium, the upper part is concave, the concavity being about the same with either base. Typical results are given below. ‘The ranges are expressed in actual thicknesses of lead and zine. 1. Lead. Brass base 1°2 cm. thick. Range (cm.). -| 0 to 0:25 0:25 to 1:27 X(em.) a 0:875 0612 0-502 | y-Rays of Thorium and Actinium. 143 2. Zinc. Brass base 1°73 em. thick. Range (cem.). ... 0 to 0°66 0:66 to 1°62 1:62 to 6:00 | Nom 0-226 0-233 0-255 3. Lead. lead base 1 cm. thick. Range (cm.). ... 0 to 0:25 0°25 to 0°62 0°62 to 6:0 | | | nN Miema. 0-608 0545 0-500 | 4, Zinc. Lead base 1 em. thick. i | | Range (em.). ... 0 to 0:34 0:34 to 1:0 1-0 to 6:0 | a We Gmn.e ..... 10238 0:251 0-270 | i But it is possible to find a disposition in which zine absorbs the y-rays strictly according to an exponential Jaw with the normal value of A, from thicknesses sufficient to absorb all the 8-rays up to the greatest thickness it was possible to use with the disposition. This was briefly referred to before (II. p. 754) and may now be more fully described. A prepara- tion of radium (7 mg.) was placed at the apex of a cone of length 11°5 cm. and base 3 em. diameter cut out of a cylinder Jee, Ike A B of lead (length 12°7, diameter 10°5). 2 cm. from the mouth of the cone was placed a short cylindrical ionization chamber of lead, 9°2 cm. inside diameter and 4:2 em. long, an electrode in 144 Messrs. A. S. Russell and F. Soddy on the the centre of which communicated with a leaf system contained in a lead electroscope. Fig. 1(p. 143) shows the disposition. Absorbing screens could be clamped tightly against the ionization chamber in position A, A. BB was a thick lead plate. Experiments were carried out with lead, tin, zine, and alu- minium, first with the short cylindrical ionization chamber and secondly with one three times as long but of the same diameter. This was done to find out if the shape of the ionization chamber had an effect on the values of the coefficients of absorption. Results are given below. A. Single Ionization B. Triple Ionization Chamber. Chamber. | Range (eq.cm.| Range (eq.cm. | r. | 100.240. Pn A. | 100 afd. nee eS terme Po 0850 to | 17-4610 | nna —o —> -- | ————— ———_—_—_ —_ — Sn || 0°355 4:90 | 0°34 onward Sn .| 0°354 4‘88 | 0:34 onward Zn .| 0:267 3:77 =| 0°21 onward Zn .| 0°264 374 | 0-21 onward oe ee ————— ———— | — 1 .| 0-080 2°83 | 0:16 onward | | Al .| 0083 2°93 | 0:20 onward Thus /d increases with d for this disposition. All the curves are exponential, except that for lead whichis convex. The length of the ionization chamber made no practical difference. These dispositions are analogous to those used in Table V. (II. p. 753). For these the rays were confined in the same cone, which however entered directly through the base of an electroscope. The distance between the base of the electroscope and the mouth of the cone was 3°5 cm. as compared with 2°0 for the present dispositions. The results, however, were quite different from those detailed above. Thus zinc had for A/d x 100, 3:0 to 4°2 over a range of 0°35 to 2°6, the absorption curve being concave. These results show how meaningless a value of X for an absorbing substance is unless a full account of the experi- mental disposition under which the measurements are made- is given. y-Rays of Thorium and Actinium. 145 Srcorron III.—The relative penetrating powers of the y-rays of the Radio-elements. It is convenient to state at the outset that, as a result of the experiments to be described, the relative penetrating power of the y-rays is, in descending order, thorium D, radium C, mesothorium 2, uranium X, and actinium C. Thorium D thus gives the most penetrating y-rays known, though the differences between the first four are not great. But although it is easy to arrange the various types of y-rays in order of their penetrating power it is a more difficult matter to assign accurate values for X to each as their values as we have shown (LI. p. 754-755) depend greatly upon the conditions, and their ratio for different rays is by no means constant when the results for several dispositions are com- pared. Thus the values of > for the uranium y-rays in four combinations, of two different metals with two different dispositions, were respectively 46, 28, 25, and 18 per cent. higher than for those of radium, and similar considerations hold good equally when the thorium and radium y-rays are compared. The radioactive preparations used in the following measure- ments were :— 1. Radium: 7 mg. of radium bromide, and in some ex- periments 0°5 mg. of radium bromide. 2. Mesothorium: a single tiny grain of concentrated mesothorium obtained from Knofler and Co., equi- valent in y-rays to about 0°31 mg. radium bromide measured through 3 mm. of lead. All these preparations are practically point sources in sealed glass tubes. 3. Radiothorium: this body mixed with moist thorium hydrate was contained in a sealed cylindrical tube about 6 cms. long and about 0°5 em. diameter. It therefore differed from the others in not being a point source. It was equivalent in y-rays to about 0°56 mg. radium bromide measured through 0°3 cm. lead. It also was obtained from Knofler and Co. Dispositions.—Sketches of the three dispositions employed are shown in fig. 2. They hardly need further description. Disposition 1 corresponds to that used in obtaining Tables II. and III. (I. pp. 644 & 646). It is easiest to use in practice Phit. Mag. S. 6. Vol. 21. No. 121. Jun. 1911. L 146 Messrs. A. 8. Russell and F. Soddy on the as the electroscope is not disturbed. Disposition 2 is similar to that used in obtaining Table I. (I, p. 633) except that the electroscope is of lead as in Disposition 1. It offers certain theoretical advantages, but is more difficult to work with. Fig. 2. O) ABSORBENT ABSORBENT The only reason for using Disposition 3 was that uranium X had once been examined with it. It was not possible to prepare uranium X for the present set of measurements, but many of the previous results can be utilized. Those of Tables II. and III. (I. pp. 644 & 646} only differ from the present Disposition 2 in that a wood stand instead of a lead stand was used for the preparation, but parallel experiments with mesothorium 2 showed that this had no effect on the values of A. The results with Disposition 1 are tabulated in Table A. After a thickness equivalent to 1 cm. of lead (total thick- ness, with base, 2 cm. of lead) all the curves are exponential. Lead is in a class by itself (Class I.) with high value for A/d throughout. The Class II. bodies have an approximately constant A/d, but the limit of density down to which this holds is greater for the more penetrating rays than for those less penetrating. The lightest bodies (Class III.) have again higher values for A/d. y-Rays of Thorium and Actintum. 147 TABLE A.— Disposition ie | Thorium D. Radium C. |Mesothorium 2.| Uranium X. ee OUNVA I Ne | WOO N/a hee OO Nal xX.” lOO n/a. Me ‘0-462 | 405 | 0500; 438 | 0620! 5-44 10-795 | 636. ica CS 0-294 | 3:34 0878 Moanin © Prise scr ti A 0-271 | 3-25 0°355| 4-25 2 ian oe 0-250 | 3-28 WOES) Hes ls dO ecco cee aeeemneeee 0:236 | 3:26 Fr co | 0305} 4-21 ao Cision ii ae |0:233 | 3:30 az | 0300) 4-24 a Siete fess 3.5 0:0961) 3:37 a] — ue S Aluminium ...... (0:0916) 3:24 = 0-119} 4-91 Cho 00886} 3°52 0-113) 448 |0122 | 4-84 Magnesia Brick... 0:062 | 3°23 0-090} 4:69 |0-0917) 4-78 Sulphur........ ... 0-066 | 3:69 | 0:078| 4:38 | 0:083| 465 |0-:0921| 5:16 Paraffin Wax...... 0-031 | 361 | 0-040} 464 | 0-050) 5:80 |0:0433) 5-02 | The nature of the absorbent has a marked effect on th relative penetrating power of the four types of rays. Taking the values for radium C as unity, this is shown in the following table, mean values for Class II. being employed. Thoriun D. | Radium C. | Mesothorium 2.| Uranium X. Mead! 2.62... 0.924 1:00 1-24 1:45 ee | | Class IT....... 0°82 1-00 1:06 1:18 In Disposition 2, with the absorbing plates clamped up to form the base of the electroscopes, two sets of experiments were performed, namely (2 A) with the preparation bare, and (2 B) with the preparation covered with 0°64 cm. lead, to see if the hardened rays produced would be exponentially absorbed by all bodies according to the density law. The result shows that this is the case except for lead. Table B (p. 148) shows the results obtained. All absorption curves for this disposition are also expo- nential after 1 cm. of equivalent thickness of lead has been traversed. The curves made by plotting equivalent thickness of metal against the logarithms of the ionizations are co- incident. So they are also in the case of Disposition I. This holds both for Dispositions 2A and 2B. This is opposite to what was previously obtained with a brass elec- troscope for the absorption by bodies over small initial eee eeeeae eee eee ecoeeee eee eee eoescee ee eeeees Sale ee he EL TS LS Se os ee pe ee Sah tee a (eS gees eee an a ee eee ie ee XvAA Ulpeeg cae anydjng eee eon ssorere eee reeesenene eeeeeeae eeceseees IL-F | ¢980:0 | — = 9%-€ | 4060-0 | %9-4 | #8400 || T¢8-€ | ze¢0.0 | Zet-¢ | coFO.0 . va oe aa = — = = = GIL-€ | £990-0 | 6¢0-¢ | 060.0 n> See ee ae | eee ee SS ee = c8-€ | 601-0 | Fr | 921-0 SFE | 11600 | G4 | 901-0 || 91¢¢ | goT-0 | ogeF | #zr.0 o SS | A S| eS NE a. | Sa POT O EE. —K_—_——_ ee es | es ee Sa bp, | 89:6 | 0930 | 9FF | GIE0 || SFE | 9FZ0 | GLE | 8960 || GeLe | F900] Fes | cog eS ——_———__. SSS ——— a S OIF | 866-0 | ser | Igo BFE | 2960 | S88 | 112-0 || 846-2 | 182-0 | Oger | ore. oD) —- | ———_ | ——_—_—_. —— | -—_—— |—_— —_——|| —— ——_— | —__._ | cl 08E | 68-0 | 88h | F880 6F-€ | 9920 | 948 | 282-0 || OFLE | G8a-0 | 9zzr | gze.0 3 as = — -- — = = = 104-6 | 608-0 | 662-4 | 6¢e0 8 ame || te eae ecm |r| | ea eer | re ee | ear a, | Saas Pees || || sees Se = 90-6 | 88-0 | OFF | 188-0 Gre | 4080 | TL | 466-0 | FI8¢ | 9680 | eizr | LLe0 fab) —— ——— ——— [| ——— | | _ s _ | —_— —_— |_| ————<—— % | gece. | ¢e90 | ce0-c | 3790 || 626 | goF0 | gece | Goro | oer | 00¢0 | cece | 210.0 Ore NX A 00K = NX =| P/N O0l |X eX OoL) NX «| poor | | 2NOOTE =X rH ‘13 ‘VS EZ VS a ‘VG o = ‘G WAIMOHLOSA PL ‘Ga WOATHOIMT, ‘O WOATAV4YT ‘AS pure yg suowsodsiq—g wavy, 148 y-Rays of Thorium and Actinium. 149 ranges of thickness of the rays of uranium X (II. p. 729). There the bodies arrange themselves one above the other in order of the density of the material, the lighter substances giving for equivalent thicknesses far more ionization than the denser. The mean values of 100x2/d are tabulated below. | Thorium D.|} Radium C. | Mesothorium 2. ase, liana Weil chaegs unit fh Ten 2 Weil area AU yaar ar Class IT. 2 A..| 376 OS Oe ee Gee HL 2B.) 348 AOE isl Mike sg ily It is seen that the values for lead are very near one another but the differences are quite real. For radium the value of X=0°517 has been obtained at least six times for Disposi- tion 2A. Taking the values for radium as unity the results are summarized in the following table :— Thorium D.| Radium C. | Mesothorium 2.! Uranium X. QA 0°79 1:00 1:24 1:27 Lead... pase eo MES AS wate’ 2B 0-82 1:00 MA _ 2 Al 0°88 1:00 1:03 — Class IT. | pe ee USAR BALI Le ae See 2 B 0°92 1:00 1:045 — In Disposition 2 A, the values of X are all about 5 per cent. greater than in Disposition 1. In Disposition 2 B lead still absorbs normally, but both Class Il. and Class III. bodies now obey the density law. Disposition 3 was only used for copper. The following values of X were obtained :— | Thorium D. | Radium C. | Mesotborium 2.| Uranium X. Metco) Conte) i) esas 0:357 0-432 | or relatively | | | 0:80 00 | TEOB5!. 1). B26 150 Messrs. A. S. Russell and IF. Soddy on the The results are collected in the following table :— | Thorium D.| Radium C.| Mesothorium 2.| Uranium X. Gil med) Oe OO 1-24 [as mn 2. | Glass Th. | Osa | 4100 «| oe a 3 |tea2a | 0O7o el 100 | 3200 Wg ee "90 100 | Io bo cue re) Weer too | 0s ae, md PA NRE: | a A Gen meg mel) Ose 1-00 1-045 a ae tees ae "Thiocon ah eto = rossi ae * Not exactly the same disposition. Thus the value of > for thorium D is from 8 to 21 per cent. less, and for mesothorium 2 from 4 to 25 per cent. greater than for radium ©. Disposition 1 might well be adopted as a standard disposition for y-ray comparison of the intensity of radioactive preparations. Preparations of widely differing activity might be compared by means of carefully prepared lead blocks of known thickness, adopting, for the value of X, 0°500 (em.)71. The y-rays of Actinitum.—Through the kindness of Messrs. Buchler and Co., of Braunschweig, who most generously lent for the purpose a very fine preparation of actinium, weighing 1°5 grams and equivalent in y-activity through 3mm. of lead to 0:23 mg. of radium bromide, it has been possible to make a more extended examination of the y-rays of this substance than has hitherto been done. The prepa- ration was in a sealed tube for these measurements, and subsequently the tube was opened and the preparation used to obtain the active deposit for the determination of the y/8 ratio described in Section I. Godlewski (Phil. Mag. [6] x. p- 378, 1905) measured the absorption of the actinium y-rays over the range from 0 to 3°5 mm. of lead and 0 to 10 mm. of iron and of zinc, and obtained nearly exponential curves, the value of A(cm.)~—1 for lead being 4°54, for iron 1°23, and for zinc 1:24. He thus showed clearly the markedly less penetrating power of these rays as compared with those of radium and the abnormally high absorption in lead. Hve (Phys. Zeit. viii. p. 185, 1907) with a stronger preparation found for » 4:1 over a similar range of lead. At 3 mm. thickness a sudden change to ) =2°7 took place and continued y-Rays of Thorium and Actinium. 15] up to 5°7 mm., the limit of the measurements. We have been able to extend the range up to 2°5 cm. of lead, or including the 3 mm. thickness of lead as base to a total thickness of 2°8 cm. Owing to the abnormally high absorption of the rays by lead, as noticed by Godlewski, Disposition 1 could not be employed, but Disposition 2 was possible. A new arrange- ment (Disposition 5) similar to Disposition 1 was used, in which the walls and base of the lead electroscope were 3 mm. thick and the preparation was mounted 83 cm. below. The absorbing lead and zine were placed directly over the preparation, and parallel experiments were done with radium for the same disposition. The results for lead and zinc are plotted in the curves (fig. 3). The abscissee are centimetres Fig. 3. Y RAYS of ACTINIUM : , DENSITY OF BODY "os 0 2 10 Tx DENSITY OF LEAD 7 ZS of thickness multiplied by the relative density of the absorber compared with that of lead. The aluminium curve would coincident with that of zine if plotted. The figure shows how abnormal lead is. Although the measurements were made through a base of lead 3 mm. thick there is still the point of inflexion in the lead curve at 0°3 cm., noticed by Eve, and a still more decided point of inflexion at 0°85 cm. After that the curve is linear up to 2°5 ems. For zine and aluminium the curves are exponential. In Disposition 2 the curves for copper, aluminium, and zine are exponential after 152 Messrs. A. 8. Russell and F. Soddy on the a small initial thickness (0°36 cm. for zinc), but for lead the curve is convex to the axis up to 0°8 cm.,as in Disposition 5, and then is straight. The results are tabulated below, the results for lead referring to the straight part of the curve beyond 0°8 cm. Disposition 5. Values of (cm.) | d (Aet.)/A (Rad.). | Actinium. Radium. 1 USEING Le eR oare se aia ea td Sa 1:20 0°547 2°20 PAWNS SES AINE LD EN | wer O21 0269 1-94 Zine (hardened by 0-6 cm. Pb), 0-420 | 0262 1:60 Aiminiann eee | 0-217 0-115 1:8) | Disposition 2. pile eA wh WO den Ate ate 1:85 0-517 3:57 NONE Es es et eed oe 0618 0:305 2:02 | Zinc (hardened by 0°6 cm. Pb) | 0495 0:264 1:87 | Arta 43. eteas Inte ee poset 0:234 0-124 1:90 cia kaa ames eae | 0722 | 0877 1-91 The separate mean values of A for different thicknesses of lead are shown below. INisposition 5. | | | Range (cm.).... 0-0:124 0°124-0'30 0:30-0:85 0:85-2:40 | NiGewe nee 33 265 | 2a2 | 120 ! J Disposition 2. Te nN Ootiorhe: 0-3-0°5 0:5-0'8 | 0-8-1°7 | Be) Uae Oe ee eR Gn. A 4-24 | 3-26 2°50 | 185 | y-Rays of Thorium and Actinium, 153, In Disposition 2 the curve is the more continuously convex to the origin. For both dispositions the ratio of the absorp- tion coefficients is about 1: 1°9 for Class IT. bodies, and the effect of hardening the zinc is the same, the value of A being reduced to 80 per cent. But the actinium rays are, rela- tively to radium, hardened more in Disposition 5 than in 2. Godlewski concluded that the actinium y-rays have only one-tenth of the penetrating power of the harder y-rays of radium. Asin the case of uranium X, however, work with more active preparations over greater ranges has reduced this difference, though actinium is still far removed in this respect from the other y-ray sources examined. Some Generalizations with regard to y-Rays. The results for actinium help to accentuate several relation- ships which exist between the penetrating power of the y-rays of the radio-elements in solid bodies and other properties they possess. There is obviously an intimate connexion between penetrating power, abnormality of absorption by lead and hardening by lead. The rule is—the greater the penetrating power of the ray, tbe less abnormal is the absorp- tion by lead as compared with that of Class II. bodies, and further the less hardening effect has lead on the rays sub- sequently absorbed by Class II. bodies. In Disposition 1, however, thorium D is an exception. Thorium) Radium | Mesotho- Uranium D. C. rium 2, d/d Lead Wee Se ee wh ee | telah Me, . sl X/d Class IT. Disposition 1 ... 1:28 1-10 1:28 1:35 r/d Lead oy X/d Class I. Disposition 2A . A/d Lead Reta Penn ie * X/d Class Ll. Disposition 2B.) 1°03 | 1:16 1:40 men A/d Class II. (Disposition 2A) | y-09 | 4492 | 119 o 1-25 d/d Class 11. (Disposition 2 B). Again, as regards penetrability, there is far more con- nexion between the y-ray and the a-rays preceding and following it in the disintegration series, than there is between Actinium 154 y-Rays of Thorium and Actinium. the y-ray and the @-ray which accompanies it. It is possible, 1 from the results that have been given, to make a rule con- | necting the penetrability of the y-ray with the period of the change in which it occurs, analogous to Rutherford’s well- | known rule for the range of the a-particle. Actinium C is | an exception throughout, and has been omitted from the comparison. In ascending order the penetrabilities of the y-rays are :— | | Uranium X, Mesothorium 2, Radium C,, Thorium D. bi This is also the descending order of their periods, and also if of their probable atomic weights. 1h If the conclusions of Hahn and Meitner (Phys. Zeit. x. | p- 697, 1909) are correct, the @- and y-rays of radium C i come from radium (©, (half-period 19 minutes), while the We a-rays come from a succeeding product, radium C, (half- | } period, about 2 minutes). This accords with Rutherford’s i} rule, as the still more penetrating a-ray of thorium C is, in il all probability, derived from a change even more rapid than 1) that of radium C,. ‘The above order of the penetrabilities of the y-rays is alse that of the a-rays which precede and follow them, whereas the order of the penetrabilities of the @-rays [ is quite different. The figures in brackets denote ranges of i the «-rays in centimetres of air. | VTAYS © oases Uranium X Mesothorium 2 Radium ©, Thorium D wy i a-rays Ti preceding :— | Uranium (2°7) | Thorium (3:5 ?) Radium A (4°83) |Thorium C (8°6) a-rays following :— | Ionium (2°8) | Radiothorium (8°9) | Radium C, (7:06) — B=rays <.3-.: Mesothorium 2} Thorium D Uranium X Radium C, Neither in penetrability, relative intensity nor homogeneity are the @-rays obviously connected with the y-rays, whereas there is a certain connexion between the y-rays and the a-rays. | Physical Chemistry Laboratory, i Glasgow University. Waisor XVIII. The Normal and the Selective Photoelectric Effect. By R. Post, Ph.D., and P. Prinesnemm, Ph.D.* ) a researches made by different authors on the photo- electric effect lead to some results which seem to be in striking contradiction to each other. 1. E. Ladenburg f and Mohlin{ have shown that the number of electrons produced by the unit of ultraviolet light increases with decreasing wave-lengths. 2. Elster and Geitel§ found qualitatively that, for the alkali metals, this number has a maximum in the visible part of the spectrum, and T. Braun || proved this to be right quantitatively for Rb and K. 3. On the contrary, Hallwachs{ denied the existence of these maxima, and from new measurements he concluded that Ladenbureg’s and Mohlin’s law is true also for the alkali metals. Contradictions of a similar kind are found in the researches on the effects of polarized light. 4. According to Pohl** the orientation of the vectors of ultraviolet light has an influence upon the number of liberated electrons only so far as the absorption of light depends upon it; this is true for solid surfaces of Cu and PtTt, as well as for liquid Hg ff. 5. But Elster and Geitel§§ had found previously for the visible part of the spectrum, that with alkali metals the photoelectric current is proportional to the absorbed energy of light at different angles of incidence only if the electric vector swings at right angles to the plane of incidence: if the electric vector swings in this plane there is only an approximate proportionality between the component of the electric vector normai to the surface, and the photoelectric current due to this component. In this second case the * Communicated by the Authors. + Jt. Ladenburg, Verh. d. D. Phys. Ges. ix. p. 504 (1907). t H. Mohlin, Akad. Abhandl., Upsala, 1907. § T. Elster and H. Geitel, Wied. Ann. lii. p. 493 (1894). || T. Braun, Dissertation, Bonn, 1905. 4] W. Hallwachs, Verh. d. D. Phys. Ges, xi. p. 514 (1909). ** KR. Pohl, 2bid. xi. p. 339 (1909). tT R. Pohl, zécd. xi. p. 389 (1909). Tt R. Pohl, zded. xi. p. 609 (1909). §§ T. Elster and H. Geitel, Wied. Ann. xi. p. 445 (1897). 156 Drs. R. Pohl and P. Pringsheim on the factor of proportionality was much larger ; hence the ratio of the photo-effect in the two main planes of polarization was found to be sometimes as high as 1:50. This has been confirmed by different authors. | 6. Finally, Pohl* found that these observations of Elster and Geitel are true only in the visible part of the spectrum, while in the ultraviolet the alkali metals behave like the others, the orientation of the electric vector being immaterial. Now we are going to show how the coexistence of these different facts, though they seem to be absolutely incom- patible, may be easily explained by assuming that there is, apart from the normal photoelectric effect, the only one known as yet, still a new selective one. The full report on this subject has been published in the Verhandlungen der Deutschen Physikalischen Gesellschaft, vol. xii.t, and all details may be found there; the numbers in [ | correspond with the divisions of these papers. We used a spectroscope with quartz-fluorite lenses, illumi- nated by a mercury arc-lamp ; the energy of the single lines was measured by means of a Rubens thermocouple, and the light was thrown through a quartz polarizer and a window ot fused quartz into a tube of the type described by Hlster and Geitel, which contained the alkali metal or some alloy. By the aid of this apparatus the photoelectric current could be measured, which was produced by the same energy of incident light in the two main planes of polarization and in different parts of the spectrum. Table I. and fig. 1 show the results for an alloy of mercury and potassium, containing 2°5 atomic percentages of the alkali [11]. These numbers and the corresponding ones for Na prove that at an incidence of 60° the emission of electrons increases continuously with decreasing wave-length, quite independently ot the plane of polarization. The effect is larger if the electric vector swings parallel to the plane of incidence (H ||) than for HL, in consequence of the greater absorption of light which follows from the optical constants [12]. But for the same energy of absorbed light the plane of polarization bas no influence upon the intensity of the emission. Hence for this photoelectric effect of the alloys, * R. Pohl, Verh. d. D. Phys. Ges. xi. p. 715 (1909). + R. Pohl and P. Pringsheim, Verh. d. D. Phys. Ges. xii. (1910) pp. 215-230, §§ 1-9; pp. 249-260, §§ 10-17; pp. 682-696, §§ 18-27 ; pp. 697-710, §§ 28-37. Normal and Selective Photoelectric fect. TABLE I. 157 2-5 atom. per cent. R ; 97°5 atom. per cent. Hg. 6=60°. w = va Gy Oo oO o S Photoelectric current 10-18 awp. ~P A0 No. Wave-length Hp. Fig. 1.—K—H_g alloy ; ¢=60°. 4 AS sae 300 Wavye-length, Photoelectric current. Bl 10-18 amp. \3 0°45 400 10-18 amp. 18 0-13 it 158 Drs. R. Pohl and P. Pringsheim on the which is due exclusively to the alkalies, the assertions 1 and 4 are true, even in the visible part of the spectrum. We call it the normal effect. All the well-known facts such as the independence of the temperature, the curves of the initial velocity and others refer to this effect, and so it may be explained by the atomistic theory of light according to Planck-Hinstein * [18]. The results are very different for pure alkali metals or Hg alloys of certain concentrations. Also in this case the normal effect exists for HE || and HL as in fig. 1, but for E || there is a second effect superposed on the normal one, Photoelectric current. A=200uu 300 400 500 Waye-length. ina short range of the spectrum ; it grows rapidly from a small value to a very great intensity, and decreases then again with decreasing wave-length to zero, suggesting a resonance phenomenon ; it is the more intense the stronger the component of the electric vector normal to the surface of the metal. The orientation of plane of polarization is of great importance for this effect, which may be called the selective one. Fig. 2 shows the superposition of the two effects [18]. Fig. 3 and Table II. contain the results which we obtained with polarized light on a sodium-potassium * Cf. R. Ladenbureg, Jahrb. d. Rad. u. El. vi. pp. 425-487 (1909). Normal and Selective Photoelectric Effect. 159 alloy (69°4 per cent. K):[10]. In fig. 3 the ultraviolet end of the selective effect cannot yet be recognized; but Fig, 3,—K-—Na alloy ; polarized light. ee LSS ee PERE EEN A= 200 uu £00 400 500 Wave-length. Fig. 4—Potassium ; unpolarized light, fo) oO o oO aaa PEE ay SEO eoe magic cals cara ee ea ee Ae eee a a NA [a lS WF el aa SN op ete PTO RTE, A= ft ey 400 500 Wave-length. cs So iS (=) [Se] So ~ Oo e Ehotoelertne current 107) amp. this is easily seen in figs, 4 4 and 5, which were observed on K [14] and Rb [15] in unpolarized light, and show again 1 160 Normal and Selective Photoelectric Effect. the addition of the two effects ; Tables III. and IV. cor- respond to these figures. Fig. 5.—Rubidium; unpolarized light. Photoelectric current 10—1° amp. Waye-length. Apis We K-Na alloy. 6=60°. (UNaiow OOR Solid K. $=60°. Solid Rb. 6=60°. TABLE LV. | Photoelectric current. | B | EL 10-}3 amp.| 107 amp. ——_—<— 546 0 O1l 436 100°0 8-0 406 159-0 12:3 138 Photoelectric .| Wave- eurrent length | 107!8 amp. ed | 546 0-9 436 60 No. There is no more difficulty in explaining the assertions 9, 3, 5, and 6 with the aid of the figs. 2-5. Wave- Photoelectric current length 10—}° amp. | pep. a a 546 0°47 (2) Elster and Geitel and Braun, using visible light at an oblique incidence, obtained the addition of the two effects, and as the selective one is much stronger their curves had a maximum in the visible part of the spectrum. (3) In Hallwachs’s experiments the light being incident Maintenance 0) Periodic Motion by Solid Friction. 161 normally had no electric vector swinging in this direction, and so the author measured only the normal effect which increases continuously with decreasing wave-length. (5) Elster and Geitel investigated the influence of polari- zation only in that part of the spectrum in which the K-Na alloy has its selective effect. Consequently they found only for EL, that at different angles of incidence the emission of electrons was proportional to the absorbed energy ; and the ratio of the effects in the two main planes of polarization had naturally values, which like 1: 50* are much too large to be only explained by the difference in the absorption of light, and which prove the existence of some special efficacy of E || £19]. (6) The same was true in Pohl’s experiments, in which above the ultraviolet end u of the selective effect (fig. 2) the singularity of H || disappears, so that only the normal effect is left. Inasmuch as all the contradictions which we spoke of in the beginning are explained now, we think that we are justified in saying that a new selective effect exists, apart from the normal one. We do not as yet know much about the true nature of this effect, but from experiments made on K-He alloys [37] and on the influence of the angle of incidence | 27 |, we conclude that it is a molecular resonance phenomenon, in which the electrons follow directly the electric vector ; at any rate it cannot be explained by the ‘“* (uantentheorie~’ of Planck-Hinstein. We are continuing the researches with the aid of a grant from the Jagor Fund (Berlin), for which we wish to express our best thanks. . Physikalisches Institut der Universitat Berlin, October 1910. XIX. On the Maintenance of Periodic Motion by Solid Friction. By ANDREW STEPHENSON f. is ‘| hee maintenance of periodic motion by solid friction demonstrates that such friction diminishes as the velocity increases through a small range at least. However the friction may vary there is always a position of equili- brium and the small motion about it is evidently of type e+ (Kk—A)e+nr7e=0, * We got even the ratio 1: 300 in K—Na alloys, in which the normal effect was comparatively much smaller than in fig. 3, + Communicated by the Author. iil. Mag. 8.6. Volo 2). Noy 121, Jan. 1911. M 162 Mr. A. Stephenson on the Maintenance of where A is a positive quantity proportional to the velocity- rate of change of the frictional force at the value correspond- ing to the relative velocity in equilibrium, v say. The position of equilibrium is therefore unstable if X is greater than «, and any slight disturbance is secularly magnified in free period. The equation continues to represent the motion for larger amplitudes if X is taken to represent the mean velocity-rate of change in the frictional force between 0 and 2. So long as & is less than v, Vis positive ; but when there is no relative motion it is indeterminate, having any value from that given by continuity down to a large negative limit. The motion therefore increases until # reaches the limit v, if « is small enough, and the steady motion is determined by the fact that =v when = (A—«K)o/n’, where \ has the greatest statical value, which is proportional to the quotient by v of the difference of the coefficients of friction at the relative velocities zero and v. Starting from this configuration the motion is approximately simple and of slowly increasing amplitude, so that the velocity v is again attained when w has a negative value numerically greater than that given above: the static friction then prevents any further change of velocity until the initial state is reached. The magnifying action of solid friction was observed by W. Froude in the case of a pendulum suspended from a rotating shaft *. 2. The instability of the position of equilibrium under friction holds evidently when the system has more than one degree of freedom. The general problem of determining the steady motion would be difficult and of little interest, but happily in the important case of the violin string simplicity is introduced through the infinitude of freedom, and a possible steady motion may readily be determined f. In the absence of damping any free motion is possible in which the bowed point has no relative motion with the bow and constant speed against it, the mean position being that produced by a constant force equal to the friction in the backward swing. This is evident from the fact that the maintenance of the static state at the point of contact during the forward swing necessitates the constancy of the friction throughout the whole motion. * Cited by Lord Rayleigh, ‘Sound,’ i. § 138. + The kinematics of the motion were long ago established by Helm- holtz in his well known experimental examination. Periodic Motion by Solid Friction. 163 If the bow is applied at a node dividing the length in the ratio p/qg where p and g are integers with no common measure, there are p-++q—1 free motions in which the point has constant speeds to and fro, the ratio of the forth and back intervals being 7/s where r and s are any integers such that r+s=p+gq. Whatever the position of the node, there- fore, two in general distinct motions are always possible, and in these the intervals are proportional to the segments of the string. When the string is subject to slight air resistance any possible steady motion must approximate to the undamped kind, and we shall prove that with the bowed point ata node that motion is possible in which the ratio of the forward interval to the backward is equal to the ratio of the greater segment to the less. Let the length of the string be 7, and the point of bowing at distance a, >7/2, from the end (1); also let y, and y, be the displacements from the mean pusitions at distances 2, and a. from the ends, both measured positive in the direction of bowing. If the maintained motion of the bowed point is of free type f= Ns Fe where the summation necessarily does not include com- ponents having a node at the bowed point, =2 ae {sin rv, (sin rt + xa cot ra cos rt)—«x, cos ra,cosrt} . . - (CJ) a A, sin r(7— a) e e ap Oper teee VoxD a a L : sin ray (sin rt +Kr—acotrm—« COs 1t) — Kxy COS Plz COSTES terms of order «? being neglected. At the bowed point, A, d d Peel Be ES QeTr>r pune COCN US ntsc. Us (3) ax, date SsIn ra The quantity on the right with the addition of some constant represents the force necessary to maintain the motion. The condition for a steady motion is that this be equal to the frictional force so long as there is relative motion, and be - not greater than the statical friction when there is no relative motion. In the case under examination the velocity is v and M 2 164 Mr. A. Stephenson on the Maintenance of —av/(a—z) for intervals —a to « and a to 27—a, so that aif Es A,= =, Sin re, T—ar aud the right hand side of (3) becomes AKT 1 SS hy Se SOS TiO! SIAL In the simpler cases 2=7/2, 27/3, and 37/4, it is readily seen that this represents a constant between « and 27—e@ 3 and in any example, since the cosecant coefticients of cos rt/7r recur and all lie below a certain limit numerically, a similar: result as to the constancy during the shorter interval may be: verified arithmetically. Let this constant value be &*, and the greatest value during the interval —a to a, k’. Then the force maintaining the motion is not greater than P+ pk’ during the forward swing, and is equal to P+pé in the backward swing, p being the tension of the string. If the pressure is F, « the coefficient of statical friction, and p’ the coetficient at relative speed wv/(7—«), then P+pk=Fwp' and P+plh'+Fu. The equation determines the mean position of the string, and. from the inequality it is evident that the pressure must not be less than p(k’ —k)/(u—p') We have shown then that the bow when applied at a node maintains that steady free motion which does not contain the components corresponding to the node and in which the point of bowing vibrates with constant speeds during intervals. proportional to the segments of the string, travelling with the bow during the longer interval without slipping: and the only condition for the possibility of the motion is that the- pressure exceed a certain limit. All further properties belong to the free vibration defined: by these facts: the equation of the motion, in accordance: with (1) and (2), is ZU UT GF where the summation includes all except the node com-. ponents. : Thus the amplitude of any component, when it occurs, is- sin rt n?—1 7? * = ; SS = = If ptg=n, > aa ae when ¢=a, so that 2 2 pa em -1 ( ie ) sree. T7—a ie SID resin @t, .\' |. 2 Periodic Motion by Solid Friction. 165 proportional to the product of the speed of the bow and the reciprocal of the smaller segment of the string, provided this product is not too large. If the bow is transferred from a more important node, such as that at a trisection of the string, to one ata distance of only a small fraction of the length, the missing com- ponents are restored without other appreciable change, since the components dropped are relatively insignificant. Hqua- tions (1) and (2) hold good so long as xa/sin re is small, and the node harmonies are therefore brought to normal ampli- tude when the fractional displacement is large compared with &/o, where o is the lowest of the frequencies involved. The rapid restoration of missing components in passing from a node has been cbserved aurally. 3. The motions examined experimentally have been found to be of the type (4): others, however, might occur. These are to be sought for among the known group of undamped motions. A sufficient illustration is afforded by the simple case when the bow is applied at a point one fourth of the length fromanend. There are then three undamped motions, and of the two in which the ratio of the intervals is equal to that of the segments, under air resistance only that is possible in which the motion with the bow occupies the longer in- terval. In the third the intervals are equal and A,= —-; sin mS Tin ; so that the series in (3) is a to demv from 0 to 7/2, and to —4xmv between 7/2 and w. ‘The motion is therefore steady if the pressure is not less than 8xvp/(u—p’'). In the ordinary case the pressure is not less than L6xrvp/(u— p'), so that for the range of pressure between these limits only that motion is maintained in which the times to and fro are equal : ym nr nit = Es Ge er tee y V/ 2 2 ( 1) (i +4n) sin 1+4nasin 1+4nt ne oe nye sin 3+4n sin 34+4n ae a vibration distinguished by the absence of all the odd octaves. Whether any possible steady motion is set up by suitable bowing remains a matter for practical trial. @etober, 1910. Peekos ol] XX. Ona Peculiar Property of the Asymmetrie System. By ANDREW STEPHENSON ™. F an asymmetric system is subject to a direct periodic force the equation of motion is of the form 2+ cknéi+n?(14+ 2/a)x=bgq’ cos gt. a being finite and 6 and « small, the steady motion is g° gee oe kuhg > r=b— n? when terms of the second order are neglected. This motion may, however, be unstable if g approximates to 2n f. Putting te 2(n+ p) and bq? =3cn*, we find that there is inset if c?>(ka)” for the range of frequency for which lp|< dn (ela)? =? The steady state of motion is then given by L= = +r cos{(n-+p)t—a} +," "cos 24(n+p)t—a«}—c cos 2(n+ p)t, cme Ce ee |k| being <1, = 8ca(1— k),\/1—(ka/ey? (xa/c)?”, and sin 2a=«xa/e, where 7/2>|a|>7/4. Thus if ¢ sufficiently exceeds xa, e. g.=2«a, the amplitude is large, being of order “ce. Any small deviation from this motion may be analysed into two simple motions one of which is reduced and the other undisturbed (so far as terms of order c are concerned). The asymmetric system then is sensitive to direct periodic force of double the natural frequency and of intensity exceeding a certain limit proportional to the motional resistance. If asymmetric oscillations exist within the molecule there would follow the possibility of monochromatic fluorescence with a frequency of emission half that of incidence. September, 1910. * Communicated by the Author. + “On the Stability of the Steady State of Forced Oscillation,” Phil Mag. December, 1907. Iie Leib XXII. The Reflective Power of Lamp- and Platinum-Black. By T. Royps, M.Se., 1851 Hxhibition Science Scholar *. ie) oe NGSTROM found in 1898 + that the reflective power of a thickly sooted platinum-black surface was from 0°82 to 1:25 per cent. for different regions of the spectrum. On the other hand, Féry f concluded from his experiments that as much as 18 per cent. of the radiation from a black body at 100° C. was reflected from a platinum-black surface, and approximately the same amount from a_ lampblack surface. Since a knowledge of the reflective power of the surfaces usually employed as receivers of heat radiation is important for the determination of the absolute radiation constants, a method suggested by Prof. Paschen § has been employed to determine the reflective power of lamp- and platinum-black for definite wave-lengths in the infra-red. The method consists in measuring the galvanometer deflexions, first when rays fall directly on a thermopile, and then when they fall on the lamp- or platinum-black, the rays diffusely reflected from it being focussed on a thermo- pile by means of a polished hemisphere of german-silver. The black surface A (fig.1, p. 168) was attached alongside the ther- mopile slit B which was situated immediately in front of the exposed junctions. An external concave mirror cast, through a narrow opening O cut in the hemisphere, an image of an illu- minated slit on to the thermopile when placed at C a little to the left of the centre D of the hemisphere. If the thermopile together with the black surface was now displaced relative to the hemisphere until the thermopile came to B at an equal distance on the opposite side of the centre, the slit image would then fall on the black surface and the reflected light would be focussed on the thermopile by the hemispherical mirror. The galvanometer deflexions in these two positions of the thermopile would measure the intensities of the incident and reflected radiation respectively. In order to interchange different black surfaces or to interpose screens in front of the thermopile only, the thermopile with the surface attached could be taken out and afterwards replaced * Communicated by the Author. A preliminary communication appeared in the Physikalische Zeitschrift, xi. p. 516 (1910). t+ Angstrom, Ofversigt of K. Vetensk.-Akad. Férhandl. Stockholm, v. p- 283 (1898). t Heéry, C. RK. 148. p. 777 (1909). § Modification of method by which Paschen made his bolometer blacker. See Ber. Berl. Akad. d. Wiss, April 27th, 1899. 168 Mr. T. Royds on the Reflective in exactly the same position relative to the hemisphere. When the external optical arrangement was altered, the age 1 U ‘f U t s 5 / t ! \ | l ! i ! 1 1 i ! t I i ' \ ‘ \ S 4 AS <. < ‘ ~ \ \ thermopile was placed in its first position to receive the light directly, and the external concave mirror so adjusted that the slit image fell on the thermopile slit which could be seen from behind through a window. In this way, it was not necessary to alter the two positions of the thermopile. These two positions were determined once for all by experiment and checked at intervals. The continuous line in fig. 2 shows the deflexions for a matt reflecting surface as the thermopile was gradually displaced in the great circle of the hemisphere. The distance between the incident slit image and its image reflected in the hemisphere was made as small as possible in order to have good definition ; this distance amounted to 2°5 mm., the diameter of the hemisphere being 5em. The thermopile slit was 1 mm. wide, the image of the illuminated slit falling just within it. A series of Langley diaphragms was placed in front of the opening in the hemisphere in order to avoid strong convection currents. The lampblack surfaces were deposited on polished silver, j mm. thick, from the fine flame of a small petroleum Power of Lamp- and Platinum-Black. 169 lamp. The most satisfactory way of measuring their thickness was to place the surfaces nearly vertical under the Fig. 2. 3 Te =7 O m77$ pone of Themen microscope after the conclusion of the experiments and to take the mean thickness of clean edges cut in the lampblack at all parts. The platinum-black surfaces were deposited according to the recipe of Lummer and Kurlbaum for different lengths of time. A galvanometer deflexion of 1 mm. corresponded to a eurrent of 0°7x107! ampere; the deflexions were pro- portional to the current. The reflective power was determined for the following radiations whose maxima lie in the infra-red :— 1. That part of the radiation from an incandescent mantle which is transmitted by a layer of water 1 cm. thick. The water absorbs wave-lengths larger than 1 py * The maximum of the transmitted radiation lies at about 0°8 pw. 2. The Reststrahlen from Selenite. A _ parallel beam from a slit illuminated by an incandescent mantle was reflected at the surfaces of three plates of gypsum before being focussed on the thermopile. The maximum lies at 8°7 p f. * Paschen, Wied. Ann. lii. p. 209 (1894) and others. + Aschkinass, Ann. d. Phys. i. p. 42 (1900) and others. 170 Mr. T. Royds on the Reflective 3. The Reststrahlen from Fluorspar. The light from a slit illuminated by an incandescent mantle was reflected at three fluorspar surfaces, the second of which was concave. The maximum of these rays lies at 2070 4. The Reststrahlen from Rock-salt. The rock-salt plates were arranged exactly as those of selenite. A more intense source, a Nernst filament, was used. The rays were very impure, but a fourth reflexion rendered the intensity too small. In order to increase the purity a clear rock-salt shutter (4 mm. thick), as applied by Rubens, was used instead of a metal one. The maximum of the pure Reststrahlen lies at 51°2 wt; In our case, however, somewhat towards smaller wave-lengths. The following simple observations serve to indicate roughly the purity of the Reststrahlen employed :— Deflexions for Reststrahlen from Selenite. |Fluorspar.|Rocksalt.| Mera MONV EO 2 isec saris cocci cot tte eereanne cen or 1330 688 100 Energy transmitted through glass (3 mm.) ...... 3 0 45 0 9% » Quartz (2°5mm.)... 7 4 62 9 %9 ” rocksalt (2°35 em.).| 1150 0 65 Several corrections are to be applied to the radiation which falls on the thermopile when placed to receive the radiation reflected from the black surface. Firstly, diffuse radiation from the incident image may fall on the thermopile. To determine its amount the slit image was placed in the centre of the hemisphere, and the energy falling on the thermopile when placed at a distance equal to that in the experiments (2°5 mm.) was measured. The curve of the deflexions so obtained for all distances of the thermo- pile is shown by the dotted line in fig. 2. It was found that 0-42 per cent. of the energy in the incident image fell on the thermopile. Secondly, energy is radiated from the black surface since it becomes heated when exposed to the incident radiation. * Rubens, Phys. Z.S. iv. p. 726 (1903). + Rubens & Aschkinass, Wied. Ann. lxiv. p. 241 (1898). Power of Lamp- and Platinum-Black. ffl This radiant energy was estimated in two ways: by measur- ing the energy received (1) when the radiant energy was absorbed, and (2) when the reflected energy was absorbed. A cover-glass, 0'1 mm. thick, placed in front of the thermo- pile slit, absorbs the energy radiated from the heated surface, but transmits the radiation which passes through 1 em. of water. On the other hand, a plate of rock-salt 0°5 mm. thick, transmits the radiant energy, but absorbs a large and measurable portion of the fluorspar Reststrahlen. The following values were found for the heating effect :— Lampblack (0-205 mm. thick) ........ 0:95 °/, of the incident energy. Platinum-black (deposited 15 mins.) .. 0-0 ys is 5 9 .) ( 7 3 ” ) Mor 0:0 1) a) 9 A part of the reflected radiation escapes through the opening in the hemisphere by which the incident radiation enters. The slit image being near the centre of the hemi- sphere in order to obtain good definition, a portion of the light regularly reflected escaped in this way. ‘This loss was considerable in the case of thin deposits, as was seen by mounting the black surface so inclined that the regularly reflected radiation could not pass through the aperture. With the thickest layers, however, inclining the surface thus did not cause any appreciable increase in the reflected energy *; further than this it was not found possible to estimate the correction for the aperture in the hemisphere. After ali other corrections have been applied, the results are to be multiplied by the reciprocal of the reflective power of german-silver, for before the radiation reflected from the black surface falls on the thermopile it undergoes a reflexion at the yerman-silver hemisphere tf. The reflective power of german-silver has been determined by Paschenf, and the corrections are made using the following values :— ee PE ee nee o ae cent. Rips: Ss NEN OG OA fF ADP YTD a ty he A IOXO) ry La MG Meira KOO) 3, * Consequently, the values given in the preliminary communication (doc. cot.) remain unaltered for the thickest layers only. For the thinner layer of lampblack, the apparent reflective powers there given are but 56:0 per cent. of their true values. The correction could not be deter- mined for the thin lampblack layers, for they were destroyed in measur- ing their thickness before this correction was found to be so considerable ; the correction is probably smaller than that found above for the medium thickness and larger for the thinnest layer of lampblack. + The polish of the hemisphere employed was not all that could be desired; this fact is of little account for the longer wave-lengths, but the correction to be applied for A=0°8 » is uncertain for this reason. t Paschen, Ann. d. Phys. iv. p. 304 (1901). 172. = Reflective Power of Lamp- and Platinum-Black. The results for the reflective power of lampblack and platinum-black are given in the following table :— Lampblack | Platinum-black | - Platinum-black (0:205 mm. thick). | (deposited 15 mins.). (deposited 3 mins.). Wave-length. | | | le eaene: | True. | Apparent. | True. sue True. asso ae | Pts! Tee | 217%, | 118%) 054% | 017%) 134%, | 1°80%| Yar Weta | 199 | 066 | 0-98 059 580 | 5°70 | 25 Se eee 204 | 067 || 135 | 093 750 | 708 | Sieve kere | 28 14 121 | 079 | 72° «olen PON sae Ls. ies Poi Shs Wael BS 74 | | iI * Rock-salt Reststrahlen using metal shutter. ? Rock-salt Reststrahlen using rocksalt shutter. As a test of the method, the reflective power of polished silver for fluorspar Reststrahlen was determined and the value 94 per cent. was found. Kurlbaum’s determinations of Stefan’s constant were not corrected for the reflective power of platinum-black. The radiation from a black body at 100° C. has its maximum at about ~=8 w, where the reflective power has been found to be 0°59 per cent. Hence Kurlbaum’s value 7-061 x 10-% ore em.” degrees * becomes # ae er (UU) 2 1 ae MS ale al kay cm.” degrees Planck’s value for the elementary quantity of electricity is consequently increased in the same ratio, and becomes, taking Paschen’s value for b *, 4-624 x 10-!° E.S. unit. A reflective power of 1:0 per cent. would bring it to 4°65 x 107", Rutherford and Geiger’s + experimental value. I am greatly indebted to Prof. F. Paschen for suggesting the investigation and for his helpful advice at all stages. Physikalisches Institut, Tubingen University. * 6=0:292, instead of Lummer and Pringsheim’s value, 0:294., + Rutherford & Geiger, Proc. Roy. Soc. A., Ixxxi. p. 162 (1908). XXIII. Proceedings of Learned Societies. GEOLOGICAL SOCIETY. [Continued from vol. xx. p. 1009. ] May 25th, 1910.—Prof. W. W. Watts, Sc.D., M.Sc., F.RBS., President, in the Chair. HE following communications were read :— 1. ‘ Dedolomitization in the Marble of Port Shepstone(Natai).’ bye Hatch, Ph.D. M.Inst.C:, BGS.) and ih. H. Rastall, MEAD EGS. The Port Shepstone marble is shown by chemical analysis to be a dolomite (the molecular ratio of calcium carbonate to magnesium carbonate being as 3: 2). It owes its marmorization to thermal metamorphism by an extensive intrusion of granite, which completely surrounds it and penetrates it in broad dykes. This intrusion took place at some time prior to the deposition of the Table Mountain or Waterberg Sandstone, and is therefore pre-Devonian. The dolomite is relegated to the Swaziland Period. The metamorphism of the dolomite under normal conditions is. shown to have produced a saccharoidal marble of coarse texture, consisting almost entirely of carbonates ; and the fact that neither periclase nor brucite has been produced in the normal marble is taken to indicate that the high-pressure conditions obtaining during the metamorphism precluded dedolomitization. In those places, however, where the dolomite contains blocks or boulders of earlier vranitic rocks, interaction took place between the magnesium and calcium carbonates of the dolomite and the silica and alumina pro- vided by the inclusions, resulting in the production, in the zone of marble immediately surrounding the inclusions, of a number of interesting silicates of magnesium, calcium, and aluminium, such as olivine, forsterite, diopside, wollastonite, and phlogopite, as well as the oxides brucite and spinel. Magnesian compounds predominate, the excess of lime recrystallizing as calcite. A noteworthy feature is the absence of minerals such as garnet and cordierite, which are especially characteristic of low-temperature metamorphism, thus indicating the prevalence of a high temperature during the meta- morphism of the dolomite. The paper concludes with a reference to the occurrence of granite boulders as foreign inclusions in other limestones, and a discussion of the chemical reactions by which the formation of the above- mentioned minerals may be theoretically explained as a result of 174 Geological Society :— dedolomitization. Comparison is made with the dedolomitized Cambrian limestones of Assynt and Skye described by Dr. Teall and Mr. Harker, from which the Port Shepstone occurrence differs in the localization of the affected areas to reaction-rims around foreign boulders, and in the part played by alumina in the formation of new minerals. 2. ‘Recumbent Folds in the Highland Schists... By Edward Battersby Bailey, B.A., F.G.S8. A description is presented of the stratigraphy and structure of a considerable portion of the Inverness-shire and Argyllshire High- lands. ‘he district considered lies south-east of Loch Linnhe, and extends from the River Spean in the north to Loch Creran in the south. | The following conclusions are arrived at :— (1) The schists of the district are disposed in a succession of recumbent folds of enormous amplitude—proved in one case to be more than 12 miles in extent. (2) The limbs of these recumbent folds are frequently replaced by fold-faults, or ‘ slips,’ which have given freedom of development to the folds themselves. (3) The slipping referred to is not confined to the lower limbs of recumbent anticlines, and is therefore due to something more than mere overthrusting. It is a complex accommodation-phenomenon, of a type peculiar perhaps to the interior portions of folded mountain-chains. In fact, the cores of some of the recumbent folds have been squeezed forward so that they have virtually reacted as intrusive masses. (4) In the growth of these structures many of the earlier formed cores and slips have suffered extensive secondary corrugation of isoclinal type. June 15th.—Prof. W. W. Watts, Se.D., M.Se., F.R.S., President, in the Chair. The following communications were read :— 1. ‘The Natural Classification of Igneous Rocks.’ By Dr. Whit- man Cross, F.G.S. The author reviews the various systems of classification which have been proposed. He discusses the origin of the difference of composition of igneous rocks due to: (1) Primeval difference, (2) Magmatic differentiation, (3) Assimilation; and points out The Natural Classification of Igneous Rocks. 175 that differentiation and assimilation are in a measure antithetical processes. If the deep-seated magmas of large volume have acquired their various chemical characters in different ways, 1t appears at once evident that this primary genetic factor cannot be used in classifi- cation, unless the characters of different origin can be distinguished in the rocks. Classification by geographical distribution of chemically different rocks is considered, and the groupings proposed by various writers are discussed; andit is shown that the rocks of the Pacifie zone of North America indicate that they possess provincial peculiarities of interest, but that these are not by any meamns identical with the features emphasized by Becke and others as characterizing the Pacific kindred. The factors of magmatic differentiation are then reviewed. The aschistic and diaschistic magmas of Brogger and the ‘ dyke rocks’ of Rosenbusch are discussed; and it is contended that certain dyke rocks of Colorado show a notable exception to the rule postulated by Rosenbusch. The conclusion is reached that the sharp distinction between the two ‘dyke rock’ groups is a purely arbitrary one, resting on an unproved hypothesis. A discussion on the classification by eutectics follows, and the writings of G. F. Becker and J. H. L. Vogt on this subject are criti- eized. The view that graphic, spherulitic, and felsitic textures are characteristically eutectic is considered to be incorrect, and it is contended that magmatic classification by eutectics is fundamentally weak—because it rests on hypothesis, because it does not apply to all rocks, and because it does not allow for the entire magma of most rocks. A classification by eutectics may, in the future, be realized ; but it seems inevitable that it must be a classification for a special interest, not for the general science of petrography. The author considers that the distinction between felspathic and non-felspathic rocks which has been so prominent in current systems is not only unnatural, but is in the highest degree arbitrary. The use of texture is then discussed, and it is shown that classi- fication by occurrence, as determining texture, or by texture, as expressing the broad phases of occurrence, is based on long disproved generalizations made from limited observation. The ‘ American Quantitative System of Classification ’ is then briefly dealt with, and the following general conclusions are formulated :— ‘The scientific logical classification of igneous rocks must apparently be based on the quantitative development of fundamental characters, and the divisions of the scheme must have sharp artificial boundaries, since none exist in Nature. ‘Chemical composition is the fundamental character of igneous rocks, but it may be advantageously expressed for classificatory purposes in terms of simple compounds, which represent either rock-making minerals or molecules entering into isomorphous mixtures in known minerals. It is probable that the 176 Geological Society. magmatic solution consists of such molecules, and that the norm of the ‘Quantitative System ” is a fairly representative set of these compounds. ‘The actual mineral and textural characters of igneous rocks are variable qualifiers of each chemical unit, and should be applied as such to terms indi- cating magmatic character.’ 2. ‘The Denudation of the Western End of the Weald.’ By Henry Bury, M.A., F.L.S., F.G.S. There are two main theories of Wealden denudation :— (1) attributing the removal of most of the Chalk to marine planation ; and (2) denying planation, and relying solely on subaérial denudation. Prof. W. M. Davis’s suggestion of a subaérial peneplain forms a sort of connecting link between the two. The evidence in favour of planation which Ramsay and Topley brought forward is inconclusive, and might plausibly, if it stood alone, be attributed to pre-Kocene causes. On the other hand, Prestwich’s arguments against planation are equally weak, while the Chalk plateau to which he draws attention strongly supports Ramsay’s views. The distribution of chert is fatal to Prof. Davis’s hypothesis, and very difficult to account for, except on the marine theory. In the case of the River Blackwater it can be proved that, long after the Hythe Beds of Hindhead were uncovered, the river-system remained extremely immature ; and this affords very strong grounds for the acceptance of the marine hypothesis. The evidence of the other western rivers is less conclusive, though the Wey and the Mole both provide minor arguments pointing in the same direction. The anomalous position of the Arun, at the foot of the northern escarpment of the Lower Green- sand on either side of the Wey, is almost certainly due to com- paratively recent captures from the latter river, and affords no ground for assuming a river-system of great age matured ona Miocene peninsula. There is no proof that any of the existing connexions between rivers and longitudinal folds are of a primitive character ; and, on the other hand, there are many alleged examples of transverse distur- bances having served as guides te consequent rivers. This again, on the whole, supports the marine hypothesis, especially if, as there are reasons for believing, the longitudinal folds are older than the transverse. THE LONDON, EDINBURGH, ann DUBLIN PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCIENCE. pa -~@— [SIXTH SERIES.] | Oi) ye 4}, FEBRUARY. 1911.\\% XXII. Hydrodynamical Notes. By Lord Rayietenu, O.M., F.RS.* Potential and Kinetic Energies of Wave Motion.—Waves moving into Shallower Water.—Concentrated initial Disturbance with inclusion of Capillarity.—Periodic Waves in Deep Water advancing without change of Type.—Tide Races.— Rotational Fluid Motion in a Corner.— Steady Motion in a Corner of Viscous Fluid. 1 the problems here considered the fluid is regarded as .4. incompressible, and the motion is supposed to take place in two dimensions. Potential and Kinetic Energies of Wave Motion. When there is no dispersion, the energy of a progressive wave of any form is half potential and half kinetic. Thus in ihe case of a long wave in shallow water, “if we snppose that initially the surface is displaced, but that the particles have no velocity, we shall evidently obtain (as in the case of sound) two equal waves travelling in opposite directions, whose total energies are equal, and together make up the potential energy of the original displacement. Now the elevation of the derived waves must be half of that of the original displacement, and accordingly the potential energies less in the ratio of 4:1. Since therefore the potential energy of each derived wave is one quarter, and the total energy one half that of the original displacement, * Communicated by the Author. Phil. Mag. 8. 6. Vol. 21. No. 122. Feb. 1911. N 178 Lord Rayleigh : Hydrodynamical Notes. ° it follows that in the derived wave the potential and kinetic energies are equal”. The assumption that the displacement in each derived wave, when separated, is similar to the criginal displacement fails when the medium is dispersive. The equality of the two kinds of energy in an infinite progressive train of simple waves may, however, be established as follows. Consider first an infinite series of simple stationary waves, of which the energy is at one moment wholly potential and half a period later wholly kinetic. If ¢ denote the time and E the total energy, we may write K.H.=K sin? nf, Poki Wicos2 7b Upon this superpose a similar system, displaced through a quarter wave-length in space aud through a quarter period in time. For this, taken by itself, we should have K.E.=E cos? nt, Pn isin? ne And, the vibrations being conjugate, the potential and kinetic energies of the combined motion may be found by simple addition of the components, and are accordingly independent of the time, and each equal to HE. Now the resultant motion is a simple progressive train, of which the potential and kinetic energies are thus seen to be equal. A similar argument is applicable to prove the equality of energies in the motion of a simple conical pendulum. It isto be observed that the conclusion isin general limited to vibrations which are infinitely small. Waves moving into Shallower Water. The problem proposed is the passage of an infinite train of simple infinitesimal waves from deep water into water which shallows gradually in such a manner that there is no loss of energy by reflexion or otherwise. At any stage the whole energy, being the double of the potential energy, is pro- portional per unit length to the square of the height ; and for motion in two dimensions the only remaining question for our purpose is what are to be regarded as corresponding lengths along the direction of propagation. In the case of long waves, where the wave-length (A) is long in comparison with the depth (1) of the water, corre- sponding parts are as the velocities of propagation (V), or * “On Wares,” Phil. Mag. i. p. 257 (1876); ‘Scientific Papers,’ i. p. 254. Lord Rayleigh: Hydrodynamical Notes. E79 since the periodic time (7) is constant, as A. Conservation of energy then requires that (heietne Gv constants... 2.9 4, CL) or since V varies as 7%, height varies as ae) But for a dispersive medium corresponding parts are not proportional to V, and the argument, requires modification. A uniform regime being established, what we are to equate at two separated places where the waves are of different character is the rate of propagation of energy through these places. It is a general proposition that in any kind of waves the ratio of the energy propagated past a fixed point in unit time to that resident in unit length is U, where U is the group-velocity, equal to do/dk, where o=27/t, k=2ar/r fF. Hence in our problem we must take ° ° ry Sede 6 heiehtavantesvasn Wenz, em ws yey) (2) which includes the former result, since in a non-dispersive medium U) =.V. For waves in water of depth J, Ge —Giebamnicl) eatin.) os. -(O)) whence 2oU/g=tanh kl+kl(1—tanh? kl). . . . (4) As the wave progresses, o remains constant, (3) determines k in terms of J, and U follows from (4). If we write Cg eee a) a G) ile Canes —Ueaee any ieee ee 4) (6) and (4) may be written Don) foie is (UNI Se ite WCU) me (6), (7) U is determined as a function of /’ or by (5) (3) becomes a kl, and therefore i’, is very great, kl=U’, and then by Coit ibn be the corresponding value of U, Po, Wij foe Fe Noe toy ees a ah (8) and in general Wohi U2) kl Bh Se we OQ) * Loe. cit. p. 255. + Proc. Lond. Math. Soc. ix. 1877 ; ‘Scientific Papers,’ i. p. 326. N 2 180 Lord Rayleigh: Hydrodynamical Notes. Hquations (2), (5), (6), (9) may be regarded as giving the solution of the problem in terms of a known o. It is perhaps more practical to replace o in (5) by Xo, the corresponding wave-length ina great depth. The relation between o and Ao being o?==2ary/Xy, we find in place of (5) P= Jali Ny Kol se eo Starting in (16) from do and / we may obtain 1’, whence (6) gives kl, and (9) gives U/Uo. But in calculating results by means of tables of the hyperbolic functions it is more convenient to start from kl. We find | kl. | P. U/U,. kl. Bas U/U,. 30 kl 1-000 6 +322 ‘964 10 kl 1-000 5 231 -855 5 4-999 1-001 4 152 722, 2 1-928 1105 3 ‘087 566 1-358 1-176 9 039 390 1-0 ‘762 1182 1 ‘010 -200 531 THEI) Mh ya (kl)? Qkl 423 HO |p ae a It appears that U/Uo does not differ much from unity between l’=:23 and l/=«, so that the shallowing of the water does not at first produce much effect upon the height of the waves. It must be remembered, however, that the wave-length is diminishing, so that waves, even though they do no more than maintain their height, grow steeper. Concentrated Initial Disturbance with inclusion of Capillarity. A simple approximate treatment of the general problem of initial linear disturbance is due to Kelvin*. We have for the elevation 7 at any point 2 and at any time ¢ n= ={ cos ku cos ot dk 7 No Bi 1 0 hi ae b Me 2) =a2) cos (kx—ot)dk+ oh cos (ka + ot)dk, . (1) in which o is a function of &, determined by the character * Proc. Roy. Soc. vol. xlii. p. 80 (1887); ‘Math. and Phys. Papers,’ iy. p. 303. Lord Rayleigh: Hydrodynamical Notes. 1381 of the dispersive medium-—expressing that the initial elevation (¢=0) is concentrated at the origin of 2 When ¢ is great, the angles whose cosines are to be integrated will in general vary rapidly with &, and the corresponding parts of the integral contribute little to the total result. The most important part of the range of integration is the neigh- bourhood of places where ka +ot is stationary with respect to k, 1. e. where ASS ree ai} SeMeN hiey rer hires Nine (2) In the vast majority of practical applications da/dk is positive, so that if w and ¢ are also positive the second integral in (1) makes no sensible contribution. The result then depends upon the first integral, and only upon such parts of that as lie in the neighbourhood of the value, or values, of k which satisfy (2) taken with the lower sign. If 4, be such a value, Kelvin shows that the corresponding term in 7 has an expression equivalent to cos (ot — kv —i 7) = ode: 7 a, being the value of o corresponding to f;. In the case of deep-water waves where o=,/(ghk), there is only one predominant value of & for given values of a and ¢, and (2) gives ky = gt? /42, CO 2 Ne dy oA) making ORG Ninos piri a ea iy oc) and finally the well-known formula of Cauchy and Poisson. In the numerator of (3) o; and fk, are functions of 2 and ¢. If we inquire what change (A) in a with ¢ constant alters the angle by 277, we find Ce NG Ro ts, Ath + (0-05 ae } ae so that by (2) A=2a/hy,, i. e. the effective wave-length A coincides with that of the predominant component in the original integral (1), and a like result holds for the periodic 182 Lord Rayleigh: Hydrodynamical Notes. time *. Again, it follows from (2) that k;v—o,t in (3) may be replaced by hide, as is exemplified in (4) and (6). When the waves move under the influence of a capillary tension ‘I in addition to gravity, G=GktTh/o, seine he p being the density, and for the wave-velocity (V) We0/Pag/k+Tk/p, so. een as first found by Kelvin. Under these circumstances V has a minimum value when h=op/To) 6) 3s ee The group-velocity U is equal to da/dk, or to d(kV)/dk ; so that when V has a minimum value, U and V coincide. Referring to this, Kelvin towards the close of his paper remarks “The working out of our present problem for this case, or any case in which there are either minimums or maximums, or both maximums and minimums, of wave- velocity, is particularly interesting, but time does not permit of its being included in the present. communication.” A glance at the simplified form (3) shows, however, that the special case arises, not when V is a minimum (or maximum), but when U is so, since then d?a/dk,? vanishes. As given by (3), 7 would become infinite—an indication that the approximation must be pursued. If k=k,+&, we have in general in the neighbourhood of i, al in cise ee (e-1e t dc t Vide t > 1.9% wees In the present case where the term in & disappears, as well as that in &, we get in place of (3) when ¢ is great cos (k,v—o;t) A 1a, tan Sf orblaaldtayt ine ere cS varying as ¢~* instead of as ¢7?, The definite integral is included in the general form [cos om daz = 1 (2) cos Og @ i * Cf. Green, Proc. Roy. Soc. Ed. xxix. p. 445 (1909). —@ Lord Rayleigh : Hydrodynamical Notes. 183 ivi: 4 +0 1 , cos a2. dz2= a/ (5) . (co: a>, Lae (s)- | (13) — 0 The former is employed in the derivation of (3). The occurrence of stationary values of U is determined from (7) by means of a quadratic. There is but one such value (Uy), easily seen to be a minimum, and it occurs when Pal/s-ljre = Idig . -.. C4) On the other hand, the minimum of V occurs when 4?=gp/T simply. When ¢ is great, there is no important effect so long as x (positive) is less than Uot. For this value of 2 the Kelvin formula requires the modification expressed by (11). When wis decidedly greater than Ugt, there arise two terms of the Kelvin form, indicating that there are now two systems of waves of different wave-lengths, effective at the same lace. : It will be seen that the introduction of capillarity greatly alters the character of the solution. The quiescent region inside the annular waves is easily recognized a few seconds after a very small stone is dropped into smooth water™, but I have not observed the duplicity of the annular waves themselves. Probably the capillary waves of short wave- length are rapidly damped, especially when the water-surface is not quite clean. It would be interesting to experiment upon truly linear waves, such as might be generated by the sudden electrical charge or discharge of a wire stretched just above the surface. But the full development of the peculiar features to be expected on the inside of the wave-system seems to require a space larger than is conveniently available in a laboratory. Periodic Waves in Deep Water advancing without change of Type. The solution of this problem when the height of the waves is infinitesimal has been familiar for more than a century, and the pursuance of the approximation to cover the case of moderate height is to be found in a well-known paper by * A checkered background, e.g. the sky seen through foliage, shows the waves best. 184 Lord Rayleigh : Hydrodynamicai Notes. Stokes *. In a supplement published in 1880 the same author treated the problem by another method in which the Space coordinates wz, y are regarded as functions of ¢, the velocity and stream functions, and carried the approximation a stage further. In an early publication { I showed that some of the results of Stokes’s first memoir could be very simply derived from the expression for the stream-function in terms of # and y, and lately I have found that this method may be extended to give, as readily if perhaps less elegantly, all the results of Stokes’ supplement. Supposing for brevity that the wave-length is 27 and the velocity of propagation unity, we take as the expression for the stream-function of the waves, reduced to rest, v=y—ae-¥ cos x—Be-* cos 2x — ye" cos 8a, . (1) in which x is measured horizontally and y vertically downwards. This expression evidently satisfies the differ- ential equation to which yf is subject, whatever may be the values of the constants «, 8, y. From (1) we find UP ~2yy = (dyp/dx)? + (dyp/dy)? — 2gy =1—2y~+2(1-g)y+2Be-” cos 2x +4ye cos bu + ae" +48? e447 +4 Oy? e+ 4aBe-*Y cos x + Gaye" cos 22-4 128ye" cosa. |.) 4 ee The condition to be satisfied at a free surface is the constancy of (2). The solution to a moderate degree of approximation (as already referred to) may be obtained with omission of 8 and y in (1), (2). Thus from (1) we get, determining w so that the mean value of y is zero, y=a(l+ 3a’) cosa—ta*cos 2a+2a% cos3xu,. . (3) which is correct as far as a® inclusive. IE we call the coefficient of cos x in (3) a, we may write with the same approximation y=a cos x—4a’ cos 24+ Za? cos3e.. . . (4) * Camb. Phil. Soe. Trans, viii. p. 441 (1847) ; ‘Math. and Phys. Papers,’ i, p. 197. SN, dag . + L. ¢. 1. p. 314. __{ Phil. Mag. 1. p. 257 (1876) ; Sci, Papers, i. p. 262. See also Lamb’s ‘Hydrodynamics,’ § 230. ar vr Lord Rayleigh: Hydrodynanical Notes. 185 Again from (2) with omission of 8, y, U’—2gy=const. + 2.1 —g—2?—a')y + «* cos 2a —4 @? cos 3.2. ais (5) It appears from (5) that the surface condition can be satisfied with « only, provided that a* is neglected and that eee Oana ia mia oi) In (6) @ may be replaced by a, and the equation de- termines the velocity of propagation. To exhibit this we must restore generality by introduction of k (=2a/n) and ¢ the velocity of propagation, hitherto treated as unity. Consideration of ‘‘ dimensions ”’ shows that (6) becomes ke? —g—a? 7h? =0, SGN aS ARLE aca (a) or Ce Gio (Meri ear yy so namie whe. nian S) Formule (4) and (8) are those given by Stokes in his first memoir. By means of @ and y le surface condition (2) can be satisfied with vel Maker of at and «®, and Hoi (5) we see that 8 is of the order a* and y of the order 2°, The terms to be retained in (2), in addition to those given in (5), are 2B(1—2y) cos 27+ 4y cos 3a+4a8 cos x =28 cos 27+ 2a8 (cos x+ cus 3x) +4y cos 84+ 4aB cos a. Expressing the terms in cosw by means of y, we get finally U? — 2qy=const. + 2y (1—g—«? — «t+ B) + (a* + 28) cos 2v + (4y—4a° —208) cos3a. . (9) In order to satisfy the surface condition of constant pressure, we must take B=— ye, Y= 192, ST a A (10) and in addition WS gaa sama nie) wh SCL) correct to a5inclusive. The expression (1) for y thus assumes the form sane an cos w+} ate cos 2u — ),a°e-*Y cos 3, . (12) from which y may be aed in terms of w as far as a inclusive. 186 Lord Rayleigh : Hydrodynamical Notes. By successive approximation, determining yr so as to make the mean value of y equal to zero, we find as far as 24 y= (4#+2 2) cos a—(ha?+ 42") cos 2x +2a%cos3u—za'tcos4z, . . (13) or, if we write as before a for the coefiicient of cos a, y=acos v—(ha?+432a*) cos 22 +a? cos3u—iatcos4a, . . (14) in agreement with equation (20) of Stokes’ Supplement. Iixpressed in terms of a, (11) becomes g=1a-—@—ta,. . . . eee or on restoration of hf, ¢, g=ke—Reei—lhate. . . . .axae hus the extension of (8) is e=g/k. kas ka), . -. ieee which also agrees with Stokes’ Supplement. If we pursue the approximation one stage further, we find from (12) terms in «’, additional to those expressed in (13). These are fo! 243 a a 125 9 COS H+ 75. C08 8H+ Fay, COS Dax ./ -Gl3} It is of interest to compare the potential and kinetic energies of waves that are not infinitely small. For the stream-function of the waves regarded as progressive, we have, as in (1), ab = — ae’ cos (vu —ct) + terms in 24, so that (dw /dx)? + (dvy/dy)?=c?e—*/ + terms in a’. Thus the mean kinetic energy per length # measured in the direction of propagation is 2 2 26 2 5 eel e~*¥dy = a Ade 78= alae —2y + 27") 9 ? “= - {a - 2 ytde } ‘ where 7 is the ordinate of the surface. And by (3) {y2da=} L(x? + a*) 4-4 na l a, Lord Rayleigh: Hydrodynaimical Notes. 187 Hence correct to at, Gti ten(lranwe Vos 8) Again, for the potential energy Pag \ pda loe(ha? + 3a’) ; or since g=1—2’, AR rasta a eet a se es, (20) The kinetic energy thus exceeds the potential energy, when a is retained. Tide Races. It is, I believe, generally recognised that seas are apt to be exceptionally heavy when the tide runs against the wind. An obvious explanation may be founded upon the fact that the relative motion of air and water is then greater thanif the latter were not running, but it seems doubtful whether this explanation is adequate. It bas occurred to me that the cause may be rather in the motion of the stream relatively to itself, e.g. in the more rapid movement of the upper strata. Stokes’s theory of the highest possible wave shows that in non-rotating water the angle at the crest is 120° and the height only moderate. In such waves the surface strata have a mean motion forwards. On the other hand, in Gerstner and Rankine’s waves the fluid particles retain a mean position, but here there is rotation of such a character that (in the absence of waves) the surface strata have a relative motion backwards, 2.e. against the direction of propagation *. It seems pos- sible that waves moving against the tide may approximate more or less to the Gerstner type and thus be capable of acquiring a greater height and a sharper angle than would otherwise be expected. Needless to say, it is the steepness of waves, rather than their mere height, which is a source ct inconvenience and even danger to small craft. The above is nothing more than a suggestion. Ido not know of any detailed account of the special character of these waves, on which perhaps a better opinion might be formed. Rotational Fluid Motion in a Corner. The motion of incompressible inviscid fluid is here supposed to take place in two dimensions and to be bounded by two fixed planes meeting at an angle a. If there is no rotation, * Lamb’s Hydrodynamics, § 247. 188 Lord Rayleigh: Hydrodynamical Notes. the stream-function wp, satisfying V/r=0, may be expressed by a series of terms m/e . / 2rja - ni |e r ' sin 70/a, 97!” sin QarO/a, sin n7O/a, where n is an integer, making w~=0 when 0=0 or 0=«. In the immediate vicinity of the origin the first term pre- dominates. For example, if the angle be a right angle, ar=7? sin 20=I22y, ... ee if we introduce rectangular coordinates. The possibility of irrotational motion depends upon the fixed boundary not being closed. Ifa m, the velocities deduced from wr become infinite. If there be rotation, motion may take ere even though the boundary be closed. For example, the circuit may be completed by the are of the circle r=1. In the case which it is proposed to consider the rotation w is uniform, and the motion may be regarded as steady. The stream-function then satisfies the general equation Vipadypjda?+ Pypldy=2o,. . . . (2) or in polar coordinates yp a Dep (3) di? “Or cer. When the angle is a right angle, it might perhaps be expected that there should be a simple expression for y in powers of w and y, analogous to (1) and applicable to the immediate vicinity of the origin; but we may easily satisfy ourselves that no such expression exists*. In order to express the motion we must find solutions of (3) subject to the conditions that y~=0 when 6=0 and when 0=a., For this purpose we assume, as we may do, that v= >. sinno/a,) |). <. . ne where n is integral and R, a function of 7 only; and in deducing V*r we may perform the differentiations with respect to 6 (as well as with respect to r) under the sign of summation, since y=0 at the limits. Thus MR, vee a aoe ee ss V? Le ae awe Mites > (= Dee Lf dr any? -) S1 e * In strictness the satisfaction of (2) at the origin is inconsistent with the evanescence of f on the rectangular axes. Lord Rayleigh: Mydrodynamical Notes. 189 The right-hand member of (3) may also be expressed in a series of sines of the form Deo) Te Sr, stamume) &,)6) 0) a (0) where n is an odd integer; and thus for all values of n we have aR GES Ay Wire Ril eles 2 n rad ie A fue en, : US dr’ ah dr oe ; { 1 ( 1) 5 (7) The general solution of (7) is Boss tla Ber eS Amar? {1 — (— at eee oo the introduction of which into (4) gives w. In (8) A, and B, are arbitrary constants to be determined by the other conditions of the problem. For example, we might make R,, and therefore Wy, vanish when r=, and and when r=7,, so that the fixed boundary enclosing the fluid would consist of two radii vectores and two circular arcs. If the fluid extend to the origin, we must make B, =0; and if the boundary be completed by the circular are r=1, we have A,=0 when n is even, and when n is odd Sax" nt ni (4a? —n?7r”) m7 (9) Thus for the fluid enclosed in a circular sector of angle « an:l radius unity ous ple 9 le nO (10 Hg ee nm (wn? —4e2) ois ) the summation extending to all odd integral values of n. The above formula (10) relates to the motion of uniformly rotating fluid bounded by statzonary radi vectores at 9=0, @=a. We may suppose the containing vessel to have been rotating for a long time and that the fluid (under the influence of a very small viscosity) has acquired this rotation so that the whole revolves like a solid body. The motion expressed by (10) is that which would ensue it the rotation of the vessel were suddenly stopped. A related problem was solved a long time since by Stokes *, who considered the zrrotational motion of fluid in a revolving sector. The solution of Stokes’s problem is derivable from (10) by mere * Camb. Phil. Trans, vol. vill. p.533 (1847) ; Math. and Phys. Papers, vol. 1. p. 3805. 190 Lord Rayleigh: Hydrodynamical Notes. addition to the latter of w= —4r’, for then w+, satisfies V(r +,)=0; and this is perhaps the simplest method of obtaining it. The results are in harmony; but the fact is not immediately apparent, Inasmuch as Stokes expresses the motion by means of the velocity-potential, whereas here we have employed the stream-function. That the subtraction of sr? makes (10) an harmonic function shows that the series multiplying 7” can be summed. In fact sin (ogi) cos(20—a) 1 ni (war — 4a? on 2 cosa 2? 8a? > so that r cos (20—«) san m/e in nO /c p/o=3r— + 82? = 2) (11) 2 cosa nT (na? —4 In considering the character of the motion defined by (11) in the immediate vicinity of the origin we see that if «<47, the term in 7” preponderates even when n=1. When a=437 exactly, the second term in (11) and the first term under } corresponding to n=1 become infinite, and the expression demands transformation. We find in this case 9,2 2 Sn TE rae 1 ¢ 2: = w= tr? + — (@—}7) cos 204 7? sin 24 — log r— — v/ 2 _ ( 4 ) = = fo) 2a 2 .. 2" sin 2nO A = : n(n?—1) the summation commencing at n=3. On the middle line 0=17, we have bons) ZZ 7 pid Plogr— a5 + gag (13) The following are derived from (13) :— ylo= yr" pe —inw 7 —inv. | 7 | —arp | | 00 | -goov0 | O-4 44112 | O08 | -13030 O1 | 02267 | 05 senor | 09 | -o7e41 | 02 | 06296 | 06 | 17306 | 10 | -coo00 03 | “10521 | 07 16210 | | The maximum value occurs when r='592. At the point p= 592, @=H7, the fluid is stationary. Lord Rayleigh: Hydrodynamical Notes. 19 A similar transformation is required: when «=37/2. When e=7. the boundary becomes a semicircle, and the leading term (n=1) is Srey: gy. 3) ylo=— 5-1 sin 0= — 5... which of itself represents an irrotational motion. When «=27, the two bounding radii vectores coincide and the containing vessel becomes a circle with a single partition wall at @=0. In this case again the leading term is irrotational, being ylo=— 2 asin Je. Leas vay ain 12 1/1 GED) (14) Steady Motion in a Corner of a Viscous Fluid. Here again we suppose the fluid to be incompressible and to move in two dimensions free from external forces, or at any rate from such as cannot be derived from a potential. If in the same notation as before y represents the stream- function, the general equation to be satisfied by wp is Wet Ohsh os, ae eaves u(y) with the conditions that when @=0 and 06=a, v=0, wanii==Us 5 5) Aon 5 2) It is worthy of remark that the problem is analytically the same as that of a plane elastic plate clamped at @=0 and @=a, upon which (in the region considered) no external forces act. The general problem thus presented is one of great diffi- culty, and all that will be attempted here is the consideration of one or two particular cases. We inquire what solutions are possible such that 4, as a function of r (the radius vector), is proportional to 7”. Introducing this supposition into (1), we get TED ak F Ca | A 2 2 2 — e e € {m nn (an — 2) ae (3) as the equation determining the dependence on @. The most general value of yy consistent with our suppositions is thus b=r™{A cos m6 +B sin m + C cos (m—2)0+ Dsin(m—2)63, (4) where A, B, C, D are constants. 192 Lord Rayleigh: Hydrodynamical Notes. Equation (4) may be adapted to our purpose by taking m=nria, . . . |<). ne where n is an integer. Conditions (2) then give At C=0; A+C cos 2e—D sin 2a=0, “= B+("™ —2)D=0, one +(2 —2)Osin Dent ("7 -2) D cos 2a=0. When we substitute in the second and fourth of these equations the values of A and B, derived from the first and third, there results C(1— cos 2a) + D sin 2¢=0, - C sin 22—D(1— cos 22) =0; and these can only be harmonized when cos 2e=1, or a=sz, where s isan integer. In physical problems, « is thus limited to the values 7 and 27. ‘To these cases (4) is applicable with C and D arbitrary, provided that we make A+0=0, B+(1-= D=0. aes Thus | p= ctf cos(28 —26) - 203 — | + Dr : a ie —26) Bu (1- sinh, ai makin g Ve= Nea. mor Oral) When s=1, a=, the corner disappears and we have simply a straight boundary (fig. 1). In this case n=1 gives a nugatory result. When n=2, we have Bie. d, w=Cr(1— cos 20) =20y?, . «ae and V*p=4C. When n=3, Lord Rayleigh : Hydrodynamicai Notes. EGS r= Cr"(cos 9— cos 30) 4- Dr'(sin@—2sin 30), . (9) V7h=8r(C cos 9+ D sin €)=8(Cx+ Dy). . . (0) In rectangular coordinates Ma MO ye or ee once a, yt i Clu) solutions which eae satisfy the required conditions. When s=2, #=27, the boundary consists of a straight wall extending from the origin in one direction (fig. 2). In e Go this case (6) and (7) give a= Or?" {cos (5n0 — 20) —cos nO} + Dr? {sin (4n0 — 20) — (1-7) )sin $n8}, (12) Wap = (2n—4)1?"""4 C cos (End — 26) sh) Sim (SG —— 2) nein athe icce”) (lbs) Solutions of interest are afforded in the case n==1. The C-solution is (C=) Ba ; y 7 2 ° ap = 412(cos 30 —cos 30) = —7* cos} sin? 40, . (14) vanishing when @=v, as well as when 06=0, 0=27, and for no other admissible value of @. The valnes of W are reversed when we write 27—0@ for 8. As expressed, this value is negative from 0 to m and positive from m to 27. The minimum occurs when 06=109° 28’. Hvery stream- line which enters the circle (*=1) on the left of this radius leaves it on the right. The velocities, ‘represented by dp/dr and r7*dy/dd, are infinite at the origin. Phil, Mag. 8.6. Vol. 20. No. 122. Feb. 1911. O i94 Lord Rayleigh: LHydrodynamical Notes. For the D-solution we may take worsing iO 2 2. (15) Here Y retains its value unaltered when 27 —@ is substituted for 6. When r is given, increases continuously from 6=0 to 0=7. On the line 6=7 the motion is entirely transverse to it. This is an interesting example of the flow of viscous fluid round a sharp corner. In the application to an elastic plate ~ represents the displacement at any point of the plate, supposed to be clamped along @=0, and other- wise free from force within the region considered. The following table éxhibits corresponding values of r and @ such as to make Y=1 in (15) :-— | Q. 9 | é. r 160° 1:60 60°. 64-0 150° 1:23 20° 10!x 3-65 120° 2°37 | TOP | Ae xe 90° 8:00 | ge co bs RRR Es SL | When n=2, (12) appears to have no significance. When n=3, the dependence on @ is the same as when nm=1. Thus (14) and (15) may he generalized: ap = (Ar? + Br?) cos46 sin? 20, . . . (16) ap =(A'r? + B'r®) sin? 10. a oe etal For example, we could satisfy either of the conditions w=0, or avr/ar—0, on the circle =I. For n=4 the D-solution becomes nugatory ; but for the (-solution we have ap = Cr?(1— cos 20) =2Cr? sin? 9=2Cy?, - . Gis) The wall (or in the elastic plate problem the clamping) along @=0 is now without effect. It will be seen that along these lines nothing can be done in the apparently simple problem of a horizontal plate clamped along the rectangular axes of x and y, if it be supposed free from force”. Ritz t has shown that the * If indeed gravity act, w=2°y’ is a very simple solution. + Ann. d. Phys. Bd. 28, p. 760, 1909, Lord Rayleigh : Mydrodynantical Notes. 195 solution is not developable in powers of w and y, and it may be worth while to extend the proposition to the more general case when the axes, still regarded as lines of clamping, are inclined at any angle a In terms of the now oblique coordinates x, y the general equation takes the form (P/da? + d?/dy?—=2 cos ad?/dady)w=0,. .« (19) which may be differentiated any number of times with respect to a and y, with the conditions w=(, dw/dy=0, Vdn@ial == (VAN omemarie 10210) w=. dw/dx=0, wieme— (i006. a2) We may differentiate, as often as we please, (20) with with respect to v and (21) with respect to y. From these data it may be shown that at the origin all differential coefficients of w with respect to a and y vanish. The evanescence of those of zero and first order is expressed in (20), (21). As regards those of the second order we have from (20) @w/dx*=0, d@w/dedy=0, and from (21) d?w/dy?=0. Similarly for the third order from (20) ad wldx =, dw/dardy =0, and from: (21) diolday? =, @w/idz dy? =0. For the fourth order (20) gives d'w/du*=0, d‘w/dardy =0, and (21) gives d*w/dy* =0, d*w/dz dy? =0. So far d4w/da*dy? might be finite, but (19) requires tliat it also vanish. This process may be continued. For the m+1 coefficients of the nith order we obtain four equations from (20). (21) and m—3 by differentiations of (19), so that all the differential coefficients of the mth order vanish. It follows that every differential coefficient of w with respect to w and y vanishes at the origin. I apprehend that the con- clusion is valid for all angles « less than 2x. That the displacement at a distance » from the corner should diminish rapidly with + is easily intelligible, but that it should diminish more rapidly than any power of v, however high, would, I think, not have been expected without analytical proof, O 2 sia XXIV. On the Recent Theories of Electricity. By Louis T. Mors, PAD., Professor of Physics, University of Cincinnati *. HE theories of matter and electricity which have been recently advanced have aroused the interest of the thinking public generally, and rather startling accounts have appeared concerning a scientific revolution. In a series of essays T I attempted to give a more or less philo- sophical discussion of these theories ; their effect on thought ; and their relation to the older atomic theories. During the writing of these essays, I became convinced that the new ideas were sufficiently crystallized to permit of a more technieul and critical discussion of their merits and defects. And in spite of an apparent advantage they may have over the older conceptions, it is quite possible that this gain is not permanent, and that the present movement is on the whole harmful to correct scientific procedure because it tends to obliterate the koundary between science and meta- physics. But I should have hesitated to raise any protest if L had not believed that a simple modification in the definition of electricity would reconcile many of the discrepancies between the new and older theories, and make unnecessary the substitution of electricity for matter and electrodynamics for mechanics. While it is not possible to draw a definite boundary line between the regions of physics and metaphysics, still we may do so in a general way by saying that the domain of physics concerns the discovery of phenomena and the formulation of natural laws based on postulates which are determined by experience and generally accepted as true; the causes of phenomena and the discussion of the postulates of science are the province of the metaphysician. This differentiation in methods of thought cannot be rigidly adhered to since this boundary line is more or less obscure, and is liable to considerable displacement as a science advances ; but the acceptance of this principle would prevent much of the con- fusion which has been introduced into the science of physics by writers who have not recognized this to be a general rule. For example, the principle of relativity is not strictly a physical law but the expression, in mathematical symbols, of the general philosophical law of the finite nature of the human mind which has been accepted for centuries. Again, the discussion of the shape of the atom or electron is not * Communicated by the Author. + L. T. More, Hibbert Journal, July 1909 and July 1910. On the Recent Theories of Electricity. 197 a phy sical problem, as it is incapable of verification by experience. This does not mean that such questions should not be discussed, but the method of their discussion and the results obtained are properly the method and results of metaphysics and are not in the category of physical phe- nomena and laws. As the purpose of this article is to show that we may retain the idea of the mechanical nature and invariable inertia of matter, and at the same time account for the electromagnetic momentum and other properties of elec- trified matter by considering electricity as an attribute of matter rather than the converse, it is convenient to state and discuss a few postulates beforehand which bear on the subject. We shall assume length, mass, and time to be the fut mental units of measure. These quantities and their deri, vatives are continuous or, at least, indefinitely divisible. The continuity of space and time is generally accepted ; without this belief it is impossible to establish the geometr ical Jaws founded on the point, line, and surface or the analytical laws of the calculus. But the divisibility of matter is not usually supposed to be infinite. Indeed, the denial of this assertion is the foundation of all atomic theories. Yet it is difficult to see how mathematics can be anything but abstract logic, or how it can be applied to physical prolilems unless this third fundamental quantity, which is as it were the con- necting link between the abstract and the concrete, be al-o indefinitely divisible. How, otherwise, can we replace finite bodies by mathematical centres of inertia? In this con- nexion Professor Sir Joseph Larmor ™® says: ‘‘ The difficulty of imagining a definite uniform limit of divisibility of matter will always be a philosophical obstacle to an atomic theory, so long as atoms are regarded as discrete particles moving in empty space. Butas soon as we take the next step in physical development, that of ceasing to regard space as mere empty geometrical continuity, the atomic constitution of matter (each ultimate atom consisting of parts which gre incapable of separate existence, as Lucretius held) is raised to a natural and necessary consequence of the new stand- point.” This is clearly an attempt to reconcile the two incommensurable antinomies of continuity and discontinuity, which are usually attached to the names of Descartes and Lucretius. This Sir J. Larmor tries to do by postulating the existence of a trwe matter. which is a continuous plenum and imperceptible to our senses, and relegating sensible 2 * ¢ [ther and Matter,’ p. 76. 198 Prof. L. T. More on the matter to the réle of a mere variation in this otherwise changeless plenum,—making it an attribute rather than an entity. If his theory denies the infinite divisibility of matter, it apparently accepts its indefinite divisibility ; the atom, as a variation limited only by our power of obsemvatiaal must become smaller with each advance in the refinement of our apparatus. Such a plenum must remain a pure creation of the imagination, and its existence is not determinable by physical or experimental methods ; it must therefore ble classed as a probiem for the metaphy sician. The distinction between atoms continually diminishing in size and the infinite, or at least indefinite, divisibility of matter is merely a question of words—the definition of what matter is. As another fundamental principle, we shall postulate the objective reality and conservation of matter. The quanti- tative measure of this matter is its mass or inertia, which is also to be taken as an invariable factor in the derived quan- tities, force and energy. M. Hannequin® expresses this idea well when he says: “Il n’existe done rien, dans le monde meécanigue, que des masses en mouvement ou, pour parler un langage rigoureux, qu’une somme constante d’énergie de mouvement et des masses sur lesquel-es elle se distribue.” Although mass is here considered to be infi- | nitely divisible, its scientific unit of measurement is at any time, the least amount of which we have cognizance ; at present, this happens to be the electron or corpuscle. Further consideration of this unit is left to the discussion of the atom. Few things have been brought out more clearly by the work of the school of ener getics than that, if we accept the doctrine of the conservation of energy, either of the two quantities, mass or energy, may be considered as the fundamental unit from which the other can be derived. This undoubtedly follows from the fact that we have no conception of mass without energy or of energy without mass. But while it is thus possible mathematically to make either of them a starting point for the explanation of phenomena, the advo- cates of energetics apparently soon develop a pronounced tendency to prefer the abstract to the concrete and to sub- tilize objective facts into metaphysical ideas. A science like physics, to be useful and not merely an intellectual gymnastics, should preserve in all its speculations a close touch with the practical and the concrete, a certain common sense. The hi-tory of the science shows these advantages have been obtained most frequently by those who maintain * Lhypothése des «tomes, p. 127. Recent Theories of Electricity. 199 mass to be a fundamental unit. The failure of the mechan- istic school has arisen from the attempt to explain the nature of matter, the cause of its forces, and the properties of atoms. However we may try to reason away the belief in the objective reality of matter, our minds persistently cling to the advantage, and even necessity, of such a postu- late ; and we consciously or unconsciously endow any sub- stitute of it with all the properties of matter, excepting its name. Hnergy therefore, although it is an inalienable property of matter, will remain a derived unit, the total quantity of which is a constant. It is customary and convenient to divide energy into two classes—kinetic and potential. The measure of potential energy is usually taken to be the product of the attractive force of two masses into a power of their distance apart. In the majority of problems we can go no further, but in certain cases, as for instance the pressure of gases, we may express a portion of the energy of the whole mass as due to the kinetic energy of small, or mole- cular, portions of it. But the internal energy of a gas must still be considered as strictly potential and incapable of further explanation. Kinetic energy is mass into a function of velocity and its formula is T=mo(v). There is gradually arising a tendency, which is well founded, to distinguish two kinds of kinetic energy. The first is that produced by the momentum of a moving body, the wis veva whose measure 1s 1/2mv? ; the other is the kinetic energy, which originates in one body and is transferred through space, apparently vacuous, to another body—in other words, the radiant energy known as heat, ight, and electricity. It has been the persistent attempt of physicists for centuries to explain this radiant energy by mechanical analogies. And this effort has fastened on the science an interminable series of impossible fictitious ethers and mechanical atoms. The most indefatigable labours of the greatest minds have been spent to imagine an atom, which would serve satisfactorily as a source and, at the same time, as a receptacle of radiant energy, and an ether which would transfer it. Not one of these models has been even partially adequate : the course of the development has been steadily from the simple to the complex, from the concrete to the abstract, from the physical to the metaphysical, until the most recent atom is a complex more intricate than a stellar cosmogony whose parts are an entity called electricity, and the esther is an abstraction devoid of any mechanical attributes. Out of all this con- troversy we have gained the following facts Heat, light, 200M ; Prof. L. T. More on the and electrical energy, originating in one body, pass through space undiminished and unaugmented to another body. We can also express this energy as kinetic energy while it is associated with matter. In transit, since our experience gives us no clue or criterion, we can assume as a formula for the energy, either a periodic motion of an hypothetical some- thing, called an ether, or a projectile motion of an hypo- thetical mass-particle. In either case, all we really do is to divide the initial or final material energy into two mathe- matical quantities, one a mass-factor and the other a velocity- factor, and give to each such a value as to make their product remain a constant. As a rule, we make the mass-factor so small that we can shut our eyes to its existen e and imagine anything about it we please. The time relation is fixed by experiment. For the purposes of theory, although this radiant energy appeals to our senses in the three forms of heat, light, and electricity, which in their qualitative respect are each fundamental and not referable one to another, we fortunately find that quantitatively all three are satisfied by one dynamic formula. We have therefore obtained an adequate quantitative knowledge of energy, but not an inkling of the qualitative coefficients in this “formula. The hypothesis of the ether is an attempt to accomplish the impossible. And while it is now generally admitted. that we cannot create such a substance as will satisfy the physical requirements of a transmitter of radiant energy, still the ether is claimed to be a useful hypothesis. This utility is said to consist in giving us a crude image, in a mechanical way, of what occurs. In other words, it lessens our innate dislike to confessing complete ignorance, and it provides a set of concrete analogies for abstract statements and equations. Thus Poincaré says in the preface to his Théorie de la Lumiere: “ Peu nous importe que léther existe réellement ; c’est l’affaire des métaphysiciens ; Vessen- tiel pour nous c’est que tout se passe comme s’il existait et que cette hypothese est commode pour Vexplication des phénoménes. Apres tout, avons nous d’autre raison de croire a Vexistence des objets matériels? Ce n’est Ja aussi qu'une hypothese commode ; 3 seulement elle ne cessera jamais de Vétre, tandis qu'un jour viendra sans doute ot! Véiher sera rejeté comme inutile.” Now the old elastic-solid and mechanical sether did afford us a concrete image and we could speak of it with some intelligence to one another, because everyone has a conception of an elastic solid. To be sure, this solid ether became a grotesque. It permitted the transference of heat and light energy, but only at the Recent Theories of Electricity. ee): expense of creating a kind of niatter entirely outside of, and contradictory to, anything in our experience. We have only to recall the properties ascribed to this ether to find that it operated equally well if it had a density indefinitely great or one sndeHnitely small ; if it were rigid or if it were collapsible, &c. As certainly as one physicist endowed it with a property, another arose who showed that just the opposite property was equally efficient. Yet we might still be staggering along with the conviction that somehow this supposititious stuff was of use to us; at least it gave us a set of words conveying some meaning. But when Maxwell proved mathematically that a third kind of radiant energy of an electrical type should be looked for, and when Hertz demonstrated its existence, no elastic solid would serve for all three kinds ; and so, for a time, we were taught simul- taneously the properties of two co-existent thers. An elastic-solid and a so-called electromagnetic ether in one space were not amicable, and the former soon acted as Lord Kelvin had suggested, it really collapsed. Maxwell’s idea produced a revolution in the theory of physics; heat and light remained no longer a form of mechanical waves but became electromagnetic waves of special periodicity. By a progressive subtilization we have now arrived at Sir J. Larmor’s celebrated definition of sether which will satisfy all forms of radiant energy. The ether* is “a plenum with uniform properties throughout all extension, but permeated by intrinsic singular points, each of which deter- mines and, so to speak, locks up permanently a surrounding steady state of strain or other disturbance.” This plenum is continuous, without atomic structure, and absolutely quies- cent. Since these points of intrinsic strain are the atoms of matter, “the+ ultimate element of material constitution being faken to be an electric charge or nucleus of permanent setherial strain,” it is evident ‘ that t the metion of matter does not affect the quiescent aether except through the motion of the atomic electric charges carried along with it.” These ideas evidently reduce matter to an attribute of electricity, and make all forces of the type called electrical forces. But if electricity is everything, we must inevitably some time explain pure mechanical actions in terms of this electrical substance. Sir J. Larmor clearly foresees this, as shown by his statement § : “The electric character of the forces of chemical affinity was an accepted part of the chemical views of Davy, Berzelius, and Faraday ; and more * © Aither and Matter,’ p. 77. POW Oe EOS Deak Gav eeps Ld. 202 Prof. L. T, More on the recent discussions, while clearing away crude conceptions, have invariably tended to the strengthening of that hypo- thesis. The mode in which the ordinary forces of cohesion could be included in such a view is still quite undeveloped.” He thus rather leaves this question in the air by concluding that a complete theory is not necessary. But the history cf science shows that we shall soon hy pothee: ite two ethers or iry to give properties to one which will include electrical, seme. and material forces ; indeed this latter is already being attempted. Evidently he fears the tendency of explaining things too exactly by mechanical analogies, for he believes * “all that is known (or perhaps need be know n) of the ether itself may be formulated as a scheme of differential equations detining the properties of a continuum in space, which it would be gratuitous to further explain by any complication of structure ; though we can with great advantage employ our stock of ‘ordinar y dynamical concepts in describing the succession of different states thereby defined.” Yet it seems that a more complicated str ueture than the modern molecule, composed of an interminable series of electrical etherial strains, conld hardly be conceived. It the conception of an elastic-solid sether was admittedly a fiction of the mind, and one impossible to align with any known kind of matter, the electromagnetic ether is so esoteric, so subtilized from all substance, that it merely provides a nomenclature for a set of equations expressing the propagation of radiant energy. Both Sir J. Larmor and Professor Lorentz give the impression in their writings that the least said of the properties of such an ether the better, since the final verdict will be that the process of radiant energy, in transit through space, is best expressed as an equation containing unknowable qualitative coefficients. We may well go still further, for I believe the time is rapidly approaching when all scientific discussion of the nature of the sether will be considered futile. But Sir J. Larmor does lay down in his treatise certuin attributes, mostly negative, which he accepts. Thus, the ether is a continuous and quiescent plenum, absolutely unattected by mechanical energy. Existing in it, are countless places of discontinuity, or electrical strains, which constitute the elements of matter. Its only positive properties are the ability 1o maintain such strains and io transmit any electro- magnetic disturbance with the velocity of light. All chemical and mechanical forces must therefore be attributes of electricity, or else referable to scme other distinct cause. * DG pal Gs Recent Theories of Electricity. 203 Professor Lorentz recognizes thé growth of the idea that it is unnecessary for the physicist to dwell on the mechanism of the coefficients introduced into our equations, but he adheres to the view* that “we cannot be satisfied with simply introducing for each substance these coefficients, whose values are to be determined by experiment ; we shall be obliged to have recourse to some hypothesis about the mechanism that is at the bottom of the phenomena.” In respect to the ether he is exceedingly vague, and so fur as a mechanism of it is concerned, gives nothing. For example, while speaking of the state of this medium when it is the seat of an electromagnetic field, he says t: ‘*‘ We need by no means go far in attempting to form an image of it and, in fact, we cannot say much about it.” In agreement with Sir J. Larmor, his ether is a plenum always at rest, capable uf maintaining and of transmitting electric strains, and containing electrons, or extremely small particles charged with electricity. Sometimes he gives tne impres- sion that these electrons are electricity only. The ether not only penetrates the spaces between the electrons but also pervades them: ‘ Weft can reconcile ourselves with this, .... by thinking of the particles of matter as of some local modifications in the state of the ether.” Here also, mechanical forces and attributes are discarded, and he holds : “ That § the phenomena going on in some (any ?) part of the ether are entirely determined by the electric and magnetic force existing in that part.” Now we are indebted to Maxwell, and to him alone, for a set of electromagnetic stresses and strains which will satisfy the requirements of the electromagnetic field. In his mind, there is no doubt, these stresses produced a material strain in the ether which was communicated physically to dielectries existing in the ether and materially distorted them. It is, of course, impossible to test experimentally the existence of these strains in the free ether, but in a long series of papers in this journal I have shown that no distortion is produeed by them in dielectrics. These experiments were undertaken when the reai existence of the setherial stresses was generally accepted. I was led to the opposite opinion, because in no other case had any static connexion been found between ether and matter, and because Helmholtz had shown that such stresses would produce motion in the ether, an effect unlikely to be true. Professor Lorentz certainly concurs in this opinion. He states || : “ While thus denying the real * “Theory of Elections,’ p. &. Pee), i! SL pe 20, | 204 Prof. L. T. More on the existence of ather stresses, we can still avail ourselves of all the mathematical transformations by which the application of the formula may be made easier. We need not refrain from reducing the force to a surface-integral, and for con- venience sake we may continue to apply to the quantities occurring in this integral the name of stresses. Only, we must be aware that they are only imaginary ones, nothing else than auxiliary mathematical quantities.” It is also, I think, safe to say that Sir J. Larmor believes in the fictitious character of the Maxwellian stresses. Does not this also lead to the idea that electrons, which are disem- bodied electricity and which produce these imaginary stresses, are themselves imaginary ? There is at present a controversy whether these electrons are rigid or deformable. The only consequence of these two views necessary to comment on now is a very pertinent remark of Hr. Abraham *, that if the electron be deformable, work will be required to effect this deformation, and to avoid contradiction with the law of conservation of energy, th» electron must possess internal potential energy. This opinion of Hr. Abraham is almost impossible to avoid. ‘To provide the electron itself with this kind of energy is to deny its character as the fundamental and indivisible unit of matter, for a body having potential energy must contain mutually reacting parts w hich may themselves be considered as units of a lower order; nor will many approve of M. Poincaré’s rather embarrassin g suggestion, that the ether may be a great and inexhaustible store-house of energy, drawn on at will by the electron each time it moves. This idea will hardly be taken seriously, as the assumption of unlimited energy existing in a fictitious ether is in no sense a scientific notion ; it contradicts the prevailing idea of the inertness of the ether and makes of ita sort of deus ex machina which interposes to help us out of difficulties. And indeed the electromagnetic ether, without material properties other than imaginary stresses, is an explanation more difficult to grasp than the phenomena of radiant energy which require explanation. Such a revolution in the nature of ezther requires a like one in our ideas of matter. ‘The most notable effort in theoretical physics, at the present time, is the hypothesis that the ultimate element of matter is not a material atom, a sort of microcosm of sensible matter, but a free electrical charge, considered to be an entity for the purpose ; added to this are the dependent ideas that inertia and all other properties of * Theorie dey Elektrizxitét, 11. Kap. 3, Leipzig 1905. - Recent Theories of Electricity. 205 matter are attributes of electricity. Tbis hypothesis can mean nothing else than that the Lucretian atom, the centres of force of Boscovic ch, the vortices of Kelvin and all the atomic models (made of weights and springs and strings), have failed and become useless as aids to the imagination. Sir J. Larmor defines this new atom as a protion*, “in whole or in part a nucleus of intrinsic strain in the ether, a place at which the continuity of the medium has been broken and cemented together again (to use a crude but effective image) without accurately “fitting the parts, so that there is a residual strain all round the place.” This strain is not of the character of mechanical elasticity, since the “ ultimate Tf element of material constitution is taken to be an electric charge or nucleus of permanent eetherial strain instead of a vortex ring: .... ‘The molecule is composed simply of a system, probably large in number, of positive and negative protions in a state of steady orbital motion round each other: .... And, moreover, we may imagine complex structures composed of these primary systems. as units, for example successive concentric rings of positive or negative electrons sustaining each other in position.” Positive and negative electrons differ only in their orbital motion from each other and their forces are all of the elec- trical type. Prof. Sir J. J. Thomson pictures the atoms of the various chemical elements as nuclei of free positive elec- tricity holding in electrical equilibrinm free negative charges, placed i in various geometrical designs. The degree of stability is determined by the radioactivity “of each element. Professor Lorentz considers the protion to be a small particle charged with electricity and probably a local modification of the eether ; but his work on electromagnetic mass leads one to the opinion that he believes electricity to be the real essence of the material universe. Professor Abraham and the modern school of German physicists are frankly endeavouring to give a purely electromagnetic foundation to the mechanism of the electron and to mechanical actions in general. Now to me, and I believe to many men of science, the chief and indeed only value of an atomic theory is to give a concrete, though crude, image of matter reduced to its simplest conditions. The word electricity gives me no such image of matter ; it conveys absolutely no ies of materiality nor even of space or time relations. What the originators of the electrical atom have done is apparently to transpose the words, matter and electricity, tacitly giving to the latter * ¢ Ather and Matter,’ p. 26. Bacup. 27. passim, 206 Prof. L. T. More on the all the ideas usually associated with the former. We may as well take the next step at once and raise the objective universe on the Liebnitzian monad or on Schopenhauer’s philosophy of * Die Welt als Wille und Vorstellung.” Again, the law of the conservation of matter has been one of the most fertile ideas in science ; according to this law at least one attribute, inertia, remains constant however all others may change, thus giving continuity to material bodies as well as to space and time. It is quite possible to imagine an element of this new electric matter to be composed of equal quantities of positive and negative electrons, whose motions are so balanced as to make all material attributes vanish and produce a quasi-annihilation of matter. Lastly, when the statement is made that the electron is merely a local modification of the all-pervading ether, some idea should be given us as to the nature of this modification. If it is of the character of a strain, no meaning is conveyed unless this strain is subject to the laws of static or kinetic mechanics. But we have no knowledge of a static strain which fulfils the requirements of matter, especially that it musi be localized at definite points and must be uncreatable and indestructible ; of kinetic strains, the only one at present oven bles abe “crt ring of Helmholtz and Kelvin. . To imply that matter is electricity and that electricity is a static strain or a vortex ring, is to make an impossible assumption and is reasoning in a circle. IJ£ the vortex ring of matter failed chiefly because Maxwell said* : “That at best it was = ona ti chet, pel ton une ter ere eee it,’ what chance has this new type to survive criticism ? In accordance with the view taken in this paper, no hypo- thesis will be made to express properties of an esther, whose existence is itself incapable of scientific proof. It is, at the same time, perfectly proper to distinguish space through which radiant energy passes by a special name, such as the ether. The amount of radiant energy in transit is best given by an equation expressing conservation of energy and containing a velocity and an inertia factor. The velocity factor of this equation, most conveniently, takes the form of a periodic motion, but no assumptions need be made as to the nature of the periodicity or of the inertia factor since they also are not subject to experimental verification. Although matter appears to us as a continuous quantity or at least as divisible far below our present methods of experi- mentation, still it is convenient to give to the smallest observable portion of matter some such name as protion. * Encycl. Brit.: The Atom. Recent Theories of Ilectricity. 207 This unit of matter must be reduced in size as refinement of observation increases so that we may always be able to discuss it mathematicully in the aggregate only. At the present time this protion is the electron, and the only attributes necessary to assign to it are inertia in the Newtonian sense, a force of gravitational attraction and a force of electrical attraction, either positive or negative in sign. No causes for these attributes can be given as they are fundamental. . If the experiments of Kaufmann, which show that an electrified particle in motion has an apparently increased momentum, are cited as supporting the view that inertia is a function of velocity and should be considered as an attribute of an invariable quantity, the electrical charge, I hope to show that it is possible to accept Kaufmann's results and at the same time the invariability of inertia. Before proceeding further with this discussion it is con- venient to assemble the foregoing ideas in a concise form. 1. The fundamental quantitative units are length, mass, and time. These are continuous functions, or at least indefinitely divisible. 2. Matter has an objective reality and its quantity is measured by its mass or inertia. 3. Mass is an invariable function whose total quantity is conservative. 4. Wnergy is a conservative function. 5. Energy is divided, for convenience, into three types: poteatial, kinetic, and radiant energy. 6. Potential energy depends on force and position and is measured by the formula, V=m¢{(f. (¢—l’)}. 7. Kinetic energy is the energy of a moving body: its formula is T=md¢(v). *, Radiant energy is the expression for the fact that heat, light, and electromagnetic energy pass through free space. It is not associated with matter and is conveniently expressed as a function of an “inertia” and a “velocity” factor, R=¢(m.v). The velocity factor will be taken to be a periodic motion with a translational velocity of 3x10” centimetres per second. ‘lhe inertia factor is twice the ainount of the energy divided by the velocity squared. 9. The ether isa name viven to a fictitious substance whose inertia is the inertia factor of radiant energy. 10. The protion, using this name to avoid confusion with the atom of chemical reactions, is the least portion of matter recognized experimentally. It is the scientific unit of mass and can be dealt with mathematically only in aggregates. At the present time this unit is the electron. ; 208 Prof. L. T. More on the So far these ideas have more or less approval and have been already discussed. ‘Those following are more novel and need to be supported. 11. The electron has an invariable ponderable mass, m, and a variable electromagnetic mass, me, due to its electrical charge when in motion. ‘Its total effective inertia is therefore, M=m +e. 12. The electrical charge, e, of an electron is an un- explainable property of matter, measured by its force of electrical attraction. Instead of adopting the hypothesis that the electric charge on an electron is constant, we shall con- sider quantity of electricity to be a function AG the velocity of matter. Electromagnetic mass thus becomes an attribute of matter somewhat analogous to hydrodynamic mass. The ditterence between positive and negative electricity may depend only on the direction of the orbital motion of the electron. 13. The electron possesses a force of gravitational attrac- tion for other electrons, expressed by the law, F, we had the creation of caloric. And now we are asked to do the same thing with electricity. It is safe to predict that history will be repeated again, and that electrical charges and their forees will also sink into the condition of an attribute of matter. It might certainly be true that two experiments showing equal electrical charges would, if we could measure the amount of matter concerned, provide us with unequal quan- tities of matter, just as conversely equal quantities of matter might show different quantities of electricity. The hypothesis of equivalence of electrical charge and matter rests solely on an analogy to electrolysis, where matter is in a quite different state and also where the equivalence may be only approximate. Matter, on the other hand, ina solid state shows no connexion between volume and den sity and electrical charge. In dealing with electricity we should not for get the immense superiority of electrical detectors in delicacy to those for mechanical quantities, so that we can appreciate far smaller quantities of electrified than of neutral bodies. There is no doubt, from the quotations given, that theorists are basing their w vork on the assumption of the electron as the Phil. Mag. § SoG. Volv 21. No. 122. Feb. 1911. P 210 Prof. L. T. More on the unit of matter. And they give to it the following properties :— its mass is wholly electromagnetic ; the motive forces are electric forces ; and the laws of mechanics are to be deduced from the laws of electromagnetics. Professor Abraham defines the electron as a rigid sphere with an electric charge distributed uniformly throughout it in concentric spherical shells. This charge is a constant, and its volume and surface densities are homogeneous. Neglecting the obscurity which occurs when we try to imagine what the sphere is, on which the electricity is distributed, and what its measure is if it be not mass, we follow him in the development of his equations for energy and momentum. The total electric or potential energy is given by the equation : U e2 [SF log 8-2 : ~ aah B -1-6 the magnetic or kinetic energy hy aR, ad or 1 4/6) 9 ° : = saeRL @ een An and the electromagnetic monrentum by Lae ee! a oe ae le |G == = ipopy |e hee (=e eae where e is the electric charge ; R, the radius of the sphere ; v, the velocity of the electron ; V, the velocity of light ; v gee, Since the electromagnetic mass factor of |G| is dependent on the velocity, we may separate it into two components—a “longitudinal ” mass due to a change in linear velocity alone, and a “ transverse” mass due to a change in direction only. The equation of motion thus becomes f= (m+ m!) f’ + (mg tm!) f", where F is any external force ; m,, the ponderable mass ; m’', the longitudinal electromagnetic mass ; m'', the transverse electromagnetic mass ; f and f’’, the corresponding accelerations. Recent Theories of Llectricity. 211 From |G], he finds that Tse +e +... } m! e 3 = dary and m! ee f(1+4)+GtbDP+G+he' t+...... a For a stationary electron, 2 ats pci ree AEE IS is 20) Ginl 0 anh 6a RV? If the velocity becomes that of light, Ses ancl ei aca For intermediate values of 8, m’>m". Kaufmann, by an ingenious experiment, tested the increase of the apparent transverse mass with the velocity of radio- active particles and found such an increase. If m,'' and m,"' are the transverse masses of two particles at different velocities, we may put a i Mo + mM, My oe band) | aaa ie My +My, Mes Then my" Pix: K(k— 1) mo K—k Now by his experiments « and & are equal, within the limits of accuracy ; since m’’ remains finite until the velocity of light is reached, m, must be excessively small and is now taken by writers on theory to be actually zero. Although no experiments have been made on the longitudinal electro- magnetic mass, the same is held to be true for it. Other theories of the nature of the electron are essentially the same. They present, however, some important differ- ences. Lorentz and Hinstein consider the electron to be a sphere only when at rest and to deform into an ellipsoid of greater and greater eccentricity with increasing velocity until it becomes a disk when the velocity of light is attained. The volume of this ellipsoid is a variable. Bucherer has suggested the same idea of the deformable electron, but in his hypothesis the volume remains constant at all velocities. Expressed in similar units, the velocity function of m”’ in the three cases is as follows :— Wo bo eee! Lee Abraham sek eis\eleheterarer ele (£8) aes a fee Garem log 75-1). Lorentz-Hinstein. (8) = (1—6?)~"”, ‘Buchereny wees @(8) = (1-67). 212 Prof. L. T. More on the In simplicity of formula the last two have the advantage, and from general theoretical principles there is little to choose between them. Here, too, the shape and other definite properties of a fundamental element of matter are not subject to proof and become finally a mere question of definition. If the electron is rigid, we must expect to obtain mea- surable and positive results in experiments, such as those of Michelson and Morley, made on the mutual relations of the quiescent ether and Spphee. These are now conceded to be impossible, either from the principle of relativity or from what, I think, is a more fundamental idea : that however finely matter he div ided, it maintains all such attributes as potential energy. So if the electron is held to be rigid and without potential energy, that attribute must be given to it from some mutual relation between it and a plenum of which it is a modification, such as is expressed by Poynting’s energy theorem. On the other hand, if the electron is deformable, work must be done to produce this deformation. This can come either from internal potential energy due to force actions of its own parts, or from an unlimited reservoir, the ether. In the first case, we are compelled to subdivide the electron, which thus ceases to be a fundamental element of matter ; and the second case leads us nowhere. Nor can the principle of relativity aid us in obtaining positive knowledge on such questions ; at best it is a principle of negation, stating in mathematical terms the idea long established in philosophy that all our knowledge is relative, and must be so, from the fact of the finitude of our minds. If now we turn to the experimental evidence to decide between these two forms of the electron, we come to no definite decision. Kaufmann * concludes with the following remarks :— “Die vorstehenden Ergebnisse sprechen entschieden gegen die Richtigkeit der Lorentzschen und somit auch der Hin- steinschen “ Theorie ; betrachtet man diese aber als widerlegt, so wire damit ack der Versuch, die ganze Physik ein- schliesslich der Elektrodynamik und der Optik auf das Prinzip der Relativbewegung zu grinden, einstweilen als missgliickt zu bezeichnen. Bane Betrachtung der Einstein- schen Theorie zeigt, dass man, um bei Beibehaltung dieses Prinuzipes dennoch Ueber einstimmung mit meinen Resultaten zu erhalten, bereits die Maxwellschen Gleichungen fiir * Annal. der Physik, xix. p. 534 (1906). Recent Theories of Electricity. 213 ruhende Korper modifizieren miisste, ein Schritt, zu dem sich wohl einstweilen schwer jemand wird entschliessen wollen.” Hupka*, by a different method, arrived at contrary results. His conclusions are :— Das wichtigste Resultat der beschriebenen Versuche ist jedoch ein Peitrag zur Ent- scheidung der miteinander im Wettstreit liegenden Theorien von Abraham und Lorentz-Hinstein. Die mitgeteilten Messungen sprechen zugunsten der letzteren. Zu demselben Ergebnis ist bekanntlich auch Hr. Bucherer gelangt.” It is not neces-ary to weigh their contradictory evidence, as it has been done most exhaustively by W. Heil ft. He shows, and I think conclusively, that it may be taken as established that the apparent inertia of an electron depends on its velocity ; but in no case is the experimental accuracy sufficiently great to decide the question of the nature of the electron. Let us now turn to the mathematical side of the dis- cussion and examine the expressions for the electromagnetic mass : we shall find contradictions between the two general ideas and further evidence that the founding of mechanics on an electromagnetic base does not harmonize with the other branches of physics. The values of U, T, and |G| all become infinite when the velocity of the particle equals the velocity of light; yet several physicists have given this velocity to the orbital motion of the electron, and others have attempted to give the same velocity to electrified light- particles. On the other hand, when the particle is at rest, 2 T and |G] become zero and U =——,. s7R that the magnetic energy and the electromagnetic momentum should then vanish, as both of them are kinetic functions of electricity ; but how are we to account for the finite values of the electrical energy and of the two forms of electro- magnetic mass? To do so would require us to assume that a free electric charge at rest still possesses inertia, a con- sequence difficult to reconcile with our experimental evidence of static electricity. It is even more convenient to turn to an analysis of Kaufmann’s { equations. He has derived the electric and magnetic deflexions of a charged particle moving through an electric and a magnetic field. It is proper * Annal. der Physik, xxxi. p. 203 (1910). + Annal. der Physik, xxxi. p. 519; xxxiii. p. 403 (1910). 1. Loewctt 214 Prof. L, T. More on the If y is the displacement by the electric field, E, z, the displacement by a magnetic field, M, Ho, the electromagnetic transverse mass of an electron for velocity zero”, V, the velocity of light, e, the electric charge on the particle, then hie. E 1 3 iy WTO and pan get Mil. Vi Tp NV BD(s)* where as before, ea alec an 3 (8) = 1B iF log 75 = ); . Abraham ’(8)=(1—f’) a . . + . Lorentz-Hinstein @(@)==(1— B?)~ 8. DUBE 8 We ig ihe ae eta According to these theories, when 8=1, O(8)=a, and y=z=0 ; and when 8=0, (2) 1 anda — i core All theories agree, that an electrified particle, moving with the velocity of light, cannot be deflected by an electro- magnetic field, and this coincides with the idea that the electromagnetic mass then becomes infinite. But for a small velocity these deflexions become very large and are infinite when the electron is at rest. Even if we should accept this result for the case of the electric displacement, we should still have the difficulty of accounting for any action between an external magnetic field and an electritied particle at rest. These results for a stationary particle ure difficult to reconcile and impossible to explain without making special and unlikely hypotheses for the constitution of the electron. It should also be noted here, that this is the only case where a small mass, whose velocity, charge, * It is impossible for me to form any conception of this quantity if it has a finite value, and yet itis one of the essential factors of the equations which follow. In the first place, an electron at rest has no electro- magnetic field ; and secondly, how canan electron with no motion have a transverse mass of any sort, when that is defined as mass due to a change in direction only? We might as well give a finite value to the centri- fuga] force of a body at rest. Recent Theories of Electricity. 215 and external attracting force are all finite, attains an infinite momentum and kinetic energy. We are compelled to say that the velocity of light is unattainable by matter, because the zther is then impenetrable or that the electromagnetic mass becomes infinite. All these theories and experiments are based on the experimental fact that — has been found to be 1°865 x 10’ approximately. It is then assumed that e is a constant and equal to the charge on the hydrogen electrolytic ion ; with this value, the mass m is about the one-thousandth part of the mechanical mass of the hydrogen atom. How can we account for this mass when the theorists claim that the mechanical mass is entirely electromagnetic ? We know what the transverse electromagnetic mass is from Abraham’s @ e e G e formula ; it is equal to m= nel where v is a constant v velocity of changeable direction. Substituting we obtain the formula already given, ml! é a a This mass mm” is evidently not the denominator of the ratio, — . Mm” To say that ¢ is a constant is an assumption based solely on an analogy to the experimental laws of electrolysis ; but in electrolysis, when we obtain equal electric charges we also find equivalent masses of matter. In the discharge of electricity through gases and in radioactivity the matter deposited is too small to be measured. This is a fundamental difference, and vitiates an analogy between the two. For example, we measure the amount of current in a vacuum tube by an electrical device, and at the same time we measure the deflexion of the current by an electric and magnetic field ; in other words, all quantities and forces are electrical, and we say that equal currents in this case require equivalent quantities of matter. But it has not been shown to be impossible or even improbable that electrons, associated with equal quantities of matter but having different velocities, might show different electrical charges; or that electrons producing equal electrical charges, might deposit different amounts of matter if it were sufficient in quantity to be detected by chemical or mechanical reactions. Thus we may imagine the following experiment :—Suppose all conditions in a vacuum tube to remain the same, except that the velocity 216 Prof. L. T. More on the of the cathode particles could be changed, it might be that this change in velocity would alter the electric charge deposited on an obstacle. As an hypothesis, I propose that, in order to make the 64. G 3 : : : ratio — agree with the experimental evidence of its value 1 and to account for electromagnetic mass, we consider m to be the mass of a particle of matter in the Newtonian sense, of constant and small value, and e, the electrical charge, to be a force attribute of matter which varies with the velocity of the particle. However novel this hypothesis may be, I have not been able to find any experimental facts more difficult to explain by it than by any of the other hypotheses which have been recently advanced ; and, on the other hand, it apparently accounts for much of the modern work in terins of the older and well-established ideas. 1. Since the quantities, e and m, occur in all cases only in the form of a simple ratio, either @ priori may be considered the variable. 2. From Kaufmann’s experiments e/m decreases as the velocity of the electron increases. This is satisfied if we assume that the electric charge has a maximum value for an electron at rest which decreases with increasing velocity until it attains a value of zero at the velocity of light. 3. The decrease in the value of e does not become notice- able until a velocity comparable to that of light is reached. 4, This hypothesis supports the theoretical value and experimental ideas of electromagnetic mass. 5. At zero velocity matter would retain its mechanical inertia and electric charge, which permits the function U to have a finite value while T and |G| both vanish. 6. Electrical conductivity increases with diminishing tem- perature and attains a large value at the absolute zero. ‘This squares with the hypothesis that the electrical charge of matter increases with decreasing veuocity. 7. If e becomes zero with the velocity of light, it is evident that the deflexions by a magnetic or electric field would be zero. 8. We are not compelled to assume an infinite momentum for a body moving with a finite velocity. 9. The difference between positive and negative electricity can still be ascribed to the nature of the orbital motion of the electron. From the very nature of my conception of the limits which should be imposed on ‘scientific inquiry, I make no Recent Theories of Electricity. Zien attempt to explain the cause for this electrical property of matter any more than [ should for its gravitational attributes. Both are fundamental phenomena to be accepted as experi- mental facts until we gain contrary knowledge. Indeed, I have ventured to indulge in this speculation rather with the idea of showing that the recent hypotheses for electricity and matter ; for the ether, protions, and corpuscular light ; for the electromagnetic and other non-Newtonian mechanics, are not necessary. We may still account as adequately for all our experimental facts by a simple addition to the pro- perties of matter and continue to base our theories on mechanical laws. So long as the measurement of physical quantities becomes ultimately a matter of measuring mechanical forces, it 1s advisable to express quantitative physical laws in terms of mechanical formule. For this reason electricity should be considered a function of mechanical energy rather than the converse. If it be possible to place mechanics on an electro- dynamic basis, it is certain that we may always explain electricity in terms of ponderodynamic laws. As both are possible, it seems far more natural and more rational to con- sider electricity as an attribute of matter than matter as a phenomenon of electricity. Before this article is closed there is a point which should be discussed. The close analogy between electromagnetic mass and the apparent increase in mass in hydrodynamic problems has been pointed out by me in a former paper *, by Professor Lorentz, and must have occurred toeveryone. But Professor Lorentz indicates a difference between the two which seems to me less essential than he considers it. To be exact, I quote from him ft. In the problem of a ball moving in a perfect fluid, “‘ we are able to determine the effective mass my+m' (or m +m’), but it would be impossible to find the values of my and m’ (or m'") separately. Now, it is very important that, in the experimental investigation of the motion of the electron, we can go a step farther. This is due to the fact that the electromagnetic mass is not a con- stant, but increases with the velocity.” This is, if true, important, as it seems to show an essential difference between these two apparent masses. But it can be shown, as follows, that such an essential difference does not exist. We know that a single electron will maintain a state of rest or of uniform motion in a straight line just as a single ball in an * Phil. Mag. vol. xviii. p. 17 (1909). t-‘Theory of Electrons,’ p. 40. 218 Messrs. Searle, Aldis, and Dobson on Revolving Table infinite perfect fluid will, under similar conditions. To produce any variation in the speed or direction of motion of an electron, other electrified particles must be present to exert an external force uponit. Thus Kaufmann determined m’” by measuring the deviation in path of an electrified particle which moved parallel to an electrified plane surface. But 1f we now call m, the true mass and m"’ the apparent, or hydrodynamic, mass of a ball moving in a perfect fluid and parallel to a fixed plane surface, can we not also separate these two masses? ‘The formula for the kinetic energy of a sphere, of density p and radius a, moving in a perfect fluid parallel to, and at a distance A from, a fixed plane surface, 1s * 3 ¢ 3 2T = = pa (1 -- ié aya We also have mo= 9 Tpa’. Since the hydrodynamic mass, m", is a function of the variable h and the true mass, mp, is a constant, we may measure the kinetic energy of similar spheres which move parallel to a plane boundary but at different distances from it and so separate m,and m". Practically, we should ex- perience great difficulties; the effect would be very small ; we have no perfect fluids; and we have not yet deduced the equations for spheres moving in a viscous fluid. But these are purely experimental ditficulties and show no essential differences between the conceptions we should make for hydrodynamic and electromagnetic mass. Both may be considered as variable quantities of the same character added to the true and constant inertia of matter. University of Cincinnati, August, 1910. XXV. On a Revolving Table Method of determining the Curvature of Spherical Surfaces. By G. F. C. SEARLE, MOA. F.RS., A. C. W. Aupis, WA., and G. Wis Dosson, B.A.t+ § 1. (ae many optical purposes it is necessary to determine the radius of curvature of a convex or a concave surface with considerable accuracy. The revolving table method, described below, has the advantage that the radius is found directly from two readings on a * Lamb, ‘Hydrodynamics,’ p. 148. ~ Communicated by the Authors, Method of determining Curvature of Spherical Surfaces. 219 straight uniformly divided scale, without corrections or calculations of any sort. We shall not attempt to give an account of all the mechanical devices which might be profitably employed in attaining the highest possible accuracy, but shall content ourselves with describing the plan of the method and the elementary geometrical prin- ciples involved in the application of the method to practical measurements. The simple apparatus which we describe has been in use for some months in one of the classes at the Cavendish Laboratory, Cambridge; the method has been found to interest the students. § 2. The principle of the method may be described as follows :— A table turning truly and without shake about a vertical axis is required. For the most accurate work the fit of the vertical spindle in its bearings must be as perfect as in a good lathe-head. The plane of the top of the table is normal to the axis of revolution, and the top carries a straight scale against which slides a carriage bearing the spherical surface (see fig. 1). We shail assume that the seale is so adjusted on the table that the straight line described by the centre of curvature of the spherical surface when the carriage slides along the scale intersects the axis of revolution of the table. The position of the carriage relative to the table-top when the centre lies on the axis of revo- lution will be called the first position. If the table be turned through any angle about the vertical axis when the carriage is in the first position, the only effect of the angular motion is to substitute one part of the surface for another. Hence, if rays from a luminous point fall upon the surface, the reflected rays will be unaffected by the motion. It is not necessary that the rays which enter the eye from the luminous point should have met the surface very nearly normally, or, in other words, that the geometrical image of the point should be observed. It is sufficient to use a pencil of rays whose axis is nearly horizontal and strikes the surface not very obliquely, and to observe the vertical focal line of the reflected pencil. If a vertical line be used as an object, the ““image”’ by reflexion will be a vertical line formed by the vertical focal lines corresponding to individual points of the object. If a microscope be used for observing the image of the vertical line, the proper adjustment of the carriage along the scale can be made with great accuracy. The microscope must, of course, be firmly supported and must be furnished with cross-wires or with a micrometer-scale, In the absence of a microscope, 220 Messrs. Searle, Aldis, and Dobson on Revolving Table a telescope with cross-wires may be used if an extra con- verging lens be fitted in front of the objective so as to allow the instrument to be used at short distances. The carriage is now moved along the scale into a second position in which the axis of the table is a tangent line to the spherical surface. If, now, the table be turned about its axis, a grain of ]ycopodium placed at the point of contact of the vertical tangent line and the surface will remain stationary. By using a microscope for observing the hy copodium, this setting can be made with great accuracy. The table may be turned backwards and forwards through an angle of a few degrees about a mean position in which the perpendicular from the centre of curvature of the surface upon the axis of the table approximately coincides with the axis of the microscope. The radius of curvature of the surface is now given at once by the difference of the two scale-readings of the carriage in the first and second positions. § 3. When the scale is not in perfect adjustment on the table, the straight line described by the centre of curvature of the spherical surface, as the carriage slides along the scale, will not intersect the axis of revolution of the table. If, howeyer, the shortest distance between these two straight Jines be small, and if in both the first and second settings the axis of the microscope point nearly to the centre of curvature of the surface, the error in each of the two settings of the carriage will be very small, provided that, in each case, the table is turned about a mean position in which the scale is parallel to the axis of the microscope. Thus a rough adjustment of the scale on the table is sufficient for any but the most precise work. § 4. The adjustments of the surface to be tested are greatly facilitated by the use of a small lathe-head to form the “carriage ’? mentioned in the preceding paragraphs; and we shall describe the method of making the measurements when the lathe-head is used. The apparatus is shown in figs. 1 and 2. The revolving table rests upon a tripod stand, and the table-top turns about a vertical rod carried by the tripod. The top of the table carries a scale graduated in milli- metres, which can be clamped in any position on the table by the screw seen in fig. 1. Revolving tables of this description are now employed in many laboratories in the determination of the fcecal lengths of lens-systems by the nodal-point method. ‘The lathe-head and its adjuncts may be regarded as additions made to the revolving table to make it available for the measurement of curvatures. Method of determinng Curvature of Spherical Surfaces. 221 The lathe-head is attached by an adjustable screw to a straight-edged board which rests upon ths top of the table with its straight edge in contact with the scale. The spindle of the lathe-head is screwed at one end to fit a brass face- plate furnished with three screws which serve to adjust a Fig. 2. =— : i) nso, 3 second brass plate to which the lens or mirror to be tested is attached by wax. Hach end of the spindle is turnel to a conical point, and the vertex of each cone lies on the axis of revolution of the spindle. One arm of a steel rod bent at right angles can be secured by a set-screw in a socket 222 Messrs. Searle, Aldis, and Dobson on Revolving Table supporting the table-top, and a small clip carrying a pin can be fixed in any position on the other arm of the rod by a set-screw. ‘Tbe bent rod also serves as a means of clamping tle board to the table in the manner shown in fig. 2. § 5. The first step is to make the axis of the spindle of the lathe-head parailel to the edge of the board. ‘The face- plate is removed frum the spindle and the tip of the pin carried by the bent rod is brought into contact with the vertex of one of the conical ends of the spindle, the straight edge of the board being in contact with the scale. The board is then removed from the table and is replaced so that the other conical end of the spindle is near the pin. If, by sliding the board along the scale, this vertex can be made to touch the tip of the pin, the axis of the spindle is parallel to the edge of the board. In order that this may be the case, it is necessary (1) that the axis of the spindle be parallel to the plane of the board, and (2) that the projection of the axis upon the plane of the board be parallel to the edge of the board. The clamping screw enables the lathe-head to be adjusted on the board so that the second of these conditions is secured. In the apparatus shown in the figures no pro- vision has been made for adjusting the axis of the spindle so that the angle between it and the plane of the board may be zero. This angle will, however, be small if the apparatus is constructed with moderate care. If, instead of being zero, the angle be @ radians, the error in the measurement of the radius of curvature (7) will be *(1—cos @) or 476, when @ is small, and thus the error will generally be negligible. § 6. After the adjustment of the lathe-head on the board is complete, the scale must be adjusted on the table-top so that the axis of the lathe spindle intersects the axis of the table. The scale and the board are first adjusted roughly by eye so that one of the conical ends of the spindle is not far from the axis of the table. A fixed microscope, with a vertical cross-wire, is then focussed on the vertex of the cone when the axis of the spindle is approximately perpendicular to that of the microscope, the image of the vertex lying on the cross-wire. ‘The table-top is then turned through approxi. mately 180° ; if the image of the vertex does not again lie on the cross-wire, the difference is halved by moving the board along the scale, and the microscope is then moved so as to bring the image again to the cross-wire. The table-top is now turned so that the axis of the spindle is approximately parallel to that of the microscope. If the image of the vertex does not lie on the cross-wire, the scale is moved at right Method of determining Curvature of Spherical Surfaces. 223 angles to itself until the coincidence is obtained. The vertex of the spindle then lies on the axis of the table. Since the projection of the axis of the spindle on the plane of the board has been made parallel to the edge of the board, the axis of the spindle intersects that of the table for all positions of the board along the scale. These adjustments may, of course, be made once for all, but for purposes of instruction it is best that each student should make them for himself. § 7. The face-plate is now attached to the spindle and the lens or mirror to be tested is fixed to the adjustable plate by wax. In the case of a lens, the back surface should, for convenience, be smeared with a mixture of lamp-black and vaseline or be coated with black varnish to stop reflexion at that surface. The spindle is then rotated and some object is observed by reflexion at the spherical surface. If the rotation cause a motion of the image, the three adjusting screws are manipulated until the image remains stationary when the spindle is revolved. The centre of curvature then lies on the axis of the spindle. During the adjustment the table- top is clamped so that it cannot rotate. A piece of ground glass witha fine vertical line drawn on it forms a convenient object ; the glass should be well illuminated. A fixed microscope may be used to facilitate the adjustment. The spindle is turned into the position in which the image of the vertical line is as far to the right as possible. The plate carrying the surface is then made to move as nearly as possible about a vertical axis by turning the appropriate screw or screws, so that the image moves to the left through the proper distance. The success of the ad- justment is then tested by rotating the spindle ; if necessary a second adjustment is made. The preliminary adjustments are now complete. If they are only nearly but not quite perfect, the errors which they cause in the radius of curvature as found by this method will be very small, since these errors of radius depend on the second powers of the errors in the preliminary adjustments. §8. The board carrying the lathe-head is now set into the “first position”? in which the centre of curvature of the surface lies on the axis of revolution of the table. The ground glass with the vertical line is set up in sucha position that the image of the vertical line which is observed in the microscope is formed by rays which fall nearly normally upon the surface. If the image move when the table is turned backwards and forwards about its axis, the board carrying the lathe-head is moved along the scale until the image remains at rest. The centre of curvature then 224 Determining the Curvature of Spherical Surfaces. lies on the axis of the table. The scale-reading of an index mark on the board is then taken and recorded. The mean of several independent readings should be taken. In the case of some lenses the accuracy of setting is limited by the departure of the surface from perfect sphericity. In these cases it is possible to make the image remain stationary in spite of the motion of the table provided the motion be so small that the part of the surface at which the rays entering the microscope are reflected is not near the edge of the lens. 2.—-, where m ¥* e is the charge in electrostatic measure and m the mass of a corpuscle. Hvyen with the smallest admissible value of 7 a corpuscle would acquire a velocity great enough to satisfy the preceding inequality by the fall through a potential difference of two or three volts ; hence it is only exceedingly close to the cathode, perhaps only inside the hole through which the rays pass, that the corpuscles are still enough to allow of any combination with a positive particle to take place. Let us take as the case most favourable to recombi- nation the one where the corpuscles are at rest and the relative motion is entirely due to the positive particles, then if v is the velocity of the positive particle for combination to take place, v?< 2 ae The smallest value of 7 permis- sible will depend upon the size of the particle: it will be of atomic dimensions. Let it equal b x 10-8 cm. mer elm— lt x LO'xX 3x 10, pealn sires 4°6 we find v< oa LOK: If the velocity is due to the fall of the charged body through V volts, then if w is the electric atomic weight of the particle, the preceding relation is equivalent to ni 246 Sir J. J. Thomson on Thus for the secondary radiation due to the dissociation of the uncharged rays, V>wxnxlix 10’, V< wae x 10°. Thus the velocity of each kind of secondary ray must be between certain limits which do not depend on the potential difference between the electrodes in the discharge-tube. If these limits are very close together the velocity of the secondary ray will be very nearly constant. JI have found that this is the ease for the secondary rays corresponding to the hydrogen atom. ; The eurves on the photographic plate which pass through the origin may arise either frem the dissociation of the uncharged Canalstrahlen—the change from an uncharged particle to a positive ray, or from the reverse process, the change of a positive ray while in the electric and magnetic fields into an uncharged particle by the coalescence with it of a negatively charged corpuscle ; a method of distin- guishing between these cases is given on p. 233. For sucha coalescence to take place, however, the velocity of the positive ray must be below a certain value; if the velocity is greater than this the ray behaves like a primary one. When the difference of potential between the electrodes in the dis- charge-tube is much greater than is necessary to produce this velocity any secondary ray must have fallen through only a fraction of the potential difference in the tube, and must therefore have been produced near the cathode. Now it is just in this neighbourhood that the positive ions in the dark space have their greatest velocity and are most likely to produce fresh ions by collision. Thus it is probable that among the ions in the secondary rays there are some which have been produced by the collision of positive ions with the molecules of the gas in the dark space, while the primary rays which have fallen through the whole potential difference have been produced by the collision of cathode particles with these molecules. There are a few, but only a few, ions which occur both as primary and secondary ; the positive atom of hydrogen with one charge is the most conspicuous example of this class ; others are the atoms of mercury and oxygen and the molecule of hydrogen. In most cases, however, the ions are quite distinct. On looking at the list of ions due to secondary radiation given on page 239 it will be seen that, with the Rays of Positwe Electricity. 247 exception of the hydrogen molecule, there is not one in which the molecule is intact, while many molecules are found among the ions in the primary rays; for example, the molecules of hydrogen, oxygen, marsh-gas, not to mention those of the monatomic gases like helium and mercury vapour. Again, the ions in the secondary rays carry in many cases more than one unit of charge ; thus, for example, we have C,,, N,,, He,,, He,,, suggesting that the posi- tively charged particles when they collide with a molecule in many cases detach more than one corpuscle from it. It is very interesting to find that in the primary rays several different types of ions are found even in elementary gases like hydrogen or oxygen, for, as we have seen, we find both the atoms and the molecules amongst the primary rays in these gases. These ions are supposed to be due to the bombardment of the molecules by the cathode rays. How is it, then, that when we expose the molecules of a gas to bombardment by cathode rays we get two types of ions, atoms as well as molecules in the gases we have just men- tioned? Itis true that in the dark space next the cathode we have cathode rays with very different velocities ; and one way of explaining the two types of ions would be to suppose that when the energy of the cathode rays exceeds a certain value, they split the molecule into atoms when they impinge against it, while the slower cathode rays only succeed in knocking a corpuscle out of the molecule without impairing the cohesion between the atoms. On this view the charged atoms would be produced by the fast cathode rays, the charged molecules by the slower ones. If this were the case, however, we should expect that the lines corresponding to the atom would be shorter than those corresponding to the molecule, as the minimum energy required to produce them is greater than that required for the molecule, and the place where the atoms are produced would therefore be further from the cathode than the corresponding place for the molecules ; as a matter of fact, the lines for the atom are often, though not invariably, longer than those for the molecule. Another way in which the different kinds of ions might arise is as follows :—Let us suppose that a diatomic mole- cule is made up of two atoms A and B, and that A is positively and B negatively charged. When the cathode rays pass through the gas they may strike either A or B, and detach a corpuscle. If A is struck, then, after the collision, A has two positive and B one negative charge, together they forma system with a total positive charge ot 248 On Rays of Positive Electricity. one unit ; the attraction, however, between A and B due to their electric charges has been increased, so that the system AB is less likely to break up into atoms than it was before the collision took place. Suppose, however, that B and not A is struck by the cathode particle, then after the collision B will be uncharged, while A has one unit of positive charge, the total charge on the system is again one unit of positive electricity ; but as B has been deprived of its charge the electrical attraction between A and B is very much less than it was before the collision took place, so that the system will be much more likely to break up into separate atoms and supply us with a charged atom A. The negatively electrified atom B will be a little less likely to be struck by a negatively electrified cathode particle than the positively electrified one A. On the other hand, when a collision did take place, it would be easier to detach a corpuscle from the negatively electrified B than from the positively electrified A. Similar considerations would apply to compounds as well as to elements. We might in certain cases get some of the atoms of a compound molecule liberated and not others. Thus, if the hydrogen atoms in marsh-gas CH, are negatively charged, and if one of them is struck by a cathode particle, it would lose its charge and he easily detached, the other hydrogen atoms which retained their charge would cling to the carbon atom. The photographs hitherto described were made with discharge-tubes whose volume was considerably greater than 1000 e.c.; I have also made some photographs when the tube was very much smaller, the diameter being about 2cm. The feature of these photographs is, that unless the pressure is reduced very low, when the potential differ- ence between the electrodes is very large, almost the only thing to be seen on the plate, whatever gas may be in the tube, is the secondary radiation, negatively as well as posi- tively charged, corresponding to the hydrogen atom ; the negative portion is not infrequently almost as bright as the ositive. When the tube is filled with air, a very faint curve corre- sponding to the oxygen atom can with difficulty be detected on the plate ; it is, however, much too faint for reproduction froma photograph. Though this line isso faint, it is remark- able that it is generally the first to appear when the photograph is developed ; the hydrogen line, though so much stronger in the end, takes a much longer time to develop. It appears as if the hydrogen atoms had penetrated much more deeply into the film than the oxygen ones, but that close to the Focal Isolation of Long Heat- Waves. 249 surface of the film the oxygen atoms produce more effect than the hydrogen ones. ‘The relative intensities on the photographic plate do not always seem the same as on the willemite screen. 7 I find from the photozraphs that the slope of the straight part of the curves corresponding to the secondary radiation, due to the hydrogen atom, negative as well as positive, for the slopes of the two are the same, does not vary appre- ciably with the potential difference between the electrodes in the discharge-tube. I have taken photographs with the tube in states for which the equivalent spark gap in air varied from *7 to 4 cm., and found only very slight alterations in the slopes of the curves. As the velocity of the particles in the rays is proportional to the tangent of the angle of slope, this implies that the velocity of the particles in the secondary rays is almost constant, a result 1 had previously arrived at by the willemite screen. I wish to express my thanks to Mr. F. W. Aston, of Trinity College, and Mr. E. Everett, for the assistance they have given me with these experiments. AXXVIT. Focal Isolation of Long Heat- Waves. By H. Ruspens and R. W. Woop *. ihe isolation of very long heat-waves, which is usually accomplished by selective multiple reflexions (Rest- strahlen method) can be accomplished also by selective refraction. It was shown in 1899 by Rubens and Aschkinass that it was possible to separate very long heat-waves from the radiation of an incandescent source by means of quartz prisms of small angle*. This method, involving the use of a spec- trometer, did not however prove to be very efficient, on account of the large loss of energy, and the isolated radiation disappeared almost entirely if a quartz plate of any con- siderable thickness was interposed in the path of the rays. The method which will be presently described is free from these objections, and has enabled us to obtain heat- waves of greater wave-length than any hitherto observed and with sufficient intensity to make accurate measurements of their properties possible. Like the other method, it depends upon the selective refraction of quartz, the separa- tion being accomplished by means of Jenses however. * Communicated by the Authors. 7 H. Rubens and E. Aschkinass, Wied. Aza. Ixvii. p. £59. 250 Profs. H. Rubens and R. W. Wood on The arrangement of our apparatus is shown in fig. 1. The radiations from a Welsbach light A pass through a Fig. 1. a eeee OO Se 5 meeese ete ies =~ =. = ~~ =. circular aperture 15 mm. in diameter, B,in a screen made of two large sheets of tin-plate, and then in succession through the quartz lens L, a second aperture I’, and the quartz lens L, which focusses them upon the thermocouple of the radio- micrometer M. The lenses have a diameter of 7:3 cm. and a thickness of ‘8 cm. in the middle and °3 cm. at the edge: their focus for light rays is 27°3 cm. The central zones of these two lenses are covered by circular disks of black paper a, and ag, 25 mm. in diameter. A movable shutter is placed between C and L, to cut off the radiation at will. The operation of this arrangement of apparatus is easily understood. The distances of the lenses from the circular apertures are so proportioned, that a sharp image of the circular source B is focussed upon F only for radiations for which the refractive index of the quartz is 2°14, the square root of the dielectric constant for slow oscillations. Inasmuch as the refractive index of quartz is between 1°55 and 1°43 for the shorter heat and light waves (fine dotted lines), which are able to pass through it, these rays form a divergent cone after passing through the lens 1. This cone is intercepted by the screen H, the circular aperture being shielded by the disk of black paper. As a result, the only rays which can pass through the aperture are the very long heat-waves (coarse dotted lines), which converge upon it, as a result of the high value of the refractive index for tliese radiations. The second quartz lens, acting in the same way, still further purifies the radiation, and eliminates completely the shorter heat waves, scattered or diffused by the surfaces of the first lens. Focal Isolation of Long Heat- Waves. 251 Details regarding the micro-radiometer will be found in previous papers *. In the present case the instrument was protected by an air-tight metal helmet provided with a quartz window. The sensitiveness was such that a candle at adistance of 2 metres gave a deflexion of 700 svale-divisions, in spite of the quartz window. The great advantage of this method of focal-isolation is that large angular apertures can be used (in the present case F 3:5), and the radiation is weakened only by retlexion from and absorptien by the two quartz lenses. Its only disad- vantage is the circumstance that a rather wide spectral range is transmitted, for all radiations for which the re- fractive index is in the neighbourhood of 2°14 are passed through the apertures. For wave-length 63 « the refractive index is 2°19, decreas- ing to 2°14asa limiting value, with increasing wave-length +. The very powerful absorption of quartz for radiations ranging from 60-80 prevents the shorter waves from getting through the system. Beyond 80 the quartz begins to show stronger transparency, 17 per cent. of the residual- rays from potassium iodide passing through a quartz plate of 18 cm. thickness, corresponding to the amount of quartz in the path of our rays f. This cireumstance gives to our transmitted radiation an energy curve very steep on the side towards the shorter wave-lengths, while on the other side it slopes down gradually, at the rate determined by the energy curve of the source of light, decreasing with the 4th power of the wave-length, if the Welsbach mantle has no selective properties in this region, as is probably the case. ‘The increasing transparency of the quartz with increasing wave-length will make the face of the energy curve still more gradual, and we should expect our isolated radiation to have an unsymmetrical energy curve with a maximum shifted towards the shorter wave-lengths with respect to its centre of gravity, and to extend from 80 towards the longer wave-lengths over a range of more than an octave. This expectation was ful- filled, as the results of our experiments will show. The 40 mm. deflexion of the micro-radiometer was reduced to zero by interposing a plate of rock-salt, 3 mm. thick, into the path of the rays, while a quartz plate of 4 mm. * H. Schmidt, Ann. der Phys. xxix. p. 1003; H. Rubens and H. Holl nagel, Phil. Mag. [6] xix. p. 764. y Hi. Rubens and E. F. Nichols, Wied. Anz. Ix. p. 418 t Rubens and Hollnagel, Joc. eit. 252 Profs. H. Rubens and R. W. Wood on reduced the deflexion to only 20 mm. To determine the wave-length of the isolated radiation and the energy distri- bution, we employed the same quartz interferometer which had been used in the investigation of the residuai rays from potassium, iodide and bromide, a description of which will be found in the paper already referred to. The plates of the instrument J were placed close to the diaphragm F, and the distance between them varied by turning the screw ot the interferometer. The micro-radiometer showed periodic increase and decrease of energy as the thickness of the air- film between the plates was increased. If we draw a curve representing the observed deflexions as a function of the corresponding thicknesses of the air-film, we can compute the mean wave-length of the radiation from the position of the maxima and minima, and from the “damping” obtain evidence of the spectrum range with which we are dealing. It was found expedient to use quartz plates of much greater thickness than those used in the former investigation. Having better surfaces and being less easily bent, the opposed faces could be brought into much closer proximity than those previously used. The following investigations were made with pairs of plane parallel plates of 2 mm. and 7-3 mm. respectively. It was found that the readings of the graduated wheel of the interferometer did not give very reliable indications of the thickness of the air-film, especially when the plates were in close proximity. The distance between the plates was accordingly determined in every case, by observing the interference fringes formed by reflecting the light of a sodium flame normally, from the quartz plates. At the beginning of each series of observations, the quartz plates were brought as nearly as possible into contact, by observing the black spot in the interference fringes obtained with white light. The black spot only appeared when the plates were pressed together with the fingers, the distance between them at the beginning of the series being determined by counting the number of fringes which passed a given point as the pressure was relieved. This usually amounted to 7 or 8 fringes only, or a very small fraction of the wave- length under investigation. The deflexion of the micro- radiometer was now taken, and the plates slowly separated by turning the screw, until 20 more fringes had crossed the mark. This corresponded to an increase in thickness of 10Ap=5°89u. The energy was again measured, and the plates again separated by ten sodium wave-lengtls, as long as the maxima and minima could be followed. Focal Isolation of Long Heat- Waves. 253 The results of two such series of observations selected from eight series, all of which were in good agreement, are repro- duced in figs. 2 and 3. Curve 2 was obtained with the J iii Nic eOD Reo, thin quartz plates (2 mm.), and curve 3 with the thick ones (7:3 mm.). The ordinates are the micro-radiometer deflexions in millimetres, and the abscissz are the thicknesses of the air-fiim, in sodium wave-leneths. 254 Profs. H. Rubens and R. W. Wood on Curve 2 shows aminimum at 46), a maximum at 85d, and a second minimum at 122°52X,. Curve 3 has its first minimum at 47°5 Ap, a maximum at 902), and a second minimum at 1282). If we calculate the mean wave- length of the radiation complex, in both cases from the first minimum, we obtain for the experimental series of fig: 2, 4 = 40 x 0589 x 4 » = 1082 ». Pomeieaaie Ay’ = 47:5 X 3°589xX4=111°8 pw. The values calculated from the positions of the first maxima in each case are >A2=100m and AJ=106 4, and from the second minima, ha Joo and), — 100: We thus see that we obtain slightly different values for the mean wave-length according to which maximum or minimum we use in the calculation. If we consider the incidence angle as 6° instead of normal (owing to convyer- gence), these values are to be reduced by about 0-6. In both cases the calculated wave-length decreases with in- creasing ‘‘ order” of the maxima and minima. The expla- nation of this lies in the unsymmetrical character of the energy curve. The first minimum gives us the value of the wave- length corresponding to the centre of gravity of the energy curve, while the following maxima and minima approach more and more near to the wave-length of the maximum of the energy curve. Theenergy curves in the two cases differ, not only in the position of the wave-length of the centre of gravity, but also in the position of the maximum and the degree of asymmetry. It was foreseen that the asymmetry of the energy curve would be greater for the thin plates than for the thick ones, for in the latter case the radiation was obliged to pass through an additional thickness of 10°6 mm. of quartz, and the short waves would be more strongly absorbed than the long ones, for which quartz is very trans- parent. The rising slope of the energy curve will be displaced towards the longer wave-lengths and made less steep by inereasing the thickness of the quartz, while the descending slope will be influenced but little. It is possible to get an idea of the approximate form of the energy curve of the radiation isolated by the quartz lenses, by a trial and error method. We know the position of the top of the curve approximately, and the terminus on the short wave-length side (determined by the quartz absorp- tion), and we can draw the curve on the long wave-length side, by assuming that our source of radiation is non-selective, and that the intensity decreases with the fourth power of the wave-length. This curve we may now divide in Focal Isolation of Long Heat-Waves. | 259 elementary vertical strips, each one of which represents nearly homogeneous radiation. We now draw the interference curves (sine curves) of the various strips, and the super- position of all these curves should give us a curve identical with the curve obtained with the interferometer, if our energy curve hvs been correctly figured. ire a and 6 (fig. 4) seamed in this way gave curves es 4, ae ae nen ue C ns ae raha ia ag ai Milas S0e Wow 120 0 UO 10 200 resembling very closely the interference curves of figs. 2 and 3, and may be considered as representing the approximate distribution of energ ey in our radiation. It is thus apparent that we have experimental evidence of the presence of heat- waves certainly 150 w and probably 200 w in length. Absorption and Reflexion of the Isolated Radiation. Notwithstanding the wide spectral range of our isolated radiation, it come of interest to determine the reflecting and absorbing power of a variety of substances for these rays, which belong to a region of the spectrum about which we know nothing, and heave up to the present time, been unable to investigate with homogeneous rays. We com- menced with the investigation of transparent solids, placing 256 Profs. H. Rubens and R. W. Wood on the flat plates, together with a 4 mm. quartz plate, in front of the diaphragm F. The introduction of the quartz plate was necessary to secure for our radiation the same energy curve as that determined by the interferometer. The follow- ing table gives our results not corrected for surface reflexion. TABLE I. Material | Dhickness @, |, Pereentage aia | peepee te | transmitted D. | mm. per cent. PLE Sy eo Ei he ae Marae ew pie. SF Ft 3°03 | O70 1 tag TS SPR pe 0-055 16:6 Efard rubber st.) 4c).55..3e | 0:40 39-0 RTA m(RyRar)3(1 + e2xXn) chon tor) (110) By the use of ne Mehler Dirichlet integral tor the zonal harmonic, if g(@) is an operation defined by 2isin?é d odd 4 Oe hate aE of where Up, U,, ... are given in slightly different notation by (34) and e=1/2z corresponding to n=0. As was indicated previously, the harmonic term of zero order in y is zero, so that n=0 may be taken as starting point instead of n=1. Writing Ut tina. . . 2.3 Then é S=z ( daUe™, 3... (ea and it is to be assumed that v'=0 at one, and only one point in the range, namely, v=vy corresponding to «=2). More- over, U is not oscillatory. In view of definiteness, vp’ will be given a positive, and vw a negative sign. ‘This has already been seen to be the case in the series most important in the Electric Waves round a Large Sphere. 283 recion of brightness, the first approximation being sufficient for a proof. With this convention, vp is a minimum value of v, and since vp is negative, v—vp must always be positive, there being no other stationary points of v. Let a new variable w be defined by Sa(O- Unsere oma ark a Cl) Then w must always be real. The ee of the ambiguity in @ is to be identical with that of «—a ), which is of necessity a factor of w*. Let w= —6 ee 10) 2. Then, since c'dx=2ada, Sane | TEE OMEN NAP (9) where V=Uw/v’ expressedasa function of w. This function will not be oscillatory. In the physical problem, of course, the value of 6 may be found on substitution of the proper values of d, and gy in v. It is sufficient for our purpose to observe that it is not small. V is finite when w=0, corresponding to c=aq. For U is finite, and the limiting value of @/v’ is that of {hu9" (vu—ay)?}8/(a—agivg'' or (2u9') 74, which is also finite, since v)’ is not zero. This limit is of course also real, for v9” is positive. But the selection of infinity as the upper limit of the integration with respect to w calls for further renrarks when the results are applied afterwards to the actual case. For when w is infinite, dx and dy both vanish, so that v—v, becomes —@x%—v), which is negatively infinite. It follows that another point has occurred, beyond x=, where v’ is zero, so that the present assumption of only one zero point in the range will not strictly correspond to the case for which its application is intended. But it was shown in an earlier section that no second point of this kind, in addition to wp, can occur when m is such that @z and ¢,, are both of order 2, or when both are of lower order than z, whether in the range of values for which they are approximately Ja or beyond. It must therefore occur when the smaller only is of lower order than z, and not the larger. In other words, it oceurs for a value of m such that kr—m is positive e and of order kr, but at the same time either (1) z—m or m—z is only of order a or less, or (2) m—z is positive and of order higher than 2s. Now let attention be restricted to points of space not close U2 284 Dr. J. W. Nicholson on the Bending of to the sphere, or to the region of transition previously defined. Since ¢, is of low order in comparison with ¢n,, the second zero point must effectively satisfy —Odbrr/On— 0= 0, or as (40) is here applicable, 6 TN Hg a a eeie —_ —~@= Sy ao des, so that m=2e=kax=kr cos 8, and mene (“28 _1), . ss ) SG a and this only tends to be of lower order than S= 2) dxUe'??, € where etal Us Wares 14 ee + eo 8 ay and by (34), U,= uv, U,= uly + yeu! tro, U, = gully + he (vu! +40"u)y3—fgur'? apy, . (183) where dei das met w' so that ves! — ww heat, = 5 e ° e . ° ° ry e (134) the accent denoting differentiation with respect to w. Let us suppose that the significant zero points can be reduced to one. Then the sum is given toa second approximation Electric Waves round a Large Sphere. 289 by (131) and (132) if the series dependent on the limit ¢ can be ignored. We have seen that both these conditions are satisfied for the series whose sum is intended, and we therefore applv the formulz in question. The process of calculation of Vy and Vz is as follows :— | In the first place, the functions y at the zero point are merely numerical. Tor @ Be mat Ao 8 —l@m*— ... A and by the definition of y, since a=cv'=0 at the zero point, A) w” Eo sy allege no Accordingly, at the zero point, Al Wi=-—% Wipe lB rere oe @ » (SD)) Secondly, it is sufficient, to the requisite order, to write U U = Uy aF =, and we require, for Vs, the first two derivates of Up at the zero point, denoted by U' and U” in (132). Now Uo) = 0/0 (wo) = uly + eu! Wri, Ug! = why + Qiu’ ol yy + cue! py — wv’? ho, differentiating the functions y according to their rule. The derivates of U, are not needed explicitly. Thus at the zero point, where suffixes denote differentiations of u and v, U,=% U, = — qu, 4+ uur, Uy =u—deurg, Uo = ug—tetyvg — Feuvg— tur. . (136) For the final result, two significant orders must be retained in V, of (132), but only one in Vg, for the function needed 7G is Vo— = Thus in calculating V,, U may be identified with Up, but this may not be done in calculating V). This indicates that derivates of U, are not needed. Finally, in (Vo, V2) we may write, in V,, ] U = u—= Gum — yyeury), and in Vo, =U, U/= w— dewve, ip cf 2 Ul = ug — tuytg— Seuv3— dur,’. 290 Dr. J. W. Nicholson on the Bending of Using these values, after some reduction, we find V ut i (2a re (205) om P= Cetus oy He where ; a EN) us Done Or wee oe cama Daal ome De and the sum of the series becomes De > ji 7 Sa) Coe cael Sg ge V9 Dz all values being taken at the zero point. The analysis of this section and that immediately pre- ceding does not apply to the present problem alone. It is generally applicable to all similar problems in which a second approximation is desired in regions of space for which incident and reflected waves are concerned, provided that an harmonic series may be obtained. Application to the special case. We proceed to apply these results to the special problem in hand. In this problem, in the region of brightness, as in an earlier section, 20 = b—bip— m0, - . . »'. Gag where @ is replaced by @ if the Mehler Dirichlet integral must be used. But when points not close to the axis are concerned, the important harmonics are those of order z, for the effect depends upon the values of the various functions, by the formule just proved, at the zero point, which is given by «=r sin 6/R in the notation of the figure, and this corre- sponds to a value of m of order z. The zonal harmonic may therefore be expanded asymptotically, if sufficient terms are retained. The usual formula of course fails, and we proceed to obtain the necessary modification. Electric Waves round a Large Sphere. 291 When sin @ is not too small, Hobson * has shown that ? he ial EES AL o a(n+$) (2 sin )5 QD ‘apne y/ 2) geal y2 098. (n+3)0 1 ) (14 * 2 on+3 (2 sin @)® ar (140) the asymptotic expansion generally quoted being the first term. We require the first two significant orders in m or n+-4, These may be obtained from the terms cos 9—T) cos (m0 + @—~= Pie 4/ - nin=s Siar? + Soda) 7 a(m) (2 sin 0)? 4m (2sin@) J Now by Stirling’s theorem, 1 a(m) = e~™m™/ 2m {1+ =5- 1D 2m Be aalg 1 1 a(m—4)=e"(m—2)"" 2/ (2m—1)a Faq ee ee Te ak ay and therefore to two significant orders in m7’, sie = (f(A) a(m) \m Zia ee The hyperbolic logarithm of this function may be expanded in the form 1 1 if 41— log m)— —m( 5 + Smt Se oe .) TRU den —¥ log m, and therefore Fa so that the second approximation to the asymptotic value of P,(~) when vn is large becomes 5 1 cos(m8+0—' = ) Ei) =A / ———{1— on oY} cos{ me = t) Li if sm sin 6 which eee be reduced to Ie) = ye eS SEY: 9 {008 (mo—7)4+ — ot 8 in( mb —=7)\ (141) * Phil. Trans. feo A, yA) Dr. J. W. Nicholson on the Bending of to the same order of approximation, and similarly, it may be shown that Ne = See aang} si0( m6 -7) “ 2 col PoE cos mb — t) - For the purpose of the present vesbloat we require the portion of dP,,/d which involves —um@ in the exponent, for this alone leads to a series with a zero point. This portion becomes m di cot 6 ee / oe jerue one fe 9 ie and this may be substituted for dP,,/dw in the summation. Second order values of R,, da, dar, and y, are also re- quired, and to these we proceed. General expressions for R, and ¢, were given in (21-24), and isolating the most significant terms when m is of order z, and equal to za, Re @2e)3, 2 2 0 eg the next term being of relative order z~?.. Thus the value of R, used in the first approximation is sufficient. The second approximation to @, 1s TH = rte} (1-2) + x sin 1 I 1 ON al i) — 3 +1 (-#) e+, 01 — 2") t. Three orders are here retained, for it must be remembered that we are in this case dealing with an Ae or Let SS emer: 30a spr: + (48) and similarly, ¢ being a/r as before, C 1 5) C74? a ae aes 13 Gesell 1)" ae and the second order value of ¢'? ‘”" becomes e (on— Orr + Ba—Bnr). where, in the second expression, the ¢’s denote the old first order values. This may be written, since the (’s are of order 27}, ) (14+18,—tBr eon Fnr), In a series involving the exponential of argument t(dn— ur) we may therefore leave the ¢’s unaltered from their first Electric Waves round a Large Sphere. 293 approximations, and multiply the non-oscillatory tactors by 1+¢8,—(Bnr or c Sas Di Gass = is) ye en?) oe : 3(1 am ee. Finally, y, is required. In the first approximation in the region of brightness, we wrote y,=0. The more accurate value is given by (27) as Xn = 2*/22(1—2)8, so that to the same order, gents 24 1+ 5 mit: La enamel GS) Using finally the asymptotic formula for the zonal harmonic and the values of the various functions derived in this section, the series to be summed may be regarded, to the necessary order, as identical with S ea = ue”, where BU = Ort OUT) iy) a ES with the same value and derivates as in the first approxi- mation, and where by (110), and (143) et seg. wzx sin? @ — ETO (Gres, (a -r tBn—tBny) Ol + e?'Xn) ZX $ 3¢cot O\ _' - (a sin® ») (1- Six y: ‘i oxve UN —— (=x? q amr! - ae ° ° 2 ° ° ° (150) provided that oe Sima. pet c= — ———— .e4, ar y? 3 cot é { C Xx Se) NT ee Ea le Ge 2(1—a?)3 Sa i 8 U(1l—a?)! (l—e?a?)s 5 Avid 5 a Ue ee eeanrh + 29 4 Dr. J. W. Nicholson on the Bending of The derivates of w are only wanted to the first order, and become Reks ox? 1 1 ‘: eee ge ee TL = art 12 Py {3+ 22? (1 +c?) — 18¢?x* + 60728 (1+?) — 2°}. If in each of these we write «=r sin 6/R, the value of YP is given by (138) on reduction. But the general result is LOO > complicated to be o£ use, and we shall confine attention to a special case, namely, that in which v/a is large, so that we are investigating the scattering of the waves at a great distance eran bs " sphere. In he case, we may write c=U, w=sin @, and deduce = sec 0, v, = a(1—2?)—? = sec? @ tan 0. vy = (14 2u?)(1—2?)—? = (142 sin? 9) sec® 6. V3[0." = tan @. 1%4/Uy” — 2 3"/v.° = 434s? 8) sec? 0. . . ea) ujfu = 4(1+2 eos’ 0) sec? 8 cosec 0. ut = 4(38+ 2 sin? @) sect 6 cosec? 0. Write C=cos 9, S=sin@ for brevity. Then by (137) M1 U v in | Wtine Ui) Nea 9—158?+ 1184 ; — {2zC8S8? * e e ° e e e (353) Moreover, by (151), Soo 1 a Se M= 5G BB? 80 AG 9 4-218? + 584 : = 948208 Ba 8 phe shoe) coeeeae (154) and the second approximation to wu 1s 2h 2, oT tr w= —( ‘sintoet {14% aq cos 6 . z Or Electric Waves round a Large Sphere. 29 The sum is by (138), Qarz\2 ur yp = S=(=) {a+ ht GAT, 2 = —aksine ean fe Ye, ah eo) where 6=At+ Lulu = (3 +8 sin? @)/12 Costs ihe Ih: (156) Thus, superposed on the effect of the oscillator in the presence of a plane reflector, there isan additional vibration of relative order z~! for which the magnetic force Is 1 a 8 26 ea kR —na y , 1 3 + 1 oF Vie e e e Y= 6° 8° O sin 8 (3 +8 sin?@) ae (157) when points in the region of transition and points close to the axis are excluded. This corresponds to an oscillator which, when undisturbed, gives a magnetic force ; WON aL Op kR F, The resulting amplitude at any point in the region is only altered to an order z~? by taking account of this vibration. For we may write but this approximation does not determine the amplitude, for terms of relative order z~? have already been neglected. A determination of the second order terms could, however, readily be made. ‘he phase ata distant point is changed from £R to kR—6/z. When the first approximation was determined, it was stated that in a case in which ka=10°, an error of relative order 107 only was involved in the assump- tion that the region of brightness is determined by the plane reflector effect. This statement is now justified. The corresponding problem for points near the axis requires a different treament, and will be investigated later. It will be shown that, as in the first approximation, the type of solution does not change near the axis. L 2 oo) C25 [eRe 7. XXXV. Note on the Derivation from the Principle of Rela- tivity of the Fifth Fundamental Equation of the Mazxwell- Lorentz Theory. By Ricwarp C. Toumay, Ph.D., In- structor in Physical Chemistry at the University of Michigan*. F we consider two systems of ‘‘ space time coordinates ”’ S and 8’ in relative motion in the X direction with the velocity v, any kinematic phenomenon which occurs may be described in terms of the variables #, y, z and t belonging to the system § or 2’, y’, z' and ¢’ belonging to the system S’. The Hinstein theory of relativity has led to the following equations for transforming the description of a kinematic phenomenon from one set of coordinates to the other f. ; 1 45 : : ore a) . 23 i. ee 1 ee ge ag er Gy gs ents Moe et ey BO a Re ey pal er On eres (where c is the velocity of light and @ is substituted for the fraction =} The content of these equations may be expressed in words, by saying that an observer in the moving system S! (S having been arbitrarily taken as at rest) uses a metre stick which, although the same length as a stationary meire stick when held perpendicular to the line of relative motion of the two systems, is shortened in the ratio of “1—,?: 1 when held parallel to OX, that clocks in the moving system beat off seconds which are longer than those of stationary clocks in the ratio 1: /1—?, and that a clock in the moving system which is a! units to the rear of the one at the centre of f ) v coordinates is set ahead by a’ 2 seconds, although the two clocks appear synchronous to the moving ‘observer. A simple non-analytical derivation of these relations has been given in another place {. Let us now take the Maxwell-Hertz equations for the * Communicated by the Author. + Einstein, Ann. d. Physik, xvii. p. 891 (1995); Jahrbuch der Radio-= aktivitat, iv. p. 411 (1907). { Lewis and Tolman, Proc. Amer. Acad. xliy. p. 711 (1909) ; Phil. Mag. xviil. p. 510 (1909). Fifth Fundamental Equation of Maxwell-Lorentz Theory. 297 electromagnetic field 1 OE curl H=47pu+ 2 Cn Rane ae (5) 1 0H curl E= here C Ot » » . > . ° (6) Gner ATEN TM pW TeiNk Sh | say 60 COD diy. H=@; Gul SACs Que a (8) If these equations are trua descriptions of electromagnetic phenomena, it is evident then by the first postulate of rela- tivity that similar equations 1 OF’ Fa ea (9) are ih) ay 9) ciety ioral teem el. deri yes td CLT) diye Hy Oye a ren ennay en tone a Gta) must hold when the phenomena are described by an observer moving with the system SN’. It has been shown by Hinstein that the following equations, together with the kinematic relations (1-4) already given are necessary and sufficient for transforming equations (5, 6, 7 and 8) into (9, 10, 11 and 12) *. Sate, ea memes pee ei el ® (13) 1 Ba ( By | He). BEN aan tS) curl H’/=47p'w’ + eurl E’ = — TERN lee lt sn ‘ | no aoe Bae Th) PANN Fi bl sles ° ° ° ° e ° e : . ° (16) * For the purposes of this transformation, it is necessary to use not only the simple kinematic relations (1-4), but also the following relations which can be directly derived from them :— Us—-V —— Ie >2 Uz! = = el PO ae ey Ne: Ut — t2/ 1-8 ee ew ! wxv ? aoe uxv ” c Tamas. eS ( xr pola’ OHy) abel (IT e) p = Ou" Oy’ Be’ = (6) Phil. Mag. 8. 6. Vol. 21. No. 123. March 1911. x 298 Dr. R. C. Tolman on the Derivation from the Principle By =p (Bt SE-) . ae) an H/= p(B. 25,) . Thus at a given point in space, we may distinguish between the electric vector E as measured by a stationary observer and the vector E’ as measured in units of his own system by an observer who is moving past the stationary system with the velocity v in the X direction. If eH is the force acting on a small stationary test charge of magnitude e, then cE! will be the force acting on the same test charge or electron when it is moving through the point in question with the velocity v, the force eH’ being measured in units of the moving system”. We are more particularly interested, however, in the vector F which determines the force eF that acts on the moving charge but which is measured in “stationary units,” thus determining the equations of motion of the test charge e with respect to stationary coordinates. Since, however, it is possible to obtain relations between the units of force used by stationary and moving observers, a method is presented of calculating F from the values of E’ already given by the transformation equations (13-18). As a matter of fact the expression for F which can thus be obtained is identical with the fifth fundamental equation of the Maxwell-Lorentz theory. Relation between the Units of Force used in Moving and Stationary Systems. Consider a body having the mass mp when at rest and moving with the same velocity v asa system of coordinates 8’. Evidently its acceleration with respect to those coordinates is determined by Newton’s laws of motion, and its acceleration with respect to stationary coordinates can be found by making the proper substitutions, giving us u i fli == Mgr — oT a3 ° ° : 4 (1 2) ! u F/=mou = eae?) > 2, an as esr cp ; 2. (l= 6") * It should be noticed that according to the first postulate of rela- tivity, if the charge of a stationary electron, for example a hydrogen ion, is e, then when the electron is in motion it must still appear to have the charge ¢ to an observer who is moving along with it, otherwise the possibility weuld be presented of distinguishing between relative and absolute motion. This justifies us in taking eK’ as the force acting on the moving electron and measured in the moving system. of Relativity of the Fifth Fundamental Equation. 299 The substitutions es yO Be) are an obvious consequence of the relations between the units of length and time used in the two systems. For example, if a body has an acceleration in the Y direction, of magnitude 1, when measured in the system S, evidently its acceleration w,' as measured in the system S! will be greater because the units of time used in that system are *“‘ lengthened” in the ratio 1: ,/1—%. Remembering that the units of length in the Y direction are the same in both systems, and noticing the time enters to the second power in the expression. Cw} Ur uw. = —— ee ios Se Gea) and 2,’ for acceleration, the relation w,/= a—A) is evident. The other relations may be obtained in a similar way. If now we define force as the increase in momentum per second we shall have, as has already been pointed out by Lewis*, where a possible change in mass as well as a change in velocity is allowed for. It has, moreover, been shown by Professor Lewis and the writert, that the two postulates of relativity, themselves, combined simply with the principle of the conservation of momentum are sufficient for a proof that the mass of a body is increased when set in motion in the ratio 1: /1—?, so that in general the mass of moving mM body m= i “ Be Substituting in the equation above, we have ise Mo du 4 d Mo = ————— U =e ye dt dt a2 4 * Lewis, Phil. Mag. xvi. p. 705 (1908). + Lewis and Tolman, Joe. cit. X 2 300 Fifth Fundamental Equation of Maxwell-Lorentz Theory. or in the case where uwz=v, ee oe Mg cae io 1 Wy 2 ea] Sas a "CdSe = Bye 1—~= 1— G ~ =p qe ity UN OSes and by the further substitution of equations (19-20-21) we obtain Beg eke) Cn en ee Vi rele a 1 re which are the desired relations connecting measurements of force in the two systems. The Fifth Equation. Returning now to the consideration of an electron which is moving with the same velocity as the system S’, we see that the transformation equations (13-14-15) together with the above equations lead to the relation ny Fy Vins me Vie By =(Ey—" H:) F.= V1-g°P/= vVI-#E, =(E.+° H,), C which is the desired equation : P=E+ "vxH. This result agrees with that obtained by Einstein in his second treatment (Jahrbuchder Radioaktivitat, iv. p.411,1907), where instead of defining force as equal to mass times acce- leration, he defined it by the a, _ A _ Mtlz tee eo _ Ab Motz _ mas ‘/1— BY ve 5 rip ieee V1—f? ay at its “Le which agree with our definition of force as equal to the rate of increase of momentum. Measurement of the Refractive Inder of Liquids. 301 The special purpose of this note is to make clear that the fifth fundamental equation of electromagnetic theory may be derived from the four field equations and the principle of relativity without making any arbitrary convention .as to the mass of a moving body. Itis quite unnecessary to place the e m transverse mass of a moving body equal to ———* and the 1 longitudinal mass equal cet The simple relation for the : van : : mass of a moving body m= °__, which was derived ye directly from the principle of relativity by Lewis and Tolman (Joc. cet.) and from ideas of light pressure by Lewis (loc. cit.) is sufficient. The fact that the fifth equation can be derived by com- bining the principle of relativity with the four field equations is one of the chief pieces of evidence which support the theory of relativity. University of Michigan, Ann Arbor, Mich., November 11, 1910. ki i XXXVI. A Note on the Measurement of the Refractive Index of Liquids. By O. W. Grirrita, B.Sc, A.R.C.S.* OR some years past the author has been setting his students, as a laboratory exercise, to determine the refractive index of water by using an ordinary spherical flask filled with water as a convergent lens. The results have always been strikingly concordant, the error in the values obtained by different observers being small and fairly constant. It was therefore thought that an inquiry into the best conditions for the experiment might prove interesting, and this paper contains the results of such an investigation and indicates two very simple methods of determining the index of refraction of liquids. It will be seen that these methods are capable of giving accurate and reliable values. As a rule the problem of refraction through a sphere either receives very meagre treatment in the ordinary text-book or is relegated to the collection of mathematical exercises at the end of the book. It is, however, usually demonstrated that the principal points of a sphere are coincident with its centre. So that if U and V are the reciprocals of the distances of conjugate foci measured from the centre of a transparent * Communicated by the Author. 302 Mr. O. W. Griffith on the sphere, and R is the curvature of the sphere whose refractive index is w, then (with the usual modern convention as to signs) le ese) ams be We may call the right-hand side of this equation the con- verging power of the sphere. If then F is the power, Pao ee be Hence, since F and R can be easily measured, mw for the sphere may be as easily calculated. In the case of a spherical flask filled with a liquid, there are two disturbing factors which vitiate the values of pw obtained from the above equation directly. They are (a) the thickness of the glass, and (b) the spherical aberration. It will be an advantage to consider these two sources of error separately. (a) Liffect of the thickness of the glass. Consider the refraction of a narrow axial pencil through a sphere of index yp enclosed in a concentric spherical shell of index w’. et the pencil diverge from a point in air the reciprocal of whose distance from the centre of the sphere is U. let V,, V2, V3, V be the corresponding quantities for the successive conjugate points after refraction at the several surfaces; and let R, and R, be the external and internal curvature of the shell respectively. Then we have the following equations for the refraction at the different surfaces, wU+V, = Ry(u'—1) BV, +e Ve = Ro(p—p') BV, +eV3 = Ro(u—p’) Vetw’V = Rip’ —1) / fe If F is the power of the system and K=R,—R,, ify nas ll 4 whence U+V=2(RA5 Re = Seas fen Poon Seeger a fe In the case of a thin spherical flask containing a liquid, equation (2) shows that the effect of the glass is to decrease sete Measurement of the Refractive Index of Liquids. 303 the power of the system, acting as a thin divergent lens if ash te placed at the centre of the sphere and of power OK Sone), Putting du for the error in the value of w introduced by an error 6F in measuring fF’, we find Spon. je or Sm ies : ; A A . : (3) But the numerical value of 6F due to the thickness of the glass is given by U] Scan 2 be Hence the error introduced by the glass envelope is °K p’—1 Sip ee sens | Pashia Id ky C iD Let ¢ =the thickness of the glass shell, and 7, the external radius, then Kn =oRe DAT BTR Se ee A I Ce a) — ° e ry fa To calculate the magnitude of the error, assume the following values: whence “03 Ty 3 2 p= a b= a {= 05 emi) then) Of and if t be taken =°04 em., “02 oft = me So that in the case of a flask 20 cms. in diameter, and of thickness ‘05 cm., the error would be -003, and this is of the order observed when using a flask of the size mentioned. Hven were this factor of no account, it appears from equation (2) that an additional small error is introduced owing to the difficulty of measuring accurately the internal radius of the flask. It would be a great advantage if it were possible to express the value of F in terms of the external radius. By slight re-arrangement of the terms in the right- hand side of equation (1) we get F=2R, 4! 9K (~ 4) Neca 4 aay fe Bp 304 Mr. O. W. Griffith on the which states that the system is equivalent to a water sphere of radius 7, and a divergent Jens of power equal to the second term. The value of this form of result will be evident when we have computed the effect of spherical aberration. | (b) Lect of Spherical Aberration. Fig. 1 illustrates the refraction of a parallel beam through a sphere. Fig. 1. The focal length, measured from the centre, is AOR sin? sin @ OL Also €=2(i:—1') and sinit=ywsini’. Substituting for 2’ and expanding in terms uf sinz by the Binomial Theorem, neglecting all terms above the second power, we get e —a bose Zeal e a e e sin @=2F 14 ee - Ze . sin? i] sin 2. p 2h Putting PM=a, OP=r, then if fis the focal length, , pr a 3y—p?—1 ap a be aus =i) Lee Ee 6 e s 6 Ae fea eee eel) (6) The second term on the right-hand side of equation (6) represents the error due to spherical aberration, and its effect is to increase the converging power of the sphere. Applying this to the case of a thin glass flask filled with water, we may Measurement of the Refractive Index of Liquids. 305 assume that the greater part of the aberration is produced by the water sphere. Assigning the value 4/3 to w in the aberration term, we see that the error introduced in the value : hia? of f is numerically Fale Now, reverting to equation (5) we see that the value of f, when the thickness of the glass is taken into account and tbe aberration is negligible, is given by ° Ta or, since K= Uae tr (po — 2 f= A + ee ) G1) 9 2S e Substituting w~=4/3 and w’=3/2 in the second term on the right-hand side of equation (7), we find that the numerical value of that term is Hence the thickness of the glass tends to increase the focal length by an amount Hence it is possible by adjustment of the diameter of the aperture of the entrant beam to arrange that these errors should just compensate. This will be the case when AG eee hablar: Dihiga, odie. Gr E28 . : aie = =\on i —4t, approximately. . . (8) La a8} ‘ 306 Mr. O. W. Griffith on the The following table gives the diameter of apertures that might be used with spheres of different radii. Diameter of aperture in em, Radius of ae las sphere in cm. t='04 cm. | ¢='05 em. | 1 “80 | -90 | “ 1 23 2 1:39 1:56 - 160 0 1-80 Small glass flasks of 6 cm. diameter and _ thickness *04—05 em., painted over with dull black varnish except an aperture of about 1°5 cm. in diameter, seem to answer the purpose very well, and good values of the index of refraction of a liquid can be obtained by their use. It is essential that the exposed parts of the flask should be spherical, and this may be tested by measuring the radius of curvature at the aperture with a spherometer and comparing it with the distance between the two diametrically opposite apertures, as measured with a vernier calipers. The parallel beam is obtained by means of a collimator, and the position of the focus determined by the optical bench eyepiece, or by means of a travelling microscope, the source of light being the slit of the collimator illuminated with radium light. By using a short length of platinum wire heated with an electric current as a source and a thermopile or bolometer as detector, the method can be applied to obtain a rough estimate of the refractive index for heat radiation of quartz in the form of a sphere, or of a liquid (such as a solution of iodine in carbon bisulphide), enclosed in a fused silica flask. It is beyond the scope of this note to discuss the general question of spherical aberration, but the accompanying diagrams are interesting in that they compare the aberrations produced in a sphere and in the usual types of convergent lenses. The effect of the aberration may be represented in two ways. If F is the power of the system, f its focal length, R the curvature of a surface and a the radius of the aperture, and mw the index of refraction, then the error produced by Measurement of the Refractive Index of Liquids. 307 the spherical aberration of a parallel beam may be written jis JEG) (7) A ea TO 2) Di ING MAGE) tN al ie ah ge LO) where ¢(u) and (uz) are certain functions depending on the type of lens employed. The curves are drawn for (1) a sphere, (2) a plano-convex lens with light incident on curved face, (3) a plano-convex lens with light incident on plane face, (4) double convex lens with faces of equal curvature. Fig. 2 represents the graph of equation (9) for values of w between cn i o(4) 1 and 2, and R*a? being unity in all cases. Fig. 3 gives the graph of equation (10) for values of pu between 1 and 2, the value of Ra* being unity in all cases. It is noticeable that the errors in the case of the sphere are considerably less than they are for any of the other lenses. 308 Aleasurement of the Refractive Index of Liquids. Fig. 3 as 7LANG~CONVEX “ANE FACE! TOWARDS LICHT Second method of determining pw for a liquid. Arrange two plano-convex lenses so as to form a telescopic system of minimum aberration, that is so that a parallel incident beam may emerge as a parallel beam. let the outer faces of the lenses be of equal curvature R, and let t¢ be the thickness of each lens, and p’ the refractive index of its material. Then the equations representing the successive refraction of a parallel beam through the two surfaces of the first lens are (B= —R(p’—1), eVo=p'V,/(14+ Vi2), V, and V, having the usual meanings, and yw being the refractive index of the medium adjoining the plane surface of the lens. This gives Hence, if d = distance between the plane-faces of the lenses, 2 Eg) Ra ~ o> Ril’ =1) Destruction of Fluorescence of Iodine and Bromine Vapour. 309 Let d=d' when p=1. Then evidently a= pd. For glass lenses of 10 cm. focus, d=20 cm., d’=27, when «=4/3. Therefore it is quite possible to measure both of these lengths to a high degree of accuracy. In the apparatus used, the two lenses are fitted on the ends of two tubes sliding one within the other through a stuffing-box. The apparatus is mounted on the table of a spectrometer whose telescope and collimator have been previously focussed for parallel light. The distance between the lenses is ad- justed so that a clear image of the collimator slit is observed in the telescope with the tube (a) full of air, (5) full of water or other liquid. Index marks are placed on the tubes and the distance between these may be measured with vernier calipers, or by means of a travelling microscope if great accuracy is desired. XXXVII. The Destruction of the Fluorescence of Iodine and Bromine Vapour by other Gases. By R. W. Woon*. [ Plate IIT. ] N extended ‘study of the fluorescence of sodium, potassium, mercury and iodine vapour has shown that the intensity of the emitted light is greatly reduced if air, or some other chemically inert gas, is present. A quan- titative study of the phenomenon, showing the relation between the intensity of the fluorescence and the pressure and molecular weight of the foreign gas, is much to be desired as a means of testing any hypothesis which may be made regarding the action of the gas upon the radiating molecules. The vapour of iodine is especially suited to the work, since its fluorescence can be observed at room tempe- rature in glass bulbs, and the conditions of pressure, density, &e. can be accurately determined, which is nearly or quite impossible with sodium vapour. A satisfactory theory of the phenomenon should not only explain the destruction of the fluorescence by the inert gas, but also the failure of bromine to show any trace of fluor- escence when under the same conditions as iodine vapour. Its absorption spectrum is very similar, and yet it usually remains quite dark even under the most powerful excitation. Some years ago I suggested the hypothesis that the molecule might be capable of storing up a certain amount of f * Communicated by the Author. 310 Prof. R. W. Wood on Destruction of the Fluorescence energy without the emission of light, but that a saturation point must be reached eventually, after which there will be an emission of radiation. If we assume that on collision with another molecule the energy absorbed by the molecule is transformed into heat, the internal energy dropping back to its original value, and the molecular velocity increasing in proportion, it is clear that if the mean free path is traversed before the saturation point is reached, there will be no fluor- escence. On this hypothesis we should explain the failure of bromine to fluoresce by ascribing to the bromine molecule a greater capacity for storing energy. In other words, the path cannot be increased sufficiently to allow the saturation point to be reached before a collision occurs. It seemed possible to test this theory by experiment. By sufficiently increasing the length of free path, we ought to be able to observe fluorescence, provided that a sufficient number of molecules remain to produce a visible illumination. A small amount of bromine vapour was introduced into a bulb, and condensed upon the wail by the application of solid carbon dioxide and ether. The bulb was then exhausted to the highest possible degree and sealed. On warming it to room temperature the bromine vaporized, and though it was so highly rarefied that it showed no colour, no fluorescence could be detected. Sunlight was now concentrated at the centre of the bulb by means of a portrait lens having a ratio of focus to aperture of 2°3. Even in a dark room with careful screening off of diffused light, no fluorescence could be detected. The outside of the bulb was now touched with a piece of solid carbon dioxide, which gradually condensed the bromine upon the wall. In two or three seconds a faint green fluorescence appeared, which vanished almost immediately, owing to the complete removal of the bromine vapour. There appears then to be one density at which bromine shows a visible fluorescence. At higher densities collisions destroy it, at lower, there are too few molecules present. This appears to be in accord with our hypothesis regarding absorption of energy, saturation point, &c. ; but more recent work, made in collaboration with J. Franck, has shown that another factor comes into play. The question will be considered again in the paper immediately following the present one. The method of observing the iodine fluorescence has been: so improved that it is now possible to demonstrate it to the largest audience. A large bulb 15 or 20 cms. in diameter is prepared by drawing down the neck of a round-bottomed flask, which should be most carefully cleaned with aqua regia of Iodine and Bromine Vapour by other Gases. oe and distilled water, avoiding the use of alcohol and ether for drying. A few small crystals of iodine are now introduced into the flask, the neck drawn down to a 1 mm. capillary at one point, and the flask thoroughly exhausted with a Gaede or other mercurial pump. 1t is most important to have a very perfect vacuum (less than ‘01 mm.), and it is usually necessary to keep the pump running for 15 or 20 minutes to secure this, if the capillary is narrow. The flask is now sealed, and can be used for demonstration purposes at any time. It requires no heating, for the iodine fluorescence is brightest at the pressure which the vapour has at room temperature. We have only to hold the bulb in the con- verging beam furnished by the condensers of a large pro- jecting lantern, or the condensed beam from a heliostat. If a lantern is used it is best to throw the beam upwards, as the reflexions give less trouble. ‘The intensely brilliant cone of yellowish-green fluorescent light can be seen from the back of the largest lecture hall. The aberration of the spherical lenses is well shown as well. The bulbs used in the photometric work were smaller, having diameters of about 7 cm. The fluorescence was excited by sunlight reflected into a dark room by a large heliostat, and concentrated to a focus at the centre of the bulb by a Voigtlander portrait objective of 12 cm. aperture and 27 cm. focus (F. 2°3). The work was done only on very clear days when the sky was free from haze and clouds. A Welsbach light was used as a standard, its colour being brought to a match with that of Higa Ll: ‘To Pump nano MANOMETEP \ @ JODINE BULB GRADUATED \ Sis VE CIRCLE _ the iodine fluorescence by a combination of pale cobalt glass and a dilute solution of bichromate of potash. The arrange- ment of the apparatus is shown in fig. 1. Between the eye and the brightest part of the fluorescent cone is mounted a small mirror m, measuring about J. x 3 mm., made by silvering a 312 Prof. R. W. Wood on Destruction of the Fluorescence piece of plate glass and breaking off thin scales by tapping the edge with a hammer, striking the blow in a direction nearly parallel to the silvered surface. The silvered scales obtained in this way usually have one edge of razor sharpness, and this edge forms the vanishing line of the photometer. © The silvered mirror reflects to the eye the light from the comparison source, which is placed in such a position that the glass sliver cuts across the fluorescent cone at its brightest point. By rotating the graduated nicol prism the intensities may be perfectly matched, the sharp edge of the illuminated mirror vanishing. The intensity of the fluorescence is measured by the square of the cosine of the angle through which the nicol has been turned, measured from the position of complete extinction. The bulb containing the iodine was in communication with a Gaede pump, a manometer, and a reservoir of the gas under investigation. The bulb was first highly exhausted and the intensity of the fluorescence measured. Air or some other gas was now introduced until the manometer showed a pressure of 1 mm., and the intensity again measured. The pressure of the gas was increased by progressively small steps, the intensity being measured for each pressure. Plotting the results, with the intensities as ordinates and the pressures as abscissee, gives us a curve showing the rate at which the intensity of the fluorescence decreases with in- creasing gas pressure. Experiments were made with air, hydrogen, carbon dioxide, and ether vapour. Several series _ were made with each gas, and the results were in good agreement. The curves are reproduced in Plate III. together with curves ebtained with other gases, which will be discussed in the following paper. An examination of these curves showed that the hypothesis of free path and saturation point would not represent the facts, and that some other factor must be taken into con- sideration. The effectiveness of the gas in destroying the fluorescence appeared to increase with its molecular weight, but was by no means proportional to it. For example, the intensity of the fluorescence was reduced from 45 to 85 by 3 mm. of ether vapour, 7 mm. CQ,, 11°5 mm. air, and 24 mm. of hydrogen. It seems clear from these results that some other property of the gas than its molecular weight must be operative. In the case of the fluorescence of anthracene, Elston found that the presence of hydrogen and nitrogen was almost with- ee of Iodine and Bromine Vapour by other Gases. als out influence upon the intensity of the emitted light. Oxygen and COs, on the other hand, reduced its intensity very rapidly with increasing pressure. This I have provisionally ascribed to “incipient chemical action” which of course in reality means nothing at all. It was found that no permanent chemical change took place, for on cooling the bulbs the anthracene condensed upon the walls, and none of the oxygen had disappeared. At high temperatures, however, the oxygen acts upon the anthracene, and it seemed possible that the first stage of the process might occur at the lower temperatures used in the fluorescence experiments, the process reversing as soon as the bulbs were cooled. This is what I called “incipient chemical action.” The probable real nature of the action will be given in the subsequent paper. The only difficulty found in measuring the intensity of the fluorescent light resulted from the slight change of colour which occurred when the intensity was considerably reduced. The colour always became slightly reddish at the higher pressures, though it was only conspicuous when seen in the photometer, the colour match not remaining perfect. The gases apparently weaken the green portion of the fluorescent spectrum to a greater extent than the red. This is a very interesting and important matter, which will be discussed in the following paper. | Experiments were also made with a highly exhausted bulb, the vapour pressure of the iodine being varied by immersing a side tube in freezing mixtures of various temperatures, or heating the entire bulb in a water-bath. The values fonnd below room temperature are given in the following table, and are shown in graphical form on Plate III. (small inset) : Temp. Intensity. 19 43 The relation is very 6° 29 nearly linear. (he 22 SO ll IF 6 =) 2 Above room temperature no perceptible increase could be detected. The intensity remained about the same up to 30°, alter which it gradually decreased. At high pressures the fluorescence disappeared entirely It is probable that increased absorption compensates very nearly for any increase that may occur above room tempe- rature. Accurate measurements were impossible on account of the change of colour due to the absorption of the iodine vapour between the fluorescent cone and the eye. Phil. Mag. 8: 6. Vol. 21. No. 123. March 1911. ¥ en AXXVIIT. The Influence upon the Fluorescence of Lodine and Mercury of Gases with different fies for Electrons. By J. FRANCK and RK. W. Woop * [Plate III. ] ARBURG f{ has shown that in nitrogen, helium, argon, and hydrogen, which have been very carefully freed from all traces of oxygen, the current obtained with the negative point discharge is much greater than when traces of oxygen are present. To explain this circumstance he made the hypothesis that in the pure gases the negatively charged electrons move with a higher velocity. Small traces of oxygen, by condensation on the electrons, increase their mass and reduce their velocity. Many other phenomena of the discharge ot electricity through gases are influenced by traces of oxygen f. Finally J. Franck made direct measurements of the mobility of the electrons in argon, nitrogen, and helium (in the latter case in collaboration with G. Gehloff—unpublished), and showed that small traces of more or less electro-negative gases operated in the manner assumed by Warburg. The affinities for electrons—i. e., the forces acting between neutral molecules and electrons—decrease as we e pr oceed from strongly electro-negative to the inert guses argon, helium, &e. In the latter the forces appear to be nil. The possible existence of a relation between the affinities for electrons and an effect upon the emission of spectrum lines has been suggested in the paper referred to. For the investigation of such a possible effect the fluorescence of iodine and mercury is especially adapted, for the excitation is caused by a single factor only, namely the impact of light-waves, which is uninfluenced by the ad- mixture of fee gases, whereas in the case of eléctrical excitation, potential gradient, current strength, and density are all affected by small traces of other gases. The reduction in the intensity and the final destruction of the fluorescence of iodine vapour by the presence of other gases has been investigated by Wood and described in the pre- ceding paper. Hydrogen showed the least influence, and taken in increasing degree, air, COQ, and ether vapour. This sequence is not opposed to the above hypothesis, for * Communicated by the Authors. + Warburg, Wied. Ann. vol. xl. p. 1 (1896). ~t See Summary by J. Franck, Verh. der Deut. ats: Ges., July 1910. Fluorescence of Iodine and Mercury. 315 the affinity for electrons increases for these gases in the order mentioned. It is not a proof of the theory, however, for the molecular weights increase in the same order. If we accept Lorentz’s hypothesis that a damping results from collisions, we can assume that the damping factor is a function of the molecular weight, and we have no means of knowing whether a second factor, the affinity for electrons, is superposed on it. The results previously obtained appeared to point towards - the existence of some other factor however. In the present paper we shall show that gases which interfere with the motions of the free electrons, by their affinity for them, interfere as well with the motions of the bound electrons, the vibrations of which give rise to the spectral lines. It may here be pointed ont that Pohl and Pringsheim have come to the same conclusion with regard to the natural frequency and damping of the resonance electrons in the selective photo-electric effect. To settle the question in the case of the emission of fluorescent light we must compare a heavy gas of small or zero affinity for electrons with a light gas of strong affinity. We have therefore examined the influence of helium, argon, nitrogen, oxygen, and chlorine upon the fluorescence. The method was identical with that described in the preceding paper, except that the light from the crater of a right-angle arc lamp was used in place of sunlight, which was not available on account of unfavourable weather conditions. An image of the are was projected upon the wall of the room by means of a lens, the position of the carbons yiving maximum illumination ascertained, and their outlines on the wall marked with a pencil. By frequent hand regulation they were kept always in the same relative position. The helium gas, carefully purified, was furnished by the firm of Siemens and Halske, through the courtesy of Dr. Holm. The argon was prepared in the Institute. The nitrogen was obtained from commercial bombs, and was not very pure, and the chlorine and oxygen prepared by heating gold chloride and potassium permanganate respectively. The results are given in the curves (Plate III.), in which the ordinates are the intensities of the iodine fluorescence and abscissee the pressures in mms. of the admixed gases. The values obtained in the previous work with air, hydrogen, CO,, and ether are given on the same plate. Check observations were also made in the present work, with air, to make sure that the results were comparable with the earlier ones. Y2 316 Mr. J. Franck and Prof. R. W. Wood on the One sees at once that helium, in spite of the fact that its molecular weight is double that of hydrogen, is much less detrimental to the fluorescence, and that the argon curve runs nearly in coincidence with that of hydrogen, though its molecular weight is twenty times that of hydrogen. Above the air curve we have a point for nitrogen, and below it one for oxygen, exactly as we should expect, for nitrogen is nearly neutral, while oxygen is strongly electro-negative. The lowest curve of all is that for chlorine, with an atomic weight of 70, much lower than the curve for ether with an atomic weight of 75. It is clear that the affinity of a gas for electrons is a powerful factor in suppressing the fluorescence. For helium, we have given two curves, one for the green portion of the fluorescence spectrum, the other for the red. ‘The measurements were made in the two cases by suitably selected colour filters. This separation was necessary in the case of helium, for it was found that the colour of the fluorescence changed rapidly from green to reddish orange, as more and more helium was added, which made photometric measurements with a standard of fixed colour impossible. As is apparent, the two curves cross, resulting from the circumstance that 7 vacuo the green portion of the fluorescent spectrum is much stronger than the red, but that it is rapidly reduced in intensity by the addition of helium, while the red portion is reduced to a much less degree. A similar effect, though to a much lesser degree, is observed with hydrogen and argon. This change of colour is probably a pure collision effect, for with electro-negative gases, such as chlorine, it is not noticeable. The very teeble fluorescence observed when we have 3 or 4 mms. of chlorine in the bulb has practically the same colour as when the iodine is in vacuo. The cause of the colour change will be given in a subsequent paper dealing with the resonance spectrum of iodine recently discovered by Wood. It seems probable that when an iodine molecule is near enough to a chlorine molecule to be influenced at all, its fluorescence is practically destroyed. The fluorescence observed when chlorine is present probabiy comes from those iodine molecules which at the moment happened to be beyond the sphere of action of any chlorine molecule. Their number will be fewer and fewer as the pressure of the chlorine increases. We have also investigated qualitatively the effects of these gases upon the fluorescence of mercury vapour, which has already been extensively studied by Wood*. Mercury * Phil. Mag. vol. xviii, -p. 240 (Aug. 1909). Fluorescence of Iodine and Mercury. 317 vapour is strongly electro-positive, and its fluorescence should therefore be very sensitive to the presence of an electro- negalive gas. Our results confirm this view, for in oxygen at a pressure of only 3 mms. we obtained no fluorescence at all, though the quartz bulb was heated to a temperature sufficient to give us mereury vapour at 30 cms. pressure. The apparatus is shown in fig. 1. A quartz bulb with a double neck was used, one sealed to a vertical tube with sealing-wax, the other serving for the introduction of the high temperature thermometer, also sealed in with wax. The quartz necks of the flasks were kept cooled with wet cotton, and the portion of the bulb containing the mercury heated with a burner. Be Ie Quartz LENS To PumP ano GAS PESEVOIRS The temperature of the mercury, the coolest part of the system, was measured, and this gave the pressure of the vapour. At the beginning of the experiment the stop-cock A was opened and the mercury allowed to sink below the lateral tube B, by lowering the mercury reservoir C. The stop-cock A was then closed and the bulb thoroughly exhausted. ‘he cock A was then opened, D being closed, and the mercury allowed to rise until the bulb of the thermometer was just covered. The cock A was now closed 318 Fluorescence of Iodine and Mercury. and the fluorescence, caused by condensing the light of a cadmium spark to a focus at the centre of the bulb, observed at different temperatures. Small amounts of any foreign gas were admitted by lowering the mercury below the junction of the tube B, and allowing the gas to enter from a reservoir through the tube B, the pressure being read with a manometer. This pressure was about doubled by raising the mercury to its original level. By raising the mercury to a higher level still greater density of the gas can be obtained, its pressure being easily determined by measuring the difference of level of the mercury surfaces in the flask and in the reservoir B, when the cock A is open. Closing the cock seals off the bulb, and heating then causes no change of level. While oxygen at 3 mms. destroys the fluorescence, helium at a pressure of 1 atmosphere scarcely affects the intensity at all! The fact previously observed by Wood, that the maximum intensity of the fluorescence, in the absence of any foreign gas, occurs at a pressure which is very different according to the nature of the fluorescent gas, can probably be explained by our hypothesis. In order to obtain a visible fluorescence we must have a sufficient number of molecules present. Their number must not, however, be so great as to cause them to disturb each other. In a strongly electro-negative gas the vibration electrons in one molecule are influenced by the presence of neighbouring molecules. In the case of bromine, therefore, which is more strongly electro-negative than iodine, we have fluorescence only at very low pressures, probably less than ‘001 mm., while in the case of iodine the maximum intensity of the fluorescence occurs at a pressure of about -2 mm. (tension at room temperature), and in the case of the strongly electro-positive mercury vapour the maximum intensity is not reached until we have a pressure of several atmospheres. The intensity of the fluorescence of mercury vapour at high pressures should be quantitatively investigated. It will not be easy, for as the pressure increases the fluorescence is confined to a layer of decreasing thickness covering the wall through which the light enters the bulb. Two facts seem to have been established by this investigation: first, that the bound electrons, which emit the fluorescent light, are effected in much the same way by the presence of electro-negative gases as are the free electrons which carry the current in vacuum tubes ; secondly, that the pressure at which ihe maximum intensity of the fluorescence of a gas occurs depends upon the electrical character of the molecule. ee ote XXXIX. The Problem of Uniform Rotation treated on the Principle of Relatvwiy. By H. Donatpson, B.A., B.Sc., Scholar of Sidney Sussex College, Cambridge, and G. Sreav, B.A., late Scholar of Clare College, Cambridge *. ie a previous paper (Phil. Mag. July 1910) we com- menced an investigation of the above problem, dealing only with the changes in the dimensions of a rotating disk. We showed that the contractions demanded by the relativity theory, due to the motion of the disk, are fulfilled if the disk buckle into-a cuplike form, whose section by a vertical plane is an epicycloid of intrinsic equation 2 : S TAB Ein, Fig. I a oO Z a ' ' ' “~ i] ait ‘ In the figure shown we have —t 2 y= sawlei=s(14 22), C where v is the linear velocity of any point on the disk (=yo). We shall find that this cuplike form of the disk gives a most useful method of visualising the processes going on during the rotation on relativity hypotheses. For, if the disk were considered as remaining plane, we should have, owing to the lessening of the circumference and the in- variability of the radius, a change in the value of the ‘constant ” a, whereby we are transformed to a “ real * Communicated by the Authors. | i i 320 Messrs. H. Donaldson and G. Stead on curved space.” This cuplike representation of our disk is a method of representing the contracted disk in ordinary space, and the results obtained by considering it will still hold good, even if the disk really remain plane. Let us now consider an observer on a fixed disk regarding the moving disk. The number of revolutions performed by the moving disk relative to the fixed disk as measured by the observer on the fixed disk will obviously be the same as the number of revolutions of the fixed disk relative to the moving disk as measured in the same period of time by an observer supposed placed on the moving disk. That this will be so follows from the fact that a number is of no dimensions in length, mass, or time. But to perform one revolution a point on the moving disk has to traverse a linear distance which is less than that traversed by the corresponding point of the fixed disk in the ratio (=e)? : 1. Hence, in order that the two numerical measures should be the same, it is necessary that the time unit on the moving system should be greater than that on the fixed system in the ratio (1—v?/c?)"? : 1. This is the result which has previously been deduced by other writers in a variety of ways for systems in linear motion. We shall now proceed to calculate the kinetic energy of our rotating disk, and to show how this leads us to the necessity for a change in the mass of the system due to its rotation. Since a ring element of area 27s.6s on the fixed disk becomes a ring element of area 27y.6s on the moving disk, and since | haa s(1—v?/c”)?, we shall have a! =o (1—v?/c?) =, where oa’ and o are the surface densities of the fixed and moving disks respectively. Now v= yo, the units being those of the fixed system, and therefore, since i 1 202 a2, y = s(1—e}e)t = s(1 +3) ; we have the Problem of Uniform Rotation. 321 The moment of inertia of the disk about its axis of rotation can now be obtained immediately, for, taking our ring element, we have CUNe él = 2ny 85.0 (14+ <5") yp? TON and y=s(14 a) 3 's 2 aes: ae ry = 2a: eatee: =) { tog (1+ 2 1 : When w is small this reduces, as a first approximation, to the ordinary expression for the moment of inertia of a circle about an axis through its centre perpendicular to its plane. The kinetic energy of the disk is given by $l’, that is, by TorG aa ct ra D —sezlog (147) When ao is infinite this becomes log( 1+ a 4 9 od The limit term is seen, in the ordinary way, to be zero. so that the maximum kinetic energy possible for the disk is 4Mc’, Now, when the disk possesses this energy its volume has become zero, as shown in the preceding paper, and therefore we arrive at the seeming absurdity that a jinite expenditure of energy is able to reduce the volume of the disk to zero. This conclusion can be avoided in two ways, or by a combination of the two. The possibilities are, first, an increase in the mass of the moving disk, due to its velocity, and, secondly, an increase in the internal (‘“ potential ’’) 322 Messrs. H. Donaldson and G. Stead on energy of the disk. Let us consider, first of all, the increase of mass. ‘To calculate this increase we have to make some assumption, and we shall adopt as our hypothesis the most general principle of mechanics, namely, the conservation of linear and angular momentum. In our particular case this means that the angular momentum of the moving disk, relative to the fixed disk, as measured by an observer on the fixed disk will have the same numerical value as the angular momentum of the fixed disk relative to the moving disk as measured by an observer on the moving disk. Now, angular momentum is given by terms of the form mr’o, and 7 is measured in the direction perpendicular to the direction of motion of the ring element considered, and is therefore the same on both systems of units. This leaves us with the fact that mm” m Do @ where the letters without suffixes denote units on the moving system and those with the zero suffix denote the units of the same quantities on the fixed system. Now, we have shown that the time unit on the moving system is greater than the time unit on the fixed system in 1 the ratio (1—v?/c?)* : 1, and therefore we have m=my (1—w/e?)~?, which is the formula previously deduced by other writers for uniform motion, and experimentally confirmed for such motion by the researches of Kauffmann and Bucherer on moving corpuscles. If now we assume, as a second fundamental principle, the constancy of Total Energy of our moving system ; that is, that the Total Energy of the con- tracted disk is the same as that of an uncontracted disk moving with the same angular velocity, we can calculate an expression for the change in the internal energy of the disk due to dimensional and mass changes which have occurred in it on the relativity theory. : We have that the kinetic energy of the disk, corresponding to an angular velocity a, is K=41e’, and therefore dK =Ilwdw + 4o7dI. If we consider our curved disk we shall be able to regard w ason the fixed system of units and shall be able to proceed the Problem of Uniform Rotation. 323 directly. Thus, when w becomes w+do, the increase in kinetic energy differs from what it would be if I were constant by 4@°dI, and hence, if we consider that the change in internal energy of the disk is $w’dI, we conserve our total energy. Hence if we write Py as the internal energy of the un- contracted disk moving with angular velocity w, and P as the internal energy of the contracted disk having the same angular velocity, we have A sheath L PP, = to? — do. 0 d@ Now for the value of I for our rotating contracted disk, taking into account the fact that the mass of each element is altered in the ratio 9.9 sa" UE ae lh ¢ we have 2..9\3 DT ON Gs, so" \? s’@ 2 dl=2ry.ds.a(1+ —,-)(1+—-)-¥ ¢ ¢ Lees Koza 2 fey? site vale ‘ 4 72 2 2 2 a4 =2ro[ 5 (5 7 1) = m—2)4 ae 3@'\ ¢? iGeu||s To shorten our calculations, we will obtain P—P,, not from the integral expression for it given above, but from the fact that P—P,=E,)—H, where Hy and E are kinetic energies corresponding to P, and P, since ("on =B and | Iw dw= Ep. v0 0 324 On the Problem of Uniform Rotation. 4 2 yee EK= = (e iia 1) (2, a 2) ae 2 O@ C (Ba 7 from our above expression for I, and, to a first approximation, this becomes lao'r® | me a) f q4 + Rh re mR ED But . Ho=tar'cw’, and therefore i 1 4.6 K)—E= sy703 ih ee Bae Hp. Hence, to a first approximation, the change in internal energy due to the contraction of the disk is given by 1 wr? 3 ¢e =r, = Ko. In conclusion we consider that this investigation shows that the relativity theory involves no contradictions when applied to the case of uniform rotation, and also that the case of uniform rotation gives a very sound and direct method of proceeding from length to time units, depending only on the fact of a number having no dimensions in mass, length, or time. Again, the maximum kinetic energy possible for the disk shows the necessity for some change in mass, though it does not give, of course, any result which is uniquely satisfied by the relativity change of mass-unit. It will be noticed that we have given no proof of the change in the length-unit of the moving system, the reason tor the omission being that the change can be directly deduced on the method given by Messrs. Lewis and Tolman (Phil. Mag. Oct. 1909) for uniform rectilinear motion. Cambridge. October 31, 1910. Eas | XL. Relations between the Density, Temperature, and Pressure of Substances. By R. D. Kureman, D.Sc., B.A., Mackinnon Student of the Royal Society *. HE writert has shown in a previous paper that an infinite number of equations can be found connecting the surface tension or latent heat of evaporation of a liquid with its temperature and density and density of the saturated vapour. These equations correspond to laws of attraction between the molecules, but none of which is necessarily the law that actually exists. Hach of these laws can be obtained by giving a definite value to the arbitrary function contained in the general law of attraction that can be deduced from surface tension or latent heat data. This general law of attraction between two molecules of the same kind is (= Ty) vim)? AN fc? T): PA the molecules and z their distance of separation, x, denotes their distance of separation in the liquid state at the tempe- rature T., =,/m, denotes the sum of the square roots of the , where T denotes the temperature of : : edt atomic weights of the atoms in a molecule, and (= i) ; i a woe denotes an arbitrary function of ;, and —. il We There exists, therefore, an infinite series of equations of mienrornml vr (o7,) 0, lial (py, yy Wa 2. , where L denotes the internal heat of evaporation of a liquid at the temperature T, and p, and p, denote the density of the liquid and that of the saturated vapour respectively. It is obvious that we can obtain from these equations an infinite number of equations containing T, p;, or p, only. Now each of the equations thus obtained must obviously be either an identity or be an equation which has an infinite number of real positive roots lying between certain limits. In practice, however, the equations found in the above way (usually by trial) do not exactly satisfy these conditions. But this is of no consequence, as the equations deduced, containing any two of the quantities pi, ps, T, usually agree very well with the facts. It is the object of this paper to pcint out and discuss some of these equations. Millst has shown that the internal Jatent heat of evaporation 18 very approximately given by the equation L= D(p;?—p3%), where D is a constant depending only on * Communicated by the Author, t Phil. Mag. Jan. 1911, p. 88. { Journ. of Phys. Chem, vol. viii. p. 405 (1904), 326 Dr. R. D. Kleeman on Relations between the the nature of the liquid. The writer* has shown that this equation can be deduced if ¢, e - i is put equal to = ): ja the general law of attraction between molecules. We then obtain D= one m! ?, where m denotes the molecular weight of the liquid, p- the critical density, and 8 a numerical constant. This value of D agrees fairly well with the facts. The latent beat is also given by the equation Tf ] [aan K, Doo og z (2), MM where Ky, is a numerical constant equal to about 1:75; which also corresponds to a particular case of the general law of attraction between molecules. Equating these two different equations for the latent heat we obtain where Dye Oe eee B= KR V/m,)?. Another expression for B which is convenient can be obtained by determining B at the critical temperature from the first of the above two equations. Writing p.=.p, we have Tome Fain B = [ ) Lv | Tali = 3 3 pul —a"*) Ltr=1 iyios and the above equation becomes p ie c If the two different expressions obtained for B are equated we get S 7p.*8 me Mea R(n) vm an equation which has already been discussed in previous papers. Equation (1) has many useful applications, and it has therefore been tested over considerable ranges of temperature for a number of liquids in Table I., which contains the 5 JUD values of a calculated by means of this equation. The * Loe. ctt. t Phil. Mae. Oct, 1910, p. 688. Density, Temperature, and Pressure of Substances. 327 TABLE I. | Ethyl Pentane, Stannic Octane, Benzene, | Heptane, ~ oxide, chloride, SFOs |\s\) Cabs SnCl,. C,H: C,H. CoH ¢. 104 10! 10! 10+ 10+ 10! aE: RB a 3B I ore T. B Se au E 7S 4-96) 213) 4-9 | 3re 496 1273) 3:02 273) 3:76 | 2731 3:35 293) 4-42 |313| 429 | 393) 499 1393) 342 |353)| 3:95 | 353/ 3-60 313 | 4-46 |333| 4:29 |413/ 501 |413| 3:44 | 373) 3:96 | 373) 3-63 333 | 4:48 | 3853) 429 | 483) 5:02 |483) 3:44 | 393) 3:97 | 393} 3°65 353 | 4:49 |3873| 429 | 453) 5:02 |453| 346 |415] 3:96 | 413) 3:66 373 | 4-42 |3893| 428 | 473) 5:02 |473| 346 | 433) 3:95 | 433) 3°67 393 | 4:48 | 413) 427 | 4938) 5:01 (4938) 347 | 458) 3:93 | 453) 3-67 413| 4:46 | 483| 4:27 |513) 4:99 |513) 348 | 473) 3-92 | 473) 3:68 433 | 4:47 | 453) 4:27 | 533| 5:04 |5338| 348 | 493) 3-91 | 493) 3°68 453| 448 | 463] 4:29 |553| 4°99 |552| 352 | 513) 3:90 | 913) 3:68 460| 4:54 | 468] 4:33 533 | 3:91 | 5383) 3°74 466] 4:56 | 469) 4°31 553} 8:93 | 539} 3:78 Hexane, Carbon tetrachloride, |—~~ Todo- Bromo- CoH... CCl,. Di- Ethyl benzene, benzene, isobutyl, acetate, C,H;I. C,HsBr. 973 3°74 273 4-15 gttis- C,H,0,,. : 343 | 3°92 1373) 494 ——_—_—-— 303 | 3°39 | 403) 3°31 | 363] 3°94 |393) 494 | 273) 3:26 | 273] 3:31 473 | 3°63 1533) 3°84 | 383) 3°94 |413|) 4:93 | 373] 3:56 | 363] 4:09 563| 3:80 | 553) 3°85 | 403) 3°94 |433) 491 | 393] 3:60 | 383] 411 583| 3°80 |573| 3°85 | 423| 3°94 |453) 489 | 413) 3-62 | 403) 4:13 603} 3°80 | 593) 3°85 | 443) 394 1473) 487 | 433) 3-62 | 423) 4:14 623| 3°81 |613| 385 | 463) 3°94 |493| 485 | 453) 3:63 | 443] 4:15 643 3°81 |633| 3°85 | 483) 3°95 |513| 483 | 473] 3-62 | 463) 4:15 499| 3:97 |533| 484 1/493] 3:62 |483] 4:16 006) 401 |553|} 4:87 |513| 3:64 | 503] 4:19 Chloro- Carbon 533 | 3°66 | 518)| 4:28 benzene, dioxide, PLO Aas Sen le O22 47 C,H;Cl. CO,. Acetone, | Chloroform, 973 3:34 243 | 8:30 C3H,0. CHC,. Fluor- Hexa- 413| 3-70 | 263) 8:38 : benzene, | methylene, 433| 3°71 | 283) 830 [273] 3:93 [293] 489 | CoHsF. | OH... 453 | 3°71 | 298) 8-44 |293/ 3-98 |313) 494 |_———___. 473 | 3-71 313] 4:02 |333/ 498 1273) 3:89 | 273] 3-70 493 | 3°71 333 | 405 393 4:11 | 363) 3:87 513 | 3°70 | Sulphur 373) 413 | 383) 3°89 533 | 3:69 dioxide! Vis gin sat Ethyl 393 | 414 | 403) 3:89 SO.,,. Ethyl propionate, |413| 415 | 423] 3:88 formate, 5H,.0,. [4383] 415 | 443) 3°87 Methyl 263! 6:12 ©) HON ee 453 ee 463 | 3°85 formate, | 283] 6-18 |——- 383| 3:85 |478| 413 | 485) 364 | O,H,0,. | 303] 6-19 [333] 435 |4u3} 388 | 493] 412/508) 383 | —— | 323626 |353| 438 |433| 390 | 513) 411 | 523) 383 | 303| 4-66 | 343] G21 |393] 440 [453) 391 | 583) 411 / 543) S84 | 3231 472 |363/ 615 |413) 440 [473] 3:92 | 998/415 | 002) 3°85 | 363| 4:73 | 383] 615 | 483) 440 |493] 3°93 Reantes bat : 403 | 472 | 403] 616 |473) 440 |513) 3-04 443} 471 | 418] 619 | 493) 441. |533) 3:98 428) 628 | 507) 447 | 541) 4:00 a 328 D.-. R. D. Kleeman on Relations between the complete data used for these calculations and others in this paper can be obtained from the tables of density and pressure data of Ramsay and Young given in papers by Mills*; part of the data (taken from this source) can be obtained from a previous paper by the writert. It will be seen that the 4 value of B is very approximately independent of the tem- perature for each liquid. Table II. contains the mean values Tasxe II. Name of B pete) : na ; Teas substance. 10% | yoo 2). | —l08e:)| Eee Ethyl oxide ......... 2°24 2°20 “930 931 1507 1496 IPENEATIC oy ctebu-Kieon 2°33 2:29 912 “916 1951 1940 Stannic chloride ...| 2:00 1-96 1-08 1°29 237 233'4 Octiney ews eee: 2°88 2°78 “916 "914 2393 2282 ZEN ZENG ses caccso sank: 2°54 2°51 "950 "939 1357 1316 He piemes: 622 5:05 080%. 2°72 2:63 914 ‘915 2246 2138 Todo-benzene......... 2°64 DY) 1:08 1:22 463-4; 4629 Bromo-benzene...... 2°60 2°56 103 1:23 621:0| 617-7 lel) tg eee 2°53 2°47 1-07 ‘914 2094 2011 Carbon tetrachloride| 2:05 2:03 1:02 1:27 403°7 3882 Di-isobutyl ......... 2-76 2°64 918 SS 2247 2109 Ethyl acetate......... 2°40 2°35 952 938 1239 1266 Chlorobenzene ...... 2°70 2°66 "983 ‘977 1049 1118 Carbon dioxide...... 1°20 1°43 “921 "972 322°7 306°8 MICCLONO pense epee ee: 2°51 Ae "953 Chloroform ......... 2°02 em 1:09 Fluor-benzene ...... 2°42 2°37 “972 "953 1001 968°7 Hexamethylene...... 2°61 2°54 1-23 "929 L661 1604 Sulphur dioxide ...| 1-61 sei 1:05 Ethyl formate ...... | 2°13 2°23 1-00 "942 1092 | 1102 Hthyl propionate ...| 2°62 2°48 "933 "934 1410 | 1447 Methy! formate...... 2°12 2°07 “969 | 951 893°9| 869-0 of s of each liquid and the corresponding values of 1 3T, EOP very good. The values of p, and T, used to calculate the value of the above expression are given in Table LV. Several results of interest and usefulness may be deduced from equation (1). It may be expressed as two equations thus : ? nal vet : pi?—F Toe i at ae a log pz = Wi(T)> where W,(T) is anappropriate function of T. Now it was found * Journ. Phys. Chem. vol. viii. p. 405 (1904). Tt Phil. Mag. Oct. 1910, pp. 679-681. The agreement between the two sets of values is Density, Temperature, and Pressure of Substances. 329 that w,(T) is approximately a constant for each substance, and that it varies very little from substance to substance. This is shown by Table ILI., which contains the values of TaBLe III. Ethyl Heyxa- Carbon Bromo- Sia B acetate. | methylene, | tetrachloride.| benzene. Reale: eee ae PEs Tea CE teh) Ny (E) Ie CL). | E |b, CD): 363| -977 | 273] 1:02 |373] 1:05 |533)1-05 1353] -933 | 353) -961 383] -969 1363] 1-07 |393] 1:04 |553/1-04 1373] -926 1373] -955 403| -962 1383] 1:17 |413} 1:04 | 573/103 1393] -921 | 393] -951 423| -957 |403| 1-24 |4383} 1:03 | 593/103 1/413] -917 | 413] -950 443| -953 |423/ 127 |453| 1:03 |613/1-02 |433/ -915 | 433] -950 463| -949 |443] 1:30 |478| 1:02 | 633/1-01 |453] -914 | 453) -931 483| -945 | 463] 1°32 |493| 1-02 473| -914 |473) -951 503| -940 | 483] 1:32 |513) 1-61 493] -913 1493] -950 518} -934 | 503] 1°32 | 433} 1-00 Todo- | 518] -911 | 513) -948 522} -932 | 523] 1°30 1553} -988 | benzene. | 533| -907 1533! -945 | cu Lak 539| -904 1553] -940 9) ey < : Fluor- Stannic oie nie benzene. chloride. 585 1:07 Hexane. Di- Octane. 603! L-06 isopropyl 353! -995 Bie) UES OP OS BSS Tei) 373| -987 | 393] 928 |393] 1:12 | 643/1-05 |363/1-11 | 323] -922 393] -982 | 413) “922 | 413) 1:10 383 1:10 | 343) -915 413] -976 | 433) 919 |433) 109 403,109 | 363) -916 433| -974 |453| 916 (443) 1:08 | Chloro- |423 1-08 | 383] -915 453| -97) |473| 915 | 473) 106 beuzene. (443 1:06 | 403) ‘916 473| -967 | 493) 913 |493| 1:06 463 1:05 | 423) -919 Mon -O57 pols| OL 513) 105 paneer sss) oe |448| -o18 513| 963 |538) 910 ]533} 1-04 | 413) “985 1499 1.09 | 463) -919 483| -959 | 503| 906 |553; 1:02 453 985 506} 1:00 | 483) ‘917 Ethyl Ethyl | 293) “979 | Pentane. | Methyl | propionate. oxide. ole on i : Die ED 383-950 |293) 945, - |” [ais] -o1g | iobaty!. 383.) 950) | 298) 1-045 358! 911 | 323| -991 | 423) -937 |353) 929 | Acetone. 1373] -911 | 323] -936 | 403 | -96g 463) -930 [413-928 393] -913 | 343] -925 | /483 | -949 500 | “926 a “O19 273 | 97] 433 ‘914 363 | OQ: | ec 538 920 330) -ug5 | 453, 913.303 016 | 6 “Q|2 2: “Q1T4 Ethyl Bn ae ety 469 -909 i B14 formate. arbon ioxide. bagels 483) -911 dioxide, Palouse pa [408] “908 933 |]. CASTS) UATE Sl RB nee | ibe baled se = Aba 243|1-01 |323| 1-03 {273/113 | formate. |——— 433|1-00 |298| 973 |363] 1:02 | 380| L-U6 507 -961 403} 1-00 __|373| -968 433) 955 | ae ‘O47 | Phil. Mag. 8. 6. Vol. 21; No; 123. Marek 1911. L 330 Dr. R, D: Kleeman on Relations between the vr, (T) at different temperatures for a number of liquids, the latter of the above equations being used for calculating W,(T). The slight deviations of W,(T) from constancy are regular, there being a slight decrease in its value with the temperature. The value of wW,(T) in terms of the critical constants, obtained by substituting for the quantities T, p;, and p2 in either of the equations their critical values, is / The above equations may thus be written: 1/3 c ae pee p a (ee . ee Py Orn log P1 Wit 3c 3 i oa gh aa ee a, Om ae QB fn log Pc)s (2) Table LI. contains the mean value of ,(T) for each liquid contained in Table III., and the corresponding values of 1/3 PEOVER Nac ioe = OL Pr) The agreement between the two sets of values is fairly good. Since the equations (2) are the same in form, for a given temperature each must have at least two real positive roots, one being equal to the value of p; and the other to the value of po. d A mAaay for P1 cM : : orc n expression for 77, is obtained by differentiating the first. of equations (2), which gives dp; pip,” log p, a eae a It will be seen that at the critical temperature ea —=O.4 a ? result that has already been established by thermodynamics. At the absolute zero Pris finite. For intermediate tempe- ratures the equation gives values of a which agree roughly with the facts, That a good agreement is not obtained is due to yr, (T) being only approximately a constant. A better agreement would be obtained by writing y,(T) =a—Tb, when we obtain dp, _ pipe’ log pi— 31 .p,6 GE, | aaa Cs Density, Temperature, and Pressure of Substances. 331 The value of 6 is best obtained from this equation by applying it to a case when all the quantities it contains are known except b. We may substitute p, for p; in the above equations. ‘Tt can then be easily shown that at the absolute zero oe =—2. Let us express the quantities p,, p,, and T in terms of their critical values thus, p,=pcn, po=pen2, T=Ten3, and substitute in equation (1). The equation reduces to ny suse ene = SS it Be ® ° ° © e 3 Por =nie—nle, . (3) which, it will be seen, is independent of the nature of the liquid under consideration. Since p; and p, are each a function of T, and n, and n, therefore each a function of nz, it follows that for equal values of nz tor a number of substances, n; and my will each have equal values. The equation thus demon- strates the theory of corresponding states, and gives a relation between tne quantities n,, 2,3. The corresponding state of substances, it will be shown later, has its fundamental reason in the occurrence of the function in the general law of attraction between molecules which has the same value for all substances at corresponding temperatures. In a previous paper* we have established the relation B(p{—p3) =T loge , Where E is a constant depending only 2 on the nature of the liquid. This equation, like equation (1), represents two different formule for the latent heat equated. It was found to agree very well with the facts. The value of BH, when the logarithm in the equation is taken to the base 10, was shown to be given by 2988 (2 Mi)? mse 2/8 A different expression for EH can be obtained in the same way as was obtained for B. Writing p.=.p,, the value of 1 at the critical temperature is T log i = T. or qT. = an = Qp2? ©” p22 x 2-303” pi tea ae Leo= 1 1 ev * Phil, Mag. Oct. 1910, pp. 686-687. LZ 2 332 Dr. R. D. Kleeman on Relations between the if the logarithm is taken to the base 10. now be written —p2)=T lo oP, 9 The equation may T. pin The mean values of E for a number of substances, given by the above equation, and the corresponding values of FIX 2303" are given in Table I]. The agreement between the two sets of values is fairly good. From equations (1) and (4) we have (pi—p3)= 692" (piP—p3?). - - = ome If we, as before, express the quantities in this equation in terms of their critical values, we have 2— ne =6 GP — ny)... «ite Ln ny Hguations (1) and (5) express approximately the relation between the quantities p,, ¢:, and T, and equations (3) and (6) the relation between the quantities m,, 7, and n3. Hquations (1) and (5) are very useful. we have = Fag \ a5 2 ° —. , Which may be used to calculate From equation (3) 6py°— 6p; Tasue IV. pc 2_ 2 3/5 1. Nameyordiqurds yt.) jp;. ep. |(Landolt & i ah (Landolt & Bornstein), [ 6p!/?—6p!? | | Bornstein). Hthyl oxide ....1.... 293 | °7135 | 00187 "2604 "206] 467°4 Octane Weir lissicsnin Which enables c Hquation (1) gives the value of B or us to calculate T, if p. is known. The values of T. or 1/3 B c have been calculated in this way for a number of sub- stances, and are given in Table IV., using the values of p previously found, and calculating B by means of the data used for calculating p.. The agreement of these values of T.. with those taken from Landolt and Bornstein’s Tables 1s as good as can be expected. It will be observed that since the cube root of p, occurs in the expression for T;, the percentage error in the value of p, introduces a much smaller percentage error in the value of T,. Further relations of interest can be deduced. We have seen that Tues ace the Oh m 2 and we therefore have from equation (1) that the value of D in the equation L=D (p}°—p3") may be written D2 Eu, —————— ee mpi Substituting the value of L from the latter equation in Clapeyron’s equation we have RY RIN Be, Gf) =) S2ORT. - gas Alea) a ase a hess MP: where P denotes the pressure of the saturated vapour. 334 Dr. R. D. Kleeman on Relations between the We may express this equation in the form of two thus: Cee mb enor2okul.. 4, dT p, = pi =F mpi3 aa at W(T), | lM eri i) be Le iy a) c | ae Now it was found that the value of y,(T) is approximately constant for the same substance. This is shown for two substances in Table V., w,(T) being calculated by means ef the latter of the above equations from the data given in the table. TABLE V. PENTANE. HEXANE. P dP P dP Po jin mm.of Hg.| dT WAT). T. | Po jin mm. of Heg| dt ¥.(T). 313 |-003390 865°3 28'81| 93:2 || 343 | 00337 784°8 24°73 | 89°7 338 |006024 1601°8 46-19| 90:1 |) 863} 00585 1409 38°80 | 87:2 358 |'01013 27421 69°37 | 93°5 || 383 | 00952 2358 57°20| 86°8 373 |:01627 4409°1 99°05} 91:9 || 403 | 01502 3723 80°55 | 85:9 393 |-02503 6740°5 136-02} 92:1 || 423} 02299 5606 109°2 | 85:2 413 |-:03861 9890 1811 | 91:4 || 443) 03472 8123 144:0 | 84:2 433 |05910 14032 235°4 | 91:0 || 463] 05155 11407 186°3 | 841 453 09354 19362 300°2 | 90°6 || 483|-07900 |. 15619 2374 | 836 The constancy of ,(T) can be tested more conveniently by means of the equation obtained by eliminating (a =) from equations (6). We thus obtain d°25RT, mpe. W(T) It follows, therefore, that if it is found that © is independent ot the temperature for a substance, then ».(T) must possess the same property. Now this is approximately realized, as will appear from Table VI., which contains the values of C at different temperatures for a number of substances. We can find another expression for the value of C, which enables us to obtain an expression for the value of .(T). Let us write p»=wp; in the above equation, and the value of C(pt®— p3? ) =p, — pa, where C= Density, Temperature, and Pressure of Substances. 330 C is given by reese Se I 4 haus l— 28 eesti The above equation may then be written oe gee =p Px + + ss 7) Cc We have also 3 Sea ail be 7RT. 4 p3 — mpl®r3(T)’ and hence yr,(1) = ———. In the case of pentane and hexane we obtain for y.(T) from this equation the values 90:3 and 90°6 respectively, which agree fairly well with 91:7 and 85°8, the mean values of aro(T) obtained from Table V. The mean value of = for each substance is given at the C bottom of Table VI. The agreement with the corresponding Ve values of = pe is fairly good. But it will be noticed that equation (7) does not on the whole agree so well with the facts as either equation (1) or (5). TaBLe VI. Methyl Di- Hexa- Carbon butyrate. ee isobutyl. Ie ENS: methylene, | tetrachloride. ee (ee a CDy De). | Cy D:| DG) | eT). | Cr). | 8 ee ee ee 883 | 980 | 273) °694 | 363| ‘861 | 333} °857 | 363) ‘897 | 273} 1:18 403 | -923 | 353] °702 | 383} °855 | 3853) °849 | 383) -890 | 373| 1:14 423 | 916 | 373| ‘707 | 403) -848 | 873] -845 |408] -884 | 393] 1:18 443 | -909 | 383] ‘707 | 423) °843 | 393] 837 | 423) -876 | 413} 1:12 463 | 903 | 393] "705 | 448] -836 | 413) 831 |443] -874 |483;) Ill 483 | -898 | 413] ‘701 | 463) 832 | 433] -826 |463)| -869 | 453] 111 503 | 894 | 433] 698 | 483) 827 | 453] °823 | 483) -866 | 473) 1:10 523 | ‘890 | 453| -677 |503| -823 | 473) 820 | 503! -863 | 493] 1:10 543 | 890 | 473] 662 |523/ -822 | 493) °818 |523) -861 | 513} 1:09 1-13 551 | -888 543| 822 |503| -820 |543| -862 | 533 Mean | -924 695 ‘837 833 “874 112 | es ee —— ee aaa —_— 31/3! -884 | 822 825 ‘R29 866 | 1:88 336 Dr. R. D. Kleeman on Relations between the General Considerations. We will now deduce some further results of a general nature from the law of attraction between molecules quoted at the beginning of the paper. It can be easily shown that the theory of corresponding states follows from the law quoted. It will first be proved in the case of a liquid in contact with its saturated vapour. The equation for the internal latent heat of uses s deduced from the law” is 4/3 4/3 p Spe 2 TN Les Aya SV) Aa gS Vu) where Hoek av, T Ai=$s(=.55) Aj= ds( — se) and 2g, a» denote the distances of separation of the mole- cules in the liquid and gaseous state respectively. The form of A, and Az, it should be noticed, depends only on the form of the expression bo(— sa) occurring in the law of attraction. We have seen that an infinite number of formulse for the latent heat can be obtained which correspond to different forms of d2, and there exists, therefore, a series of equations of the form iA; PL (% ¥m)?—A aS my)? mils =A AP (SV m4)? — te: Vm) mils or Bas L G4 (% ay t) Dy o( = - ds a “T= | (Oop —¢,(~, Wy. } pi =... where —~ = © , and ogee eee Le Pc Xe Pe It will be readily seen that if we express the quantities py, po, and Tin terms of their critical values in the same way as before, the equations will contain n,, m, and ns, only. Since it must be possible to express n; and nm, in terms of 7, therefore if n; for a number of substances has equal values, m,and nz will each have equal values. This proves the theory of corresponding states from the general law of attraction for the quantities p;, pe, and T. * Phil. Mag. May 1910, pp. 793-794. Density, Temperature, and Pressure of Substances. 337 The above series of equations are the general fundamental equations connecting ‘T’, pj, and p2, for a liquid in contact with its saturated vapour. Since they also contain. p, and T., they are the fundamental equations for determining these quantities from any convenient data. The equations we have been discussing, obtained by equating different formulee for the internal latent heat, will readily be recognized as belonging to this type of equations. Making use of the above result, the equation for the latent heat may be written 4/3 i L=B?2 (Saf n0,.)?, mils when B is a constant which is the same for all liquids at corresponding states. It can then be easily shown * by means of this equation that L=IL,7, where Ip is the latent heat at the absolute zero, and 7 is a constant which is the same for all liquids at corresponding states ; which establishes the required result in the case of the internal latent heat. By means of the foregoing results and thermodynamics the law of corresponding states can be established for the pressure p of the saturated vapour of a liquid. If the value of L from the former of the above two latent heat equations is substituted in Clapeyron’s equation, it can be reduced to the form —., 7/3 Ay WE my (2) =p—aepet. where W is the same for all liquids at corresponding tempe- ratures. At the critical point the right-hand side becomes zero, and since p, is finite at the critical point, W must also patpol[Nc W may be written p,. V., where V, is a numerical constant, and the equation at the critical point becomes LE Be Po= a (= my)? (*:) Combining this equation with the above equation we have 7/3 T V, {e) sede Se Wp pe Pag ie or p=N, pe, Where n, is the same for corresponding tempera- tures, which is the required result. * Phil. Mag. June 1910, p. 845. } Phil. Mag. Dee. 1909, p. 908. become zero. The limiting value of the ratio 338 Dr. R. D. Kleeman on Relations between the The equation p=n, p, is of theoretical importance, as 4 is a function of nj, mo, ns, and the equation therefore gives the relation between p,p;, p2,and T. It could at once be written out if the exact form of the arbitrary function in the law of attraction were known. It will be of interest to develop the equation as far as possible. The value of m, is (V.Wni?+n3), from one of the above equations. Referring again to the paper quoted above, we have W=N— eat i c where N, is a numerical constant, and 2 Ne 22 i d = — —_—— n 71/38 Ji.8 39 ni an; where if il Ny— No A = 3 lo Ny MjNo changing some of the symbols to those used in this paper. The quantity B? which occurs in the equation for the latent heat is a function of 7, ng, and n3; its form would be known if that of the arbitrary function in the law of attraction were known. From the way it is obtained it is likely to be a series*. Writing B?=.W; (7, ng, 3), the above equation reduces to 5/3 abr3(11, Ng, 23) « Non?! fe p= {v.| = dn, + 1— NeVen{® 3 Pew (14 —N,)nz This can be changed into an equation involving pj, p2, and T, by means of the equations . py=7 Pe, P2=N2 Pe, =ngl. Thus the part of the right-hand side of the equation not under the integral sign at once reduces to ~ (1 —N.YV, ( eB") The quantities nj, , under the integral sign can be expressed in terms of n; by means of the resuJts already obtained and the expression integrated, and LL then substituted for n;. T. This cannot yet, however, be effected in practice since we do not yet know the exact form of the function ¢y. Without performing this operation, the equation connecting the * Phil, Mag. May 1910, pp. 793-794. Density, Temperature, and Pressure of Substances. 339 quantities p, p1, P2) T, may be written Pi po | jk sh puay Ue Noe pe 7) “P20 ae p=nP (1—n.v.(2 ) he 5/3 > TP : ans Pe Pe (pi- Po) The law of corresponding states can also be proved for all possible states of matter. From thermodynamics we have a) an dp T const: . p const. where dQ is the amount of heat given out by a mass of matter of volume v ata temperature T, when the pressure changes by dp. Now Q=pvtu, where wu is the internal energy of the substance. ‘The internal energy we will take (as usual) equal to the internal heat of evaporation at constant temperature into a vacuum. According to the formula for the latent heat used in a previous part of the paper (deduced from the law of molecular attraction), this 1s for unit mass equal to $s (oa i aE “m3 (vm), where 2@ is the distance of separation of the molecules of the matter and p its density. This expression for the internal — energy we will write for shortness HCp**, where We have then d() dv dH ot 4 oo eee =O Oe oo Da Cae Gab at ale jen (Sr), =) Ge) {r+0(5, yi/3 ~ Fa Let us express the quantities p, v, and T in terms of their critical values thus p—=Ppc, V=20e, T= T;,. The above equation may then be written [++#.(%9) + pan(ig): (Cae 8®) 1A), CD nr ty ip =H, where D and E are functions of «, 8, and ¢. weer cm reer + ee oe 340 Dr. R. D. Kleeman on Relations between the At the critical point, when «=1, B=1, g=1, let the numerical limiting values of D and E be denoted by D, and E,, so that ce =H,. Eliminating v,, C, and p,, we have fier i een which contains the quantities a, 8, and ¢@ only. This proves the law of corresponding states in Meslin has shown (C. R. 116, 135, 1893) that if the equation of state contains as many constants as there are varialles (7. e. three—volume, temperature, pressure), it can be reduced to an equation involving «, ¢, and @ only, or the law of corresponding states applies in that case to substances. Since we have established the Jaw in a different way, we deduce that the equation of state must contain three constants. CD. The equation ee : Pete on substituting for p,, V-, and C, from the equations Nios _O vm)? p=pe; C= Aly, and) c= = 2 eee may be written p\73 le p=MW (2) (3 Vm), where M? is the same for all substances at corresponding states. This equation we have previously obtained for a liquid in contact with its saturated vapour. We now see that it applies in general. The equation 4/3 a, T= He (2) /m,)% where H? has the same value for all substances at corre- sponding states, was also previously shown to apply to a liquid in contact with its saturated vapour*. Now if we substitute for the quantities T,, p,, from the equations pi=ap, T;=0bT, where T and p refer to any given state of matter, we obtain the equation T=Ha(E) vn), * The equation was deduced with the help of thermodynamics from the general law of attraction quoted at the beginning of the paper. Density, Temperature, and Pressure of Substances. 341 where H, is the same for corresponding states. This follows since the above two equations may be written and therefore Ns Ge, ‘and b=, a B or 6 and a have the same values at corresponding tempera- tures. Thus the equation in question is proved for all states of matter. ; didaedl From this and the above equation we obtain pu= See which applies to all states of matter, where I is the same at corresponding states. The general equation of state can be deduced from one of the foregoing equations, but it is of a form which is not of any use. We have obtained dv OU where Wel AL Atel ae ( dv vii 3 ae) and is therefore a function of v and T. This is a linear partial differential equation, and the Lagrangian subsidiary equations are therefore iy ant An independent integral of these equations is T =s\. Aine let the other be denoted by w;(T, v, p)=B. Then an integral of the partial differential equation is given by ~bo( i, i(T; 2 r)) =), where we is an arbitrary function. One of the conditions which determines the form of the arbitrary function is that the r equation must approximate to pu=—— when the matter is Dd in the gaseous state and the pressure is lowered. London, December 19, 1910. mee || XLI. The Problem of the Uniform Rotation of a Circular Cylinder in its Connexion with the Principle of Relativity. By W. F. G. Swann, D.Sc., A.R.CS., Assistant Lecturer in Physics at the University of Sheffield *. HRENFEST (Phys. Zeit. Nov.1909) considers the problem of the uniform rotation of a circular cylinder in its relation to the principle of Relativity. He remarks that according to that principle, each element of the circumference moving ze with linear velocity v should contract in the ratio ur a : C being the velocity of light, while since each element of the radius is moving in a direction perpendicular to its length, the radius should not alter in dimensions. He points out that the co-existence of these two conditions is impossible. In the Philosophical Magazine (ser. 6, vol. xx. no. 115, p- 92), Stead and Donaldson have suggested that the explanation is to be found, at any rate in the case of a thin disk, by assuming the disk to become cup-shaped when rotating, and they have calculated the shape of the cup as a function of the velocity of rotation which is necessary to ensure non-violation of the above conditions. They suggest, moreover, that in the case of a cylinder of appreciable length, the cylinder more probably becomes strained. It must be observed, however, that the assumption of the cup- shaped body involves a displacement of the particles of the disk in a direction perpendicular to its initial plane, the relative displacement of the particles varying with the distance from the axis of rotation; and it seems that this displacement of the particles would be as inconsistent with the assumed principle of relativity as a contraction of the radius would be. Further, suppose that instead of con- sidering a flat disk, we consider a disk which when not a Cameron A . & rotating has a shape the section of which is shown in fig. 1, A. If this disk is set in rotation, and there is to be * Communicated by the Author. Uniform Rotation of a Circular Cylinder. 343 no alteration of its dimensions perpendicular to its median plane, and if say the right side is to buckle so as to satisfy the assumed principle of relativity, the left side will have to take the shape shown in section in fig. 1, B, so that the condition of the particles on the left side will not be consistent with the principle. Again, the solution of the problem is not to be found in the disk becoming strained, for a strain would involve the alteration of the radius, and the final state of the disk would be one inconsistent with the conditions assigned above, which conditions are required by the principle of relativity if this problem is one to which the principle of relativity can be applied in the form which it has been applied. The object of the present paper is to show that there is no fundamental difficulty in the question as far as its applications to ordinary matter is concerned, and in fact that the apparent inconsistencies arise from neglect of considering all the phenomena involved. In the first place we notice that though each point of the circumference of the cylinder is moving with constant velocity, the problem is not one of constant velocity in a straight line: each particle has an acceleration towards the centre. Asa matter of fact, we know that if the disk were set in rotation, it would not contract at all, it would expand owing to centrifugal force, to an extent depending on the elastic properties of the material. It would seem useless to endeavour to limit the case to that of a rigid body, for the very alterations of dimensions which we are discussing are alterations which are necessary to secure equilibrium of the internal forces and motions when the body is set in motion as a whole, and the alteration in dimensions due to centrifugal force ts as necessary for the maintenance of internal equilobrium as is the contraction of which we are speaking. If we look upon all internal forces, cohesion included, as being of electromagnetic origin, a complete electromagnetic solution of the problem, if it could be obtained for our rotating cylinder, should show us that the cylinder actually does expand, the centrifugal force phenomenon being in fact involved in the electromagnetic scheme. It would be im- possible, however, to obtain such a solution, unless we knew the nature of the electronic distributions and motions in the cylinder when at rest, because the solution would, as we know from experience, involve the elastic properties of the material, which on our View are totally determined by the nature of the electronic distributions and motions. I think the real solution of the apparent inconsistency is to be found in the following. 344 Dr. W. F. G. Swann on the If we set a piece of matter in motion in a straight line with constant velocity, the electromagnetic theory leads to the conclusion that whatever may be the nature of the internal electronic, atomic, and molecular motions, the ( 2\3 system as a whole will contract in the ratio Ce in the direction of motion in such a way to insure that at corresponding times the z component of any electron shall be (1—(z) times the x component of the electron in the fixed system *. Of course, so far as we are concerned in our notions of the constitution of matter as founded on the electromagnetic theory, all differences of elastic properties, &e. are to be attributed to differences of the strengths, distribution, and motions of the electrons. The curious thing is that the above contraction is absolutely independent of the nature of the elastic properties of the matter con- sidered, 7. ¢., it is independent of the nature of the electrons and of their mctions in the system ft. Now when we come to consider problems of motion other than motion in a straight line with uniform velocity, it may be, and in fact certainly will be, that the final condition of affairs will depend not only on the motion imparted to the system but also on the motions which the electrons had in the system when at rest. It is interesting to attempt to treat the problem of uniform rotation on the principle adopted by Sir Joseph Larmor for uniform translatory metion; for in spite of the fact that the principle of relativity is a principle which is postulated independently of its complete verification from the electromagnetic theory, it must nevertheless be looked upon as being suggested by direct argument, and it is of interest to trace the course of an analogous line of argument for the case of uniform rotation. Let us first briefly review the problem of uniform translatory motion in the manner given by Larmor (‘ AXther and Matter,’ pages 167-177). ~ Strictly speaking this is not true: the electromagnetic theory really only leads to this conclusion when the system discussed is one in which all the electrons in the fixed system are absolutely devoid of motion (see final paragraph of this paper). + Really this fact is not as curious as it seems when we realize that the conclusion as derived directly from the electromagnetic theory only strictly holds for one particular kind of system, viz., one in which the alectrons have no orbital motions. It is the postulation of the principle as holding for all systems which makes the result seem so startling. 407 Uniform Rotation of a Circular Cylinder. 345 Suppose our system is moving parallel to the axis of a with velocity v. Let f,g,h be “the satherial displacement vectors, a,b, ¢ the magnetic induction vectors, and C the velocity of light. The electromagnetic equations when referred to a system of axes at rest in the ether are of the form pee 0° 0? — (402) 108 = OF _ OF (1) yt Oy 02 eR by ey HOT VO Oo OC with similar equations for aan on at When we transform to axes a’, y’, 2’ moving with the system so that TH eh 6, ve ell TON ined Ui mati we obtain, in view of the fact that — Z becomes a —vV sa ; dt dt Ax 2 Og ue h ee aay) a= d= Sailr = (() 0 a set of equations of which a typical pair are vb \ \ Of _ de _ 4mry) ee (ec) yu ~ Oy Oy (4arC ) Ot! ran, Oy’ + Oy! ve i (2) _ db _ d(4avh) Lee L o(ca) | Oz’ (oe 02’ 02’ Now it will be seen that the terms on which 2, operates may be grouped together, a similar remark applying respectively to the terms on which as and 5s! operate in the other equations. On writing a’, b,c = (a,b+Aavh,c—A4mvg) 2 .. ) : vb (3) iMG h’, =(L9-¢ Ao oe h+ =) 2 ° | our equations take the typical form Di 45, OC) Ou, Ay ao oy Neen oe oy oe Phil, Mag. 8. 6. Vol. 21. No. 123. ee OLDS) 2 A A ue Oh’ _ Og’. 346 Dr. W. F. G. Swann on the In virtue of the relations (3) the elimination of f,g, A from the equations may be completed, for example g=ot+ es (c' + 47vg), 2 ite Uv : so that on writing e-! for 1- ce «owe obtain / a I UC ~ € =O) TEE : : - = . : (5) / Finally, on writing —= / / ’ 2 Peet 7} GT, 1 = ere 6 bc Jo Gu i = © 2 jf ge (tee h=et(t!— Sea’) C2 our equations revert to the original type (1), the typical equation becoming Dic OG Ob; Hy O01 _ Ohy eae 9 47 = — ~\4A1 = = 2 eee ee 0 eG OY; aie Cas ) Ot; onl 021 ( ) The remainder of the argument then follows as given in ‘ Aither and Matter’ (pages 174-177), the above being as much as we need abstract for the purpose in hand. Now let us turn to the problem of the uniform rotation of a cylinder, and let us choose coordinates 0, 7, z. Let f, g, h, a, b, c be the corresponding etherial displacement, and magnetic vectors. Maxwell’s equations may then be written down in terms of 6, 7, z. It will only be necessary for our purpose to write down one pair Oe Oe "Ob elie Oe ee O4_ oh_ og fee (oye ee oon — (4rC’)7} Tf we. transform to a system of coordinates 6’, 7’, 2’, moving with the cylinder, so es becomes og : ry; ae 8a” o being the angular velocity, then in view of the fact that ‘Oy O97) Oh Oa 0b 3 Ac ‘ pars see ef) pote ae as ———— 100; t By ee” HOO! Or! * Ba! "2 also Uniform Rotation of a Circular Cylinder. 347 the above equations give of ie 0c ? 0 : 19\ —1 Aa Ooh y foe UO Sea an ale: mC 2) aa ma Siler) b fe 09 ; ~aieriee? ~ 39 452 (Gate corresponding equations of course holding for Og Ooh Ob Oe De wou Ol Ok. When we reached this stage in the case of uniform rectilinear motion (see equations 2), we were able to put the terms on which —; operated together, and similarly for oy the terms on which 2 and ~./ operated. Considering “ Z the corresponding case in our present problem, however, we see that we cannot for instance put ly —7! 2, (darvgq) in the fol ae form a unless we know g as a function of the variables ; r ‘ and similarly for the other equations it would be necessary to know f and h, a, b, and ¢, as functions of the variables in order to accomplish this, 7. e., it would be necessary to know the strengths, distributions, and motions of the electrons in the system. Again, even supposing that these were known so that we could put our equations in the form corresponding fomser s(t) a@,b3¢, 759 W, being, however, functions of Pe Gla ee ap nen involve the strengths, distribution, and motions of the electrons in the system, we should, Aerie he less, be unable in general to perform the step indneaied by (9) in such a way as to obtain for example g in the form g= Ad + Bed, where A and B are constants. A and B would have to be functions of the variables depending on the conditions peculiar to the particular system we were studying. If A und B cease to be constants, of course the whole endeavour to perform the step analogous to that performed in (6) breaks down, and the equations cannot in general revert to the type for the fixed system of coordinates. It may be remarked that even in the case of uniform rectilinear motion 2A 2 ee aan on . = See 348 Messrs. J. D. Fry and A. M. Tyndall on treated by Larmor, it is necessary, in order to show that the moving system which he discusses is the system which the fixed system becomes when set in motion, to restrict the problem to the case in which all the electrons in the system at rest are absolutely devoid of motion. Such a system could not of its own accord be in equilibrium, without imposed constraints, and the introduction of such constraints of non-electromagnetic origin presents a serious difficulty to the mind, in view of the fact that the whole theory rests on the assumption of the completeness of the electromagnetic scheme. The difficulty of imagining the constraints may to a certain extent be overcome, and this fact, combined with the definite experimental fact to hand in the Michelson and Morley experiment, provides us with the confidence we need to postulate the principle of relativity for uniform rectilinear motion ; but it would seem that in view of the difficulties encountered in this case, the postulation of the principle for non-rectilinear motion can hardly be expected not to lead us into mutually contradictory eonclusions. AXLIL. On the Value of the Pitot Constant. By J. D. Fry, Lecturer and Demonstrator in Physics, and A. M. TYNDALL, M.Se., Lecturer in Physics in the University of Bristol *. 'HE method of measuring the velocity of a stream of gas by means of a Pitot tube is well known. A “ pitot ”’ is generally a fine tube of which one end is open and directly facing the stream, and the other end is connected to a pressure-gauge. A pressure is thereby set up in the pitot, the value of which is a function of the velocity (v) of the stream at its end. If P is the excess pressure over the static pressure at the open end of the pitot (referred to in what follows as the ‘ pitot pressure ”?) and if p is the density of the gas, then 0=Ky / aF P K is the Pitot Constant, and previous experiments have shown that its value is not far from unity and constant over a considerable range in velocity. Thus among the more recent determinations of K are those by Threlfall +, who * Communicated by the Authors, + Threlfall, Proc. Inst. Mech. Eng. 1904. the Value of the Pitot Constant. 349 obtained 0:974 between the limits 10 and 60 ft. per sec., and Stanton *, whose value was 1:03. Stanton’s number was obtained for use in his work on the resistance of plane surfaces in a uniform current of air, and was known by him to be a few per cent. too high. During 1903-05 a long series of determinations of K in air by two different methods was carried out by the authors ; but as discordant results were obtained, which at the time received no explanation, the work remained unpublished f. The authors have since reinvestigated the question, and it will be shown below that by applying certain corrections the results by the two methods may be brought into agreement. In the first method, the pitot was moved at constant speed through stationary air. For this purpose, it was fixed to the end of a revolving arm and connected to one side of a pressure-gauge by a tube leaving the arm at the centre of rotation. The pitot and the arm were at right angles to one another so that the former always pointed along the direction of motion. Due to the relative motion of the air and the pitot there will be a pitot pressure P given by a= KK 2P p where v is in this case the velocity of the pitot. At the same time, however, due to centrifugal force the hydrostatic pressure in the arm at the centre of rotation will be below that at the pitot end by an amount p. ap . 13 = is the slope of the pressure in the arm at a radius r, then dp=p wr dr, where is the angular velocity of the rotating arm whence R 2 p=po* { ey 0 where v has the same meaning as before. The resultant * Stanton, Proc. Inst. Civil Eng. 1903. t These experiments were first uudertaken in collaboration with Professor A. P. Chattock, who identified himself so much with them that his name is omitted from this publication only at his express wish. ‘Che authors take this opportunity of thanking him for the part that he took in the earlier work and for his guidance in its more recent stages, 350 Messrs. J. D. Fry and A. M. Tyndall on pressur 3 age indicated by the gauge is therefore pu? __ pv? ere Oo": From this K may be calculated. It will be noticed that if its value is exactly unity there will be no gauge deflexion, and for this reason this method possesses an advantage over the second method. In the second method the pitot was placed parallel to the axis of a pipe of radius R through which air was Howing at a constant rate, and a series of pressure readings obtained by moving the pitot along a diameter of the pipe. If p, is the pitot pressure and v, the velocity of flow at a distance 7 from the centre of the pipe, then as before peg), 2 a The quantity of air flowi ing per second through a ring of thickness d2 at this radius is Meee given by = 2p, gas —! 2a rdr. p Integrating this across the section of the pipe, we have “R a2, Q) — i K a / 20 rdr, Ah a where Q is the total quantity of air flowing through the pipe per second. Unless K has a constant value at all velocities it cannot be taken outside the integral, since the velocity of the air in the pipe varies at differ ent points along a diameter. If K is not constant and this is done, the values of K obtained are purely artificial and depend not only on the quantity of flow through the pipe but also on the distribution of that flow across the section. The symbol K, will be used in what follows for the artificial values obtained in this way. The only justification for using this method to measure K itself lay in the fact that previous experimenters had found that K had a constant value over a limited range, and foe appeared to he no theoretical reason why any marked change should occur when the range was extended. The observed values of K obtained by the centrifugal method and of K, by the pipe method are shown in the Value of the Pitot Constant. dol Curves I, where K (circles) and K, (crosses) are plotted with the velocity of the pitot relative to the air, the velocity in the case of the crosses being the mean rate of flow of the air along the pipe. CurvES I. * 4-02: a ee a ee a saa ee ie 200 co 6co 80d 600. (200. «ston. shoo«Boo.S« DODD CM GEC” velocity. The results of the centrifugal method show that except for a 1 per cent. variation at low velocities, where the ex- perimental difficulties were considerable, K was 1-002 at all the velocities investigated. K,, on the other hand, although about 1:00 at higher velocities, rapidly decreased with decreasing rate of flow. Now K, only differs from K when the latter is not constant, so that assuming that the centrifugal value of K is correct the two curves should be identical. There must, therefore, have been some source of error in the pipe method which was not present in the centrifugal method. No information could be obtained on this point from the work of others, because the velocities at which the dis- agreement was appreciable were far below those which had been previously investigated. In view of this fact and of the general importance of the question of the flow of air through pipes, a reinvestigation of the pipe method was undertaken. es { ti i) 352 Messrs. J. D. Fry and A. M. Tyndall on it Experimental Details. thi Centrifugal Method.—One of the chief difficulties of il putting this method into practice arises from the fact that 1 the rotating arm inevitably produces a vortex in the air of | the laboratory, with the result that error is introduced into \, the observations in two ways :— i (a) Owing to the motion of the air, the relative ii speed of pitot and air is not so great as the rate of iH rotation of the arm would imply. Mh (b) The static pressure of the air falls towards the Hi centre of rotation, and the pressure at the open end of Mi the pressure. gauge is not necessarily the same as the Hil static pressure along the path of the pitot. Mh It was, therefore, important (a) to bring the air to rest Hil along the path of the pitot, and (6) to connect the open end iI] of the gauge in such a way that the pressure upon it was the | average of the static pressures encountered by the pitot. | | Fig. 1. ! | (Q) tl | | Hl A i i Hi | As far as could be ascertained after many trials, the following arrangement fulfilled these two conditions (fig. 1, a and 0), the Value of the Pitot Constant. 353 The pitot P was fixed on the end of a tube leading along an arm a and passing at ihe centre of rotation through an oil joint o to one limb of a sensitive pressure-gauge. The oil vessel was mounted so that the arm a and the pitot could be rotated in a horizontal plane. ‘The pitot was horizontal and at right angles to the arm and moved round in a circle of 5 feet radius close to a cardboard rim 7, the outer surface of which was covered with a series of paper flutings. Above a and over the greater part of the apparatus was a horizontal cardboard sheet c¢ to which were attached at equal intervals along radii 16 vertical cardboard structures s; the pitot passed through holes in these during its rotation. Oonsistent with free motion of the pitot these holes were made as small as possible. The fintings and screens were effective in breaking up the air-flow in the neighbourhood of the pitot. A horizontal band of card e, 3 inches wide was placed round the rim 7 to stop as much as possible motions under the sheet ¢ from affecting the air outside rv, For this reason also muslin curtains were hung from r down to the ground and horizontally across beneath the rotating arm a. The other end of the pressure- gauge was connected to a static pressure, tube, which in this case was a 2-inch cardboard tube < encircling the apparatus and fitted with 16 openings. In this way the static pressure taken was the average of the static pressures around the pitot’s path. In order to obtain as uniform a motion of the arm as possible, a viscous resistance to the motion was supplied and the arm subjected to a constant turning couple. In the figure, W is a heavy iron wheel from which the metal vanes D were supported ; these vanes dipped into an annular dash-pot containing oil and supplied the necessary friction. The device for obtaining a constant couple is shown in fig. 1, b. Hach half of the belt from the pulley 6 passed to the turning table d over two fixed pulleys between which was a movable pulley carrying a weight. These two floating weights w,; and w, were unequal, and the table was rotated at such a rate as to keep them suspended freely at the same level. When they were balanced in this way the pitot moved at a constant velocity ; the rate of turning could be varied by altering the difference between the weights. The pressure-gauge employed was of the Chattock-Fry type used in the work of Stanton and described in his paper (loc. cit.)*. This is a form of tilting U-tube gauge in which “ An improved form of this gauge but modified so that its water surfaces are trapped by mercury, has also been described in Phil, Mag. |6] xix. p, 461, { ? | | B54 Messrs. J. D. Fry and A.M. Tyndall on a bubble of water in benzene is used as an indicator, the line of separation between the two liquids being viewed through a microscope and kept coincident with the cross wire in the centre of the field of view by motions of a screw. It is not advisable to allow the indicator surface to be much displaced, because, although it is restored to its original position by tilting, slight changes in surface tension cause it to take a temporary set. Hence it is necessary to insert in the gauge a clamping-tap which is not in general opened until the gauge has been set at the anticipated reading. The particular gauge employed when working well would in- dicate a pressure difference of one ten-thousandth of a millimetre of water. The results already summarized in Curves I show that for velocities 600-1400 cms. per sec., K is quite constant and equal to 1:002. At lower velocities there was a one per cent. variation in K which the authors never succeeded in eliminating by modification of screens and so forth. There was, however, always a certain very small amount of pumping action at the oil joint during rotation due to imperfections in the pivot A upon which the rotating system turned, and it is possible that this was responsible for the observed change in K. The pressure effects of such action would be proportional to the first power of the velocity of rotation, and hence compared with pitot readings would only be appreciable at low velocities. It seems highly probable, at any rate, that this change was due to experimental errors, and that one is, therefore, justified in taking 1:002 as the true value of K. Pipe Method.—It is by this method that the recent re- investigation has been made, but except in the case of very small mean velocities the apparatus only differed from that of 1903-05 in a few particuiars. In this method one is met with the difficulty that the mean pressure of the gas is above or below the atmospheric pressure. This has been met by other experimenters in various ways, the one adopted by the authors being to use a differential pressure-gauge, one limb of which went to the pitot and the other toa tube in the side of the pipe, and in the same plane as the end of the pitot. It was assumed that if this side tube ended flush with the inner surface of the pipe, it would measure the static pressure of the gas in this plane, except when the velocity was considerable or when the pitot and side tube were too near to an end of the pipe. In the earlier experiments the pitot and its static pressure tube were placed near the mouth of the pipe, and it was the Value of the Pitot Constant. 300 possible as a result that the static pressure was not eliminated from the recorded pitot pressures owing to a curvature in the equipressure surfaces across the pipe. The pitot.in the recent work was placed as in fig. 2 A, 8 inches down a Rio) 2A. A Het ARE por an - To Pressure Qeuge = a Dp a a pipe AB. The pipe, which was 2 inches in diameter and 18 inches long was fitted with 12 side tubes ¢,, 4, &c., and was screwed to a funnel at its lower end, the joint being made as smooth as possible on the inside. The static pressure slope down the pipe could thus be obtained by connecting a gauge to any two of the side tubes, the rest being kept closed. It was found that at the highest mean velocities used, this slope was only uniform down the pipe when a piece of wire gauze was introduced across the section of the pipe atits base. For work at low velocities the gauze was not necessary. The pitot P could be set at any position along a diameter 356 Messrs. J. D. Fry and A. M. Tyndall on by a micrometer head M, and was kept on that diameter by the arm R passing through a guide G. The same type of pressure-gauge as above was used for low velocities, but for higher velocities a much less sensitive tilting gauge of a simple U type was employed. The pipe and its funnel were fitted to the top of a gasometer consisting of three zine cylinders A, B, and C, each 4 ft.6 ins. in height and arranged as in fig. 2 B, so that C could slide on rollers R,, Ry, &e., in the annular space between A and B. This space was filled with oil. The diameter of C was 1 ft. 6 ins., those of A and B differing from this by about an inch. The gasometer could be raised about 4 feet and then allowed to fall, its velocity being varied by altering the balancing weight W, which was attached to C over the pulleys P, and Py. Piston rods immersed in dash-pots D; and D, containing treacle tended to render the fall uniform, and the gasometer could also be accelerated or retarded on its downward path by hand application to the rod H. When a uniform velocity had been attained, the air was expelled through the experimental pipe at a uniform rate which could be determined from the dimensions of the gasometer and its rate of fall. The latter was measured by clipping to the suspending wire a paper strip S which passed under a recording pen. This was made to mark seconds on the paper by an electro- magnetic device connected with a standard clock. To eliminate those parts of the fall during which the velocity was not correct, a second recording pen was placed side by side with the first. It was arranged that by com- pleting an electric cireuit, the observer could indicate with this, those parts of the fall which were uniform enough to be used in calculation. The procedure was then as follows :— The pitot was placed in a known position and the gauge was clamped and set at a reading which by the process of trial and error had been calculated to be right for the velocity required, and for the particular position of the pitot ona diameter. The gasometer having been drawn up was set free; when it had reached a more or less constant velocity, the gauge was opened and the line kept on the cross wire either by accelerating or retarding the gasometer by means of the handle H. The experiment was then repeated with the pressure gauge commutated. After a little practice it was generally possible to keep the the Value of the Pitot Constant. oom gasometer at the right velocity for the greater part of its fall, and the required rate of this was given by the average distance between the seconds marks on the paper strip. Determination of K,.—Having determined by the process of trial and error the distribution of pitot pressure across a diameter for a given mean rate of flow, the value of K, at that velocity can be determined from curves drawn i 4/ P as ordinates and positions of the pitot along a radius as abscissee, as mentioned above. Some examples of these are given in Curves II for mean velocities 400, 120, and 50 cms. Curves II. Seen ea ‘Oy = "O50 oO MMS Rosition of Pr bok per second respectively, P being measured in centimetres of water. The values for the ordinates of v=120 and v=50 are doubled and trebled respectively for convenience in plotting. The pitot in these cases was a circular metal tube of radius 0°59 mm. The results verified the general form of 358 Messrs. J. D. Fry and A. M. Tyndall on the curve given by crosses in Curves I, and it appeared, therefore, that ussuming that there was a source of error in the original work with this method, it had not been eliminated. It being desirable to check these readings in some way, attempts were made to obtain them at still lower mean velocities when the motion in the pipe would no longer be eddy but stream line. With stream-line motion the distribution of velocity across a diameter is parabolic. Assuming then that K is constant and of value 1:00 (as obtained by the centrifugal method), the curve of root pitot pressure across a diameter must also be a parabola. The forms of these parabolus of root pressure for various mean velocities (referred to in what follows as “ theoretical parabolas’*’) may be obtained by calculation and compared with the experimental curves. Pitot pressure readings, however, for mean velocities lower than 50 ems. per second could not be obtained with any accuracy, because the motion of the gasometer caused an irregular bending of the rubber connexions from the pipe to the pressure-gauge, which at low velocities resulted in fluctuations of the observed pressures comparable with the readings themselves. In the hope of obtaining stream-line flow for such velocities a one-inch pipe was substituted for the two-inch, but it was found that the motion of the gasometer was then too slow to be maintained regularly. The falling gasometer method was, therefore, abandoned, and the stream of air obtained by fixing the pitot and a surrounding one-inch pipe to the top of a 50-gallon tank and gradually filling the latter with water from the mains. The rate of flow of the air-stream was obtained from the rate of change cf the water-level in the tank, and this remained fairly constant provided no water was being drawn elsewhere in the building. The pitot pressures were obtained by the change in gauge reading on commutation. The absence of moving rubber and the more constant mean stream velocity increased the working sensitiveness of the gauge over fitty-fold, so that fairly accurate readings could be obtained to as low a mean velocity as 6 cms. per second. It was not possible now to allow for the presence of water vapour with certainty in calculating the density of the air in an experiment, because it was probably not accurate to assume that the air was saturated or that its humidity was that of the atmosphere ; this, however, was a small correction and negligible compared with the discrepancy investigated. As it was necessary to know whether the motion was really stream-line at these low velocities, the static pressure the Value of the Pitot Constant. 309 slope along the length of the pipe was measured for different mean rates of flow by means of the side tubes ¢,, t, &c. shown in fig. 2A. The results are given in Curve III. Curve III. 1-50 Pressure S lope in 107 3 yom ura ter It is not easy to say when the curve leaves the straight, that is to say when stream-line motion ceases, but it seems justifiable to state that of the various rates used stream-line motion was certainly present at v=6-3 cms. per second, possibly present at v=28°5, and not present at v=76. It is now possible to compare the experimental »/P curves with the theoretical parabolas for these three mean rates of flow. This is shown in Curves IV, where \/P is plotted with the position of the pitot on a diameter. The crosses are the experimental points and the dotted lines are the three corresponding theoretical parabolas obtained by assuming that K=1:-00 and constant. lt will be seen that in general there is no coincidence of an experimental curve with the corresponding parabola ; in fact in the former it is only by neglecting the readings near the side of the pipe that a smooth curve through the origin can be obtained at all, and there seems to be no reason why these readings should be neglected. The results at v=6°3 cms. per second, where the motion was certainly stream-line, show that the pitot pressures across Ht || Hl! 360 Messrs. J. D. Fry and A. M. Tyndall on a diameter do not fit the theoretical values, and are moreover too high to give a pitot constant of 1:00. The same is probably true of the other velocities, though the curves for v=76 cms. per second give no information on this point because the flow was evidently eddy. CurvEs LY. Wy rE cenhe of pcp] Sa SS SS ee 2 le 6 8 10 mms as, tron of Pat When working at the lowest velocities it was important to eliminate any errors due to temperature effects which might be comparable with the small pressures observed. Thus if the connecting tubes leading from the pressure-gauge to the pitot are not at the same temperature, a spurious pressure will be present which will only be eliminated by commutation if such takes place close to the pitot; at first it was assumed that uniform temperature was obtained when the connecting tubes were twisted together, but in all the later work the commutator was placed as near the pitot as possible the Value of the Pitot Constant. | 361 (about 6 inches beneath it). This change had, however, no effect on the results. Referring to the part PQR of the pitot tube and its, connexions in fig. 1, an uncommutated pressure could arise, if the arm P was at a different temperature to the arm R. Such a difference was possible because the air in contact with the water in the tank could quite conceivably differ in temperature from the general air of the room, in which case the arm P would then be affected during the flow. This was tested by taking direct and reversed zero readings of the pressure-gauge immediately after the flow had ceased, and before the temperature of the arm P could have changed appreciably. Sometimesa slight effect seemed to be present, but it was variable in its direction. It might possibly have caused some of the fluctuations which always occurred in the readings. Analysis of Results. Curves IV as they stand do not bring out the aa nature of the discrepancies between the observed and the ideal pitot pressures ; a simplification is introduced if, instead of taking the square roots of the pitot pressures, the actual pitot pressures themselves are compared with the squares of the corresponding points on the theoretical parabola—that is to say, with an ideal set of pressures For K = 1-00 and constant. The results of this analysis are shown in the following table, | r=0 eas) iret | pase | a6) |e 72 Gee 1-4 C2) el Mockmoce ios oan Pemsees) | oP. ...6 1-0 1:0 O07 OA enn 0:0 ome 04 | o8 | 03 | 02] 03] 03 | 030 peo, | Pee. 3:5 32 | 25 | 15 | 09 | 05 ae 3°1 27 | 20] 121 05 | 00 Ft aot 0-4 05 | 05 | 038 | o4 | 08 | 0-48 ei (Suh aon 6-4 59 | 47] 291 15] 08 Peel, 61 53 | 40 | 23 | 09 | 00 ee (03) | 06 | 07 | 06 | 06 | 08 | 0-65 ose 122 | 112 | 86] 56] 27] 12 ae 124 | 108 | 81] 48] 19 | OF v....../(—02) | 4) | (05)| 08 | o8 | 1-41 | 0-90 Pee Sey) Dee 195 | 174 |137 | 90 | 41 | 15 Bias 2003 | 175 |133 | 79 | 30] 02 Dee Oeyn COL) ie Os) Tad Pah pois be P17 Phil. Mag. &. 6. Vol. 21. No. 123. March 1911. 2B where P, stands for pitot pressures measured in thousandths i of a millimetre of water, P, squares of corresponding il parabolic pressures, and « the difference between them for five | different mean velocities. These are arranged in vertical columns, each giving the values for a given position (7) of the pitot measured from the centre of the pipe along a radius in millimetres. For the two lowest velocities, in which it will be seen from Curve III that the flow is stream-line, the value of zis roughly constant for all positions of the pitot and increases as the mean rate of flow through the pipe increases. This is true also for v=15°6 cm. per sec. (if the reading at r=0 is omitted). F v=22-0 and e=28°3 this does not appear to be the case, although it may be so for positions close to side. Now Curve IIT shows that the statie pressure velocity line leaves the straight gradually, that is to say the transition from | stream-line to eddy motion is also gradual. It would appear, | ! 362 Messrs. J. D. Fry and A. M. Tyndall on ——— therefore, that with increasing velocities of flow, eddy motion, which must first appear at the centre of the pipe’s section, gradually spreads towards the walls. Consequently it is reasonable to suppose that, at the velocity »=15°6 cms. per sec, stream-line motion existed in the pipe except near its centre, but that at v=22 and »=28:3 eddy motion had | extended to a radius of at least 5 mms. This effect would account for the low and sometimes negative readings of w in SSS SSS = Se TE i that region. In these cases, therefore, we may assume that i the value of w near the side is that which would have He held across the pipe had stream-line persisted. The average values of « (x) for different velocities are given in the last column, those values of « in brackets being omitted on the eddy motion assumption in taking means. ‘The relation between Z and the mean rate of flow of air through the pipe is shown in Curve V, where the values of 2 are plotted as ordinates and rates of flow as abscisse. Considering the extreme smallness of e—the largest value obtained being only just over a thousandth of a millimetre of water—the results are fuirlv definite and show that @ is proportional to the mean rate of flow—at any rate at these low velocities. The authors have not succeeded in obtaining a satisfactory explanation of this phenomenon, but the following con- il Wetec present themselves :— "t The x effect could not be accounted for by an un- a. o a lel . Wl ea an temperature effect such as is mentioned on Hit! page 361 because the difference of pressure which such would Mii! produce would, unlike x, be constant for all velocities of flow. ny H| the Value of the Pitot Constant. 363 2. When a pitot is placed in a non-uniform flow the pitot pressure which is measured is an average taken over the Curve V. to) 20 30 Cmsec! mean Velocity of flow area of the end of the pitet, and differs from that at its centre by an amount depending upon the area of the pitot and the distribution of velocity over its end. The amount of this error was calculated assuming that the velocity changed linearly across a diameter of the pitot, but it was found to be quite negligible at low rates of flow for the size of pitot used. 3. It seemed feasible that superposed on the main disturbance of the flow in the neighbourhood of the pitot there might be a smaller secondary disturbance which, if it depended on a power of the velocity less than the square, might become comparable with the observed pressure when the velocity was very low, even though it were negligible at higher velocities as seemed likely trom the work of other experimenters. Definite evidence, however, has been obtained against this view for the case of the round pitot used in the above. Thus the pressure readings were unaltered by threading over the pitot a piece of rubber tubing so that the whole pitot stem was thickened about 4 diameters. 2B2 364 Messrs. J. D. Fry and. A. M. Tyndall on 4. Lastly, the presence of # might be the result of a slight suction at the static pressure tube. It is well known that if a piece of thin-walled tubing is placed in a stream so as tolie with its long axis perpendicular to the flow, such a suction does occur, and it may even amount to as much negative pressure as a pitot placed at the same point would give of positive pressure. It has, however, generally been supposed that this is removed by either providing the tube with a wide flange at its mouth or by placing it, as in the present work, with its end flush witn the inner surface of the pipe. In support of this there is the work of Heenan and Gilbert (Proce. Inst. Civil Eng. vol. exxiii.) and Threlfall (doe. cit.), and also the fact that in the present work K is approximately the same as K, within the range of velocity used by most experimenters. It is conceivable, however, that suction is not completely eliminated in these ways. If, for instance, a residual suction is present which is a function of a power of the velocity less than the square, it might easily only be comparable with the pitot pressures at very low velocities, and might therefore have escaped previous detection. For instance, making the very arbitrary assumption that is proportional to v at all velocities—the values of K calculated by correcting all the pitot pressures accordingly does not differ more than 1 per cent. from 1:00 throughout the whole range from 6 to 2000 cms. per secend. On the other hand, if there were suction at the static pressure tube, a change in the size and shape of that tube mizht be expected to affect the value of @ But no difference of pressure was observed when the gauge was connected to any two of the following holes made at the same level in the side of the tube—circular holes 1 mm. and 4mm. diameter, rectangular holes 1:5 x3 and 3x15 mm. respectively. On the whole, therefore, the suction hypothesis does not seem feasible, though the fact that @ is proportional to the rate of flow drives one to locate the effect at the walis of the pipe and not at the pitot itself. It is possible that experiments on the pitot pressures in pipes of different sizes may throw some light on the subject. Pitots of different shapes. Before it was realized that the pressure readings across a diameter were not in accordance with theory, an attempt was made to examine the motion of the air at the side of the pipe. Even when eddy motion is present the flow close to the the Value of the Pitot Constant. 365 walls must still be stream-line and must break up into eddies at some unknown distance from the walls. It seemed that the place where this change occurred and its relation to the mean velocity of the air, could be determined by using a pitot which was narrow enough in section to approach the walls very closely. The above results show this is not possible unless the pressure readings are corrected for the « effect, which in practice is impossible to do very accurately ; but at the same time, the attempt brought out a new and interesting fact. The results of previous experimenters with various pitot tubes have led to the generalization that the size and shape of a pitot has no effect upon the value of the pitot constant K. The authors found, however, that a very fine pitot does not show the same pitot pressures as a larger one say 1 mm. in diameter. For instance, values of the pitot constant at various velocities have been taken with a very fine rectangular pitot 0°2 mm. by 2°0 mm., and at high velocities this gives a mean value for K of about 1:10. This pitot was obtained by flattening the end of a brass tube and grinding down its edges to a thickness of 0-05 mm. A pitot with about the same opening 0°1 mm. by 2:0 mm. bnt thicker walls each 0:2 mm. gave, however, a norinal value K=1-00 at high velocities. A circular glass pitot 177 mm. diameter and with walls °027 mm. in thickness, also gave a value of K several per cent above unity. Now close to the edges of the pitot, the pressure due to the current of air will fall off rapidly and will be considerably less than that at its centre. This edge effect has been thoroughly investigated by Stanton (loc. cit.) for the case of plates placed in uniform currents of air. In an ordinary pitot it has no effect on the observed pressure, which is the average pressure over the internal area of the end of the pitot tube, because either the walls of the pitot are relatively thick or its area is considerable. In the above pitots with high constants, the area influenced by this edge effect was evidently a measurable fraction of the whole area and the resultant pitot pressures were, therefore, all too small. In support of this view the authors found that if a small piece of mica was fitted on flush with the end of one of these pitots so that the pitot was then a circular plate 2 mm. diameter with a slot at its centre, the value of K obtained was normal 1-00. Further, when the wedge pitot was touching the side of the pipe, the pitot pressure was actually greater than when it 366 On the Value of the Pitot Constant. was 0'1 mm. from the side. This was evidently due to a change in the disturbance of the flow ; in the former case the flow was restricted to one side of the pitot only, while in the latter it was more uniform. This effect was still more marked in the case of a pitot formed by making a hole in the side of a tube stretching across a diameter of the pipe: the flow past such a pitot close to the wall of the pipe is still further modified, Owing to the constriction introduced into one side of the gauge connexions by these narrow pitots, it was difficult to obtain accurate results because the two sides of the gauge never filled up at equal rates ; a small disturbance, such as a slight change in velocity or a sudden draught, thus often produced a temporary displacement of the gauge-surfaces which, at low velocities, was comparable with the pitot pressure itself. In fact all the low velocity work, even with the large pitot, could only be undertaken on days of atmo- spheric calm. ‘The disturbing effects were decreased, but never eliminated, (1) by placing a capillary tube in the circuit on the other side of the gauge, and (2) by making the _ volumes of the connecting tubes and of the air spaces in the gauge as small as possible. Many more readings wiil, however, be necessary before a quantitative discussion of the readings with wedge pitots can be presented, but the results as yet obtained seem to show a similar apparent drop in K, with decreasing velocity. SUMMARY. 1. The value of the pitot constant obtained by a centrifugal method is 1:002 and is constant between velocities 60 and 1400 cms. per second. 2. The method of determining the pitot constant by measuring the distribution of pitot pressure across a pipe is unreliable at small velocities. 3. If a small correction proportional to the mean rate of flow through the pipe is added to all the pitot pressures across a section of the pipe, this method also leads to a value 1:00 between the limits 6-2000 cms. per second. 4. A very small pitot possessing very thin walls gives in general a pitot constant several per cent. above unity. This is readily explained in terms of effects known to be present at the edges of plane surfaces placed normal to a current of air. Ranson. XLII. On Restricted Lines and Planes of Closest Fit to Systems of Points in any Number of ghee on By K. C. Snow, W.A.* STATEMENT OF THE PROBLEM. FPXHE theory of the lines and planes of closest fit to systems of points when no restriction is placed upon those lines and planes has been developed by Prof. Pearson in various papersy and is of frequent application. The connexion of these lines and planes with the formule of the theory of multiple correlation is indicated in those papers. If the criterion of “closest fit’’ is that the sum of the squares of the deviations from the line or plane measured in the direction of the “dependent” variable is to be a minimum, the equation of the line or plane is identical with the corre- sponding multiple correlation formula. If the sum of the squares of the deviations measured at right angles to the line or plane isa minimum (and this, from the purely geometrical point of view, isthe more satisfactory criterion), the result is not of sucha simple form, but the determinant from which the mean square residual is obtained is similar to the multiple zorrelation determinant. While working on certain vital statistics, it was desired to obtain a formula connecting the ‘‘dependent” variable with the “independent” variables when the values of all the variables were known at the beginning and end of a certain range, and the correlation between “ dependent ” and “independent” variables at all intermediate points was amaximum. ‘Thus, if zp denote the “dependent” variable, EUINOL Be Bey 5 oi tel the ‘“‘independent”’ ones, we require to make the correlation of «#) with 2, %,...2@, a Maximum, with the condition that when a, &,...a, take up the values p11, Por, +++ Pnis Pic) Po2-++Png Yespectively, x9 is to take the values jo, and pp. A similar problem occurs in certain branches of Physics, especially in connexion with solutions and alloys. A property—e. g., the freezing-point—of a pure substance may be definitely known, and it is required to investigate the behaviour of that property as certain amounts of some other substance or substances are added. Fixed conditions will be imposed upon the law which is to be investigated by the known properties of the pure substance. The law, then, “ Communicated by Prof. Karl Pearson, F.R.S + See Phil. Mag. Nov. 1901, pp. 559 et segq. ; Phil. Trans, vol. elxxxvii. A, es 301 et seqq. —— = = eS ee ee ee se SSeS ———S—= TAS Sa 368 Mr. E. C. Snow on Restricted Lines and has to give the best fit to the observations made of the property as definite quantities of the other substances are added. Two examples of such cases are given from the figures of certain alloys (§7 below). The idea is capable of generalization, and the theory for the general case will be investigated. Looked at from the point of view of correlation, we shall require to assume a linear law connecting x with a, #,...%n, and shall make the sum of the squares of the deviations of the actual observations from this linear law measured in the direction of £) a minjmum, making use of the exact conditions which are imposed on the law. This will be first investigated. Buta better geometrical fit to the observations will be obtained by measuring the deviations perpendicular-to the ‘ plane” given by the linear law, and this will be worked out sub- sequently. ANALYTICAL INVESTIGATION. First Method. 2. Let there be » “independent” variables and (£+1) conditions connecting them with the ‘‘ dependent” variable. (k+1) is necessarily less than n. Measuring from one of these fixed conditions, we can assume our law is Xo — A,X, + ApXy+ wi idkens + AnXn, ° e . e (1) with the & conditions Por = 4 Prt ePort 26+ HAP, 7 Por = % Pig bp Pog t+ «+2. +FEnPna, (2) Pok = Put AoPat -..> +AnPur- J Then we want to make Vv = S (29 — 44.4] — Apa — eeecd =the eae the sum of the square of the deviations in the direction of 2, a minimum, subject to the above conditions. Hence we must have 0 = S(vy— a2, — ApXg— e@eoe — Ay &n) (x, . da, + He) dap + sso +, + ay. dag with the conditions Og nO@lnc alOa so<. st Dpewda. (S == 12a k.) <4) / 7 ae a Planes of Closest Fit to Systems of Points. 369 Multiply equations (4) by As (s=1,2,....%), and add to (3). Then, by the ordinary theory of maximum and minimum values, we know that the coefficient of dat (t=1,2,....n) in the equatior so obtained must vanish: This gives the n equations Pa + Az Pre eS ketene + Nx Pix = Sxi(a— At — dg%g— 6... —AyXn) oar Roz— a, Ry,—a, Ry; — ese — din Rt, ea wie (5) ino e epi, tor all positive intepral values of wu and v. (5) gives n equations connecting the (n+) unknowns, ae. An, NaaAgs2- Ak. (2). gives, &) other equations between the a’s, in which, however, the X’s are absent. Thus we have the following set of equations from which to determine the a’s and the X’s : — QRytagRoe+ .... tanRutripatrAspet .... trAcPHa = Roe. ie Ruel (i=1, ) - Q Pist AgPos+ ..-. +Onpns = Pos (seh 2c ie. ) Let A denote EY) LS cial ate Po P02 ++++ Por Ro Ee see Ey Pil Pig -*e6 Prk Ron Jatsys 0 0"0 Ey, Pni Pn2 «+++ Pnk Po. Pu eeee Pri (0) (0) O00 0 ; Po2 P12 o2ee Pn2 0) ) cece 0 e Poe Pik «+++ Pnk OMA MERA ca HO) a determinant of the order (n+h+1). ‘Then the solutions of equations (6) are : Aw “ —_ _ —_—_ (es ll 2? eaeae 1 a Aoo ( ang ") ©) ane Ag = — Aeute.o (Seu 2. s e@ & @ k) ) sd e (8) Ao, 370 Mr. E. C. Snow on Restricted Lines and A,» being the first minor of the constituent of the (w+1)th column and (v+1)th row, and being positive or negative according as (w+v) 1s even or odd. Thus the coefficients in ihe required formula can be found at once by the evaluation of a number of determinants of order (n+k)*. Particular Cases. 3. The simplification of the above results in a few par- ticular cases will be useful. (A) k=0.—In this case we have a plane passing through a single fixed point and closest fitting to a system of points. Here all the d’s disappear, and the equation of the plane becomes Ly = AX, + Agkgt «1... +AnXn, where A being line @: » ©. Ke. y 6: fie” Je, ke sec ve 6 ne R,,» being equal to R,,, and is the sum of the products &y.&, taken throughout the system of points. If z,; and o, be the mean and standard deviation of 2,, and 7,, the correlation between the coordinates x, and a@, we _ have ie — S (uty) = No,Ocw -~ Nz, 2p if U = = vy and RK, = Neo,?+Nz 1 a=». The analogy between A and the determinant used in the Od * It is not difficult to show that, by first finding the a’s in terms of the d’s from the first x of equations (6), and substituting these values in the last / of the same set—thus giving / equations for the \’s,—the a’s can be found in a form involving only deteiminants of order nm, though the number of determinants it is necessary to evaluate is increased. If & is large, this increase is considerable. The general result in this form will not be given, but it is exemplified in a particular case below ($ 3). Z Planes of Closest Fit to Systems of Points. oud theory of multiple correlation is now clear, for a; can be / written ———, where A’ = 2 L @ LoLn = o%1 1+—, Miko fee Ton + awe 0 0091 Ton Be: ae i 2 oe Lor} vy x Mor -+ 2 dee mas | ae _ 9 ° LyXg Vs ss Yo2 ate Fey Sy, eh ey set ree il + Taye 3 ae ee 7 Ton + SRD ST SSirentiey) pei se, Vel -e)hce e e¢ @ @ e aL + TES 0Fn On | Thus this determinant can be derived from the multiple correlation determinant by increasing 7 in the latter by V,. Vz, and by increasing the constituents of the leading term by the corresponding V,?, where V, and V; are the coefficients of variation of the coordinates x, and a. If the fixed peint is at the mean of the system of given points, A’ becomes at once the ordinary multiple correlation deter- minant. Putting n=1, we derive the two-dimensional case of a line passing through the origin and giving closest fit (mea- suring in the direction of y) to a system of points. ‘he equation of the line is easily seen to be Ryo ie S@eg) YS Ree ee eaaNg aunts . ° ° e (a) Putting n=2, we reach the three-dimensional case of a plane passing through the origin and giving closest fit (measured in the direction of 2) to a system of points. Its equation is RyeRoo— Ro Roo RR — RoR RyRgo eat Rys” Riko» —R,.” S(ay) .S(yz) —S(awz) . 8(y’) S(ay) .S(vz)—S(yz) . S(2?) ) SY") — {Bley) For values of n>2 it is more convenient to derive the coefficients direct from the determinant, and there is no need to write them in full. (B) k=1.—Here we have the case of a plane-—in n dimen- slons—passing through two fixed points and fitting most ut (8) 342 A Ron Eves op tio ss +s Rie Pia ese Mr. E. C. Snow on Restricted Lines and closely a system of other points. We have in this case R,0 Po e Ry pr race wees Ran Pn Po Pi . Pn 0 Po» Pi» +++» Pn being the coordinates of one fixed point relative to the other, which is taken as origin. The coefficients in the required equation can be found from the above in the form of determinants of order (n-+1). But in this case the first n equations of (6) become ayRyt cece + anRut = Roe— Ape C=. i): Solving these for aj, d,....@, in terms of X, we have a _ Sor Gan On, where i Roo Ro » Ro, Ro Ap, Ru Saves Ron APn ap 0 Ran | =o Noe where 3 = Roo Ro ° Ron Ro Ru Se Ron Rin . lay and 3” —s i Roy ° Ron P Ry 5 Bip P2 ; Pn ini . ies so that : Oo . | a= — Sou Oat (eat 2 san ee : ae ‘eg Planes of Closest Fit to Systems of Points. 0) When these values are substituted in the equation giving the fixed condition, viz. Po = HPit «2+. £A:pit --.- +anpn, »X can be found, and from this the a’s can be completely determined. This method will be used in an example below. No other particular cases of this method need be worked out in detail. We see that it is always possible to obtain a plane in n-dimensional space to pass through any number (less than n) of fixed points and to be such that the sum of the squares of the deviations of any number of other points from the plane measured in a fixed direction is a minimum. Second Method. 4. We have now to investigate the equation of the corresponding plane when the criterion for “closest fit” is that the sum of the squares of the deviations from the plane measured at right angles to the plane is the least possible. From a purely geometrical point of view this will give a closer plane to the system of points, but it will not give the regression of one variable on the others. Let J,..../, be the generalized direction-cosines of the plane, and take one of the points through which the plane has to pass as origin. Then the equation of the plane is Lm ie ait eal ies es 8 co Ba ee ee Xe) the total number of variables being taken as n for convenience of notation. There will also be the conditions Lips crates “te Unga meme eat). 2. ))ica | Gl 0) and by Pit leport+ ceo. tetra = om Li prot ly poo + eee. +l Pno-= 0, Ly Pre + lo pox + eoee flak = 0, J (k+1) being the total number of fixed conditions. The criterion for closest fit is ve S(l,a)+ Rh sre ats ee to be minimum, subject to the above conditions. Hence O=SCait .... Hhan)(aydly+ .... + andl), together with cay) Lidl, a oko kbar Lndln = 0, pudl; Sates alleg + Dridln = 0, pirdl, + ere + Dg lr = (), 374 Mr. E. C. Snow on Restricted Lines and The conditions for a minimum give Xy toe + py put lePet ..+. bee Pu = 0, RM er bel tose ete a Vo. ; (12) Xn t+ Mn + py Prat M2 Pra oes Fhe Par = 0, where 4, j4,.--. yuk are undetermined constants, and De Oak ace bln ln) dis pecs / fr) ase Conner eV ay Pa see where, as before, ie. — Se Ga = liven Multiply equations (12) by 4, l,,...., respectively, and add. Remembering the conditions (10) and (11), we at once obtain +l, X,+loXo+ eeee +1,X, => 0, where J,....J, have the values which make V a minimum. But then /,X;+(,X,+ .... +1,X, becomes Vn, the minimum value of V. It follows that i ay Substituting this value of w in (12), we shall have with (1J) (n+) equations between /;....1,, wy.... x and V,,. Hence we can eliminate the /’s and the y’s and obtain an equation to determine Vn. Since . XR eae ee this eliminant can be written in the determinantal form Ru-—Vn Roy ween 0 Rat Pu Piz «+++ Prk D == itias Roo— Vin écece Rae P21 P22 areuete P2k Ran Ren coos Rin— V;, Pri Pn2 osee Pak Pu P22 ecee Pal 0:0 ee ee Q Pi2 P22 soon Pr2 0 0 2a Pik P2k ceee Pnk 0. 0 Gee This is an equation of the (n—k)th degree in V”. Its roots are necessarily positive (being the sum of a number of =e Planes of Closest Fit to Systems of Points. ato squares of real quantities), and’ the smallest of these must be taken. When this value of V,, is substituted in (11) and ae ; (n+k) homogeneous equations in J; l,.... ln, py Me - are obtained, Teas 1) of which, together Sieh nA). ae to determine all the ?’s and the [l’s. ~ As before, this can be done in determinant form, the order of the doheninans involved being (n+k—1). Particular Cases. >. Useful particular cases of the general formula are obtained by taking k=0 and k=1. (A‘) If k=0, we derive the case analogous to (A) above. The equation te determine V,, takes the w am known form Rii— Vin Ro; ahve (ef, it JBtag Roo Ve, eeee lies C ° > — 0. Rin Ren eieireice Rin— Vin Putting n=2, we have the two-dimensional case, and V, is the least root of the quadratic Rui Vin Roy | ay 0 i Ris Roo— Vin | a i so that 2V,, == (Ri + Roe)—{(Ru— Ro»)? +4R.” a , aS since Ii) == Ee,. 2(Ry—V,,) = (Rii— Rep) +{(By—Ry)? + 4Ryp ae Put Be a OR Cog) sim Then 2? 2(Rii— Vn) = p(1+cos @) = 2p cos? 5 and 2Ri2 = 2p sin € 603 Z The first of equations (12) now gives l, cos = e 5 tbs sin °= 0, since the p’s vanish when k=0. 376 Mr. E. C. Snow on Restricted Lines and The equation of the line, therefore, is ; 6g x sin Fy 0s sag Oe where DAliwe tani ee ee Ry — Ree 25 (ay) and the notation is altered to agree with the usual form. This value of tan @ gives rise to two values of - each less than 180°. In a particular numerical example, however, it is not difficult to pick out the value required ; while it can be verified that the other value corresponds to the “ worst- fitting ”’ line. In cases of n>Z it is better to substitute the values of the R’s direct in the determinant above, and to find V,, by the usual methods of approximating to the roots of an equation. (B’) If k=1, the equation to determine V» is ee Ru-— Vin Roy cuekeiia Rut, P1 ca Rye (Roo— ey rece Rw, Pa nee ee | Ri, Ron SOO 6 (Ran— Vin), Pn Pi po Pete fae 0 the origin and the point (p;po....p») being the fixed points. In this case, equations (12) take the form L(Ru— Vin) == 1,Roy+ Scis ae L,Ryi = —Npr, L, Rin a lpRoa+ eieie ssh [Rn —Vin) = — Ap. Hence J; is proportional to d;, the first minor of the con- stituent in the ¢th column and bottom row of d, and V,, is given the value which is the least root of d=0. Using (10), the actual values of the /’s can be found. 6. It will be seen from the foregoing analysis that the work involved in determining the “ closest fitting ” plane by Pianes of Closest Fit to Systems of Points. (3F7 the second ¢riterion is much greater, at any rate in all cases ‘of n> 2, than that necessitated by the first-criterion.- In the second method, if (n—*) is three or more, the smallest root of an equation of the third or higher degree has to be approximated to. This in itself is no light task, and is not necessary in the first method. The methods do not necessarily lead to results at all alike (see Example 3, below), and only the terms of the particular question in hand can decide which method is to be used. The second gives the best geometrical fit, considered in a direction perpendicular to the plane. The first gives the “ regression’ plane—i. e., the most probable value of one variable in terms of the others. This is the most frequently needed in practical cases, as is exemplified in Examples 1 and 2. 7. ILLUSTRATIONS. -- J, The second column in Table I.* gives the temperatures (Centigrade) at solidification of a series of alloys of iron and TABLE I. als 25 Temperature b Temperature b orc. che aoe a ety lst method Ea 2nd method ne prccentl aad ReNoa: nearest degree. nearest degree. 02 1470 1501 1501 2, . 1470 3483 1483 16 1465 1476 1476 a7 1450 1474 1474 ‘24 1448 1461 1461 °38 1416 1436 14386 - D3 1404 1409. 1409 ‘61 1394. 1395 1394 °80 1351 1360 1359 81 1351 1358 1357 1:31 1286 1268 1267 151 1244 1231 1230 1°85 1179 1171 1169 2°12 1110 1122 1120 2-21 1107 1105 1103 carbon. The percentage of carbon in the various alloys is given in the first column. The solidifying temperature of pure iron is 1505° C. Any curve, therefore, which attempts _* The table is taken froma paper on “The Range of Solidification and the Critical Ranges of Iron-Carbon Alloys,” by H. C. H. Carpenter, ‘M.A., and B. F. E. Keeling, B.A., in the Journal of the Iron and Steel Institute, No. 1, for 1904. Phil. Mag. 8. 6. Vol. 21. No. 123. Mareh 1911, 2 C Sa ane ie eee ee — ee eT Se a SS i et Ti SE» 378 Mr. E. C. Snow on Restricted Lines and to express a relationship between the percentage of carbon present and the solidifying point of the alloy should pass through the fixed point (0 %, 1505°C.). Up to 2 °/, of carbon a straight line seems to be the most likely fit. The two methods will therefore be applied to get a line to pass through the point (0 °/,, 1505° C.) and to fit the series of observations. It will be seen from the figure that up to a percentage of carbon of 0°5 °/, the results are ir- regular, but from that point up to 2 °/) the irregularities are small and within the limits of experimental error. Measuring 2 positively from zero and y negatively from 1505, we find Se yo OG: S(zy) = 3438-41, S(y?) = 624986. By the first method the equation of the line (measured from 0 %/, and 1505° C.) is = "a 2 io) fone 2) — 180-30ien: wv E The relationship between the temperature of solidification and percentage of carbon present is therefore T = 1505 —180°801 z. If we apply the second method, we find tan @ = —°0110035, whenee DS —==089" 41’ yp -23 2 SS and tan : = 181°80843, which gives the relationship T = 1505—181°808 2. The actual temperatures obtained from the two formule are given in columns 3 and 4 of Table I. _We see that, to the nearest degree, there is no difference in the results up to 0°5 %Jy of carbon, and the difference beyond that point is small. The two lines cannot be distinguished on a diagram of the size shown. ‘The line OP in the diagram represents, Planes of Closest Fit to Systems of Points. 379 therefore, the best fit to the system of points of a line through O by both methods. +P II.* In this case we will take the figures of an alloy of three metals—copper, aluminium, and manganese. ‘The percentage of mangatese present varies from 0 to 10°24, and of aluminiuin from 0 to 7-40. Itis not possible to tell from the mere figures if the distribution is approximately coplanar or not, but the material seemed good enough to work upon. The freezing-point of pure copper is 1084° C. ; in this case, therefore, we require to find a relationship of the form T—1084 = p.xc+q.y, where T is the temperature at solidification of an alloy con- taining v°/y of aluminium and y°/, of manganese. Taking our origin at zero percentages of aluminium and manganese and 1084°C., we require to obtain a plane through the origin and fitting most closely the observations in the second, third, and fourth columns of Table II. * The figures for this example were taken from Table 43 (p. 229) of the “Ninth Report to the Alloys Research Committee’? to the Inst. of Mechanical Engineers, by Dr. W. Rosenhain and Mr. F. C. A. H. Lantsberry. 202 t | iM 380 Mr. E. C. Snow on Restricted Lines and ; i tesie TT: Actual | Deviation | Percent elP reentage | Deviation Deviatio | vi reentage| Pe v eviation i Freezing-| of Temp. amo - of 5 7 met a from ae iy from } Point. from |Aluminium|Manganese| ~°7""? | Actual Orn eerual . i °C. | 1084° C.|° present. | present. (y). Vemp. 7Ke): Temp. | 1077 7 1-19 114 10678 | — 92 | 10677 9) 93 1045 39 1:37 2°75 1053-2 + 82 1052-9 + 79 1051 33 1:04 4-92 10366 —~14°4 10866 | —144 1023 6) 1-42 5:38 1030°7 + 77 1030°2 -+ 7:2 1015 69 0°94 6:48 1024-0 + 9:0 1023°4 + 84 1007 17 1-43 dia 10108 + 33 1010°8 + 38 1011 | 73 O91 8:16 10121 + 11 1009-2 =— 18 985 99 plod 10°24 987-2 + 22 985°9 + o9 1075 9 BIOS Wh eteeaecs 1069°3 — &7 1069-2 — 58 1057 27 2°36 1°95 10546 — 2°4 10544 — 26 1045 39 2°31 3°82 1039°0 — 60 1038-6 — 64 1010 74 Died 7-80 1006-5 — 35 1004-2 — 52 996 88 2°37 9°76 988°4 = RO 987°5 — 85 1059 25 3°26 0:97 1058-0 — 10 1057°8 — 12 KOA ake 2 371 2°97 10886 — 34 1038'2 — 38 1043 41 3°93 2°99 1037-2 — 58 1036°8 = 62 1024 60 3 29 4°80 1025°4 + 14 1023°1 + 09 1022 62 3°57 564 1016°7 — 53 10161 — 59 1015 69 3°08 6:88 1008°9 — 61 1008:2 ears 5) 89 3°95 7:95 9951 + 01 994-3 — U7 1067 Lie ASO dpiweealiplatseleroe 1058°5- — 8&5 1058-4 — 86 1054 30 4°14 Meee 1046°4 — 76 104671 — 79 1031 53 4°62 3°26 1031°1 + 01 1030°7 — 03 1036 48 4°48 3°86 1026°8 — 92 1027°4 — 96 987 4 451 (Rie 993°3 | + 68 993°0 + 6:0 975 109 412 9°60 980°2 + 5:2 979-2 + 42 1050 34 5°21 0-98 1047°3 — 27 1047-0 — 30 1035 49 5°66 2°58 10312 — 38 1030°8 — 42 1018 66 5°21 4°86 1014-4 — 36 10138 — 42 1001 &3 5°86 6:00 1001-2 + 02 10055 — 05 998 86 mld 6°72 998-7 + 07 998-0 0:0 986 98 5°62 8:50 982°7 — 33 980°4 — 56 | SiGe Bn) 2S 5°99 9:50 970°8 — 52 969°8 — 62 | 1025 59 6°88 0:98 1038°1 +13:1 1037°8 +123 | 1030 54 6°54 1:98 1031-5 + 15 1031+1 + Il | 1022 62 6°29 3°50 1020 0 — 20 1019°5 — 25 fs s28) 89 6°62 .-482..-1 10070 | +120 1006°4 +114 | 978 106 6°26 8:02 9R1°9 + 39 981-0 + 30 DS ealy L 6°91 9:06 969°5 +12°5 968°5 +115 1042 | 42 740 ee 10436. |: + 16 10434 + 14 If z denote deviation of the temperature from 1084° C,, we find | i ss 8(a) = _779°1838, — S(yz) = 15421280, i : OB’) = 13360155, a | S (ez) = 10617-960, i ~~ §(22) = 190296°0, S(ayy= 751-017. Planes of Closest Fit to Systems of Points. ool Using equation (8) above, we quickly reach : 2 = 54605 a+ 84732 y, and therefore T = 1084—5°46052—8°4732y. . . . (y) The values of -T obtained by this formula are given in column 5 of Table II. The differences between these values and the experimental results are shown in the next column. It will be seen from the figures that the fit is a good one except at the ends of the range. Had the last seven obser- vations been omitted, 7. e. had the amount of aluminium present in the alloy been less than 6 °/o, a linear Jaw suchas. the above one would have agreed quite well with the observed ~ results. “As the authors of the original paper state that “‘the precise temperatures given in the table possess no very great significance,” it seems quite reasonable to assume that the observations, up to 6 °/) of aluminium, follow a linear law. The sum of the squares of the deviations from the observed temperatures in this case is 1641:0210, and the ‘‘ root mean square” is 6°41 *, The sum of the squares of the deviations: measured perpendicular to the plane can be obtained from the above figure by dividing by }{(5°186)?+ (8:392)?+1}, 2. €. 102°6120. It is found to be 15:9924. ~ When the second method is used, the equation in V,, is Howto AT One 10617:96 751-017 1336 OLD VE) Was d2ie ae = Oe | 10617-96 15421:23 190296 —V,, This when expanded becomes Vin— 192411 V7, + 52425892 V,,— 786607941 =0. We want the least root of this cubic. It is quickly seen to be in the neighbourhood of 15, and by successive approxi-. mations is found to be Vn = 15-9362, very nearly. * The second decimal place was taken into account in finding this figure. ‘This was done in order to compare with the results of («), which do not yreatly differ from (y). 382 Mr. E. C. Snow on Restricted Lines and If le+my+nz = 0 j is the equation of required plane, the equations to find the i ratios of 1, m, and n are i 763°1981-+ 751-017m+ 10617-960n = 0, 4 751-0171 + 1320-079m + 15421-230n = 0. iW From these we find i i l m n 54903 85582 —1" The equation of the plane is 2 = 549082 + 8°5582y or Tae a eS ++ - ---~ aca and | T = 1084—5°49082—8°5582y. . . . . (€) The temperatures given by this formula are shown in column 7 of the table, and the deviations from the observed values in column 8. They do not differ greatly from the results given by (vy). The sum of the squares of the devia- tions in the table is 1658-9718, which is, of course, greater than the corresponding number given by the first method. The “‘ root mean square ” is 6°44, not greatly different from the first method value. Also ?+m?+7n? becomes 104°3923. The actual sum of the squares of the deviations perpendicular to the plane is therefore 15:8917, which is less than the value given by the first method, as it should be, but is not a very great improvement on it. Thus in this example, as in the last, the two methods lead to very similar results. III. For a third example we will take the case of a plane in three dimensions to pass through two fixed points and to be closest fitting to a series of other points. The data for this case are taken froma railway time-table. The two fixed points are two terminal stations, and the variables are «, the distance (in miles) from one of these stations to some other station ; y, the scheduled time (in minutes) allowed fora train between those stations; and z, the first-class single fare (in pence) between the stations. ‘The figures are :— -—-—- = 2 a es eC ee SS eee SSS. 7S es ee aT ae a ee | ae Ye S, Hi 30 10 ie) 69 |} 52 80 117 mn 60 97 135 i) 69 145 156 Wt 81 136 182 hi 100 164 224 Planes of Closest Fit to Systems of Points. 383 the corresponding figures up to the other terminus being 114, 187, and 244 respectively. The figures should be expected to be fairly coplanar, and any formula obtained to represent them ought to give a good “fit.” Four cases can be worked out here, viz. those obtained by making the sum of the square of the deviations in the directions of 2, y, z and perpendicular to the plane respectively a minimum. We find : S@?)= 28526, S(yz) = 105264, Sy? )= 16827, S(zx) = 64160, (ee la S(ay)= 46801. For the best fit in the direction of z, the equation of the plane will be 2 = axt by, with the condition 224 = NAG VSO ae ee) (Dh In this case we have A 1 64160-114% 105264-1872r 64160-1142 28526 46801 105264-137 2 46801 16827 Using the relations Aro As GQ=——, b=—= Aoo oo we obtain a = 2°2376 +:°00526 2X, 6b = °90706—:00077 2X. Substituting in (), % becomes —27:2290, and therefore @ ==) 20943. = O28. ana the best fitting plane in the direction of z is 2= DOM Se- O28lynals ye 3 (8) In a similar manner we find the best fitting plane in the direction of y is Pe O D2 (ait COSOS. a yest an tory nin GOD and in the direction of « it is er Oy Oneal 4 Ce eR) 384. - Mr. E. C. Snow on Restricted Lines and When the deviations are measured in a direction perpen: dicular to the plane, the equation to determine V,, (the least’ sum of the squares of these deviations) is, by § 5 above, 0 114 187 244 114 28526—V,, 46801 64160 bd 187 46801 76827—Vin 105264 ae 244 64160 105264 144311—V,, the determinant being reversed for convenience in evalua- tion, 2.é. 107501 V;,—19413930V,, + 191936824=0, giving Vix= 10°49664. Then — vu da 1 114 187 244y 114y 28515°5 46801 64160 187 46801 76816°5 105264 244y 64160 105264°°->" J4430iE5 and 7 | mane. _ __ do WAAL eh gan n(Zy ha, b=, b=, where Let lay + lsz = 0 is the equation of the required plane. Since only the ratios of 1;, l.,.and /; are required, it is sufficient to find dj, do, and dz) (each of which contains pw asa factor). When we find these ratios we must divide each by {li+ 12+ 12}? in order to have the sum of their squares unity. In this way we find the equation of the plane is *88430—"4632y—-0582-=0. . . « (¢) The deviations of the results given by the formule (6), (f), (€), and (@) from the actual values are (the deviation being positive when the formula gives a value greater than the actual value) :-— Planes of Closest Fit to Systems of Points. 389 (@) (Gina, ©: (2) (?) Deviation Deviation Deviation Deviation in direction in direction in direction perpendicular of z*. of y. of 2. to plane. ~ —4:795 + °227 + 285 — 181 —5°850 +5:319 — 2:200 +2°122 | —6°619 + °445 — ‘178 + 275 —8°265 —1:°787 + 1°592 —1°325 — 8:544 —3101 +2:281 —1°952 —9:966 + :070 + °796 — °063 The sum of the squares of these deviations are (8) 341°6570, (£) 43-2458, (£) 133240, (P) 10°4942 (the exact value here should be. 10°4966, the value of V,, above). The sum of the squares of the deviations given by (8), (€), and (€) in directions perpendicular to those planes are found to be (by dividing the above values by the sum of the squares of the coefficients of the various equations) 63°4239, 11°7969, and 10°5767 respectively, all these, of course, being greater than the corresponding value given by (@). Hquation (¢) can be written in the three forms : z= 15:20482—7:°9644y . . . . () pe le OG oe gis 5238y-e OCSBe My al o/s CE) These equations should be compared with (@), (€), and (€) respectively. The sum of the squares of the deviations * At first sight it seems remarkable that all the deviations given by (9) are of the same sign, but a moment’s consideration will show that this is quite possible. For a line in the plane is fixed, and the plane can only swing about this line. All the points may be on one side of the plane, but on either side of the line. Swinging the plane about the line to become closer (measured in a particular direction) to some of the points, therefore, may take it farther from some others. ‘To verify (@) the results given by the planes 2=2°140427 and s=27+°'0856, one on either side of (@), were found. The sum of the squares of the deviations given by these formule were 341°6991 and 341°8112, both greater than ‘the corresponding number for (6). . Thus (@) gives a true minimum. 386 Mr. T. R. Merton on a Method of given by these equations in the directions of z, y, and 2 respectively are 3102°9568, 48°9069, and 13°4189. Comparing these with the results given by (@), (Q), and (€) above brings out clearly the fact that the plane which satisfies one criterion for closest fit may give a very bad fit if measured by another criterion. This is particularly the case with (@) and (6'), though (&') is not greatly inferior to (€) as the best fitting plane in the direction of a. The Sir John Cass Technical Institute, London, E.C. December 1910. XLIV. A Method of Calibrating Fine Capillary Tubes. By Tuomas Rate Merton, B.Sc. (Oxon.)*. fi ees methods commonly used in the determination of the bore of capillary tubes are direct optical measurement of the bore at the orifice, or weighing a drop of mercury which occupies a known length in the capillary. When very fine capillaries, having an internal diameter of the order of ‘I mm., are to be measured, these methods present serious difficulties. For many purposes it is necessary to obtain a value of the mean bore, and as no glass capillary is uniform for any ecnsiderable length, 2 measurement of the bore at the orifices is liable to.seriouserror. The weight of a column of mercury 10 cm. long contained in a capillary tube of -1 mm. bore is about 0:01 grm., so that to obtain an accuracy of 1 per cent. the weighing must be correct to 0'1l mgrm. The following experiments were performed with the object of investigating the accuracy with which a measurement of the electrical resistance of a fine glass capillary filled with mercury can be made. From this a mean value of 7” (where v is the internal radius) can be calculated. The first series of experiments was conducted in a large water-bath, containing about 30 litres, kept at 18°C. by an electric-filament lamp which was governed by a large’ spiral toluene regulator; and other experiments were performed in a bath kept at 25° by a small gas-flame governed by a fluted toluene regulator. In both baths the temperature could be kept constant to 0°01 C. * Communicated by the Author. Calibrating Fine Capillary Tubes. 387 The measurements of resistance were made with a metre bridge with resistances lengthening it to 10 metres and a resistance-box that had been carefully calibrated. The accuracy of the measurement far exceeded the accuracy with which the resistance could be kept constant. The arrangement of the capillary tube is shown in fig. Ll. Fig. 1. An airtight joint with the side-tube A and B is made by means of the two rubber plugs, RR. To fill the tube, the limb A is haif-filled with mercury and closed with a rubber plug. Itis then turned upside down, fig. 2, and exhausted To aur bumb A through B by means of an air-pump. On inverting the tube and removing the plug from A, the mercury was forced through the capillary into B. In making the measurements, connexion is made by means of stout copper wires dipping into the arms A and B, which are half-filled with mercury. The two rubber plugs RR were dispensed with in later experiments by grinding the tubes roughly together, and then cementing them with a small quantity of Schoenbeck enamel. i 388. Mr. T. R. Merton on a Method of ! | The tubes were cleaned by drawing through them con- | centrated sulphuric and chromic acids, followed by distilled i water, and were dried by heating to about 160° C.and drawing F air through them. The results obtained for three tubes are given in the following tables. Tube A at 18°C. Length of Capillary = 14:08389 cm. Comparison Resistance of Resistance. Capillary Thread. stealing ee eee. 18°509 ohms 18614 ohms 18-609 18613 18°709 18611 2nd Filling......... 18:709 18-612 18°609 18611 18°509 18613 ord Filling ......... 18-509 18-605 18:609 18°605 eset 7 hence the potential a, required for zero leak is rather less in these experiments than in Hull’s. This accounts, in part at any rate, for the difference between the values 2°18 and 2°33 volts. Comparing the effects of the light from the two sources when transmitted through quartz, it is seen that the hydrogen discharge produces a greater proportion of faster electrons than the are. Also the maximum velocity of tae electrons due to the light from the hydrogen discharge and trans- mitted through quartz, corresponds toa potential of 2°18 volts ; while for the light which comes from the are through quartz, the potential is 1°43 volts. According to Lyman * , the shortest wave-length transmitted by thin quartz from a hydrogen discharge is 11450, which corresponds in these experiments to the potential 218 volts. Ladenburg’s law states that the maximum velocity of electrons due to ultra- violet light is proportional to the frequency of the light. The velocity i is proportional to the square root of the potential required to stop the electrons, so that we have ~V « ATl. The wave-length corresponding to 1:43 volts will therefore be 41780. The experiments show that there is no appreci- able radiation from the mercury are in the region between * Lyman, Astrophysical Journal, xxy, p. 45 (1907). 400 Mr. A. Lil. Hughes on the 1450 and 21780. The greatest velocity of electrons pro- duced by the light from the are corresponds to 3°0 or 3-1 volts, which means that the shortest appreciable wave-length in the spectrum of the mercury arc is 1230. It is perhaps necessary to justify the application of Ladenburg’s law to this investigation. From considerations ot Planck’s theory of radiation, one would expect the frequency of the light and the velocity of the electrons to be connected linearly. Sadenburg expressed his results in the form 7 ViA=const.; but Joffé * has shown that if Ladenburg’s results are plotted, the curve showing the relation between the frequency and the velocity is a straight line which, however, does not pass through the origin. -As the linear relation has the better support in theory, we may regard Ladenburg’s law VA = const. as an empirical relation which is satisfied over a limited range. The work of Ladenburg and Hull shows that this relation is true over the range A 1230 to 72600. Since the wave-lengths dealt with are found to be within the above limits, the application of Ladenbure’s Jaw to this investigation is justified. The only substance available whose transparency was known was thin quartz, and this sufficed to determine the constant in the equation VA = const., while in Joffé’s form there are two constants to determine. The experiments of Professor Lyman show that the spectrum of hydrogen extends as far as 11030. He suggests, however, that perhaps still shorter wave-lengths may be emitted, but that either the photographic plates used in the experiments are not sensitive to shorter wave-lengths, or that the grating used for giving the spectrum does not act efficiently beyond about 71000. The electrical method of detecting the shortest wave-length has the advantage that the shorter the wave-length, the more sensitive is the test. The experiments described above show that the mercury spectrum ends at about 11230, while Lyman’s work shows that the hydrogen spectrum extends further, viz. to » 10380, 2. The following is a short account of an experiment on the nature of the photoelectric effect. Several views as to the mechanism of the effect may be considered. 1. Light may be regarded as molecular in structure and Joffé, Ann. der Phys, xxiv. 5, p- 939 (1907). Ultra- Violet Light from the Mercury Are. 4()1 each unit of light releases an electron from a molecule in the surface upon which the beam is incident. The velocity of the electron depends solely upon the energy in the unit and consequently upon its frequency. 2. The photoelectric effect may be of the nature of resonance. There may be many different atomic systems in matter each of which can be rendered unstable by light of suitable frequency *. Here, as before, the velocity of the ejected electron is determined by the frequency ot the light. 3. The electrons emitted from a surface may be in tem- perature equilibrium with the beam of light. The greater the frequency of the light, the higher is its radiation temperature. This also Jeads to the known experimental result that the greater the frequency of the light, the greater is the velocity of the electrons emitted. The principle of the experiment is as follows. The distribution of velocities of electrons released by light of the shortest available wave-length is measured, then an intense beam of ultra-violet light of longer wave-length is superposed and the distribution of velocities is again obtained. On the first and second views of the photoelectric effect indicated above, the final distribution of velocities would simply be the sum of the distributions due to the components of the beam. On the third view, however, this would not be the case. The temperature of the mixed beam would he higher than that of its long wave-length component and lower than that of its short wave-length component, and the final distribution of velocities would not be obtained by the simple addition of the distributions due to the two components. The simplest method of carrying out the experiment is to measure the maximum velocity of the electrons due to a beam of short wave-length ultra-violet light; then superpose longer wave-lengths, and examine whether the maximum velocity is the same as before or less. This was done, using the upper portion of the apparatus in fig. 1 to measure the maximum velocity. An apparent diminution in the maxi- mum velocity was found, but this could be explained, to some extent, by considerations of the same kind as those brought up in the discussion of fig. 3. It was therefore necessary to devise a method in which the disturbing effect of reflected light was completely avoided. In the method * Sir J. J. Thomson, Phil, Mag. xx. p. 258 (1910). =e OE ES esa 402 aT ACSI Hughes on the adopted a magnetic field was used to measure the velocities. The apparatus is shown in fic. 4. Fig. 4. to eleckromebir A flat cylindrical box is divided into quadrants by thin brass partitions. In three of these, slits 10 mm. by 2 mm. were cut. The radius of the circle passing through the centres of the slits was 1:1 cm, The illuminated plate P was connected to an electrometer and capacity by means of which the total leak from the surface could be measured when required. The surfaces inside the box were all covered with soot. The source of light of short wave-length was a discharge in hydrogen in the tube C. This was separated from the brass box by a fiuorite window F which transmitted light down to 1330. ‘The superposed long wave-length ultra- violet light was obtained from a mercury are in a fused quartz tube placed close to the quartz window Q. The shortest wave-length transmitted by the fused quartz was KAS: The distribution of velocities for each of the two sources of light was first obtamed. This was done by measuring tlie charge acquired by M for different magnetic fields per pendicular to the plane of the box. Although the light was as intense as could be obtained under the conditions, yet the charge communicated to M was always small. In some eases the leak was measured with the tilted electroscope at a sensitiveness of 600 divisions per volt. The velocities were obtained by the formula 52 Her = = “= He ioc ett 1-11 om, The distributions of velocities are given in fig. 5, the total | Ultra-Violet Light from the Mercury Are. 403 leak from the surface P being the same for both sources of light. It is seen that the electrons are on the whole con- siderably slower when produced by the light from the Fig, 8. ! 2 3 Ie 5 b 7 & qQ {1D HM atria joee. mercury arc than when produced by the light from the hydrogen discharge. The result of the superposition experiment is given in the following table. TAB ian Uitk H: effect due to H, discharge alone. ‘Ay: i » . Hg arc alone. H+ (add.): sum of two previous results. H-+A (exp.): effect due to H, discharge and Hg are superposed. Velocity of electrons. 18, aAN H+A (add.). | H-+A (exp.). 6°33 X10" cm./sec. 84:3 11°8 46:1 46°3 7°38 5 26°6 1:8 28°4 28°1 8:44 m ‘3 15:6 0 156 16:2 | Quoin ene Be ee 0 84 so | LOD: A 2°8 0 2°8 3:2 | The last two columns agree within the limits of experi- mental error, 404 Profs. Richardson and Cooke on the Heat liberated It was unnecessary to carry out the experiment for smaller velocities as the effect, if it existed, would scarcely show itself in this region. A few experiments were also tried when the total leak due to the arc was twice as great as that due to the H, discharge, but they still led to the same result. This result points to the conclusion that each electron emitted from a surface illuminated by ultra-violet light is associated with only one frequency in the beam of light, and its velocity is quite independent of the presence or absence of waves of different frequency in the beam. Summary. 1. The ultra-vielet spectrum of mercury, investigated electrically, extends to about A 1230. 2. There is no appreciable radiation from the mercury are between 2» 1450 and A 1780. 3. The hydrogen discharge has relatively more energy in the short wave-lengths than the mercury arc. 4. The velocity of the electrons due to one wave-length is independent of the presence of other wave-lengths in the beam of light. I have much pleasure in thanking Professor Sir J. J. Thomson for his suggestions and interest during the course of the work. Cavendish Laboratory, Cambridge. ; Jan. 9th, 1911. — XLVI. The Heat liberated during the Absorption of Elec- trons by Different Metals. By O. W. Ricnarpson, Professor of Physics, and H. L. CooKe, Assistant Professor of Physics, Princeton University*. [> a recent paper} under asimilar title the authors showed that when slow moving electrons were received by a platinum strip, part of the heat developed was independent of the kinetic energy of the electrons. This production of heat was explained as the thermal equivalent of the difference in the potential energy of the electrons inside and outside the metal. The present paper describes the results of similar * Communicated by the Authors, + Phil. Mag. July 1910, during Absorption of Electrons by Different Metals. 405 observations on strips of the following metals and alloys :— gold, nickel, copper, silver, palladium, aluminium, phosphor bronze, and iron. The method of experimenting was practically identical with that already described (loc. cit.). The only important change made was in the method of balancing the disturbing effect of the thermionic current. This was made simpler, both in theory and practice, by the following arrangement. The resistance Rs (fig. 1, loc. cet.) was replaced by a constantan wire of resistance 4°7 ohms wound ona circular drum and placed in an oil-bath. A movable contact-maker enabled the thermionic current to be tapped off at any point of the wire. The contact-maker was connected directly to the point E in the figure referred to and the resistances R, and R; were done away with. From the principles of the method of balancing the thermionic current already described, it is clear that if it is tapped off at the right point of the resistance R; it will produce no effect on the galvanometer G. To balance the effect of the thermionic current it was also necessary first to oppose the two batteries C; and C, so as to destroy the electromotive force in the Wheatstone’s bridge circuit, and rotate the contact-arm until the galvanometer spot was in the same position with the thermionic current ‘‘ on” or “ off.” The reliability of the method was tested by dummy expe- riments, using the electromotive force from a battery in a manner precisely similar to that used in testing the original method, and it was found quite satisfactory. Liffect of Pressure. Tn the former paper it was suggested that the results might perhaps be affected by the pressure of the residual gas present in the apparatus. At that time experiments were: made to test the point. So far as they went they seemed to indicate that, if the pressures, such as occurred, exerted an influence on the results, it was an unimportant one. It was felt, however, that these experiments were not very satis- factory; so fresh experiments have been made under better conditions. In these experiments an iron grid was used, and the osmium filaments had been heated continuously for along time. The conditions were generally very steady. On account of the continuous evolution of o gas by the hot metal, it was ee use a continuous “pump in these experiments. This made it somewhat diffieult to control the pressure. The obvious way of producing a desired change in the pressure is to vary the speed of the pump. This was 406 Profs. Richardson and Cooke on the Heat liberated found to be no good with the Gaede pump used, as, in order to obtain a pressure appreciably higher than that given by the pump when working at its most efficient t speed, it was necessary to work it so slowly that the changes in the tempe- rature of the apparatus caused by the periodic variations of pressure entirely precluded any attempt at measuring the effect. Several methods were tried, and that finally adopted con- sisted in connecting the apparatus to a side tube provided with an indiarubber joint which leaked slightly. The connexion was made through a good eround-glass cock. In this way two different pressures were available, namely, the limiting values obtained when the side tube was, and was not, connected with the apparatus. These pressures were about 2x107-* and 5x107-* mm. respectively, and were quite free from the slow periodic variations which had previously been found so objectionable. With this arrangement several experiments were made, of which the following are typical examples. With 8 volts applied potential-difference, and the side tube shut off, the pressure both before and after one set of observations was 20x107*mm. The effect in scale-divisions per unit thermionic current under these conditions was found to be 1°680. After connecting the side tube to the apparatus the pressure at the beginning of an experiment with 8 volts applied potential- difference was 56x 1074 mm., and at the end 46 x 1074 mm. The effect in this case, in the same units as at the lower pressure, was 1683. Thus changing the pressure from 20 10—- mm. to oll oc) ae mm. produces no change in the effect per unit thermionic current. The same was true at other voltages. Thus with 18 volts the pressure was 20 x 107* mm. at the beginning of an expe- riment with the side tube shut off and 19x 107* mm. at the end. ‘The effect per unit thermionic current was 3°01 scale- divisions. After connecting, the pressure was 46 x 107* mm. at the beginning and 42x 1074 mm. at the end of an expe- riment under. the same potential-difference. In this case the effect was found to be 2°98 in the same units as before. This is identical with 3°01 within the limits of observational error. These experiments prove conclusively that the measure- ments are quite unaffected by small fluctuations in the pressure of the surrounding gas; so that such irregularities as have been found must be attributed to other causes. Experiments with the Different Metals. A general review of the results which have been obtained shows that they are much more inconsistent than those yielded during Absorption of Liectrons by Different Metals. 407 by the former experiments on platinum. We believe that we have discovered the main cause of this inconsistency but, so far, unfortunately, we have not succeeded in regulating it. The cause seems to lie in an instability in the thermionic emission of osmium itself, and we are investigating the phenomenon in detail in the hope of being able to control it. Tie experiments which follow were carried out, so far as we are able to judge, in the same way and under the same con- ditions as those previously made with platinum. In view of the considerable range of the value of the effect for each one metal. and also of the peculiar effects first observed when experimenting with iron (see below), we sball only give the final numbers obtained in each case and shall omit the details of the measurements which led to them. In every case, the strips used were of the purest specimens of the metal obtainable. The gold, silver, palladium, copper, and nickel were obtained as pure from Messrs. Johnson, Matthey & Co., London. This specimen of copper was compared with one rolled from commercial magnet wire, and did not exhibit any notable difference. The aluminium was rolled from commercial aluminium wire and then cut by hand, and the phosphor bronze was a strip such as is used for galvanometer suspensions. As arule, a considerable nuniber of experiments were pede on each material, The final results of all those which appeared to be satisfactory are given in the following table :— Iwerehteat| Metal Corrected Values of ¢. Mean. iene | : Volts. Volts. | Wolts: | EWGNB Uh Goune salen a sues 654, 716, 6:83, "8:04, 17:86), (6716: | 7TOl | BOG PEON Oe dk... 519, 538, 561. PRDON I iNesan WO PEE nesses .s ee (305) oh) O86, 1O;SGO FON Ole) 16504 | ie Phosphor Bronze ...| 5°82. | 582 | — Palladium. 3050). 6:04, 5:44. 5 72 | 56 eilvere V0 dee 5:06, 583, 416, 5:46, 5-26 SN Is 22 Aluminium \.0..cs-. TORR MODI LE Ord 49.4 86 74 | = Tron (low) ...ceccece+. 495, 418, 4:39, 485, 5:54, 544.| 4-88 | eae iron (hiph)) ye.eo OQ eons. too!) 52. 6-72 ae IU epee eee ees - ~ Se 408 Profs. Richardson and Cooke on the Heat liberated The weighted means were judged by inspection of the experimental points. It is questionable whether they are much more reliable than the others, as it is our opinion that the experiments are affected by causes, which we are not able to control, which may remain constant during a single set of experiments. In the case of silver and aluminium the results of the experiments seemed less trustworthy than in the other cases. Often it was impossible to get the same value twice, for the same thermionic current and the same potential-difference, owing to some change with time which was going on. Moreover, in several experiments with these metals the heating effect did not turn out to be a linear function of the applied potential-difference. This was pro- bably due to some parts of the grid being different from others, and the heating current moving from one part to the other as the -potential-difference was changed. Probably most of the difference from one part of the grid to another was due to the effect on it of the heating and sputtering. This suggestion is supported by the appearance of the grids. The aluminium ones, after they had been used a few.times, were quite changed in texture and were so much alteréd that they crumbled to pieces as they were unwound. ‘They were also very badly discoloured. The experiments made with iron grids led to a discovery which we think likely to account for a considerable part of the irregularities which have been noted with all the metals. It seems that under the conditions of the experiments on iron, and probably on the other metals, there is a kind of instability in the thermionic emission of the osmium filaments. The nature of this instability is best described by considering the way the thermionic emission changes as the temperature of the filament is altered. Starting at a comparatively low temperature, the emission increases rapidly with increase of temperature following the usual inverse exponential law. This goes on until temperatures of a certain value are reached, when the current shows a tendency to sag off. If the tempe- rature is now raised and maintained at a certain value, there is asudden drop in the thermionic emission. In favourable cases the new current may beas little as one-thirtieth of what it previously was at the same temperature. When the tem- perature is raised further the current is found to be stable again, and increases with temperature according to an inverse exponential law. The behaviour of the emission as the temperature is lowered is the reverse of what has been described. We have first a guick but regular decrease in current following the inverse during Absorption of Electrons by Different Metals. 409 exponential law. At a certain stage there is a sudden increase in the emission. This is followed by a third region in which the current again diminishes with decreasing temperature according to the regular law. There are thus two ranges of temperature in which the thermionic current isstable. These are separated by a region of instability. We shall refer to the two stable ranges as the low-temperature range and the high-temperature range respectively. The heating effect in the case of iron has been examined, both when the osmium was on the low-temperature range and also when it was on the high-temperature range. The numbers are given in the last two rows of the table. The mean of the six values for the low-temperature range is 4°88 volts, and of the four values for the high-temperature range 6°72 volts. There is thus a ditference of nearly two volts in the effect given by the two ranges. We next attempted to make experiments upon the other metals under such conditions that we knew whether the osmium was on the low- or the high-temperature range. At the outset we obtained values for platinum in the neigh- bourhood of 7 volts as against the value 5°5 volts obtained in our previous investigation. We believed at the time that we were working on the high-temperature range ; but that par- ticular filament burnt out, and later on we found it impossible to get a filament which would develop the two ranges. This situation compelled us to desist from the direct line of attack for the moment ; as it is necessary to make a more thorough examination of the thermionic properties of osmium, in order to be able to control the conditions which determine its . thermionic emission. We hope to be able to report on this matter at an early date. If we return for a moment to the table on page 4()7 it is a striking fact that four of the mean values, viz.: gold, 7:26, copper 7°1, aluminium 7:4, and iron (high) 6°72, are equal, within the range of experimental error, to a common mean value 7°11 ; whilst the other five, viz.: nickel 5°3, phosphor bronze 5°8, palladium 5:6, silver 5°15, and iron (low) 4°9, are equal within the same limits to the common mean value 5°35. It looks as though the values for gold, copper, and aiuminium had all been obtained with the osmium on the high range, and the values for nickel, phosphor bronze, palladium, and silver with the osmium on the low range. It would follow from this that the value of the heating effect is independent of the nature of the metal which receives the electrons, being determined almost entirely by the metal which emits them. Phil. Mag. S. 6. Vol 2t0 No. 124. April 1911. 2 HE 410 Heat liberated during Absorption of Electrons by Metals. For the reasons which have been stated we have found it impossible to test this question directly, up to the present. The present theory of these phenomena requires that the heating effect should be practically independent of the metal receiving the electrons, as one of the authors has already pointed out*. It is satisfactory to know that our experi- mental results, so far as they go, do not conflict with the theory. In our previous paper we omitted to take into account the possibility of the existence of an electric field between the osmium and the platinum arising from an intrinsic electro- motive force ; and this omission led us to identify the heating effect with the work done when the electrons escape from hot platinum. The identification should be with the work done when the electrons escape from hot osmium. This quantity has not been measured yet, but we propose to measure it in the course of our investigation of the thermionic properties of osmium already referred to. The most important facts which we have so far established are :— (1) The heating effect due to the difference of the potential energy of an electron inside and outside of a conductor, which we previously established for platinum, occurs in the other metals. (2) The effect is of the same order of magnitude in all cases, the measured values ranging from about 4°5 to 7°5 volts. (3) The values are influenced very considerably by the nature and state of the thermionic emitter. (The experiments do not preclude the possibility that the true effect is almost independent of the metal receiving the electrons.) (4) The measured effect is not influenced by changes in the pressure of the residual gas in the apparatus, provided this be reasonably low. (5) Under certain conditions, not yet completely deter- mined, the thermionic emission from osmium becomes un- stable ; and there are two ranges of stability, one at low and the other at high temperatures. We are glad to take this opportunity of thanking Messrs. Baldwin, Carter, Critchlow, Ferger, Frederick, and Gibbs for again assisting in taking the very numerous observations. Palmer Physical Laboratory, Princeton, N.J. *O, W. Richardson, Rapports du Congres Int, de Radiologie, Bruxelles, 1910. are) XLIX. Interferometry with the Aid of a Grating. By Cart Barus, Ph.D., DL.D., Professor of Physics at Brown University, Providence, U.S.A. Part I.—INTRODUCTION. 1. Remarks on the Phenomena.—lIn the earlier papers ¢ I described certain of the interferences obtained when the oblique plate gg, fig. 1, of Michelson’s adjustment, is replaced Fig. 1. dM by a plane diffraction-grating on ordinary plate glass. Some explanation of these is necessary here. In the figure, L is the source of white light from a collimator. Such light is therefore parallel relative to a horizontal plane, but con- vergent relatively to a vertical plane; M and N are the usual silver mirrors. A telescope adjusted for parallel rays in the line GE must, therefore, show sharp white images of the slit. As the grating is usually slightly wedge-shaped, there will be (normally) four such images, two returned by M after re- flexion from the front (white) and rear face (yellowish) of the plate gg, and two due to N. There will also be two other, not quite achromatic slit images from N or M, re- spectively, due to double diffraction before and after reflexion. These will be treated below. In the direction GD there will thus be a corresponding number of diffraction spectra, more or less coincident in all their parts, and therefore adapted to interfere in pairs throughout their extent, If * Communicated by the Author. + Abridged from a Report to the Carnegie Institution of Washington, U.S.A. See also Am. Journ. Sci. xxx. 1910, pp. 161-171; Science, July 15, 1910, p. 92; and C. & M. Barus, Phil. Mag, July, 1910, pp. 45-59, 2H 2 412 Prof. Carl Barus on Interferometry the two white and the two yellow images of the slit be put in coincidence and the mirrors M and N are adjusted for the respective reduced or virtual path difference zero, the interferences obtained are usually eccentric ; 7. e. the centres ot the interference ellipses are not in the field of view. The effective reflexion in each of these cases takes place from the front and rear face of the grating at the same time. Hence the interference pattern includes the prism angle of the grating plate and is not centred. The air-paths of the component rays are here practically equal. In addition to the ellipses, this self-compensating position also shows revolving linear interferences, and (as a rule) a double set, is in the fieldat once, consisting of equidistant symmetrically oblique crossed lines, passing through horizontality in opposite directions t together, when either mirror M or N is suitably displaced. If either pair of the white and yellowish images of the slit be placed in coincidence when looking along EG, the interference pattern along DG is ring-shaped, usually quasi- elliptic and centred. The light returned by M and N is in this case reflected from the same face of the grating, either from the face carrying the grating or the other (unruled) face. The corresponding air-paths of the rays are in this case quite unequal, because the short Bapetn is compensated by the path of the rays within the glass plate. Hence these adjustments are very different, in one instance GM, in the other GN, being the long path. For the same motion of the micrometer-screw, the. fringes as a whole are displaced in opposite directions. In one adjustment there may bea single family of ellipses; in the other there may be two or even three families, nearly in the field at once. If the grating were cut on optical plate glass, the adjust- ment for equal air-path would probably be best. But with the grating cut as usual on ordinary plate, or in case of replica gratings on collodium or celluloid films, the adjust- ment for unequal paths is preferable. Here, again, one of the positions is much to be preferred to the other, owing to the occurrence of multiple slit images from one of the mirrors, as above specified. In fig. 2, for instance, where the grating face is to the rear, there are but two images, 1 and 2, from M if the plate is slightly wedge-shaped ; but from N, in addition to these two normal cases (not necessarily coinciding with 1 and 2), there are two other images, 3/ and A’, i, 3 and 4 are spectrum rays, resulting from double diffraction, with a deviation @, and angle of incidence ik respectively @I1, in succession ; or the reverse. with the Aid of a Grating. 413 As the compensation for colour cannot here be perfect, the two slit images obtained are very narrow, practically linear spectra, but they are strong enough to produce interferences like the normal images of the slits with which they nearly agree in position. Other very faint slit images also occur, but they may be disregarded. The doubly-diffracted slit images are olten useful in the adjustments for interference. Among the normal slit images there are two, respectively white and yellowish, which are remote from secondary images. If these be placed in coincidence both horizontally and vertically along HG, fig. 1, the observation along DG through the telescope will show a magnificent display of black, apparently confocal ellipses, with their axes respec- tively horizontal and vertical, extending through the whole width of the spectrum, from red to violet, with the Fraun- hofer lines simultaneously in focus. The vertical axes are not primarily dependent on diffraction, and therefore of about the same angular length throughout ; the horizontal axes, however, increase with the magnitude of the diffraction, and ence these axes increase from violet. to red, from the first to the second and higher orders of spectra, and in general as the grating space is smaller. It is not unusual to obtain circles in some parts of the spectrum, since ellipses which have long axes vertically in one extreme case, have Jong axes horizontally in the other extreme case. The inter- ference figure occurs simultaneously in all orders of spectra, and it is interesting io note that even in the chromatic slit images shown in fv. 2, needle-shaped vertical ellipses are quite apparent. It is surprising thatall these interferences may be obtained with replica or film-gratings, though not, of course, so sharply Al4 Prof. Carl Barus on Interferometry as with ruled gratings, the ideal being an optical plate. With thin films two sets of interferences are liable to be in the field at once, and I have yet to study these features from the practical point of view. If the film is mounted between two identical plates of glass, rigorously linear, vertical and movable interference fringes, as described * by my son and myself, may be obtained. 2. Cause of Ellipses—The slit at L (fig. 1) furnishes a divergent pencil of light due (at least) to its diffraction, the rays becoming parallel in a horizontal section after passing the strong lens of the collimator. But the vertical section of the issuing pencil is essentially convergent. Hence, if such a pencil passes the grating the oblique rays relatively to the vertical plane pass through a greater thickness of glass than the horizontal rays. The interference pattern if it occurs is thus subject to a cause for contraction in the former case that is absent in the latter. Hence also the vertical axes of the ellipses are about the same in all orders of spectra. They tend to conform in their vertical symmetry to the regular type of circular ring-shaped figure, as studied by Michelson and his associates, and more recently by Feussner f. On the other hand, the obliquity in the horizontal direction which is essential to successive interferences of rays is furnished by the diffraction of the grating itself, as the deviation here increases from violet to red. In other words, the interference which is latent or condensed in the normal white linear range of the slit, is drawn out horizontally and displayed in the successive orders of spectra to right and left of it. The vertical and horizontal symmetry of ellipses thus follows totally different laws, the former of which have been thoroughly studied. The present paper will therefore be devoted to the phenomena in the horizontal direction only. At the centre of ellipses the reduced path-difference is zero; but it cannot increase quite at the same rate toward red and violet. Neither does the refractive index of the glass admit of this symmetry. Hence the so-called ellipses are necessarily complicated ovals, but their resemblance to confocal ellipses is nevertheless so close that the term is admissible. This will appear in the data. If either mirror or the grating is displaced parallel to itself by the micrometer-screw, the interference figure drifts * Phil. Mag. J. c. + See Prof. Feussner’s excellent summary in Winkelmann’s Handbuch der Physik, vol. vi. p. 9858 et seg. (1906). with the Aid of a Grating. A15 as a whole to the right or to the left, while the rings partake of the customary motion toward or from a centre. The horizontal motion in snch a case is of the nature of a-coarse adjustment as compared with the radial motion, a state of things which is often advantageous. The large divisions of the scale are not lost, in sibee words. Moreover, the dis- placements may be used independently. The two motions are coordinated inasmuch as violet travels toward the centre taster in a horizontal direction, 7. e. at a greater angular rate, than red. Hence the ellipses dritt horizontally but not vertically. Naturally in the two positions specified above for ellipses, the fringes travel in opposite directions for the same motion of the micrometer- screw. As the thickness of the grating is less the ellipses will tend to open into vertical curved lines, while their displacement is correspondingly increased. With the grating on a plate of glass about e="68 cm. thick, and having a grating space of about D=-000351 em., at an angle “of incidence of about 45°, the displacement of the centre of ellipses frvem the D to the E line of the spectrum corresponded to a displacement of the grating parallel to itself of about "006 em. It makes no difference whether the grating side or the plane side of the plate is toward the light or which side of the grating is made the top. If the grating in question is stationary and the mirror N alone moves parallel to itself along the micrometer-screw, a displacement of iN Ollem. roughly moves the centre of ellipses trom D to H, as before. This “displacement varies primarily with the thickness of the grating and its refraction. It does not depend on the grating constant. Thus the following data were obtained with film gratings (on different thicknesses ¢ of glass and different gratirg spaces D) for the displace- ments, N, of the mirror at N, to move the ellipses from the D to the E line, as specified :— Glass grating, ruled......... é="68 em. A/D=:168 N=:010 em. Film on glass plate ......... i) ie "352 008 _,, Film on glass plate ......... C= 24h "352 003, Film between glass plates cami ie: 852 003 _,, é = 24. 9 Reduced linearly to e='68 cm., the latter data would be N’=:010 and N’=:009, which are ae the same order and as close as the diffuse Meorbetenee patterns of film gratings permit. The large difference in dispersion, t together with 416 Prof. Carl Barus on Interferometry some differences in the glass, has produced no discernible effect. An interesting case is the film grating between two equally thick plates of glass. With this, in addition to the elliptical interferences above described, a new pattern of vertical interferences identical with those discussed in a preceding paper™ were obtained. These are linear, per- sistently vertical fringes, extending throughout the spectrum and within the field of view, nearly equidistant and of all colours. Their distances apart, however, may now be passed through infinity when the virtual air-space passes through zero ; and for micrometer displacements of mirror in a given direction, the motion of fringes is in opposite directions on different sides of the null position of the mirror. I have not been able, however, to make them as strong and sharp as they were obtained in the paper specified. 3. The Three Principal Adjustments for Interference.—To compute the extreme adjustments of the grating when the mirror N is moved, fig. 5 (below) may be consulted. Let y, be the air-path on the glass side, whereas y, is the air-path on the other, e the thickness of the grating, and p its index of refraction for a given colour. Then for the simplest case of interferences, in the first position N cf the mirror, if I is the normal angle of incidence and R the normal angle of refraction for a given colour, y,+ evleos R=y, for equal paths, Similarly, in the second position of the mirror, y= y', + en/cos R. Hence, if the displacement at the mirror be N, as the figure — shows, Yay = 2e tan Rsin I, N=2ewcos R. The measured value of this quantity was about 1°88 cm. The computed value would be 2 x °68 x 1:53 x °71=1°84 cm. The difference is due to the wedge-shaped glass which requires a re-adjustment of the grating for the two positions. The corresponding extreme adjustments, when the grating gg instead of the mirrcr N is moved over a distance z, are in like manner found to be Cb 2 eam tanr. cos Lcos R * Phil. Mag. J. c with the Aid of a Grating. AIT The observed value of z for the two positions was about 1°3.em. The computed value for [=45° was the same. On the other band, when reflexion takes place from the same face of the grating while the latter is displaced ¢ cm. parallel to itself, the relations of 1 y and < for normal incidence at an angle I are obviously y cos l=c. For an oblique incidence 2, where 1—I =a, a small angle, the equation is more e complicated. In this case sina . y=e(142 u ues sin) Joos ik ‘ COs 2 This equation is also true for a grating of thickness e, whose faces are plane parallel. For the direction of the air-rays in this case remains unchanged. Finally, a distinction is necessary between the path difference 2y and the motion of the opaque mirror 2N which is equivalent to it, since the light is not monochromatic. This motion 2N is oblique to the grating, and if the rays differ in colour further consideration is needed. For the simplest type of interferences, in which the glass path difference, as in fig. 5, is eu/cos R, and for normal incidence at an angle I, let the upper ray ¥,, and the grating be fixed. The mirror moving over the distance AN changes the zero of path difference from any colour of index of refraction “yp to another of index p,,, while y,,, passes to y,,. Then a simple computation shows AN =e(#,, cos R,, —p;, cos h,,). This difference belongs to all rays of the same colour difference, or for two interpenetrating pencils. If reflexion takes place from the lower face the rays are somewhat different, but the result is the same. If the plate of the grating is a wedge of small angle ¢, the normal rays will leave it on one side at an angle I+6, where 6 is the deviation 3=(u—1¢. The mirror N will also be inclined at an angle I+, to return these rays normally. We may disregard di=¢dp, if is small. 418 Prof. Carl Barus on Interferometry Part IJ.—Dtreotr Caszs oF IXTERFERENCE : DiFrFRACTION ANTECEDENT. 4, Diffraction before Reflexion.—lf in fig. 1 the diffracted beams (spectra) be returned by the mirrors M’ and N’ to be © reflected at the grating, interference must also be producible along GD. Again there will be three primary cases: if reflexion takes place from both faces of the grating at once, the air-paths must be nearly equal, the grating itself acting asa compensator. The interference pattern is ring-shaped, but, as usual, very eccentric. If the reflexion of both com- ponent beams takes place from the same face of the grating, the interference pattern is elliptical and centred, and the air-paths are unequal. The case is then similar to the pre- ceding §$1, 25; but there is no direct or normal image of the slit, as M and.N are absent. In its place there may be chromatic images of the slit (linear spectra) at C due to the specified double diffraction of each component beam in a positive and negative direction, successively. But these chromatic images are nevertheless sharp encugh to complete the adjustments for interference by placing the slit images in coincidence. It is not usnally necessary to put the spectrum lines into coincidence separately (both horizontally and vertically), as was originally done, both spectra being observed. Again, along GE (fig. 1) approximately, there must be two successive positive diffractions of each com- ponent beam, which would correspond closely to the second order of diffraction. The advantage of this adjustment lies in the fact that there are but two slit images effectively returned by M’ and N’, and hence these interferences were at first believed to be stronger and more isolated. As a consequence I used this method in most of my early experi- ments, before finding the equally good adjustment described in the preceding sections. 5. Elementary Theory.—To find the path differences fig. 3 may be consulted. the grating face is shown at gq, the glass plate being e em. in thickness below it, and n is the normal to the grating. M and N are two opaque mirrors, each at an angle © to the face of the grating. Light is incident on the right at an angle 7 nearly 45°. In both figures the rays y,, and y, (air-paths) diffracted at an angle © in air, are reflected normally from the mirrors M and N respectively, and issue toward p for interference. ‘The rays y, pass through glass. Both hgures also contain two com- ponent rays diffracted at an angle @ in air, where 0Q—O=a, with the Aid of a Grating. 419 and reflected obliquely at the mirrors M and N, thus enclosing an angle 2 in air and 2f, in glass and issuing Fie. 3. | toward g. ‘These component rays are drawn in full and dotted respectively. ‘lhere may be two incident rays for a single emergent ray, or, as in fig. 3, a single incident ray for two emergent rays interfering in the telescope. The treatment of the iwo cases is different in detail; but as the results must be the same they corroborate each other. The notation used is as follows :—Let e be the normal thickness of the grating, e’ the effective thickness of the compensator when used. Let distance measured normal to the mirror be termed y, y, and y,, being the component air-paths passing on the glass and on the air side of the grating gg, so that y, >y,. Let y=y,,—y, be the air-path difference. Similarly let distances z be measured normal to the grating, so that z may refer to displacements of the erating. Let 2 be the angle of incidence, r the angle of refraction, and yw, the index of refraction for the given colour, whence sin i=, sin 7. Similarly, if © is the angle of diffraction in air of the y rays and ©, the corresponding angle in glass, Sal (vir sisit a) Saas Maes aaa 8) and if @ is the angle of diffraction of any oblique ray in air and 6; the corresponding angle in glass, sin@= sin (O@+a)=pysinO,. . . « (2) 430 Prof. Carl Barus on Interferometry Here @>0, so that «=0@—O© is the deviation of the oblique ray from the normal ray in air. Finally, the second reflexion of the oblique ray necessarily introduces the angle of refraction within the glass, such that sin(@—a)=p,sin 3, . . inves If D is the grating space, moreover, sinz—sin@=),/D and sinz—sinO@=~rA /D. (4) 6. Equations for the Present Case—From a solution of the triangles preferably in the case for single incidenee, as in fig. 8, the air-path of the upper oblique component ray at a dev riation, a, 1S 2y,, cos ®/ cos (O—a). The air-path of the lower component ray is 2 cos O(y, —e sin O(tan @,— tan @,))/ cos (@—z). The optic path of this (lower) ray in glass as far as the final wave-lront in glass at @, is ,e(1/ cos 6, +1/ cos 8;). The optic path of the upper ray as far as the same wave-front in glass is sin o fe, Sin By a a7 (Ope y, +e sin O(tan @,—tan @;)) —e(tan 6,—tan B,) } - (9) Hence the path difference between the lower and the upper ray as far as the final wave-front is, on collecting similar terms, if the coefficients of y,,—y,=y (where y is positive) and of e be brought together, as far as possible, and if the path difference corresponds to n wave-lengths A, Nd, = —2y Cos a+ Zep, cos a(1/ cos @,— cos (A, — G;)/ cos O;) +eu,(1+ cos(@,—8;) We cos: G),: |. . -euaneas which is the full equation in question for a dark fringe. It is unfortunately very cumbersome and for this reason fails to answer many questions perspicuously. It will be used below in another form. The equation refers primarily to the horizontal axes of the ellipses only, as e increases vertically ahove and below this line. The equation may be abbreviated, apd in case of a parallel compensator of thickness ¢ (y being the path difference in air) may be written ni,= —2ycosat(e—e’)(Zo—Z,) . . - TW) with the Aid of a Grating. If e=e', i. e. for an infinitely thin plate e=0, nd, = 2y COS & 421 (8) Again, for normal rays y,, and y,, «=0 and 0,=0,= A). Hence, for a parallel compensator of thickness é’, DN ae —e Ne cs)... . . Co} If wo =1=p,, equation (6) reduces to nr=2 cos a(y+e/ cos @), clearly identical with the case e=0 for a different y. In view of the complicated nature of equation (6) I have computed path differences for a typical case of light crown olass T. as shown in Table I., for the Fraunhofer lines B, D, H, Ff, G, supposing the H ray to be normal (a=0). Table for to mirrors. Ga 35° Spectrum lines... Path Difference * cms e=1 em. The same for 2y=3'2648 cm. ; ¢='68 cm. Path alee AVANTE EN ee Se et COS ate sep, i rays normal Difference, =—2ycosa+eZ. Light crown glass. Grating space D=-000351 cm. B. 68°7 15118 30° 46' Oo Ah —3° 5! —1° 44’ 232) 20) —3° 38’ 1:9970 "0383 32100 —1:9970 +3°2483 — ‘0079 — ‘0107 D. 58:93 15153 32°) 38) 20° 51’ —i8 187 —0° 40’ De ils! —1° 26' 1-9996 ‘0176 32404 — 19996 +3°2580 — 0042 — 0056 K. 52°70 1°5186 30° 51’ rarely O20; OS @ 212304 O10} 2:0000 ‘0000 32618 — 2:0000 +3:2648 +0 +0 F. 48 61 15214 34° 40' DA S| 0° 49! 02 26! PALSY ss! 0° 44’ 1:9998 — ‘0095 3285 —1:9998 +3:2710 +0040 +0049 The [= 45°. G. 43 08 15267 30° 46' 22° 30! oo! O%59! 20° 49! Lo 4’ —-0219 | 33058 | 2 | ay + 0120 +0156 | cm. cm. em. clu. em. cm. * ‘The purpose of these data is merely to elucidate the equations. AN refers to the displacement of either opaque mirror, M or N. t+ Taken from Kohlrausch’s ‘Practical Physics, lth edition, 1910, pe Ake: 422 Prof. Carl Barus on Interferometry Table is particularly drawn up to indicate the relative value of the functions Z, and Z,. It will be seen that Z=Zoe+Z,, in terms of X, is a curve of regular decrease having no tendency to assume maximum or minimum values within the range of A, whereas ycose passes through the usual flat maximum for «e=0. The path difference, which is the difference of the ordinates of these curves, thus passes through zero for a definite value of e and y, and this would at first sight seem to correspond to the centre of ellipses. That it does not so correspond will be particularly brought out in the next section. It is obvious that for a given e the value of y (air-path difference) which makes the total path difference zero, varies with the wave-length, hence on increasing y continuously the ellipses must pass through the spectrum. It is also obvious that if the grating is reversed, the path difference will change sign, cet. par., and the ellipses will move in a contrary direction, for the same dis- placements y of the micrometer-screw, at the mirror M or N respectively. . If e=1 em. and y=3:2648 em., the path difference will be zero for the E ray. The Table shows the residual path differences in the red and blue parts of the spectrum. The ellipses will be larger as e is smaller, the limit of enormous ellipses being reached for e=0 or e=é’ of the compensator. The same result may be obtained to better advantage by reconstructing equation (6), and making the path difference equal to zero for the normal ray. This determines ¥ =e[L,/ COS in terms of e, and the path difference is now free from y». 7. Interferometer.—If, in equation (6), y alone is variable with the order of fringe n, while a, 0, ©, @;, 1, B,, e, r, pm, are all constant ; 2. e.if the number of fringes n cross a given fixed spectrum line like the D line, when the mirror is displaced over a distance y, it appears that a) ie (10) dn 2cosa adn where « refers to the deviation of A, from A, normal to the mirror. If «=0, dy/dn=X/2, the limiting sensitiveness of the apparatus, which appears for the case of normal rays. The displacement per fringe dy/dnor dz/dn, varies with the wave-length. Hence if the ellipses are nearly symmetrical on both sides of the centre, 2. e., if the red and violet sides with the Aid of a Grating. 423 of the periphery are nearly the same, or the fringes nearly equidistant, the smaller wave-length will move faster than the larger for a given displacement of mirror. It is at first quite puzzling to observe the motion of ellipses as a whole in a direction opposed to the motion of the more reddish fringes when these are alone in the field. $. Discrepancy of the Table—The data of Table I. are computed supposing that the path difference zero corresponds to the centre of ellipses. This assumption has been admitted for discussion only and the inferences drawn are qualitatively correct. Quantitatively, however, the displacement of about AN =:003 em. should move the centre of ellipses from the D to the E line of the spectrum, whereas observations show that a displacement of either opaque mirror of about AN=-‘01 em. is necessary for this purpose ; 7. e. over three times as much displacement as has been computed. ‘The equation cannot be incorrect ; hence the assumption that the centre of ellipses corresponds to the path difference zero is not vouched for and must be particularly examined. This may be done to greater advantage in connexion with the next section, where the conditions are throughout simpler, but the data of the same order of value. Part I1J.—Drrect Case: ReFLEXION ANTECEDENT. 9. Hquations for this Case.—If reflexion at the opaque mirrors takes place before diffraction at the grating, the form of the equations and their mode of derivation is similar to the case of paragraph 6, but the variables contained are essentially different. In this case the deviation from the normal ray is due not to diffraction but to the angle of incidence, and the equations are derived for homogeneous light of wave-length X and index of refraction u. In fig. 4, let y, and y,, be the air-paths of the component rays, the former first passing through the glass plate of thickness e. Let the angle of incidence of the ray be I, so that y, and y, are returned normally from the mirrors M and N respectively, these being also at an angle I to the plane of the grating. Let 7 be the angle of incidence of an oblique ray, whose deviation from the normal is 7—I=a. Let R, r, and 8, be angles of refraction such that Sin? cine ieeo)— sings) sin ly sin Re: sin (L—«)=wsin §). NN = —2y COS a+ pe i= = 424 Prof. Carl Barus on Interferometry The face of the grating is here supposed to be away from the incident ray, as shown at gg in fig. 4. Fig. 4 Then it follows, as in paragraph 6, mut. mut., that the path difference is Gf y=y,,—¥Y,,) coke 2eoisee (— 2) a COS 7 Cos 7 = —2ycosa+e(Z,—Z,+Z;), ° ° e e where n is the order of interference (whole number). ‘This equation is intrinsically simpler than equation (6), since (=p as already stated, and since « is constant for all colours, or all values of wand > in question. R and 7 replace ©, and @,. | In most respects the discussion of equation (11) is similar to equation (6) and may be omitted in favour of the special interpretation presently to be given. If w=1, equation (11) like equation (6) reduces to the case corresponding to e=0, with a different 7 normal to the grating. All the colours are superposed in the direct images of the slit, R, and R (fig. 4), seen in the telescope, and the slit is therefore white. This shows also that prismatic deviation due to the plate of the grating (wedge) is inappreciable. The colours appear, however, when the light of the slit is analysed by the grating in the successive diffraction spectra, D, or D, respectively. In equation (11), w is a function of X, and hence of the deviation @ produced by the grating, since sin(I[—a)— sin@=A/D. (11) with the Aid ofa Grating. 425 The values of equation (11) for successive Fraunhofer lines and for =3° have been computed in Table IT. for the TaBLE II. Path Difference, —2( Yn y,) cosa+eLZ,—~eZy+eZ3. Light erown glass @= 3° throughout. Grating space °000351 em. JT=45°. e=1 cm. Spectrum lines...) = B. D. iE. F. G. RSG E =a 68:7, 58°93 52:70 48°61 43:08 | cm. poe | olls yy dtosy |) Lolse | 152145) 5267, R20 1d3) WAT 49" WI 27L 45! | DFO 42? 272135) | OA 2O > 2a W 207 tS. |, 2OOMA EO 8. Br =| 26° 1G) | 260 12 2693") 9605" 2600": a =| 8°0' | 3°0' | 300" | s°0' | 8°0" aR =| 1ossh |) 10337) ye337 |Pres3e | Teas! =e LOM | Go OMe eae NO! 3° 9° Bo (oy , =| 34160 | 34218 | 34272 | 3:4318 34403 em. Z Th. ee ABEND (ae | 16" 803. ie 896 | em. Zeme=GSOn Tags \l 700m ay 928 om. Path Difference | al pa —1-9992 | — 1:9992 | —1-9992 | —1:9992 |—1-9992 = ie ue (+3°4198 | +3°4252 | 4-3-4304 | +3:4352 |+3°4436 | The data are merely intended to elucidate the equations. The values Z are nearly equal, So also the cases for a=3° and a=0°. same glass treated in Table I., the data being similar and in fact of about the same order of value. The feature of this Table is the occurrence of nearly constant values, e=7—I, r—R, and r—f,, throughout the visible spectrum. Hence if the following abbreviations be used : A=cosa='9986, B=cos(r—R)='9996, Oz 1+ cos (r— 81) = 1°9984,. | , A, B, ( are practically functions of « only and do not vary with colour or A Furthermore, if the path difference is annulled at the E line, equation (11) reduces to nr = 2eA(u/cos R—p cos R,) +e(C— 2AB)p/cosr, where 2eA and (U— 2A B) are nearly independent of »X and Phil. Mag. 8. 6. Vol. 21. No. 124. April 1911. .2F | | af | | | | 426 Prof. Carl Barus on Interferometry the last quantity is relatively small. The two terms of this equation for e=1 em. show about the following variation :— ine...) B: D, E. p. G. —O111 =0052 §-0 +-0048 -Uieamam — 000009 —-000004 -0 +4 000005 -000013 Hence for deviations even larger than «= 3°, the pi: ath ditference does not differ practically from the path difference for the normal ray. ‘Ihus it follows that the equation ni = —2y + 2eu/cos R is a sufficient approximation for such purposes as are here in view. . Finally, for e= °68 cm. the actual thickness of the plate of the grating, y and N, the semi-path difference and the displacement of the opaque mirror, will be :— B. D. E. F. Ge. Y= 11630 11650 11668 11686 Ligie Ay,= — 0033 —-0918 0 +0016 +0045 SN,= +0014 +0007 0 ~ 0005 ~ 0018 AN,= --0052 —-0025 ‘0 +0021 +-0063 where oN, is the colour correction of Ayy, and AN» =Ay—8No determines the corresponding displacements of the opaque mirror ; or, more briefly, No =%—ev tan Rsin R= 2ep cos R. The data actually found were (the B line being un- certain) :— Lines......... 1), Seema B. ANG — 0153 — 0078 ‘0 +°0057 These results are again from 2 to 4 times larger than the computed values. True the glass on which the grating was cut is not identical with the light crown glass of the tables ; but nevertheless a discrepancy so large and irregular is out of the question. It is necessary to conclude, therefore, that here, as in paragraph 8, the assumption of a total path difference zero for the centre of ellipses is not true. In other words, the equality of air-path difference and glass- path does not correspond to the centres in question. “It is now in place to examine this result in detail. 10. Divergence per Fringe—The approximate sufficiency of the equation : j nv = Zep feos R—2y ee makes it easy to obtain certain important derivatives among with the Aid of a Grating. 427 which d@/dn (where @ is the angle of diffraction corresponding to the angle of incidence I, and » the order of interlerence) is prominent. If e and y are constant, n, X, w, and R variable, the differential coefficients may be reduced successively by the following fundamental equations, D being the grating space, w and r corresponding to wave-length X and angle of dif- fraction 8 :— Gita We ayal ee (1B) dX=—Dcos@. dé, aeanema ii, | 2st 0! CLAN) nee Ne ence tha wel Westley ey |) LO),) where (as a first approximation) a = ‘015 is an experimental correction, interpolated for the given glass. Incorporating these equations it is found that dé a cos R Th O10 Gis Ge areas REM ATEEr I, ae If the path difference » is annulled, doe MeN cos R dn 20 cos@ ueu(1—tan? R)’ * (17) which is the deviation per fringe, supposedly referred to the centre of ellipses. These equations indicate the nature of the dependence of the horizontal axes of ellipses on 1/D (hence also on the order of the spectrum), L/e, 1/u, and 1/a, where the meaning of a, here an important variable, is given in equation (15). If instead of the path difference y the displacement N of either opaque mirror is primarily considered (necessarily the case in practice), the factor (1— tan? R) vanishes. Table I1I. contains a survey of data for equations (16) and (17). The results for d@,/du would be plausible, as to order of values. The data for d@/dn, however, are again neces- sarily in error, as already instanced above, paragraph 8. They do not show the maximum at EH, and the A-elfect is overwhelmingly large. Since equation (17) is clearly inapplicable, giving neither maxima nor counting the fringes, it follows that in this equation y>eu/cosR; i. e., the centres of ellipses are not in correspondence with the path difference zero. In other words, the air-path difference is larger than the glass-path difference in such a way that d@/dn is equal to © for centres (here at the E line), but falls off rapidly toward both sides of the spectrum. 2’: 2 SS eee ee yy pg yy, pl 428 Prof. Carl Barus on Interferometry There is another feature of importance which must now be accentuated. In case of different colours, and stationary mirrors and grating, y is not constant from colour to colour, whereas 2N=2y—2epsin Rtan R is constant for all colours, as has been shown above. Thus the equation to be differentiated for constancy of adjustment but variable colour loses the variable y and becomes nN =2enw cos R—2N. N here is the difference of perpendicular distances to the mirrors, M and N, from the ends of the normal in the glass plate, at the point of incidence of the white ra Performing the operations : 2 ‘ a nih cos R G8) dn 2Dcos@eu(1+a)—ycosR Hence maximum d@/dn=~x at the centres of ellipses which occur for y= (1+a) cos R The value of d6@/dn for the different Fraunhofer lines above, if a=:015 is still considered constant and e= in Table III. ‘68, is given Tape ITT. Values of d@/dn and displacements of mirror, N,. e=°68 em. N, and y,, etc., refer to centres of ellipses. | Spectrum lines...| = res sae E. By G. | _1 Se i ae | ol | Equation 17: | d6,/dm =| Wav” | V5" | 53” | 467 | 35m ee icles = a1 | sii 03\ 4/20" |. opaaal ee , 18: | d0/dn =| —6'53"|—10'38"] oo | +847"! 43°51" | |—du/dd =| 281 445 623 793 | 1140 | 19: | d@/dn =| —1'54"|—1'50"| oo | 42/21”) 440" er: 30) | d0/da =| = 14a) Synag| | oe | 3 en | y, =| 11779 | 11811 | 11921 | 1-1981 | 1-2090 | | ay, =| —0142/--0110} 4:0 | +-0080 | +-0169 N, =| 9285 | 9274 | 9391 | -9156 | -957 AN, =| —0156)/—-O117 | +0 | +-0065 | +0187 (observed) AN, =| —°0153; —:0098 | +:°0 + -0057 se Equation | d0/dn = | —1' 54} — oo. Zag eld 48" ! * Interpolated between D and G by p=a+ lA+cX?, where 6= — -00273, c= "0000197. with the Aid of a Grating. 429 These results show that the distance apart of fringes on the two sides of the centre of ellipses is not very ditferent, though they are somewhat closer together in the blue than in the red end of the spectrum, as observed. There is thus an approximate symmetry of ovals, and d@/dn falls off very fast on both sides of the infinite value at the centre. The observed angle between the Fraunhofer lines D and E for the given grating was 4380’... The number of fringes between D and BH would thus be even less than 4380''/638" =6'7 only, which is itself about 4 times too small. The cause of this is then finally to be ascribed to the assumed constancy of —a= (du/p)/(dr/X), a discrepancy still to be remedied. We may note that a does not now enter as directly as appeared in equation (16). By replacing a by its equivalent, equation (18) takes the form GON UNE i) dn 2D cos@ dp ; cos oR (u— x ) ne and a definite series of values may be obtained by computing dujdr; but as all experimental reference here is, practically, not to path differences but to displacements of the movable opaque mirror N, the form of the equation applicable is 2 Soe : =i. (20) dn 2D cos@ pee r gH) aX e( pcos icosiwian To make the final reduction, I have supposed that for the present purposes a quadratic interpolation of mw between the B and the y lines of the spectrum would suffice. Taking the Hi line as fiducial, I have therefore assumed an equation for short ranges, corresponding to Cauchy’s in simplified form, (19) [y= fy = (APM —1/rg), in preference to the more complicated dispersion equations. From the above data for light crown glass we may then put roughly, 6=456 x 10~” and dufdrn= = 25/3. Thus I found the remaining data of Table III. The results for d0/dn agree as well with observations as may be expected. The ovals resemble ellipses, but are somewhat coarser on the red side, as is the case. The centres of ellipses are thus defined by the semi air- vath equation e dj 2b 9 Yo Sel Naa =—— 5 (ut+2 bin?) nearly, (21) 430 Prof. Carl Barus on Interferometry or the corresponding equation in terms of N,. The trend data for AN, agree fairly well with observation, except at the D line, which difference is very probably referable to the properties of the glass, since the grating was not cut on light crown. “The number of fringes between the D and E lines now comes out plausibly, being less than 4380"/104"=42. It is difficult to count these fringes without special methods of experiment ; but the number computed is a reasonable order of values, about 25 to 30 lines being observed. Some estimate may finally be attempted as to the mean displacement of mirror 6N per fringe, between the D and HK lines. As their deviation is @2=1° 13/ and the displacement from D to E AN,, ee _d9 AN, | a2 ae = a dOjdn) dn 0 ~ 2Dcos0 AN, Ge if the value of d@/dn for the D line be taken. Thus SN > ?/(2D cos 8) =A7/D sin 26, nearly. Hence 6N is independent of the thickness, e, of the plate of the grating, as I showed* by using a variety of different thicknesses of compensator. Since A= 000059 cm., D=:000851cm., &N>:00031 cm. The values found were between °00033 and 00039, naturally difficult to measure, but of the order required. 11. Case of dr/dy, and d@/dy, etc.—If, in equation (12), e and n are constant while w, R, y, and A vary, the micro- meter equivalent of the displacement of fringes may be found. Here dp/dy=(dp/dr) . (dr/dy) dR/dy = (dR/dp) . (didn) . (drjdy), and which coefficients are given by equations (13), (14), and (15) Centres correspond to N,=epn cos R— anak on ae cos R dd * American Journal of Science, xxx. 1910, p. 170. with the Aid of a Grating. ABt Tase LV. Values of dx/dN, dO/dN. e='68 em. | Spectrum lines... = B. D. E. 15 Ge | | pales ANSON EE PDL Fp | OC, | Equation 22:*|d\/dN=| ... | —0055| oc | 4+-0081 | +-0031 | 53 22: | d\/dN=| — 0044 | —-0050 wa) +:°0075 | +0023 | em. | ie 22: | d0/dN=| — 146 | — 17:0 0 +260 |+ 81 | rad.| | * Constants interpolated between D and G by p=a+bA+ed?, where 6= — ‘00273, c=-0000197. Thus a 2 . (22) dN N—ewcosR + er(du/dr)/cos K ae so that if dA/dN = ©, the maximum at the centres of ellipses, the simultaneous effect at ’ will be (as the mirror has not moved) aN. r’ aN ear node Re GD alien \" Ayes Ta este) él caaht dX ~= cos R! If the centre of ellipses is at the E line the values of Table TV. hold. The motion on the blue side of the E line is thus larger than the simultaneous motion on the yellow side, conformably with observation. 12. Interferometry in Terms of Radial Motion.—KHither by direct observation or combining the equations (20) and (22) for dx/dn and dd/dN, the usual equation for radial motion again results : dN X ie ae where N is the displacement of mirror per fringe. This equation is best tested on an ordinary spectrometer by aid of a thin compensator of microscope glass revolvable about its axis and placed parallel to the mirror M. The change of virtual thickness e’ for a given small angle of incidence i may then be written : »sin RdR dP]? é +, nearly. aa Sena. co? Ro 21-P/p 432 Prof. Carl Barus on Interferometry Tf l=0, d’=(dl)?. Theretore 2de’=edl?/n?. Ina women trial for e =-0226 em., dl="053 radian, w=1"53, one) tema reappeared. Hence | dN/dn = pde' =4 x 1:53 x 27 x 10-8 = 21 x 10-6 em., which is of the order of half the wave-length used. 13. Interferometry by Displacement.-—In a similar experi- ment the displacement of ellipses due to the insertion of the above glass, e’=°(1226 cm., was from the D line to about the G line. If AN is the displacement of the mirror N, to : bring the centre of ellipses back to the same line, D or E, | we may write w=1+AN/e’. I found at the Hf line, «=1°53, | at the dime; w= 1°53. Special precautions would have to be taken to further deter- | mine these indices. Thus there are two methods for measuring yp, either in terms of the radial motion of the fringes, or, second, in terms of the displacement of the fringes as a whole. Moreover, 1 the preceding paragraph 10 may be looked upon, reciprocally, | as a method for measuring dyu/dn, directly. Part [V.—INTERFERENCES IN GENERAL AND SUMMARY. 14. The Individual Interferences.——In figs. 5, 6, 7, gq is the face of the grating, M and N the opaque mirrors, and I i the incident ray. i As the result of reflexion from the top face, the available | air-path being y,, and y’,,, there must be two images of the slit seen in the telescope directly, viz. a andc (fig.5). Of these ¢ will be more intense than a, which is tinged by the long © path in the glass. These two rays together, on diffraction, | will produce stationary interferences whose path corresponds | to the equation t nA=2en cos R. The optical paths of the two rays are / reflected-refracted, I’, 2y,,+eu(2 cos R—sec R), refracted-reflected, II’, 2y',, + 3eu sec R= 2y,, + eu(3 sec R—4 sin R tan R), If the plate of the grating were perfectly plane parallel, the slit images a and ¢ would obviously coincide. | The directly transmitted rays, however, after reflexion | from N give rise to four images of the slit. In case of a with the Aid of a Grating. 433 slightly wedge-shaped plate, the one at a (fig. 5) being white, that at ¢ yellowish, the distances apart being the same as in the preceding case. Besides this there are two images of the slit at b, figs. 6, 7, which result from double diffraction Fig. 6, JOS Ue at the lower face, the case shown corresponding to 6 i thereafter, while the other image corresponds to @>i and @, and the age-distribution then follows from F(a, )=p(@)B@—=a)..' . J 4 Gee Problem in Age-Listribution. 437 4, From the nature of the problem p{a) and B(a) are never negative. It EoOe: that (4) has one and only one real root 7, which is = = 1, according as (atarplasdaz 1 ERO) Any other root must have its real part less than, For if r; (cos 9 +i sin @) is a root of (4), i= es GOO et (Gi) 0 "Y It follows that for large values of ¢ the term with the real root 7 outweighs all other terms in (3) and B(Z) approaches the value 5 PG) se Ase ar arte cies bo tating ete CH) The ultimate age-distribution 1s therefore given by EY Gey Ge Np (@) 1 a eet a tng |: ()) eA (aie) (a net pam on |) Formula (9) expresses the “absolute” frequency of the several ages. To find the “relative” frequency c (a, t) we must divide by the total number of male individuals. ore ot Apevia ah pi@er” ( F(a, t da Ae) e” “p(a)da \ e” *n(a)da e 0 0 =e." "1 (a) une aa eae aicenie (LOS ia where =I) cp aida ee a CEL b 0 The expression (10) no longer contains ¢, showing that the ultimate distribution is of “fixed” form. But it is also ‘“‘stable;”? for if we suppose any small displacement from this “fixed” distribution brought about in any way, say by temporary disturbance of the otherwise constant conditions, then we can regard the new distribution as an “initial” distribution to which the above development applies: that is to say, the population will ultimately return to the “ fixed” age-distribution. * Compare Am. Journ. Science, xxiv. 1907, p. 201 438 Dr. J. W. Nicholson on the Damping of the | Tt may be noted that of course similar considerations apply to the females in the population. The appended table shows the age-distribution calculated according to formula (10) for England and Wales 1871-1880. The requisite data (including the life table) were taken from the Supplement to the 45th Annual Report of the Registrar General of Births, &. The - mean value of r’ (mixed sexes) for that period was -01401, while the ratio of male births to female births was 1°0382. It will be seen that at this period the observed age-distri- bution in England conformed quite closely to the calculated “ stable ” form. TABLE. MALEs. FEMALES. Persons. AGE [= eae a en (Mears): Gale, ||) Obs || Cale, | Obs. |) Cale. NOE Qo05) | se 139 136 132 138 136 510-2) ats 123 115 117 116 120 10-15...| 107 110 104 104 106 107 15-20...) 97 99 95 99 96 97 20525...) 8S 87 8&7 9 87 89 25-35... 150 144 148 149 149 147 35-45...| 116 112 116 115 116 118 45-5d...| 86 84 8&8 87 87 86 d5-65...| 57 59 62 61 59 59 65-75...| 30 31 39 39) Vb 3e 23 75-00 11 | 12 15 15 13 13 LI. On the Damping of the Vibrations of a Dielectric Sphere, and the Radiation from a Vibrating Electron. By J. W. Nicuoxson, M.A., D.Se.* aig? (las note is supplementary to two short papers in the Philosophical Magazine for October and November last, dealing with the initial motions of conducting and dielectric charged spheres. The object of these papers was chiefly to indicate that if an electron could be regarded as not subject to contraction when in motion, the most useful property to * Communicated by the Author. Vibrations of a Dielectric Sphere. 439 ascribe to its interior would be that of a dielectric of high specific inductive capacity, fur other specifications in terms of conduction would introduce difficulties of an indeterminate type into the discussion of its initial motion under the action of a small force, when the Newtonian mass tends to zero. The present note is devoted to an examination of the rate of decay of the free vibrations of a movable dielectric sphere in the general case in which its motion starts from rest and is not large at any instant considered. Certain conclusions of anegative character are drawn with respect to the radiation from a non-deformable electron possessing the dielectric property, when plane harmonic waves are incident upon it. Like the papers mentioned above, which with some of the present considerations were communicated to the British Association at the Sheffield meeting, this note consists, to a great extent, of a detailed examination of certain points raised by the recent memoir of G. W. Walker, a memoir which constitutes the most comprehensive and_ successful attempt yet made to set the theory of the accelerated motion of electrified systems upon a rigorous dynamical foundation, without an appeal to the method of the quasi-stationary principle, or to special assumptions, such as that of rigidity of electrification, which cannot be formally justified when the motion of the system ceases to be uniform. When a dielectric sphere, whether small or not, is fixed or uncharged, or when its Newtonian mass is very large in comparison with that of electromagnetic origin, the period equation for its free vibrations is that given by Lamb*. With a change of notation, itcan be written in the form (tanh &:))/eh=1+«eN(1—A) /{(e—-I)(L—A)+«r*}, (1) where « is the dielectric constant, and if w+ ev be any root of this equation in A, the corresponding vibration has a period 27(a/vc)* and contains a factor e~-“, where k=wC/a. The radius of the sphere is a, and C is the velocity of light in the free zether outside. Lamb has discussed this equation when the sphere is of atomic size, and « is extremely large, but does not give the decrement of the vibrations explicitly. Walker, in his paper, gives a formula for the first root as N= 444938 e724 (4493)eF . . . . (Q) from which, for values of « of the magnitude «= 108 used by * Camb, Phil. Trans., Stokes Commem. Volume. Sz 440 Dr. J. W. Nicholson on the Damping of the ‘Lamb, the vibrations would be very persistent when started. Walker quotes the result without proof, and perhaps only a misprint has occurred, for the true formula, as will appear below, is of the form A= +4493 ie-3-+ (4:493)4e-3,. 2 2. (8) and the resulting decrement is thus increased in the ratio 10”, so that the free vibrations would not have such a pronounced degree of permanence. When the sphere is capable of free motion while the vibrations on it are taking place, and has a surface charge distributed over it, the field will set it into motion, and the electromagnetic inertia, given for small motions by m' = 2e?/3ac?, where e is the charge, will enter into the question. The more general formula valid in this case is found by Walker to be i, | | (tanh «*d)/X=1+«r? (1+ = —x)/ {@-H( ire —x) m m’ —KXr en tot ne “ih te (4) and if m’ is negligible in comparison with m, this is Lamb’s formula(1). Weshall calculate the decrement corresponding to any root of this equation when « is large. In this case, the main term of A, expressed as a series of inverse powers of «2, is of order «2. The roots are therefore, to a first approximation, those of tanh AiN=@N, ¢-.0 | ee whose first root is well known to be given by «A= 4+ 44932, being purely imaginary. Let «’=-+¢p denote any pair of roots, and let «#X=+ip 4 a be the corresponding roots of the equation (4), « denoting merely the leading term of the real portion. That o must be positive is known from the physical consideration that the vibration must die out ulti- mately, and the equation being real, the roots must occur in pairs in this way. It is evident that o will be of lower magnitude than p. Then i ae | ei: ! os 4) tanh (7p +a) : =|+ ! . ! j ie («—1) (14 — =) (ip+o)— a+ (ip +a) K2 Vibrations of a Dielectric Sphere. 441 where p is of no order in «7. Expanding in powers of a, a me |e — 10, itanp tasec?p=ip +o—ipr(1+ n—ipxé*) /{ (ek—1)1+4 n—ipk®), — inp? —ip*« 3h, rejecting a, and even o in the last term on account of the smallness of the denominators. ‘This is justified by the final result. If a=(k—1)(1+n), B=p(Ki—Kk-2=4+nki+tp'«-2); . (7) then i (tan p—p) +o tan? p= —ip3(1+n—ipx-*)/(a—if). But the same equation must be true for —p, so that —i(tan p—p) +o tan? p=7p3(1+n+ipx-*)/(2+78); whence on addition 2a tan? p= p®(Ba,—2aB,) /(e SI) liens Veni eel) where g4,=1+n, 6,=px=. Writing now tan? p=p?, and retaining only the most significant orders in «, a? + B= («—1)*(1+n)?+p{iLtn) +e *(p?—1)}? ==iG (| m’[m)?, Ba,—a8,;=(1+n)p §e?— KE + nk? + p°« 2+ —p(1+n) (K?—«7~*) = (1+ m’/m)(p?/«+ m’|m)pK* (the rejection of lower orders was unsafe before), and therefore o=p?(m! + mp?/«)/(m + Tvs Kan pana (9) from which N= +7pK72+ p?(Km! + p?m)/(m+m')xn, . . (10) giving the formula already quoted when m/ tends to zero. For the case treated in the earlier paper, when there is no Newtonian mass, m=0, so that Nr UOe repel Rae, vent sia! (IL) Thus the vibration for the fixed sphere contains a factor e-** where k,=p*C/ax*, and that for the charged and movable sphere with no Newtonian mass contains e~’2* where foo Clare. lt will be sufficient in applications of these results to restrict attention mainly to the first vibration, for which p=4t4e3. Phil. Mag. 8.6. Vol. 21. No. 124. April 1911. 2-G 449 Dr. J. W. Nicholson on the Damping of the. We apply the results in the first place to the model atom used by Lamb to illustrate selective absorption. In that model, it is found that in order to obtain a fundamental vibration which shall fall in the ultra-violet, with a sphere of atomic dimensions, « must be of order 10° The actual values taken by Lamb are «=5.108, a=1°3.10-°, in ¢.e.s. units. In this case, it is found from the above formula (3) that 4, =1°5, which is not small, and the fundamental vibration is decreased in a ratio 1/e in 2/3 of a second. The vibrations are therefore not very permanent. With the uncorrected formula of the decrement given by Walker, the ensuing value of fk, is of order 10~™, leading to oreat permanence. The difference in the results 1 is therefore considerable. But the comparatively rapid dissipation of the free vibrations is perhaps not sufficiently rapid to impair the efficiency of Lamb’s suggested model of a gas exhibiting selective absorption. The positive particle may be supposed to be of atomic size. Moreover, we may write for this particle, m=10~*4, e/e=10~” as approximate values, where e is its charge. Its electro- magnetic mass for small motions is therefore m ‘= 26/30c UF 5 .10-*3, so that m'/m=5.10-%. Thus with <=5.108, the value requir ed to brin g its vibration also within the visible spectr um, it is possible, with p equal to 5 approximately for the funda- mental free vibration, to ignore m’/m altogether in the expression (m/ + mp?/k) I(m+m) of (10). Accordingly, the decrement may be given its value for the uncharged fixed sphere of atomic size, and is again k=1°5. This is the decrement of the fundamental free vibration of the positive particle if it can be regarded as a superficially charged dielectric of constant form and of such a character that this vibration comes within the visible spectrum. The decrement of the higher vibrations is of course greater on account ofp. The second root of tauh «A=«*h is given by X= +7°7251, and m'/m being negligible more and more i the higher vibrations, the decrement is proportional to p%, and becomes k’/=13:1. For the third vibration. p= 10-904, leading to k’’=52°2. The increase in k is therefore rapid. Proceeding to the case with which we are at present more immediately concerned, of a hypothetical spherical electron without deformation, we may write, in accordance with current estimates of approximate size,a=10-8. As Walkes has remarked, in order that a vibration from a sphere of this size shall appear in the visible spectrum, the dielectric con- stant must be of order 10'*. This appears at once from the ex- pression for the period as 27r(a/ve)?, where iv is the imaginary Vibrations of a Dielectric Sphere. 443 part of X. The period is accordingly 27a*«:/(pc)?, where p =4'493, and this leads to the value in question. We may now consider two cases: firstly, that in which the mass is entirely of electrical origin ; and, secondly, that in which m and m! have an ordinary finite ratio. The second is the case favoured by Walker from his analysis of the results of Kaufmann’s experiments. In the first place, when the mass is wholly electrical, the decrement reduces from (10) merely to ky =p?c/ax? as in (11). With a=10-, and «=10", this gives 4, =6.10-", indicating avery permanent vibration. This vibration would inevitably persist throughout the time during which the equations for the motion of the sphere under an applied force can be re- garded as furnishing good approximations to that motion. An exception is of course presented to this statement when the applied force is of a periodic character, so that the forced motion is vibratory, and the sphere never deviates far from its initial position, and therefore the equations for the dis- placement & of the sphere at any subsequent time, and the function defining the state of things outside, never tend to become less accurately representative. They only do so, for example, in the problems discussed in the earlier papers, of motion under a uniform force, on account of their assumption that the sphere remains approximately at the origin. But in the face of this consideration, the periodic applied force must not be regarded as exceptional. Tor, the decrement being of order 10-7, it isa matter literally of months,and not seconds, before the free vibrations could be neglected, and therefore all kinds of new agencies would have introduced further free vibrations in the meantime. We may conclude, therefore, that it is never possible to regard the free vibrations as in any way less important than the forced motion. This persistence is rather more pronounced on Walker’s view of the negative electron, for if m' and mare of the same order, we may ignore mp*/«, but not m, in comparison with m’,so that k,=p?m'/«’? (m+m'). The effect of including m is therefore to decrease fy in the ratio m'/(m+m’'). For example, the actual ratio of m to m’ derived by Walker is about unity for Kaufmann’s second set of experiments, so that k. 1s halved. In other words, the oscillations may be said to be twice as permanent as they would be if the mass were wholly electrical. Finally, then, we see that if a free period of the negative electron, regarded as dielectric, is to come within the visible spectrum at all, it is necessary to suppose that its vibrations are extremely permanent, and therefore that a constant force 2G 2 444 Dr. J. W. Nicholson on the Damping of the can never produce a constant acceleration in an electron. This result is, however, not necessary in the case of the positively charged particle, nor in the case of an atom con- sisting of an ag yolomer ation of electrons. The essential basis of this conclusion is the lar ve value of the dielectric constant which is forced upon the electron. A small dielectric sphere could have free vibrations which would vanish very rapidly for extremely large values of the inductive capacity, if its periods were not in the visible spectrum. It would therefore speedily develop a constant acceleration under a constant force. For example, with an electron of the same size, and a value of « equal even to 10”, we should have £,=60, indicating rapid damping. But the necessity for «=10'® determines the matter. For a conductor, on the other hand, there is only one vibration, which Tne period 47ra/c(3 +4m’/m)? and a modulus c/2a of decay. This modulus is extremely great (of order 10). For a dielectric of large inductive capacity this vibra- tion is,as Walker pointed out, an approximation to an isolated vibration of the dielectric not mentioned by Lamb, but when the Newtonian mass is zero it is absent, as appeared in the earlier paper. Its presence in other cases does not of course interfere with the argument above regarding the proper series of vibrations, the first of which we called the funda- mental. Their amplitudes must be of the same order as that of the forced vibration under a periodic exciting force. The foregoing considerations of the damping effect have certain important consequences, and more particularly in the theory of the radiation from an electron executing forced viorations under an incident periodic force. A small sphere in vibratory motion is usually understood to emit radiation, as the Poynting vector indicates. Walker showed in his paper that the ordinary neglect of the exciting field in the determination of that vector is not justifiable, as well as an assumed relation y=e&/c between the quantities y and & below, and that the proper expression from which to deter- mine the radiation is a dissipation function D given by eS (y—eE/c) | 30, i. where £ is the velocity of the vibrating sphere, and y is a function determining the external field at points of space which the vibrations have had sufficient time to reach. The radiation should then be 2D. A calculation which Walker mmakes in the case of the perfect conductor verifies that the result thus obtained is in accordance with that derived by the Poynting flux method, when the proper relation between Vibrations of a Dielectric Sphere. 445 y and € is maintained. This of course confirms Larmor’s formula for the radiation from a vibrating electron, and the force being mechanical and equal to F cos nt, the mean rate of radiation is 244.2 / 2.2.2 7 504 pee ey. ae) 3 mc C ¢? or merely ¢K?/3e%(m +m’)? since anfe can ordinarily be neglected. But we notice that this is the case of a perfectly con- dueting electron, and we have just seen that the decrement coefficient of the free vibration is of order 10%. There is no trouble, therefore, in neglecting this vibration in the deter- mination. Consider now the radiation from a dielectric electron, where « is of the order already found necessary. In this ease Walker has found, by the same method, that for the whole range of periods possible to an harmonic vibration falling on the electron, there are regions in which the forced vibration of the electron leads to absorption of radiation, separated by others in which it leads to emission. Thus for a negative particle, “If m'/m.is greater than 2, there is emission from infinite wave-length to very far out in the ultra-violet. If m!/m is less than 2, there is absorption from infinite wave-length to a certain wave-length which depends on the ciosenss of m’/m to 2. Unless m'/m is very nearly 2, it will be in the ultra-violet.” These results are obtained by supposing that the free vibrations of the electron have died away, and we have seen that the decrement factor ky is of order 10~7 for any relative values of electrical and Newtonian mass of the same order of magnitude, the most favourable case being that in which the mass is entirely electrical. Accordingly, the free vibrations must not be ignored, and the expression for the radiation must be greatly modified, and will probably lose its special characteristics. A precise determination of the matter is difficult for two reasons. In the first place, the distribution of the free vibrations among their several periods presents analytical difficulties, and moreover, on account of the smallness of the damping factor, many vibrations are of equal importance with the fundamental, for in the equation (10) the value of in the approximate root X= +upk7? must be such as to make the damping coefficient p?m'/(m-+m’')k? small, or in other words, op must be of order x, or 10! at least. As the successive values of p only differ by about 3, for example, 446 Mr. J. Crosby Chapman on Homogeneous the tenth is only 29°812, the number of important vibrations is enormous, and an analytical solution by the method of Walker does not appear to be possible, and what the exact result would be cannot apparently be predicted. But one assertion may be made with certainty. ‘he amplitudes of the free vibrations of y and & are of the same order as those of the forced vibrations, in any time for which the periodic force would not be disturbed by other agencies, and it is quite likely that the dissipation function defining the radiation may not be capable of a negative value, with absorption as a consequence instead of emission. These remarks apply of course only to the dielectric electron with such a coefficient that its free fundamental vibration shall be in the visible spectrum. for a conducting electron, the expression for the radiation under a periodic force, with its consequence of absorption for a certain range, is not affected. But arguments against the possibility of the non-deformable conducting electron can be found on other grounds. It seems probable that when x is nearly infinite, the sum of the amplitudes of the forced vibrations (excepting that corresponding to the conductor) would tend to zero, leading to the results for a conductor. But when the vibration is constrained to be in the visible spectrum, no such conclusion ean be drawn. ‘ | | LII. Homogeneous Réntgen Radiation from Vapours. By J. Crosspy CHApMAN, B.Sc., Layton Research Scholar of the University of London (King’s College) ; Research Student of Gonville and Caius College, Cambridge *. pee bodies when exposed to Réntgen radiation emit secondary X-rays. It has been shown + that these secondary rays consist of two types—a scattered radiation having the same penetrating power as the primary beam and resembling it in that it is heterogeneous and an X radiation characteristic only of the element used as radiator and independent of the penetrating power of the exciting primary beam. The elements belonging to the group with atomic weights from hydrogen to sulphur have been shown to give out, when excited, a great preponderance of the first type of radiation termed scattered radiation t, while those in the group from * Communicated by Prof. C. G. Barkla, M.A., D.Sc. 7+ Barkla & Sadler, Phil. Mag. Oct. 1908. t Barkla, Phil. Mag. June 1905. Réntgen Radiation from Vapours. 447 chromium onwards emit almost wholly the characteristic radiation which, on account of its homogeneity, suffers equal percentage absorptions when transmitted through equal thicknesses of aluminium. By determining these percentage PN absorptions the values of a (where [=Ije~™ and p=density of aluminium) have been obtained for the different elements giving characteristic radiations. Previously, in experiments performed for determining the coefficient atthe elements used have been, for purposes of convenience, in the solid state, either pure or in the form of compounds. The following Viiteg, Bie experiments were undertaken at the suggestion of Prof. Barkla with a view to showing that the same type of homogeneous radiation is emitted by the elements, whether they are in the solid state or in the form of vapour. un 43 i i ih Bit Wh d i) ee ee SS ee SaaS aakinal 448 Mr. J. Crosby Chapman on Homogeneous The apparatus used to determine the nature of the Secondary ~ Radiation emitted by the vapours when exposed to the rays, consisted of an iron box which contained the gas: this was fitted with aluminium windows and was placed as shown in the diagram (p. 447), so that the radiations reaching the electroscopes could come only from the vapour inside the chamber. That such is the case with this arrangement is indicated by the dotted lines in the figure which mark the path taken by the extreme scattered rays. The process of the experiment was as tollows. The radiation from the vapour inside the chamber was allowed to pass into both electroscopes, which were of the ordinary gold-leaf type described by Prof. Barkla. The deflexion in the electroscope M was observed, while there was a certain deflexion in the standardiser 8. An aluminium sheet of required thickness was then placed in front of the electroscope M, and the deflexion again read while the standardising electroscope suffered the same deflexion as before. In this way the percentage of the radiation which had been absorbed could ; xX be determined, and thence a value for —. Owing to the difficulties in obtaining and working with gases containing elements of atomic weight greater than 52, in the experiment it has been impossible to determine the en r coeficients — for more than two of the elements. The vapours of ethyl bromide and ethyl iodide were employed on account of their, comparatively speaking, high vapour pressures at the temperature of the experiment. Secondary Radiation from the Vapour of Ethyl Bromide and from Solid Bromine Compounds. Air which had previously been bubbled through ethyl bromide and afterwards passed through a glass spiral immersed in water at 4°C., so that it was saturated ata temperature lower than that of the room, was drawn through the chamber by means of a filter-pump and a steady condition was obtained. The X-ray bulb, which was per- manently connected to a pump, was kept at a suitable degree of hardness in order to obtain a maximum intensity of radiation for measuring purposes. The following typical results were obtained with the bromide, proceeding as previously described. Réntgen Radiation from Vapours. A49 Radiation from Vapour of Ethyl Bromide. (Time of observation 2 to 3 minutes.) Percentage absorption by Percentage absorption by Aluminium previous to Aluminium (‘00626 cm.) after absorption in other column. absorption in column 1. 0) 24-1 24 24-4 OA : 24-1 56 23°6 | 67 | 24:0 75 | pu | Mean value...... 24-2 | | | me ae = 16-4. p As the value of ™ for solid bromine Lad not previously been determined, the gas-chamber was replaced by a plate of sodium bromide obtained by sticking the powder to a thin aluminium sheet The same observations were repeated with Radiation from Solid Bremine Compounds. Percentage absorption by Aluminium |Percentage absorption by (00626 cm.) after absorption in column J. Aluminium previous to absorption in other columns. Radiation from Radiation from Sodium Bromide. Bromyl! Hydrate. 0 | 24:8 24-0 24 24-3 | 24-5 42 2374 24°2 | 56 | 24:0 | 24°3 67 24-1 | 24-1 75 | 23:6 | as Mean value.... 24-0 | 24:1 | | 5 (fox NaBr}= 16:2. * (fox Br(OH))=16'3. 450 Mr. J. Crosby Chapman on Homogeneous the radiation from the plate as with that from the bromide vapour. A plate of bromy] hydrate was also used in this way. It will be observed that the value of us obtained from the vapour has, within the limits of error, the same value as that from the solid, thus showing that the two radiations are identical in character. In order to find the nature of the curve e e e . Xr e s connecting the atomic weights with —, in the neighbourhood 2 nr e e e of bromine the values of — for the radiations from selenium strontium, molybdenum, were determined. Element used as Value of is for radiator. : at | radiation. | Selenium corse .ee eee 18°5 Strontium = sehr lel Molybdenum! 22222" 4°88 Using these values combined with others known before to plot as against atomic weights, a smooth curve results, on which the value of Dor abr orine lies, showing that the latter both in the solid and vapour state gives out a characteristic radiation the absorption coefficient of which follows the law determined for solid elements. Radiations from Vapour of Methyl Lodide and from Solid lodine. The apparatus was similar to that used with the ethyl bromide with the exception that, in this case, a quantity of the iodide almost sufficient to saturate the space inside the chamber was poured into an aluminium dish in the box ; the filter pump and saturation bottle were dispensed with. In addition the X-ray bulb was hardened by abstracting a little air with the pump, in order that the rays given off might excite the hard iodine radiation. The intensity of the rays from the ethyl iodide was very much less than that from ethyl bromide owing to only a Réntgen Radiation from Vapours. Ad51 small part of the incident primary being sufficiently pene- trating to excite the characteristic iodine radiation. This combined with the fact that hard rays do not ionize to any large extent made the times of observation much longer than in the other case. Radiation from Vapour of Methyl Iodide. (Time of observation 15 to 20 minutes.) Percentage absorption by Al | Percentage absorption by Al previous to absorption in (0377 em.) after absorption column II, in column I. | 22 21:6 hy 20°8 a2 22°95 | Mean value...... | 2AEG Xr eo Pp A plate of solid iodine was constructed and the radiation from it examined in the same manner as the bromide plate, with the following results :— Radiation from Solid Iodine. Percentage absorption by Al | Percentage absorption by Al previous to absorption in (‘0377 cm.) after absorption column IT, in column [, 22 20°3 ag Dalen o2 200 Mean value ...... 20:9 ‘ r ‘ | Again, the value of —for the solid and vapour was the a a el a 452 Mr. J. Crosby Chapman on Homogeneous same. By continuing the curve before mentioned, it will be seen that this value lies approximately on it. Although it has only been proved in these two cases that elements in the solid and vapour state emit the same type of radiation, yet it is safe to conclude that what applies here holds generally ; especially considering that the atomic weights of bromine and iodine are well separated. It is evident that this similarity of character in the radiations is what would follow from the fact that the phenomena of secondary X-rays are atomic in their nature. Bombardment of Atoms by Ejected Corpuscles. In a previous paper* facts have been brought forward indicating that the characteristic secondary radiation does not result from the subsequent bombardment of atoms by ejected corpuscles. A slight adaptation of the above experi- ment shows this. For if carbon dioxide and hy drogen are used separately under the same conditions as the gas in which the vapour of ethyl bromide is passed into the “chamber, i in the former case it is the carbon dioxide gas which is chiefly bombarded by ejected electrons, while with hydrogen it is the ethyl bromide itself which has tur the most part to stop the expelled corpuscles. Therefore, if subsequent bombardment causes the characteristic radiation, we should expect greater intensity with the hydrogen than with the carbon dioxide as the gas. This point can be investigated experimentally. The apparatus used was practically identival with that previously described. ‘The tin box was moved farther back and the primary rays were cut down by a lead tunnel in place of the slits then used. At the same time, the electro- scope S was moved into a position to receive radiations from a thin stick of selenium so placed in this tunnel that a small part of the exciting radiation from the X-ray bulb fell on it. The electroscope M was brought nearer to the box, this was possible owing to the alteration in breadth of the primary beam. Since the atomic weight of selenium is 79 while that of bromine is 80, the intensity of secondary radiation from the stick of selenium in the funnel standardizes the power of the incident rays in exciting the homogeneous bromine radiation. In the first part of the experiment, the carbon dioxide gas obtained from a cylinder was saturated with ethyl bromide at 3°°5 C., and was passed through the chamber till a steady state was reached. The deflexion in the electro- scope receiving radiations from the chamber, while the other * Chapman & Piper, Phil. Mag. June 1910. Rontgen Radiation from Vapours. 453 electroscope suffered a certain deflexion, was noticed. The carbon dioxide cylinder was replaced by a hydrogen “ kip,”’ and hydrogen saturated at the same temperature was.passed under similar conditions through the box, and the deflexion of the chamber electroscope, while the standardizer under- went the same deflexion, was noticed. ‘The results are shown :— Intensities of Radiations in the two cases. Temperature of saturation = 8°°5 C. flexi Deflexion of Deflexion of De ee chamber electroscope chamber electroscope of Sa with carbon dioxide as with hydrogen as electroscope: saturated gas, saturated gas. 50 38°0 38°6 50 37°38 38°4 50 38°2 38:0 50 38°) 50-4 Mean values...... 38:0 38 4 Amount of secondary radiation with H, as gas saturated with C,H;Br : Amount of secondary radiation with CO, as gas saturated with C,H,Br — 1-08, This slight difference in the intensities is of the order of magnitude which would result from the excess of absorption in the heavier carbon dioxide gas. Knowing the vapour pressure at 3°°5 C., some relative idea of the magnitude of the difference can be deduced. For, assuming that the absorption of the 8 particles varies directly as the absorbing mass, we get :— Case I. Amount of corpuscular radiation absorbed by ethyl bromide _ aaa a =e Amount of corpuscular radiation absorbed by CO, — Casa II. Amount of corpuscular radiation absorbed by ethyl bromide 18:8 —10 O. Amount of corpuscular radiation absorbed by H, ‘Thus if the expelled electrons do by bombarding the bromine atom make it emit its characteristic radiation, the above calculations show that there must be a most noticeable difference in the intensities of radiation in the two cases. ADA. Mr. F. W. Jordan on the Direct The experimental results obtained show, however, that there is practically no difference in the intensities for the two gases, which proves that the bombardment theory is quite untenable. 3 In conclusion my best thanks are due to Professor Barkla for his interest and encouragement during the carrying oat of these experiments. Wheatstone Laboratory, King’s College. = LIU. The Direct Measurement of the Peltier Effect. Hoyt WV eu ORDAN, eALity.C.S., aSGe ‘| ae Peltier coefficient may be measured directly by the calorimetric methods of Le Rouxt and Jahnf, or it may be deduced from the thermoelectric power by using the thermodynamic relation PHT 3 where P=Peltier coefficient, a =the thermoelectric power at the absolute tempe- rature T’, The methods of Le Roux and Jahn are tedious, and can ouly beapplied when one of the junctions is isolated thermally from the other. The apparatus described in this paper was designed in 1909 for the direct measurement of the Peltier coefficient between copper and a short specimen of crystallized bismuth in the limited space between the poles of an electro- magnet. Pellat§, in 1901, suggested a somewhat similar method, but he does not appear to have made an experiment to test its accuracy. He discusses the case of a compound bar of iron and zine traversed by a current of 20 amperes in such a direction that heat is absorbed at the junction. He suggested that the Peltier absorption of heat might be com- pensated by the heat evolved by a current through a fine insulated wire embedded in the iron close to the junction. ~ The Peltier effect will produce in each bar a temperature and * Communicated by the Author. + Le Roux, Ann. de Chimie et de Phys. i. p. 201 (1§67). { H. Jahn, Wied. Ann. xxxiv. p. 755 (1888). § Pellat, Comptes Rendus, exxxiil. p. 921 (1901). Measurement of the Peltier Kifect. 4.55 gradient towards the junction, and therefore, to compensate this, the current through the fine wire would be adjusted so that the temperature gradient, as indicated by insulated thermojunctions, vanishes in each bar. The Joule effects in the bar and leads 4 mm. thick were considered to be negligible, and to have no disturbing effect on the tempera- ture gradients in the bar. If [=the current through the bar, 1=the current through the heating coil, e=the potential difference at terminals of heating coil. Then el e7- At the ordinary temperature of the air, the heat evolved per cm. length of the iron lead is approximately one-third of the heat absorbed at the junction for a current of 20 amperes, and therefore the neglect of the Joule effect is scarcely justifiable. The dimensions of the apparatus are large for an experiment in a limited space, and owing to the large thermal capacity of the bar a considerable interval would be required before the steady state and the final adjustment of the compensating current were attained. In the following method the two junctions of the copper with the bismuth could not be isolated, and the apparatus is equivalent to a duplication of Pellat’s arrangement with many subsequent advantages. If a current be sent round a circuit of two metals, then the temperature difference between the junctions, arising from the Peltier effects, can be made to vanish by a con- tinuous supply of heat to the cooler junction. In the ideal case, when the junctions are equal in all respects, the rate at which heat must be supplied to the cooler junction is equal to twice that at which it is evolved at the warmer junction. The thermal conductivity of copper is much greater than that of bismuth, and therefore it 1s preferable to supply the heat by the passage of a current through an insulated fine wire embedded in the copper close to each junction. The presence of a heating coil at each junction renders it possible to eliminate from the final result the electrical resistances and the different thermal emissivities of the junctions. The apparatus was also dimensioned so that, at each junction, the Joule heat for the average current of one ampere was about one-tenth of the Peltier heat. In the experiments of Jahn with a Bunsen ice-calorimeter the Peltier heat was only a small part of the measured quantity of heat. aoe SE NENA Sen SS A TL alia Po a i SE 456 Mr. F. W. Jordan on the Direct The bismuth rod e was cut from a crystallized* mass, prepared by Dr. Lownds, so that its axis was parallel to the principal cleavage plane of the metal. Two copper cylinders c,d, were bored out to receive the heating coils, and their =| t) i“ eurved surfaces were planed down to within a millimetre of the coils. The end of the bismuth rod was pressed against and fused to the flat surface of the cylinder. . After solidifi- cation the excess of fused bismuth around the rod was removed. ach heating coil consisted of a spiral of fine double-silk-covered eureka wire, which was soldered at each end to thicker copper wire. The upper copper lead was soldered to a circular copper disk a, and the lower one was attached to the cylinder by electroplating with copper. A copper wire & soldered to the upper lead served as a potential lead for measuring the resistance of the coil, and also as a lead for the heating current. The coil and copper disk were insulated from the cylinder by paraffin wax. The object of securing the coil in this way was to maintain as far as possible an equality of temperature between the copper cylinder and the terminals of the coil, and so minimise the loss of heat by conduction along the leads. Measurement of the Peltier fect. 457 The difference of temperature between the cylinders was indicated by four thermocouples of copper and constantan wires. These junctions were laid in grooves and insulated from the cylinders by thin strips of mica. The grooves were filled in with cotton-wool, and this together with the junctions was tied to the cylinders with silk thread. The available galvanometer was of the suspended-coil type having a re- sistance of 45 ohms, a period of abont 10 seconds, rand a figure of merit of 350 mm. scale-divisions per microampere. The galvanometer, when connected to the neeane Tne: gave a deflexion of about 10 mm. per 0°01 ampere from one cylinder to the other. The difference of temperature between the copper-bismuth junctions may also be observed by interrupting the currents through the rod and heating coil and connecting the coppers with the galvanometer. Stray thermoelectric forces in the use of this method would be relatively more important since the thermoelectric power of this junction is only about one- third of that of the four copper-constantan couples. This method would be suitable with a galvanometer of low re- sistance and short period. The temperature of each cylinder was measured by a copper-constantan junction f, g, soldered to the copper. The various leads were insulated from one another in glass tubes and passed through a copper tube, p g, which was soldered to the coyer of the enclosure. The constant-tempe- rature enclosure was a small rectangular copper vessel with walls 2 mm. thick. This was designed for the small gap between the poles of an electromagnet, and in this position the temperature could be reduced by immersing tbe copper rod extension 7 in ice and water. The currents through the junctions and the heating coil were both sent through the same copper lead, n, 0, and therefore, if necessary, could he easily interrupted simul- taneously. The currents were supplied from two separate batteries of accumulators and could he varied almost inde- pendently. Hach of these currents could be measured to about 1 part in 500 by moving coil ammeters. In making an experiment the current through the heating coil was kept constant, and that through the | junctions was adjusted so that the thermo-junctions ~ gave a small steady deflexion. The exact compensating current through the junctions was deduced from the change of deflexion of the galvanometer, produced by a small change of current through the junctions. Since both the currents traverse the same copper lead to cylinder, it is necessary to reverse the heating Piul. Mag. 8. 6. Vol, 21. No. 124, April 1911. 2 0 458 Mr. F. W. Jordan on the Direct eurrent through the coil in order to eliminate its Joule effect in the lead. ‘This was done, and the mean compensating current through the junctions was observed. The unequal emissivities of the two copper cylinders were eliminated by reversing the current through the junctions and passing the heating current throu oh the other coil. To obtain the temperature of each junction, one of the thermo-junctions 7 was connected in series with another and the galvanometer. ‘he second junction was immersed in water, and this was warmed until the galvanometer was undeflected. The temperature of the water, as read on a thermometer, was then the temperature of the copper cylinder and each junction. The compensating current through the junctions was also determined by using the copper-bismuth junction as an indicator of the temperature ditterence between the cylinders. It was found that, in the two methods, the compensating eurrents differed by 0 005 ampere approximately for a given heating current. It follows that the insulated thermocouples could be relied upon to indicate the temperature difference between the cylinders. It is assumed, in calculating the Peltier coefficient, that the temperature of each copper cylinder is practically uniform ‘throughout, and that possible Peltier and Thomson eifects external to the surface of contact are negligible. ‘There is no doubt that the fusing of the bismuth rod to the copper damages the crystalline structure at the end of the rod, and so produces a thin transition layer through which the Peltier effect is distributed. The sum of the Peltier effects across this thin layer is measured in this experiment, and according to the laws of the thermoelectric circuit this total Peltier effect is equal to the Peltier effect between the copper and the crystallized bismuth. Let Q=rate of supply of energy by heating coil ; (’=mean compensating current through junctions for a given current in one heating coil : P= Peltier coefficient ; p=effective resistance of the metals about a junction ; k=rate of loss of energy from a copper cylinder per degree excess of temperature above the en- closure ; : {=excess of temperature of each cylinder above the enclosure. Then the following equations held for a current C, through Measurement of the Peltier lfect. the junctions during the steady state : Q, + C77, — PC, = Ay, ies Cyr. at PC, = Koty Site and for a reversed current C, : ie keQy + h,Qs OR -+ PC = hit ° Qe +- Cy" cane JEG = hots ° ° These equations give +(C,—C;) if ky — Zr ky + Ky In the experiment Q, was made nearly equal to Q). vr ee i + 5 459 ai (1) oie eee (3) (4) (ony) ; : (5) In ° ° ° ky 2 © ‘e this case the maximum value of the second term, as deduced from the observations, was about 0-001 of the first term, and is therefore negligible. The second term may be made to vanish by making C,=C,; but this necessitates an approxi- tate value of the ratio of k, to £, to evaluate the first term. Thus jeu QO As Qs. 2 (Cr Cn a The tables below contain some of the results of experiments with different currents and at shghtly different temperatures. The thermoelectric powers cf the copper-bismuth junction as ealeulated from the relation are also given. Heating Current. a+0:°125 amp. a—O0'125 5+0°130 b—0°1380 a+tO-UTU a—O-070 b+0:072 b—0:072 a+0O°'l0U a—O0'100 4+0°102 b—0:102 Junction Current. c—1:466 amp. » 1431 d—1-366 sy) Lda0 c—0'461 ,, 0455 d —O°426 », O-411 | e—1:040 ALORS) ad—O:892 5, 0868 Poy dE dl Mean Temp. of junction. Deas Oe rca | Peltier Coefficient. 0-01590 volt ——. -—_ | 001573 volt 0-01462 volt 29 9 QCA a ae The resistanee of coil a=2°780 ohms. b= 2692 (6) Thermoelectric Power. 538X107” volt | egane shag I53-6X10 © volt | | | —6 ‘52:0 X10 volt 460 Mr. I’. W. Jordan on the Direct As an instance of the calculation of P :— At 22°°4 C., | Daz 021252527 30-1307 x 2°692 el 66S 143 1366S ikaale P= 0:01590 volt from bismuth to copper. In.this case the excess of temperature of cylinder above enclosure = 3° nearly. The resistance of the bismuth rod between copper cylin- ders—i7o0 < 10s ool: Length of rod=1°76 em. Mean cross-sectional area of rod=0°154 sq. em. Specific resistance at 19°°5=151 x 107° ohm. The effective resistance of the copper Jead to cylinder was deduced from the change of the compensating current through the junction on reversing the current through the heating coil, Tet ©, and C, be the values of the junction currents ; ¢ the current through the heating coil; rz the effective resistance of the copper lead which is traversed by both the junction and_ heating currents; . r, the effective resistance of the bismuth rod at this junction; 7, the total effective resistance of the copper lead and bismuth rod at the other junction. Then the following equations hold for currents C, and ¢ in the same direction through the copper lead :— Q4+ (Ce)? Cp. — FO =i) | en Gre + PC =Hhot. 2 ee and for a reversed current ¢ :— Q + (Cy —¢)?73 4+ Co?7,— PC,=ht. . . See Orr +PC.=ht. . 2 St These equations give hy + he rs(C— Cz + 20) (C, +0.) = 4 ro(Cy?—C,”) + P(C,—C)} (14+ 72) Cie ee i aE . : : 1 Re To obtain an approximate value of 7s, 7 may be p written =2: and since the results in the table show that the Measurement of the Peltier Effect. 461 effective resistances of the two junctions are nearly the same, r, may be written =73+7,. Then BCC, Shae Ce) 3 = ROO) . HM Me titre) a> Phe (11) Taking the results from the first part of the table, —_ 0:0159 x 0-035 OS ig yO Ee r,—.0°0015 ohm nearly, The maximum possible resistance of each junction will be r, together with half the resistance of the bismuth rod. In the complete equation for E ©); 7” =1,=0°0015 + 0 0009 =0:0024 ohm nearly. k Se The approximate value of a as obtained by substitution 2 in (1) and (2) is 1:08. Substituting these values in equation (&) the second term shy a = I | (a-O) 4 (FER pa R ee”) f = —(1'45—1°35) f(s =r] 0:0024 + ot = —9°2 x 10~ volt nearly. 108+ 1 Since P=15900 x 10-® volt by equation (6), it follows that the second term in (5) is much less than the errors of observation and can be neglected. Thus the calculation of P by equation (6) is correct to 1 part in 1000, even when Q; differs from Q, by 4 per cent. When errors of observation are taken into account the final result is probably correct to 1 part in 200, and this order of accuracy was aimed at in designing the apparatus. The bismuth, which was supplied by Griffin and known as Kahlbaum’s pure bismuth, was carefully crystallized by Dr. Lownds in the following way. A crucible was over- wound with a coil of eureka wire and surrounded on all sides with sand and asbestos. About 1000 gms. of bismuth were melted in the covered crucible, and the whole mass was very slowly cooled by gradually diminishing the current through the coil. The whole operation of cooling lasted 36 hours. The crucible was broken and the lump of bismuth removed. A blow with a hammer near the upper edge 462 Mr. F. W. Jordan on the Direct divided the lump along a plane inclined to the axis of figure of the crystallized mass. The brilliant surfaces of cleavage could be traced by chipping and cutting to the middle of the lump, and were found to be nearly plane and parallel to each other. The rod of bismuth in this experiment was cut from the middle of the crystallized mass so that its axis was parallel to the principal cleavage plane of the bismuth. The surfaces of contact of the bismuth rod with the copper were at right angles to the principal cleavage planes. This method of preparing the specimen of crystallized bis- muth is, with the exception of the electrical heating, similar to that employed by Perrot*. He prepared several crystals from the same mass of bismuth and measured for each the thermo- electric power with copper in two different positions. In one of these positions, the surfaces of contact with the copper were parallel to the principal cleavage plane of the crystal, and in the other position they were at right angles to this plane. The first of these positions was designated by the symbol Il and the second by the symbol 1. He found that the thermoelectric power in the position Il was approximately twice as great as in position 4 at a temperature of 55° ©. The thermoelectric powers at a temperature of 55° C. for four of his crystals in position 1 are given here for the purpose of comparison with the value derived from the Peltier coefficients in the table above. Thermoelectric Power at 45° C. Crystal PF) 4 53°4 x 10-6 volt per degree. 39 Gt coe od 9 29 ”? 99 A tL ose 58:3 39 39 39 99 M 2) Sa 5d°d 39 39 9? The lengths of these crystals varied from 19 mm. to 30mm. In the table above the thermoelectric power as derived from the Peltier coefficient would at 55° ©. be equal to 576 10~* volt per degree. Considering the widely different results that Perrot obtained, this value is quite acceptable. The bismuth used by Perrot was analysed by three chemists and found to be pure, with the exception of an. undetermined trace of iron. The different results obtained by him are probably due to slight irregularities in the structure of the crystals. The bismuth rod in this apparatus has not yet been analysed, and consequently no accurate comparison of the results can be made. * F. L. Perrot, Arch. des Sciences Phys. et Nat. Aug. 1898. Measurement of the Peltier Effect. 463 In Perrot’s experiments a large temperature-gradient was established between the two contact surfaces, and the thermo- electric power would include possible thermoelectric. forces arising from slight changes in the crystalline structure. In the apparatus, described in this paper, the mean of the Peltier forces near the ends of the crystallized rod is measured. and therefore the derived value of the thermoelectric power might differ from that of Perrot’s. Differences, arising from im- purities in the bismuth rod, are also possible. To apply this method to the measurement of the Peltier coefficient for any two metals, the copper leads n, 0 would be replaced by one metal and tne bismuth rod e by the other metal. To dissipate the Joule effect and to reduce the effective resistance of each junction, it is preferable to take each metal in the form of a thin strip. Hach strip should be approxi- mately dimensioned so that the total flow of heat per unit temperature-gradient is the same as for the bismuth rod. In this case the resistance of the junction would be much less than that of the copper-bismuth junction in this apparatus. An additional lead for the current through each heating coil must be soldered to each copper cylinder. The same sensi- bility can be attained, in the case of small Peltier effects, by increasing the number of thermo-junctions attached to each cylinder, and also by reducing the size of the copper cylinders. This modified apparatus would be suitable in the case where only a short length of one of the metals was available. The Peltier effect could also be measured at various temperatures by varying the temperature of the enclosure. An attempt was made to measure the thermoelectric power of the copper-bismuth junction by passing a current through one of the coils and measuring the H.M.F. of the junctions. The temperature difference thus produced between the junctions was small and could not be measured with sufficient accuracy owing toa defective attachment of the junctions 7, g. One junction of the two wires projected about 1 mm. from the cylinder, and although the other appeared to be satis- factory the results showed that this was slightly defective. Owing to the brittleness of the bismuth rod, these defects were not remedied. The following results indicate the existence of these defects. E At 22°°5 C. = =64'8 x 10~° volt per degree, with faulty junction at the higher temperature. i 464 The Direct Measurement of the Peltier Fffect. When the temperature-difference was reversed, - =56°3 x 10-® volt per degree, ‘ with faulty junction at the lower temperature. The fault of the junction arose from the temperature gradient along the wires (0°l mm. diam.). This could be remedied by soldering a junction of thinner wires in a groove and winding the insulated leads around the cylinder, in order to minimize the temperature gradient. A junction was attached in this way to a similar copper cylinder. ‘The defective junction and the insulated junction were also imitated approximately. The latter was laid in a groove and separated from the cylinder by a thin strip of mica 0°02 mm. thick. The rest of the groove was packed with cotton-wool and overwound with silk thread. The cylinder was heated electrically inside a copper enclosure, and the following results were obtained :— Excess temperature of cylinder over enclosure=8° nearly. Temperature difference between the pertect |: avjo sea . . . uJ S = 0 25 eG amc Mere ChIYe a UDE TIONS gree-paa sents aoe et The temperature difference between the perfect and insu- lated junction varied from 0°°3 to 0°45, according to the tightness of the filling of cotton-wool in the groove. The results indicate the precautions that must be taken in fixing the thermo-junctions to the cylinder for this particular measurement. Itis to be noted that this slight defect in the junctions /, g will not disturb the temperatures in the third column of the table by more than 0° 1. The high results for the direct measurement of the thermoelectric power are just what are to be expected when the junctions are faulty and the rod isa short one. It is intended to conduct experiments with this apparatus in a magnetic field to determine how the Peltier coefficient depends on the strength and direction of the field across the surface of contact of the bismuth with the copper. South-Western Polytechnic, Chelsea, S.W. September 18, 1910. [ 465 J LIV. On Condensation Nuclei produced by the Action of Light on Iodine Vapour. By Gwitym Owen, M.A., D.Se., and Haroup Prauine, W.Sc., University of Liverpool * ee experiments made by G. Owen and A. L]. Hughes (Phil. Mag. Oct. 1907, June 1908) it seems that the gas evolved by a soliditied mass of carbon dioxide previously condensed in a dry and dust-free condition contains large numbers of nuclei, the presence of which can be shown by their ability to act as centres for the condensation of super- saturated water-vapour. This fact suggests that the sub- limation of solid CO, consists not merely in the escape of separate gaseous molecules, but also in the liberation of large numbers of relatively large molecular aggregations. The question arises as to whether the same is true for other subliming substances, and it was during the course of experiments designed to test this point that the effects described in the present paper were observed. Experiments were made with camphor, naphthalene, benzoic acid, and iodine—the method being to pass a current aes cir through a tube containing one of these volatile substances, andithen to test dhe air ton nuclei by suddenly expanding it in a bulb containing distilled water. In fig. 1, A is the glass tube containing the substance to lo expansion apps be tested. The air swept through this tube was first rendered dust-free by a plug of cotton-wool as shown. ‘Lhe cloud- chamber B was a glass bulb some 6 cms. in diameter and was sealed on tox a Wilson expansion apparatus t, the piston of which worked in distilled water. ‘The whole apparatus to the right of the tap C was made of glass. Any clouds produced in B were ae is visible by foot ussing on the bulb the light froma Nernst lamp. If we take as a * Communicated by the Authors. + C.T. R. Wilson, Camb. Phil. Soc. Proc. ix. p. 888 (1807). 466 Dr. G. Owen and Mr. H. Pealing on Condensation measure of the expansion the pressure-fall—that is, the difference between initial and final pressure of the gas in the apparatus,—then, as is well known *, there is no con- densation in the body of the gas when the pressure-fall is less than 15 ems. of mercury. When the expansion is between 15 cms. and 20 ems. a few scattered drops are observed (“ rain-like ” condensation), the nuclei in this case being the few ions always present in the gas. On subjecting the gas to a pressure-fall over 20 cms., a dense fog is obtained. The nuclei on which this fog forms are generally regarded as being minute drops otf water continually being formed from the saturated water-vapour. In the present paper we shall refer to the above effects as the ‘¢ Wilson effects.” Results of the Tests made with Camphor, Naphthalene, Benzoie Acid, and Lodine. With camphor, naphthalene, benzoic acid, the showers or clouds obtained on expansion were identical with the usual Wilson effects. Evidently, then, these substances do not sublime in the form of molecular aggregations sufficient large to act as condensation nuclei. C. Barus ft had previously obtained a similar result with a somewhat different apparatus for camphor and naphthalene. With iodine, how- ever, we obtained very marked effects, as is shown in the following table. The figures in the columns marked Tei le Cloud-chamber filled with pure moist | Cloud-chamber fiiled with air dust-free air. | which had passed over iodine. eee | (Ordinary Wiles effete) | cuca | | O=S=laam 15:0 | 0 | 15:0 Few drops. 085) | Few drops. | 16°5 Thin shower. 175 | Thin shower. ico | Good shower. 185 Good shower, | 18°5 | Tinted cloud. | 19°35 | Very dense shower. | 20°5 Fog. | * ©. T. R. Wilson, Phil. Trans. A. vol. exxxix. (1897). + C. Barus, ‘Condensation of Vapor as induced by Nuclei and Ions.’ (Carnegie Institution of Washington, May 1907.) Nuclei produced by Action of Light on Iodine Vapour. 467 pressure-fall ” are centimetres of mercury, and, as already explained, may be taken as a measure of the expansion. For the purpose of comparison the Wilson effects obtained with our apparatus are first given. The above table shows that the presence of the iodine vapour in the cloud-chamber produces a considerable increase in the density of the clouds obtained, and that the effect of the iodine is specially marked for pressure-fall of 18°5 cms. The influence of the iodine was found to be very persistent, sweeping in fresh air through the by-path D for several hours failed to get rid of the effect. In fact, it was found necessary to take the apparatus down and wash it thoroughly to make it give once more the ordinary Wilson effects. Later experiments (described below) showed that the in- fluence of the iodine ultimately disappears if the apparatus is allowed to stand in bright light for three or four days. When the above effect was first obtained, we were naturally led to believe that iodine, in contradistinction to the other substances tried, does sublime in the form of particles sufficiently large to act as condensation nuclei. But this conclusion was upset by the following modification of the experiment. Between the iodine reservoir and the cloud-chamber a tube containing a long plug of glass-wool was inserted, in order to see if the iodine particles could be trapped and prevented from reaching the cloud-chamber. After sweeping air and iodine vapour through this tube for two minutes, the clouds obtained with a pressure-fall of 18°5 were now much denser than before. Further, the passage of the iodine vapour was observed to produce a discoloration of the wool, and, finally, when the latter was coloured through its whole length, heavy clouds were obtained for quite small expansions corresponding to a pressure-fall of less than 10 cms. This last effect was at first regarded as showing that when the iodine is dispersed in the form of minute erystals in the interstices of the wool the mechanical action of the air-current results in small solid particles of iodine being dislodged from these crystals and carried over into the bulb. This view, however, was shown to be erroneous by the following subsequent observations :— (1) The glass-wool very soon lost its power of giving rise to the larger nuclei. ' (2) On expanding without previously illuminating the gas in the cloud-chamber the results were just the same as the Wilson effects. Thus the effects obtained with the iodine are not mechanical at all, but photochemical. 468 Dr. G. Owen and Mr. H. Pealing on Condensation A large number of observations have since been made (a) when the iodine vapour was placed inside the cloud- chamber itself ; (6) when iodine vapour was swept through glass-wool before admission into the apparatus. It will be convenient to discuss the two cases separately, although, as will be shown, they are probably intimately connected. Experiments with Lodine placed in the Cloud-Chamber itself. When studying the effect of iodine placed inside the cloud- chamber, the apparatus shown in fig. 2 was found convenient. Fig. 2. The apparatus could be filled either by filtered laboratory air drawn in through A or by air from boiling liquid air stored in B. Asa matter of fact, both these sources of dust- free air gave practically the same results. The apparatus was first calibrated with pure air and then iodine vapour was introduced by placing a few crystals in the hollow tap C. The taps D, E allowed fresh distilled water to be admitted when desired. In order to study the effect of light, the cloud-chamber and expansion apparatus were wrapped in black cloth. The covering of the cloud-chamber was pro- vided with two vertical slits, one on each side; the one slit Nuelei produced by Action of Light on Iodine Vapour. 469 admitted the light, and the other permitted an inspection of the cloud obtained. The slits were provided with flaps, which could be closed when it was desired to keep the apparatus in complete darkness. Table II. shows the effect of light. The results given in the last column were obtained by producing the expansion in the dark and then admitting the light immediately afterwards. TABLE IT. ] | s BUhae i Oecd! Todine in Apparatus, | No Todine in the | Iodine in is Expansion made in the | = aioe Pent oe areas Lig on dark. Light put on ulb continuously. | ulb continuously. immediately afterwards. Pressure-| (9). Saves | Pressure- Obs gies Pressure- Obsoteation fall. rs Sy) eefalle Duin eee eesti 3, 2 145 | Few drops. | ee 0 155 | Few drops. DOME ait Sl0 wietean ie || nasser ai) eniate 17°5 Thin shower. | 17:5 | Dense shower. 17-5 Thin shower. | i| | 18:5 Good shower. | 185 | Coloured cloud.| 185 | Good shower. | | MS Ge || The above table shows clearly :— (a) That the addition of the iodine causes an immense number of fresh nuclei to appear. (6) That the majority of the nuclei are small and con- sequently require a large expansion corresponding to a ressure-drop of 18°5 cms. to catch them. (c) That all these nuclei are wholly produced by the action of light, for when the expansions are made in the dark the effects observed on admitting the light imme- diately afterwards are the same as in the aleerice of iodine. These effects were obtained with a Nernst lamp as the source of illumination. Other modes of illumination—are lainp, fishtail burner, Nernst light screened by red glass, and diffused daylight—were tried with the same results. In the last case the Relonce were observed by illuminating the bulb with a Nernst lamp after the expansion had “been made. 470 Dr. G. Owen and Mr. H. Pealing on Condensation Lxperiments on the Growth of the Nuclei in the Light and their Decay in the Dark. Table III. shows how the density of the cloud depends upon the duration of the illumination. Pasian hh, Result of the Expansion. Duration of illumination. Prcceune-fll= 1 ee | 0 | Good shower (= Wilson effect). | About 4 second. | Dense shower. | ! 1 second. | Cloud. | 2 seconds. _ Cloud (same as above). Thus the nuclei grow under the influence of the light and attain a maximum size in less than one second. Table 1V. shows that the nuclei disappear very quickly after their formation. The bulb was illuminated in each case for the same period and then kept in the dark for various intervals before expansion. TABLE IV. Period for which the Nuclei were kept in the dark | Result of the Expansion. before expansion. (Pressure-fall = 18°5 ems.) | 3 minutes. Good shower. 4 | Same as ‘ Wi eo 1 minute. Good shower. ilson effects. | 80 seconds. | Very dense shower. | 10 seconds. | Cloud. | | 0 Cloud. Thus nearly all the nuclei live for ten seconds, but all have disappeared in one minute. This may be due to the nuclei diffusing to the walls of the vessel, or to their actual break up. If the latter is the correct explanation, evidently the substance forming the nuclei is very unstable. } Nuclei produced by Action of Light on lodine Vapour. 471 Influence of the Gas in the Expansion Apparatus. Air, hydrogen, carbon dioxide, coal-gas, and oxygen were tried in turn in the expansion apparatus. In each case the apparatus was first calibrated before introducing any iodine in order to study the normal Wilson effects in these gases. With hydrogen, CO,, and coal-gas, the introduction of the iodine produced absolutely no change. On the other hand, in the case of oxygen, the iodine gave rise to clouds similar to (possibly a little denser than) those obtained with air. Thus the presence of oxygen is necessary to the formation of the nuclei. Alcohol in the Expansion Apparatus. Some experiments were tried in which the water in the expansion apparatus and in the cloud-chamber was replaced by alcohol. With this liquid the normal Wilson effects begin at a pressure-fall of 10 cms., and fogs are obtained for a pressure-fall of about 12 cms. No increased effect could, however, be detected on admitting iodine. In fact, when the iodine had been in the apparatus for a day or two so that the alcohol had developed a bright yellow colour by the solution in it of some iodine vapour, the Wilson effects at any given expansion were then distinctly smaller than they were initially. Behaviour of the Nuclei in an Electric Feld. In order to ascertain if the nuclei are charged, a cloud- chamber containing a horizontal platinum disk was con- structed, the distance between the disk and the water surface being about 1°5 cms. A potential difference of 230 volts could be established when desired between the disk and the water. We could, however, detect no evidence of any motion of the nuclei under the electric force, for their rate of disappearance in the dark was the same with the field on as without. It may be concluded, therefore, that the nuclei do not carry an electric charge. Dinunution of the Iigect with Time. We noticed early on in the course of the experiments that the coloured clouds obtained with the iodine in the apparatus become less and less dense as time goes on. By the second day the result of an expansion of 18°5 is only a “dense shower.” This decay continues from day to day, until by the fourth or fifth day the effect obtained is actually smaller than the normal Wilson effect for the same ex- pansion. It was found, in addition, that the effect of the 472 Dr. G. Owen and Mr. H: Pealing on Condensation iodine decreases much more rapidly when the apparatus is kept unshielded in bright diffused daylight than when kept in the dark. If the apparatus be taken down when in this non-sensitive state and thoroughly rinsed out with distilled water, the large effects already described once more make their appearance when iodine is admitted. The following possible causes of the disappearance of the effect naturally suggested themselves and were investigated In turn :— (1) That some change took place in the properties of the iodine itself, say through its becoming damp. Intro- _ ducing fresh iodine, however, failed to bring back the original clouds. (2) Thinking that possibly the nuclei might be due to an action of the iodine vapour on some impurity brought into the apparatus when it was filled with air, and that the disappearance of the effect in the course of a few davs was due to this impurity being all used up, we tried admitting a fresh supply of air. On two or three occasions fresh air did partly bring back the clouds, but generally this was not the case, both air from boiling liquid air and dusty laboratory air being equally in effective. (3) Again, it was thought possible that the original clouds were due to an action of the iodine on vapours evolved by the vaseline lubricating the taps. But we found that introducing fresh vaseline into the apparatus pro- duced no change. Various other possible sources of impurity, such as tap-grease, indiarnbber, red wax, cotton-wool, were placed in the apparatus. In every case there was no increased effect. (4) Again, we thought that the film of iodine which naturatly forms in time on the sides of the cloud-chamber might possibly cut off the effective part of the light entering the bulb. But driving off this film by gently heating the glass failed to bring back the original clouds. (5) Again, it is well known that the value of the expansion required to catch nuclei of a given size depends upon the nainre and condition of the liquid in the cloud- chamher. Now, after the iodine has been in the apparatus for two or three days, the water in the cloud- chamber develops a bright yellow colour owing to the solution in it of some of the iodine vapour. That the disappearance of the clouds is not due to this change in the water was shown in two al a (a) The coloured water in the cloud-chamber was drained off through the tap E and fresh distilled water admitted through D. This process, however, had no effect. Nuclei produced by Action of Light on Lodine Vapour. 473 (6) A small quantity of radium was placed near the cloud-chamber and_the minimum expansion required to catch the ions so produced carefully determined, first before any iodine has been introduced, and after- wards when the water in the apparatus had become strongly coloured by the iodine. The condensation on the ions was found to start at exactly the same expansion in both cases*. Hence the vapour-pressure of the water is not appreciably altered by the iodine. (6) It has already been mentioned that the clouds after their disappearance are brougat back by rinsing the apparatus out atresh with distilled water. We have also stated that admitting fresh water into the cloud-chamber through the tap D (fig. 2) is without effect. The two statements may appear contradictory. But an inspection of fig. 2 shows that there is a difference between the two operations. As’may be seen from the figure, the water enters the apparatus through a nozzle F projecting into the bulb, and consequently settles in the bulb without flowing down the walls. With an earlier form of cloud-chamber, in which the nozzle F was absent, the water on admission ran down the sides of the bulb, and in this case the clouds were found to have been partly brought back. But the experiments with coo) the cloud-chamber of fio. 2 show that this increase in the effect was due not so much to the changing of the water in the apparatus as to the rinsing of the glass walls by the water as it flowed into the bulb. * Taking as the measure of the expansion the ratio of the final to the initial volume of the gas, we found that in our apparatus the minimum expansion required to catch the ions produced by radium was 1:22. C. I. R. Wilson (Camb. Phil. Soc. Proc. vol. ix.) gives the same value for the same form of apparatus. It has already been mentioned that the ordinary Wilson effect was observed in our experiment to be smaller when the apparatus had had iodine in it for some days than it is for the same expansions in an apparatus free from iodine. And yet the experi- ment with the radium shows that the ionic condensation commences at the same point in the two cases. This fact suggests that the spontuncous wenization in a closed space is reduced by siturating the space with iodine vapour, A similar effect was noticed in the experiments with alcohol. We propose investigating this point further. It is possible that there may be a connexion between this decrease in the Wilson effects and the result obtained by Henrv (Proc. Camb. Phil. Soc. 1897) in his experiments on the “ Effect of Ultra-Violet Light on the Con- ductivity of lodime Vapour.” Henry found that the discharge of ions from a metal plate when illuminated by ultra-violet light was ereatly reduced by admitting iodine vapour into the jonization-chamber, but he was uncertain as to whether this was a real effect or a spurious one due to the weakening of the light in its passage through the vapour. Plul, Mag. 8. 6. Vol. 21. No. 124. April 1911. 21 474. Dr. G. Owen and Mr. H. Pealing on Condensation From a consideration of the above we are led to regard as | follows the disappearance of the clouds and their reappear- ance on washing the apparatus. As will be seen later on, the nuclei probably result from the production of some substance which is deposited on the walls of the apparatus, and it is also likely that the action ceases when a certain amount of this substance has been formed. The flow of water over the glass would remove this deposit and allow the action to proceed as before. The view that the glass plays a part in the production of the clouds is supported by the results of experiments made with glass-wool. A passing reference to these experiments has already been made. We shall now consider them in greater detail. : Heperiments with Glass-wool. It has already been stated that when the iodine has been in the apparatus for a few days the coloured clouds obtained for an expansion corresponding to a pressure-drop of 15°5 ems. entirely disappear, and that the introduction of a number of foreign substances into the apparatus failed to bring the clouds back. We found, however, that the intro- duction of a plug of glass-wool gave rise to a cloud at a pressure-fall of 18°5. But after the wool had been in the apparatus for some twenty-four hours, the effect had again disappeared. Evidently, then, something was introduced with the glass-wool which for a time facilitated the production of nuclei. In order to ascertain whether this was accidental or whether it was a general property of glass-wool, we obtained through Messrs. J. H. and 8. Johnson, of Liverpool, three different lots of pure glass-wool, guaranteed to have been supplied to them by three different makers. We found, however, that plugs of each specimen when placed in the apparatus brought back the clouds. Clearly, then, the property is general. But the most effective way of making evident this action of the glass-wool is shown in fig. 3. If a slow stream of air be drawn into the cloud-chamber through this arrangement, a dense fog is obtained for a pressure-drop of 18°5. J urther, when the glass-wool has become coloured by the iodine along its whole length, clouds are obtained for much smaller expansions. Referring to Table III., it is seen, too, that illumination for so short an interval as one second is sufficient to preduce the nuclei caught with a pressure-drop of 18-5. On the other hand, we found that to produce the large nuclei caught by small ee Nuclei produced by Action of Light on Iudine Vapour. 475 expansion, the light had to be kept on for half a minute or more. This shows that the larger nuclei result from the growth of the smaller nuclei under the influence of the light. Fig. 3. ( ) ts Capansicr Afparalis p e No Nag Sone 7% Adine ie Eee) eee ss ie we (x Ne “oh ye! or ne fi, a df Sicsg Heol e Yp Gl ie 9 A gs] A iy (ey oS tea Cees / The clouds obtained at large expansions become less dense as time goes on, showing that the glass-wool gradually loses its efficiency. Further, the decrease in the density of the clouds at large expansions is accompanied by the total dis- appearance of the clouds cf small expansions. This shows that the nuclei only succeed in attaining a considerable size when their number is very large. Inscussion of the Results. These experiments with the glass-wool suggest that the production of the nuclei when the iodine is put directly into the cloud-chamber (fig. 2) is influenced by the condition of tne glass walls of the apparatus. It now remains to consider the source of this influence. In the first place, it is not probable that the clouds obtained when the iodine is placed in the cloud-chamber are due to the presence of organic impurities on the glass, for they were invariably produced when the apparatus had previously been carefully washed with nitric acid and distilled water. And if this were the case it is difficult to see how the process of rinsing the apparatus with distilled water could bring back the clouds after they had once disappeared. On the other hand, the glass-wool itself is rendered less efficient as an aid 212 476 Dr. G. Owen and Mr. H. Pealing on Condensation to the formation of the nuclei by soaking it in strong nitric acid and then washing it with distilled water. A small plug of glass-wool so treated did not cause a cloud when placed (in a dry condition) in the cloud-chamber. This same treat- ment, however, did not prevent the process of sweeping the iodine vapour through a long plug of it (as in fig. 3 from greatly increasing the density of the cloud at 18°5 pressure-fall, although it entirely stopped the-formation of the large nuclei caught at small expansions. If the nuclei originate asa surface action at the glass surface, this effect of the nitric acid on the glass-wool may be regarded in two ways:—(a) That the acid alters the catalytic properties of the surface of the glass-wool, or (6) that it partly removes from it some substance the presence of which aids the formation of the nuclei. One would not expect glass-wool to contain an “impurity ” in the ordinary sense of the word. It is well known, however, that there is an action between glass and pure water resulting in the forma- tion of minute traces of NaOH at the surface. Supposing for a moment that the production of the nuclei depends upon this action, it is not unreasonable to suppose that artificially increasing the alkalinity at the glass surface would increase the number of nuclei produced. To test this point the apparatus was rinsed out with a weak solution (5 per cent.) of NaOH and then worked with this same solution in the bulb and expansion cylinder instead of distilled water, as in the previous experiments. Under these circumstances we found the iodine to be entirely without effect, for the nuclea- tion was identical with the Wilson effects which we studied before introducing the iodine. Thus the nuclei are not due to the minute traces of NaOH usually present on moist glass; or, at all events, the degree of extra alkalinity that we happened to try, instead of facilitating the production of nuclei, prevented it completely. We also found that treating the apparatus in the same way with a dilute solution of H,SO, prevented the production of the nuclei. We thus find it difficult to say definitely what the nuclei are. ‘They are evidently minute particles of some unstable compound requiring the co-presence of iodine-vapour, oxygen, and water-vapour for its production. The effect is clearly photochemical in character, and in all probability belongs to that class of phenomena studied by Tyndall *, Aitken f, and C. T. R. Wilson t. Tyndall found * Tyndall, Phil. Trans. vol. clx. p. 383 (1870). + Aitken, Proc. Roy. Soc. Edin. vol. xxxix. i. p. 15 (1897). t Wilson, Phil. Trans. vol. excii. p. 403 (1899). Nuclet produced by Action of Light on Iodine Vapour. 477 that ultra-violet light caused clouds to form without ex- pansion in air containing traces of amyl nitrite, iodide of allyl, and other vapours. | Aitken found that clouds are produced in NH;, H.SO;, H,8, HCl, and Cl, when expanded after exposure to sun- light. C. T. R. Wilson showed that the action of ultra-violet light on pure moist air or oxygen produced condensation nuclei which grew under the influence of the light, and with very Intense light he obtained clouds without any expansion at all. Wilson suggested that the clouds obtained by him are due to the formation of H,O,, which, by dissolving in the drops as they form, lowers the vapour-pressure and thus makes it possible for the drops to grow where drops of pure water would evaporate. Bevan * again, in his experiments on the “‘ Combination of Hydrogen and Chlorine under the Influence of Light,” found that some substance is produced by the action of the hight on the chlorine and water-vapour which acts as condensation nuclei, and that the formation of this intermediate substance is necessary for the production of the HCl. | With regard to the nuclei described in the present paper, if is possible that the reactions going on under the influence of the light are as follows :— H,O+I1, —> HI+HIO0, H10+0,—+ HI+0,; or it may be a reaction in which the iodine is oxidized directly. Moreover, it is difficult to say which of the products of such reactions would actually form the nuclei. Ozone, of course, is known to give rise to clouds, but, according to Meissner {, only as a consequence of reactions by which some of the ozone is destroyed. The above reactions, being reversible, would cease when a certain amount of the products had been formed. ‘This fact would explain why no clouds are obtained when the iodine has been for some days in the apparatus. It would also account for the reappearance of the clouds on washing the apparatus, for such treatment would remove the produce ts of the reaction and allow it to go on once more. [rom this point of view the effect of introducing a plug of glass-wool into the cloud-chamber is readily understood, for the fibres * * Bevan, Phil. Trans. vol. ccii. A. p. 847 (1903). Ft Quoted by C. T. R. Wilson, Phil. Trans. vol. excii: p. 428 (1899). 473 Nuclei produced by Action of Light on Iodine Vapour. will present a considerable area of fresh glass surface at which the action may proceed. On the other hand, the part played by the glass-wool in the experiments where the iodine vapour is swept through it before admission into the cloud- chamber is not quite so clear. It either means that the glass fibres by some preparatory catalytic action on the iodine- laden air facilitate the subsequent formation of the nuclei, or it means that there is some unexpected “imparity” in the glass-wool, which, reacting with the iodine, produces the results described. Since clean glass is known to affect certain chemical reactions, we are inclined to adopt the former view. Summary of the Main Results. Tn contradistinction to the results previously obtained with solid CO,, we find that (1) Camphor, napthalene, benzoic acid, and iodine do not sublime in the form of particles sufficiently large to act as condensation nuclei for water-vapour. But (2) When moist air (or oxygen) containing iodine vapour is wlluminated, nuclei are produced, possessing the following properties :— The nuclei are very unstable, disappearing in a few seconds in the dark. They do not carry an electric charge. They are not obtained except in the presence of oxygen and water- vapour. They grow under the action of the light, but, generally, do not attain a size greater than that requiring a pressure-fall of 18°5 cms. in order to catch them. The light required for their production need not be very intense nor of a high degree of refrangibility. (3) No nuclei are produced after the iodine has been in the apparatus for some days. The cessation of the action is probably due to a state of chemical equilibrium having been attained. The equilibrium is destroyed by rinsing the glass walls of the apparatus with water. (4) Glass-wool possesses the peculiar property of facili- tating the formation of the nuclei, the number produced when iodine-laden air is admitted into the apparatus through a plug of glass-wool being much greater than the number obtained on placing iodine directly in the cloud-chamber. This property becomes less and less marked as the wool gets more and more saturated with the iodine. The action of the glass-wool is regarded either as being of a catalytic nature or as evidence of the wool being an unexpected source of Potentials required to produce Discharges in (ruses. 479 contamination for the iodine vapour during its passage through it. In conclusion, we have pleasure in expressing our best thanks to our colleague, Dr. H. Bassett, for much valuable information on points pertaining to the chemistry of the subject. Holt Physics Laboratory, Liverpool University. Feb. 8th, 1911. LV. Investigation of the Potentials required to produce D13- charges in Gases at Low Pressures. By A. B. MusErvey, New College, Oxford”. Sparking Potentials between Concentric Cylinders. 4 Me theory of sparking has been completely worked out for uniform fields only, but it would appear that some additional light might be thrown on the processes of ioniza- tion that take place at very low pressures from experiments on sparking in a non-uniform field. These experiments were undertaken with this end in view, and the field between concentric cylinders was selected as the simplest form of a field that is not uniform. Some preliminary experiments J, which were made in this laboratory, with concentric cylin- ders, showed, as might have been expected, that the sparking potential varied with the direction of the field, and the curves obtained in the two cases crossed at a point near the minimum sparking potential. Janea Ab: IBatteries and | Galvanemeter Pump, McLeod Guege,ete. |. Voltmeter The apparatus used in the present experiments is repre- sented in fig. 1. . Band) Bi, * i ‘a ee “SAS van mn Overicl C', a ‘A 8 rh barges. 95 In fic. 4, curves A and A’ 7 45 s - 25. >, Band B, + * ie SSO! Lays ” ”? ” C and C’, si s x ARN Ba se 484 Mr. A. B. Meservey on the Potentials required A few sets of readings were taken with aluminium and platinum electrodes inside, but they showed no consistent differences from those obtained with brass. It seemed as if there might be some effect due to difference of material, but if so, it was different in air from what it was in hydrogen. Fig. 4. In fig. 5 the sparking potential is plotted against the diameter of the inner cylinder for a constant. pressure of 1:5 mm. of mercury. The sparking potentials for parallel plates at distances equal to those between the cylinders for to produce Discharges in Gases at Low Pressures. 485 the different diameters of the inner cylinder, have also been plotted (curve c). The sparking potential with the inner 68 uasbueONU beet OS ee cae i _ eee ear eer a oO 2 ¥ & UV IO LY ib Pirn, OR gee. electrode positive (curve a) is below the parallel plate value, for the largest diameter, but becomes relatively higher as the diameter decreases ; with the inner electrode negative (curve 6), the sparking potential for the largest diameter is lower than with either parallel plates or positive inner cylinder, and becomes relatively lower yet for the smaller 486 Mr. A. B. Meservey on the Potentials required diameters. The positive and negative curves therefore make larger and larger angles with each other and with the parallel plate curve as the diameter decreases. Figs. 6 and 7 show similar curves (a and a’ for positive, b and 6’ for negative inner electrodes, ¢ and c' for parallel plates) for several different pressures in air, and exhibit a progressive change in the forms and relative positions of the different curves. As the pressure becomes less, the positive curve lies higher as a whole with reference to the parallel plate curve ; while in the negative curve the divergence from the positive becomes less, and finally, instead of bending away from the positive, the negative actually rises and approaches it. The rise in the sparking potential for large diameters with negative inner electrodes at pressures of *} and samme is due to the fact that for these diameters the pressure 1s below the critical value, so that the pressure-sparking poten- tial curve is rising very rapidly. The curves for hydrogen, of which two pairs are shown in fig. 8, exhibit similar general characteristics, but are somewhat more confusing owing to the fact that the critical pressure is higher in hydrogen than in air, and the rise for the large diameters is much more marked for certain pressures. In fig. 9 the diameters of the inner cylinder are taken as Fig. 9. Ret eee aie : 1 | Rane ACHE Ree eer Simwis ota aa Vo 26 Savane USER eae) 6 yy 9 2 162 Ign Laer Lae) 26 36 Diameter in Mm. ae abscissee, and the corresponding values of the minimum sparking potential as ordinates. For the largest diameter employed the minimum sparking potential in air is 380 volts with the inner cylinder positive, or about 30) volts above the parallel plate value ; when the inner cylinder is negative, the minimum sparking potential is 330 volts, or about 20 yolts below the paralle! plate value. As the diameter is to produce Discharges in Gases at Low Pressures. 487 decreased, the minimum sparking potential rises for the first direction of the field, and becomes still lower for the second, the change becoming more and more rapid .as we approach the small diameters. But finally a critical diameter is reached for the negative and a different one for the positive, and when the diameter is decreased below these critical values the effect on the minimum sparking potential is reversed, so that a sharp drop appears in the positive and a rise a ee negative curve. Fig. 9 also shows curves for hydrogen which are some- what ae to those for air, with a maximum in the positive and a minimum in the negative . The number of points for the hydrogen curves is not such that we can consider the latter as definitely established, but they are probably ap- proximately correct. If the curves in fig. 9 are produced, as seen by the dotted lines, oy come to horizontal positions at approximately parallel plate values of the minimum sparking potential, and at the diameter of the outer cylinder. Additional readings might cause the curves to be drawn somewhat differently, but this ought to hold true in any case, since the field between concentric cylinders approaches that between parallel plates as a limit, and the minimum sparking potential for parallel plates is constant. The phenomena of the discharge between concentric cylinders may be explained along lines sugyested by Prof. Townsend in connexion with the collision theory * Above the critical pressure, all the main effects may be explained on the supposition that the ions are produced by the collision of positive and negative ions with the molecules of the gas. But below, and possibly in the region of the critical pressure, some other process of ionization in addition to ionization by collision evidently comes into play, and Prof. Townsend suggests that some ‘form of non- -penetrating radiation, due to the impact of negative ions against the positive inner electrode, may be the cause of the effects obtained. Such radiation: dependent upon the velocity ot the negative ions, would affect only the ;ositive curves in figs. 3 and 4, since only when the inner cylinder was joey e would the negative ions appro oach the metal under a large force and impinge on the electrode with a high wale The radiation would naturally be absorbed at “the higher pressures, but when the pressure first reached the point at which radiation would be present, the curve for positive inner electrode would begin to fall below the position which * ‘The Theory of Ionization of Gases by Collision,’ pp. 67 ef seg. and 76. 488 Mr. A. B. Meservey on the Potentials required it would occupy if this effect were not present, and would fall farther below the position as the pressure continued to diminish. If the effect were sufficiently great, it would eventually balance the effect of the non-uniformity of the field upon the ionization by collision, the two curves would cross, and below that point the radiation effect would keep the positive curve below the negative. This process would correspond exactly with what takes place in every case in the present experiments, for the small diameters ; for the larger diameters the tendency is the same, though the actual point of crossing does not come within the limit of available potential difference as the field becomes more and more nearly uniform and the curves approach that for parallel plates. The fact that the curves cross in different instances at very nearly the same pressure might also be explained on the radiation hypothesis. As the diameter of the inner cylinder decreased, the drop of potential near the positive inner electrode would become more abrupt, so that on the average the negative ions would strike the metal with greater velocity. The amount of the radiation effect tending to lower the positive curve would therefore be greater. On the other hand, the initial difference between the curves, due to the regular process of ionization by collision, would be greater in the less uniform field. The relative increase in the radia- tion effect with the diminution of pressure would probably remain the same, so that the larger amount of radiation might balance the larger initial difference at the same pres- sure at which the smaller amount of radiation would balance the smaller initial difference. In addition to the fact that the radiation hypothesis fits in with the characteristics exhibited by these curves, it may be mentioned that the preliminary readings of some experiments now being carried on indicate the presence of some kind of radiation, at the pressures under consideration, the amount of which varies directly with the force and inversely with the pressure. The curves in fig. 5 are just what the collision theory would lead us to expect for pressures somewhat above the critical pressure. Owing to the concentration of force near the inner electrode, the sparking potential when that electrode is negative should be lower than when it is positive and lower than the parallel plate value. Decreasing the diameter of the inner cylinder produces a still greater concentration of force near that electrode, and a corresponding weakening of the rest of the field. Such a decrease, therefore, should to produce Discharges in Gases at Low Pressures. 489 tend to increase the sparking potential, as compared with the parallel plate value, if the inner electrode is positive, and to decrease it if that electrode is negative. That is, the positive curve should rise, with reference to the parallel plate curve, as the diameter decreases, and the negative should drop, the positive and negative therefore diverging toward the smaller diameters. This corresponds exactly with what takes place in fig. 5. The positions of all the curves may be somewhat affected by the fact that decreasing the diameter of the inner cylinder increases the distance between the electrodes, but their relative positions probably depend mainly upon the distribution of force in the field. The curves in figs. 6 and 7 show the same characteristics in a general way, but there are certain steadily progressive changes which have been already pointed out. As the pressure diminishes, tle position of the positive curve as a whole becomes higher with reference to the parallel plate curve, and the curvature increases ; while the curvature of the negative is at first reduced and finaily reversed as the pressure diminishes. The fact that these changes are con- tinuous points to some continuously changing factor or factors in the determination of the sparking potential as the eause. Such continuously changing factors are to be found in 2 and 8, the coefficients of ionization for negative and positive ions respectively. Since « and ® depend upon both X and P, where X is the electric force and P the pressure, a change in P will necessitate a change in X in order to satisty the conditions for sparking, and therefore in general a change in the total potential difference \x dr. The change of # and @ with P may therefore very possibly be the cause of the change in the relative positions of the curves in figs. 5 to 7. In fig. 9, where the minimum sparking potentials are taken as ordinates, and the diameters of the inner cylinder as abscissze, the negative curve is very much like the curve in fig. 7 for a pressure of °5 mm. In fact, for the two larger diameters the critical pressure is near enough to °5 so that the potentials for these diameters lie practically on the °5 curve, while for all the other diameters the critical pressure is just over *3 and the curve for °3 very nearly coincides with that for “5 at all these diameters. The negative curve for minimum sparking potentials is therefore practically a constant pressure curve like those in figs. 5 to 8, and is probably to be explained in the same way. For the positive curves the critical pressure does not remain so nearly constant, and the positive curve in, fig. 9 Piil. Mag. 8. 6. Vol. 21. No. 124. April 1911. 2K 490 Mr. A. B. Meservey on the Potentials required does not follow any one curve of figs. 5 to 7. But it is of the same general form as all of them except for the drop at the two smallest diameters, and undoubtedly the change in distribution of force in the field would have the same kind of effect on the minimum sparking potential as on the others. Judging from the relative positions of the curves in figs. 5 to 7, the decrease in the critical pressure with the diameter should muke the positive curve in fig. 9 less steep than the other positive curves, as it is, but this does not seem to account for the drop which occurs at the smallest diameters. But these two diameters are the only ones for which the critical pressure falls below the pressure at which the radiation effect seems to become large, that is, below the point at which the curves cross in fig. % ihe presence of the radiation would explain the lowering of the minimum sparking potential at these diameters for positive inner electrode, and therefore the sudden change in the curve in fiz. 9. If, then, we consider the continuous change in the relative positions of the positive and negative curves in figs. 5 to 7 as due to continuous changes in a and , we can explain all the curves obtained from these experiments by the collision theory and the presence of radiation at pressures in the vicinity of the critical pressure. The results of these experiments may be summarized as follows :— 1. For every value of the diameter of the inner cylinder, two different sparking potential curves are obtained, cor- responding to the two directions of the field. For smaller diameters, the two curves cross near the critical pressure, and that for a negative inner electrode is higher on the side of the lower pressures, but they tend to coincide as the field approaches that between Sraltel plates. 2. The minimum sparking potential depends on the diameter of the inner electrode, and is always lower when that electrode is negative. 3. The minimum sparking potential is always higher than that for parallel plates when the inner electrode is positive, while the lower values when the inner electrode is negative are below that for parallel plates. The lowest values obtained were 311 volts for air and 240 volts for hydrogen. 4, Curves drawn to show the relation between diameter and sparking potential at constant pressure exhibit a rise as compared with curves drawn for parallel plates if the inner electrode is positive. The negative curve is lower, and at the higher pressures diverges sharply from the positive for the smaller diameters, but the divergence lessens with the pressure. to produce Discharges in Gases at Low Pressures. A491 5. Curves drawn to show the relation between diameter and minimum sparking potential are of the same nature, but the positive drops sharply for the two smallest diameters. 6. The results obtained at pressures down nearly to the critical pressure may be explained on the theory of lonization by collision. Below that, the results are entirely in harmony with the supposition that at these pressures some kind of radiation is produced by the impact of negative ions 8 on the electrode under the action of a fairly strong force. The Effect of a Continuous Discharge upon Sparking Potentials at Low Pressures. Some time ago a series of experiments was carried out by J. A. Brown to determine the relation between the sparking potential and the potential required to mainiain a current in a gas*. These experiments show that above the critical pressure, the potential required to maintain a current is less than the sparking potential, the difference increasing with the current, while below the critical pressure the effect is the reverse. The collision theory, which explains perfectly so many phenomena above the critical pressure, indicates that the potential required to maintain a current should be less than the sparking potential, regardless of the pressure, except for very small currents. Some other factor evidently comes into prominence at low pressures, in addition to the ordinary process of ionization by collision, and it is suggested, as an explanation of the relatively greater rise of the main- tenance potential below the critical pressure, that the gas is heated by the discharge and part of it driven out from between the electrodes. The effect of such a diminution in the quantity of gas between the electrodes would be to lower the maintenance potential above the critical pressure, and raise it below the critical pressure. In the former case, the difference which should theoretically exist between the curves would be increased ; but in the latter case the effect would be exactly the opposite, and if large enough would more than balance the difference demanded by the collision theory, so that the maintenance curve would be above that for the sparking potential. The present experiments were indertaken in order te test directly for the existenee of any such expulsion effect. R. I’. Harhart has since published a paper embodying the results of experiments concerning the effect of temperature upon the potential required to produce a lumineus discharge, * Philesopbical Magazine, September 1906, 2K. 2 492 Mr. A. B. Meservey on the Potentials required : and upon that necessary to maintain it*. These experi- ments indicate that, at least for the potential required to produce the luminous discharge, the effect at ordinary temperatures is similar to the heating effect mentioned above, that is, the curve for a higher temperature is higher below the critical pressure and lower above it than that for a lower temperature. In Brown's experiments, all the points on a sparking potential curve would be found at approximately the same temperature, while those on the corresponding maintenance curve might be at higher temperatures owing to the passage of the current. If the effect of temperature change on the maintenance curve were like that on the other curve, and sufficiently great, it might cause the maintenance curve to cross the other and run above it when the pressure was below thie critical value. But the effect of temperature on the maintenance curve, in Harhart’s experiments, does not seem to be very determinate, and at any rate is not large, so it would hardly seem that we could explain the peculiar relation between the two curves on the ground of temperature effects. However, the ex- periments were not undertaken for the purpose of settling this point, and did not give definite information on it, so it was decided to continue a method of direct investigation. Fig. 10. y ee | Stevage eelisand - Galvane meter Efectrosteétie | Usitueter FAL ed Gan yo In the present experiments the spark chamber used (see fig. 10) was one of those used by Brown, and the rest of the apparatus was the same as his, with a few additions. The spark chamber consisted of two aluminium disks, set into * Physical Review, September 1909. ¢o produce Discharges in Gases at Low Pressures. 493 thicker supporting disks of brass, and separated by a ring of glass one centimetre thick, an arrangement which eliminated discharge from the backs and edges. of the plates. Small holes in the centres of the disks enabled the gas to be pumped through the apparatus. A battery of small storage- cells provided “the necessary difference of potential, and the passage of the current was detected by a galvanometer of the D’Arsonval type. The difference of potential was measured by a Kelvin multicellular electrostatic voltmeter reading to 600 volts. In place of the straight glass tubes leading from the spark chamber in Brown *s apparatus, capillary tubes were inserted, with bends in them which could be filled with mercury and used as valves. The gas used was hydrogen. It was allowed to stand for at least a day in a drying-tube before being admitted to the apparatus, and there was another drying- -tube in the apparatus near the spark chamber. The sparking potential was found for a given pressure by increasing the potential difference until the galvanometer gave a deflexion. The current was then run for a few minutes, with the mercury in the position shown in the figure, so that gas heated between the electrodes would have an opportunity to expand along the tubes into the rest of the apparatus. The mercury was then raised far enough to block the bends in the tubes, and the current instantly shut off. After an interval a second determination of the potential was made. And finally, the valves were opened, the pressure permitted to adjust itself, and the sparking potential again found, the mean of the first and third values being taken as the value under normal conditions. The results of this method are shown in Table I. V,, Vo, jeep: I ie Nie Va: R | P Ney We wa Wen ies 388 403 380 19 | = 3-92 487 507 487 20 omy G08 487 508 48) 22 (?) | 6:09 in 806 325 308 1S AE) | 61 | 333 338 334 5 | 60 325 361 325 36 (?) 60 4}1 446 410 yD) 4°55 | 425 458 426 OT, {58 | 269 280 271 10 1-03 319 342 320 22 (?) 63 320 342 321 A bo 32] 342 | 323 20) 63 354 366 | 850 14 8 350 362 348 18 58 358 582 564 ot S25 494 Mr. A. B. Meservey on the Potentials required. V3, are respectively the first, second. and third values of the sparking potential ; R is the difference between V», and the mean of V, Ane V;, reckoned positive when V, is the greater: P is the pre-sure in millimetres of mercury. There was usually more or less variation in the value of the sparking potential, so several readings were generally taken for each of the quanties V,, Vo, and V3. ‘The values given in the table are in each case those that seemed most steady and reliable. A table was also made out in which the first reading obtained in each case was used, another in which the second was used, and a third in which the last reading taken was used. In these tables the values of R were sometimes quantitatively different from those given in Table I., but qualitatively they were the same, and in many cases differed by only a volt or two. A few sets of readings were rejected because “normal”? conditions were not steady, apparently, as there were large differences between V, and V3. Three sets of readings are marked doubtful because of filme to record at the time the point in the experiment at which the spark chamber was opened, so that it was necessary to depend on memory for this. ‘The doubt, however, as to the correctness of these readings is very slight indeed. The experiments show that R was positive in all cases, that is, that there was a rise in the sparking potential at all pressures, both above and below the critical pressure, with an average value of about 20 voits. The critical pressure was about 1 mm. A further test was made by running the current while the spark chamber was closed. The initial sparking potential was found either before or after closing the valves, experi- ment having shown that the closing of the valves made no difference in it. The current was then run as_ before, but with the valves closed, the gas was given an opportunity 10 cool, and the sparking potential was again determined. Finally, the valves were opened and a third determination made, for comparison with the first. The results are shown in Table IT., in which the same notation is employed as in Table I. ah one case a small decrease in sparking potential is recorded, in all the others an increase. In no case does the sparking potential remain the same for V,. as for Vy and V3. By jurae the current with the valves open all the time did not seem to have any appreciable effect on the sparking potential. Two or three sets of readings were obtained which showed an increase of two or four volts from the average value before the current passed, to that afterward ; to produce Discharges in Gases at Low Pressures. 495 but this is very small compared with the differences shown for the same pressures in Table I., and is no greater than the variation in the values without regard to current. TABLE II. ) | | | Vi. Vie Ve | R. P. | ig : t | ey 488 gg | 8 5°69 | | 480 493 | 4800 | 13 Bete | azo Ee CN eee Ana 6-0 | 480 g9500 1) 470 16 6-0 oR 588 | a6) 8-25 | 330 BOON Wit 328 | uly 60 | There seems to be no definite relation between R and the pressure in these experiments, nor much uniformity in the values of R themselves; but this is not surprising, as the conditions were not the same in all cases. The length of time the current was run was different in different instances, the currents used were of different strengths, and the dis- charge between the plates, though steady, was not always uniform over the surface of the plates. The experiments are of a qualitative rather than a quantitative nature, but qualitatively the results are the same, regardless of variations in the current. It is evident from these experiments that the fact that a higher potential is required to maintain a current than to produce a discharge below the critical pressure, is not due to the expulsion of gas from between the electrodes. If in the first method (Table I.) some of the gas is driven out from the spark chamber by the passage of the current, ths second value of the sparking potential should be lower than the normal above the critical pressure, and higher below it, since a smaller amount of gas is then occupying the same space that was previously occupied by a larger amount at the same temperature. For the gas cools to its original temperature very rapidly in contact with so much cooling surface, and the diminution of volume due to the rise of the mercury in the bend of the tube is entirely negligible. Table I. shows that, although the results obtained below the eritical pressure are in harmony with the supposition that gas is driven from between the electrodes, those obtained 496 Mr. A. B. Meservey on the Potentials required above the critical pressure are the direct opposite of what would result from the expulsion of gas. In the second method (Table I1.), so far as expansion is concerned, the gas should be in exactly the same condition at all three times. The valves are closed before the current is started, and are kept closed, so that no gas can escape; and after the current has stopped, and sufficient time elapsed to allow the tempera- ture to regain its original value, the gas should be exactly the same, with regard to temper ature, pressure, and actual amount present, as it is at the start. Dad since it makes no difference at the start whether the sparking potential is found before or after the valves are closed, so at the end it should make no difference whether it is found before or after the valves are opened. That is, all the values should be the same, so far as any expansion of the gas originally between the plates i is concerned, and the value of R should be zero. But in Table II., which gives the results of this method, R is never equal to zero. Table I1., therefore, like Table L., shows plainly that the effects obtained in these experiments are due to some other cause than the expulsion of gas from between the electrodes. In considering what may be the real cause of the dif- ferences in sparking potentials observed in these experiments, the first thing to be noticed is that, whatever its nature, it is confined to the gas. If the valves are left open during and after the passage of the current, a proceeding which leaves the gas free to circulate but which cannot have any effect upon the electroles, running the current does not affect the sparking potential. This fact, and the fact that as soon as the gas is allowed to pass, by the opening of the valves, in the methods used in obtaining Tables I. and II., the potential drops to practically its original value, show that the cause of the effect is to be found in the gas. The first thing which suggests itself as a cause is the driving out of gas from the electrodes themselves by the passage of the current. The gas which is most frequently met with in such cases is hydrogen, but the addition of a little hydrogen to that already present dces not suffice to explain the results obtained. With the spark chamber closed, an increase in the amount of hydrogen should increase the sparking potential above the critical pressure, that is, the value of R should be positive ; while below the critical pres- sure, R should be negative, and numerically er eater because of the steepness of the sparking- potential curve in this region. Table II., taken by itself, is more or less in accord with this supposition, since the value of R is positive in each case to produce Discharges in Gases at Low Pressures, 497 above the critical pressure, while in the single reading we have below the critical pressure it is negative. It may be noted, however, that the negative value of R is numerically smaller instead of larger than the positive values, and if compared with V3; instead of with the average of V, and V3, is only three volts. Furthermore, this negative value was obtained immediately after the set of readings j in Table I., which gave the unusually small value of 5 volts for R, and it is possible that the conditions were in some way abnormal for both. Evidence of « much more definite nature is found upon examining the results obtained when the spark chamber is not closed till it is time to stop the current. In this case, the pressure has a chance to adjust itself all the time the current is running, so that when the valves are finally closed and the current stopped, the pressure and the kind of gas are the same inside the spark-chamber as outside, even if some hydrogen has been driven from the electrodes. Thus opening the valves should make no difference, or at most a small dif- ference, if the pressure has not become quite equalized, and ihis small difference should be of opposite sign above and below the critical pressure. Table I. shows that the results are the exact opposite of these. Instead of a zero value of R, or small values which change sign at the critical pressure, we have much larger values of R on the average than with the spark-chamber closed all the time, and the sign of R does not change at any point. From these considerations it appears that we cannot explain the results by an increase in the amount of hydrogen between the plates any better than by a decrease. If an explanation is sought in the presence of some other gas, it must be one whose sparking-potential curve lies above that of hydrogen at all points, as R does not change sign at any pressure, above or below the critical pressure. Since air fulfils this condition, it was thought possible that there wasa slight leak in the black-wax joints of the spark-chamber, and that the air thus admitted, though not enough to change the pressure appreciably in the whole apparatus, might be sutfi- cient to have an effect in the small space a the spark- chamber. Such an effect, however, would be independent of the presence of the current, and the fact that the sparking potential is sensibly the same whether the chamber is open or closed, so long as the current is not run, but changes atter the passage of the current, disposes of this ‘possibility. If the effect were due to some gas of generally higher sparking potential than hydrogen, such as air, driven from the electrodes s, one. would naturally expect that the effect 498 Potentials required to produce Discharges in Gases. would be much ee when the valves were closed all the time than when they were open during the passage of the current, as in the former case there would be no opportunity for diffusion. But asa matter of fact the readings indicate the opposite. As has already been ed out, the readings are not of great value quantitatively, but one might reason- ably expect that on the whole those taken with “the valves closed would average higher than those taken with them open, while actually they are considerably lower. Another possibility is the existence of some sort of electric fatigue in the gas after the passage of the current. At first thought it would seem that the existence of such an effect could not have escaped the observation of so many previous observers. But the conditions of previous observations have been such that the gas between the electrodes has been in free communication with that in the rest of the apparatus at all times, and under those conditions in the present experi- ments no change in the sparking potential is observed. On this supposition, however, the effect should be at least as great when the valves are closed during the time when the current is run as when they are closed only at the end of that time, thongh not necessarily greater, since the effect might be produced as rapidly as iresh gas could diffuxe into the chamber. This supposition, therefore, is open to the same objection on the ground of quantitative results as is that mentioned in the last paragraph, but to a less extent. It would have been well if a few more readings had been taken at certain points, but the apparatus was taken down immediately after the present readings had been completed, in order to make room for other apparatus, and the results were not considered in detail until later, so that the desira- bility of further readings was not at the time observed. As it is, the conclusions arrived at are :— 1. Running the current under the conditions of the ex- periments produces a rise in the sparking potential, and the cause of the change lies in the gas. 2. As to the main object of the experiments, the investi- gation of the suggested heating of the gas and expulsion from between the electrodes, with the resulting effect on the sparking potential, the effects obtained are distinctly i incon- sistent with this hypothesis. The explanation of the relation between the maintenance and sparking potential curves below the critical pressure is evidently to be sought on different grounds. 3. One possible explanation of the effects obtained is that air is driven out trom the electrodes by the passage of the On the Pressure Displacement of Spectral Lines. 499 current; or conceivably the current has some small temporary effect on the gas itself, as is the case with oxygen when ozone is formed. The quantitative evidence is contrary to both of these methods of explanation, though less strong against the second. Quantitatively, however, the readings are not very reliable, and further investigation may show that one of these explanations is correct. In conelusion I wish to express my gratitude to Prefessor Townsend for the kindness he has shown in giving assistance and advice in the course of these experiments. The Electrical Laboratory, Oxford. LVI. On the Pressure Displacement of Spectral Lines. By R. Rossi, M.Sc.* UIA ae wINCH the discovery of the pressure displacement of spectral lines several theories have been put forward by difterent authors to explain this phenomenon. Schuster 7 first suggested that it ought to be ascertained whether the displacement of the linesis due to pressure only, 2. e. to molecular impacts, or due to the proximity of molecules vibrating with equal periods. FitzGerald t and Larmor §, treating the atom as an electro- magnetic oscillator, found that an increase of specific inductive capacity (due to an increase of pressure) would cause a dis- placement of the spectral lines; while Richardson ||, con- sidering the forced vibrations set up in an atom by the surrounding atoms, arrived at the same conclusion. Humphreys §, on the other hand, does not consider a change in the specific inductive capacity of a gas to be the main cause of pressure displacement. If the pressure displacement is a linear function of the specific inductive capacity of the radiating vapour, the experi- mental methods so tar available are not accurate enough to prove it on account of the small differences of the specific inductive capacities of ditferent gases; butif it is proportional to (w°—1), as claimed by some authors, (u being the refractive index of the medium surrounding the vibrating atom), con- siderable changes in the displacements ought to be detected o by surrounding the source of light with different gases. * Communicated by Prof. E. Rutherford. t Astrophysical Journal, vol. iii. p. 292. { Astrophysical Journal, vol. v. p. 210. § Astrophysical Journal, vol. xxvi. p. 120. || Phil. Mag. [6] vol. xiv. p. 557 (1907). §] Astrophysical Journal, vol. xxvi. p. 18, 500 ~ Mr. R. Rossi on the Pressure Passing from air to carbon dioxide, for instance, the dis- placements ought to be very nearly in the ratio 2 fa Be Some work on spark spectra in different gases under pressure has already been done by Hale and Kent * and by Anderson f. The latter found larger displacements in carbon dioxide than in air; but, as pointed out by Humphreys f, his results were not eencliee as to the effect of specitic inductive capacity on the pressure displicement. In the following work the displacement of some lines of the spectrum of an iron are burning in air and carbon dioxide under pressure are compared. Hydrogen was also tried ; but the are burns so poorly in that gas under pressure, that too long exposures of the photographic plate would have been needed, and the results would have been spoiled by leakages of the apparatus and changes of temperature of the room Photographs were taken at 15, 30, and 50 atmospheres with the 214 feet concave grating ‘spectrograph of this laboratory. A small region of the spectrum containing some sharp lines was chosen, thus enabling the measurements to be made more accurately. The arc was found to burn in carbon dioxide just as well as in air. The accompanying table gives the displacements at the various pressures. Figures in brackets denote doubtful readings, the letter R inchientes that the line was found to be reversed at that pressure. It can be seen that, with a few exceptions, the displacements are the same in the two gases within the limits of experimental error. The mean displace- ment per atmosphere ofall the 23 lines studied is found to be ‘00411 and -00401 Angstrom unit for the are burning in air and carbon dioxide respectively. There also are no very noticeable differences between the appearances of the spectra in the two gases. There are afew more reversals of lines when the are burns in carbon dioxide than when it burns in air; and there seems to be a slight tendency of the lines to be broader and more diffuse in carbon dioxide than in air. The relative intensity remains the same. So far then as this evidence from only two gases is worth, it points to the fact that the specific inductive capacity 1s = but secondary importance in the cause of the displacement * Publications of Yerkes Observatory, vol. iii. Part II. (1907). + Astrophysical Journal, vol. xxiv. p. 291, { Astrophysical Journal, Vol xxvinDlS: § At 15 atmospheres the exposure necessary when the arc was burning in air or CO, was on the average | or 2 minutes, while it was estimated that for the same are burning in hy drogen an exposure of 3 hours would have been needed. : Displacement of Spectral Lines. SOL Displacements in thousandths of an Angatrém Unit at 15 atms. 30 atms. 50 atms. Wave-length. oe | Ae Wy Ait COR ha eau CO; Air co, 4422-67 40 SORT aa09) 67 89 97 27°44 68 GEN AD 108 190 196 42°46 90 9] 138 105 195 200 43°30 62 73 108 90 100 124 47°85 67 80 145 120 193 173 54-50 46 56 86 el 108 NGS 59-24 64 68 147 126 245 240 61°75 4() 53 R 69 78 103 169 66°70 40 45 R 84 [eano-e 108 104 76:20 4] 38 79 76 116 he 82°35 66 70 120 119 225 237 94:67 82 G3 ee ay, 146 250 250 4528°78 (71) R 72R | (170) 142 260 255 S1e2D 45 47 88 86 126 118 47°95 45 355) 90 83 124 148 92°75 50 39 ee I Gs! 130 142 4603:03 44 39 94 89 Lely) 133 47°54 52 40 96 83 110 114 5+:70 25 22 58 44 85 81 91°52 68 66 104 94 129 eS y¢ 707°45 191 180 Sle a) Bl) (538) (520) 10°37 638 69 104 112 127 129 36°91 179 189 R 319 302 (570) (544) Way. s2.0e cca: 675 Gel 124-2 113°7 184°3 188°6 Mean displacement \ Wh 4-5 Feil 3.8 3:7 3-8 per atmosphere. of spectral lines. This is also in accordance with experiments made by using pure metals and alloys of those metals or carbon poles with metallic impurities, when the same pressure displacements were practically found in each case*. All these facts seem to show that the pressure displacement is not a density effect, but is due to the total pressure of the radiating vapour, 7. e. to the compactness in number of the atoms, irrespective of their kind. In conclusion [ wish to thank Professor Rutherford and Professor Schuster for the interest they have taken in this research, Physical Laboratories, Manchester University. * Tlumphreys, Astrophysical Jonrnal, vol. xxvi. p.18. Duffield, Phil, Trans. Roy. Soc, A. vol. ccix. p. 218. it ‘ig ’ i :] i | LVII. The Common Sense of Relatimty. By NorMAN CAMPBELL ”*. HE Principle of Relativity has been discussed so often and in so many ways that it is perhaps presumptuous to attempt to add anything to the discussion except by offering original deve a ae But it appears to me that the needs of ‘the man in the laboratory ”—to paraphrase a convenient modern expression—have been insufficiently considered by expositors. He has been offered profound mathematical investigations, which are intensely important and interesting, but tend to obscure the fuadamental points at issue in the mind of one who thinks physically r ailnen than mathematically. And on the other hand he has been offered collections of apparently paradoxical conclusions deduced from the Principle, which are sometimes elegant and entertaining, but more often fallacious. As a natural result he is inclined to think that this new dev aes of science, the most important, in my opinion, since the days of Newton, is extremely abstruse and ‘tea HSIbIG: In the following pages I desire to attempt to remove this misconception and to show that the view of the relations of moving systems adopted by the Principle of Relativity is very much simpler than that which it displaces, and that all its apparent difficulties are due to confusions of thought and mis- apprehensions. A few of the observations offered may be of interest to those who have studied the matter deeply, but it is to those whose knowledge is superficial that these remarks are primarily addressed. As a basis of the discussion the admirable authoritative summary given by the original propounder of the Principle himself will be used (Einstein, Jahrbuch der Radioaktivitat, Bd. iv. pp. 411 &e., 1907). The notation used there will be adopted without explanation. I. The Nature of the Principle. 2. Let us first inquire exactly what the Principle of Relativity is and what it asserts. The Principle i is what is more often termed a ‘“ theory ”— that is te say, it is a set of propositions from which experimental laws may be logically deduced. It can be proved to be true or false in a manner convineing to everybody only by comparing the laws so deduced with those found experimentally ; ; but a theory which never conflicted * Communicated by the Author, The Common Sense of [elativity. 503 with experiment might yet (as I hold) be judged objectionable on other grounds, and, conversely, a theory which was not in compiete accord with experiment might yet be judged satisfactory. The special laws which it is the business of the Principle of Relativity to explain (that is, those which it is specially important to be able to deduce from the theory) are those which are met with in the study of the optical and electrical * properties of systems in relative motion, bunt in this case, as in most cases, it turns out that laws other than those con- templated originally are deducible from the theory. It is important to notice that there is another theory, that of Lorentz t, which explains completely all the electrical laws of relatively moving systems, that the deductions from the Principle of Relativity are identical with those from the Lorentzian theory, and that both sets of deductions agree completely with all experiments that have been performed f. It, then, anyone prefers one theory to the other it must be either on the ground of differences in the laws not con- templated originally which are predicted respectively by the two theories, or because of some general grounds independent of experimental considerations. 3. The fundamental propositions of which the theory consists will now be enumerated and a few remarks made upon each. For the sake of brevity, I shall call asystem the parts of which are all relatively at rest a “quiet”? system, and one of which the parts are in relative motion a “disturbed” system. The terms are convenient to distinguish quiet and disturbed systems from those which are moving as wholes relatively to each other. Two quiet systems may be in relative motion as wholes. (A.) The first assertion of the Principle of Relativity concerns quiet systems only. Ii asserts that any law and, consequently, any theory from which laws can be deduced, which has been found to hold for one quiet system which includes all the particles of which mention is made in the * “Optical” and “ electrical’? will be employed throughout as equivalent terms, one or the other being used according to the context. + As given in the Eneyclo. d. Mathemat. Wissen. { This statement is only true if quantities of a higher order than the second in v/c are left out of account. Since such terms cannot be detected experimentally the conclusion given is not affected, and the terms will be neglected throughout our argument. In saving that both theories agree completely with experiment, I do not wish to offer any opinion as to the value of the experiments of Bucherer and others whicn have been subjected to critisism: I merely assume that the results announced in them will be accepted ultimately, because, if they are not, the Principle of Relativity would seem to be unworthy of further discussion. if a ba 3D as _ ae tereen enceeetiawn as tar dO4 Mr. Norman Campbell on the laws, will hold for all quiet systems which are not accelerated nee to that oe system *. In particular the theory expressed by the funamental equations of the electron theory has proved perfectly satisfactory so long as it is applied to a quiet system. So long, that is, as all distances are measured relatively to axes fixed in the earth, all times are measured on clocks fixed in the earth, and the phenomona considered are those of charged bodies fixed in the earth in magnetic fields produced by instruments fixed in the earth, or those of sources of light fixed in the earth observed by ‘instruments fixed in the earth, no conclusions have ever been obtained which are not consistent with that theory. The Principle asserts that, if the whole system of axes, scales, clocks, charges, magnets, light-sources, telescopes, and observers were placed on a ship’ moving uniformly relative to the earth and the experiments repe: ated, exactly the same relation would be found between the quantities measured. This proposition is known as the First Postulate of Relativity. The justification for it is that it is in accordance with all known experimental facts : it is, moreover, directly implicated in the usual formulation of the theory of dynamics. Now it is the main object of the Principle of Relativity to establish a connexion between the laws of a quiet system and those of a disturbed system. In order to establish such a connexion Hinstein shows that it is only necessary to introduce a small number of additional fundamental propositions into his theory, and he shows also—it is this which makes his work so brilliantly ingenious—that the additional propositions which must be introduced do not concern the laws of any quiet system. Consequently, if the theory is true, it will tell Pik. difficulty arises from the fact that no system where anything “happens ” can possibly be quiet, for almost all the changes which physics investigates involve relative motion of the parts of the system investigated. The logical questions raised by this difficulty will be discussed more fully in another place: for the present we may regard “laws for a quiet system” as pri opositions toward which the Eee ot actual systems tend as the relative velocity of the parts of those systems tends to zero. Thus the fundamental equations of the electron theory, involving the terms pv, cannot be applied to quiet systems, since they consider “the motion of an electron (which is part of the system) relative to the rest of the system: they may be regarded as the limiting form of the equations v alid for disturbed sy stems, : as the velocity v tends to zero. There is no practical difficulty about this extrapolation, for the form of laws is not found to change with the relative velocity of the parts of disturbed systems over a large range limited, on the one hand, by a relative velocity of about 10° em. /sec., aud, on the other, by the smallest velocity which can be detected experimental ly. The laws for a quiet system are, then, the laws which hold within this range of velocities. Common Sense of Relativity. 505 us how to change from the form suitable for a quiet system to that suitable for a disturbed system not only the laws which it is the special business of the theory to investigate, but any laws whatsoever. Instead of working out, as heretofore, the transformation necessary for each special law, we shall arrive at a transformation which is valid for all laws. The three chief * propositions necessary for this purpose are as follows :— (B.) “Space is homogeneous and three-dimensional : time is homogeneous and one-dimensional.” Mathematically this means that the transformation of the space and time co- ordinates is to be linear. It would take us too far afield to inquire what it means in terms of observations, and since no difficulty appears to have been felt in connexion with it, it is unimportant for our present purpose. (C.) “If the velocity of a system S’ relative to 8 is determined by an observer on 8 to be v, then the velocity of S relative to S! determined by an observer on S’is v” +. The necessity for introducing this proposition is generally over- looked. But it is a proposition which can be reasonably doubted and of which the truth can be tested by experiment only. Of course, all the evidence that there is is favourable : if it were not, we should not speak of “relative velocity.” (D.) “* The velocity of light determined by all observers who are not accelerated relatively to each other is the same, whatever may be the relative velocities of the observers.” This proposition is known as the Second Postulate of Relativity and more will be said about it hereafter. 4. The result which represents the attainment of the primary object of the Principle of Relativity is deduced from these fundamental propositions by purely mathematical argument. It may be stated as follows :— Suppose that the disturbed system consists of two parts, A and B, each of which, regarded as a complete system, is quiet : let A be the part which contains the observer and his instruments for measuring 2, y, z, t, and let the relative velocity of A and B bev. Then if A and B formed together * The other propositions are those which are implied in all physical measurement and in all theories. ‘The propositions given are not those given by Einstein explicitly, but those which seem to be implied by his argument, ¥ The sign attributed to v is a matter of pure convention ; so long as each observer adheres to his own convention throughout, it does not signify whether the same or contrary signs are attributed to the relative velocity by the two observers. Phil. Mag. 8. 6. Vol. 21. No. 124. Aprit{ 1911. 2 L 506 Mr. Norman Campbell on the a quiet system, the known laws for quiet systems would state a relation between the time and the coordinates of the various parts of the system on the one hand, and some quantities P, Q, R, representing the physical state of the system on the other (forces, for example). Let this relation be represented by A Wil, Geest. ley QR. ao.) ae It is shown, as a consequence of the Principle of Relativity, that the analogous relation for the disturbed system is obtained by substituting for each set of coordinates 2, y, ¢, ¢, belonging to a particle of B, the quantities a’, y’, 2’, t', where ; F r UL (a 195% > ')=(B(w—e0), Y, &; B(t— )). It must be noted that the quantities P, Q, R,... will often involve implicitly the coordinates and the time, that is to say the values of P, Q, R,..., will be determined by certain measurements of a, y,2,¢ for certain identified particles. In fact there will be a relation of the form PCE AOL steps. 0,125) = Os In this case the substitution of a’, y’, 2', t for x, y, z, ¢ must be carried out consistently, and for P,Q, R,... must be sub- stituted P’, Q’, R’,..., where these latter quantities are given by f(P',Q’, Rk... 757, 2,t)=dC, Q; Ree In this manner the relation is obtained fies ete 0 We Oo giving directly the relationship which holds for the disturbed system between 2, y,z,t on the one hand, and P,Q, R,... on the other, all measurements being still made by the observer on A with the measuring instruments which form part of his quiet system. ‘This relation is that which we set out to seek, the Jaw for the disturbed system as observed by an observer who forms part of it*. So far surely nobody can find any difficulty : anything more beautifully straightforward it would be hard to conceive. Not only is the result magnificently simple, but it furnishes us with a mathematical instrument of extraordinary power. In place of the elaborate calculations which have hitherto been necessary in dealing with moving systems, all that we * The best examples of the process are, of course, those worked out by Einstein in the paper referred to, Common Sense of Relativity. 507 have to do now is to solve the problem under consideration for the limiting case of infinitesimal velocity, and then effect a mere algebraical transformation. The only objection thai seems likely to be raised is that the Principle proves too much, that it appears impossible that such far-reaching conclusions can be drawn from such simple assumptions: the only difficulty, in fact, is that the thing 1s too easy. 5. That the ar uments by which the conclusion is attained are valid can, of course, only be proved by examining them, but I think a few remarks of a general nature may remove one cause of uneasiness. It is felt that the universal im- portance attributed to the velocity of light is strange, when it is proposed to apply the principle to laws which have nothing to do with optics. Why, it may be questioned, do we drag in the velocity of light rather than that of sound or of the trains on the twopenny tube? Some part of this un- easiness may arise from the unfortunate way in which Einstein introduces the Second Postulate in his paper: he seems almost to try to deduce it from the [first Postulate. In describing the First Postulate he says :—“ In particular the same number must be found for the velocity of light in vacuo for both reference systems.” It is very pertinent to ask here why, then, the velocity of light rather than that of sound. Of course the Second Postulate cannot really be deduced from the First. What the First Postulate asserts is that all laws must be the same for all quiet systems having no relative acceleration : in particular, for all such systems, the velocity of light or the velocity of sound determined from a source which forms part of the system to a receiver which forms part of the system must be the same. But the proposition which is necessary for the argument is quite different from this. It is that the velocity of light from some source common to two systems will be found to be the same by observers on both systems, even if those systems are in relative unaccelerated motion. Since the source is common to the two systems in relative motion, it is clear that both systems, if they both include the source, cannot be quiet, and, therefore, that the First Postulate, which refers only to quiet systems, can have nothing to do with the matter. The Second Postulate is really made up of three distinct propositions *. The first is that there is some velocity which * The complexity of the Second Postulate appears very clearly in Miukowski’s ‘Raum und Zeit.’ Minkowski’s treatment is somewhat different from that of Kinstein and involves an entire rejection of the conceptions of space and time. . 2L2 508 Mr. Norman Campbell on the is found to be the same by all relatively unaccelerated observers ; the second is that this velocity has been measured ; the third is that it is the velocity of light. Only the first proposition is implicated in the result stated in the last paragraph: the quantity c is this universal velocity, whatever it may turn out to be. The second and third propositions are not introduced until the result is applied to the deduction of the optical and electrical laws for a disturbed system from those for a quiet system. The first proposition is that which is really characteristic of the Principle of Relativity, and is the feature which distinguishes it from all other theories. It seems at first sight rather startling, but perhaps it may be made to appear more plausible, if it is pointed out that it means some velocity must be ‘ physically infinite ”’—that is to say, it must be such that the addition to it or subtraction from it of finite velocities do not change its magnitude. Ifits magnitude had turned out to be represented on the scale of measurement of velocities ordinarily adopted by a mathematically infinite number, no difficulty would have been felt with regard to it: itis the fact that the second part of the Second Postulate proposes to repre- sent the physically infinite velocity by a mathematically finite number which causes surprise. There is, however, nothing more difficult in such a representation than there is in the representation of the physically infinite low temperature by the mathematically infinitesimal number zero ; both repre- sentations are merely consequences of the definitions of velocity and temperature adopted, and physical and mathe- matical infinity could be easily brought into agreement by a change of definition *. But if the first and second parts of the Second Postulate be accepted, there can be no doubt about the third, for if we are going to identify the physically infinite velocity with any velocity which has ever been observed, there is, on general grounds, no doubt as to its identification with the velocity of light. For this velocity clearly cannot be less than any velocity which has been measured : to suppose that it could would be self-contradictory. Hence if weare to identify the physically infinite velocity with any velocity which has been measured, it must be with the greatest velocity which has been measured, the velocity of light in vacuo. The agree- ment of the propositions deduced from the Principle of Relativity with the aid of this identification with the experimental work of Bucherer is strong evidence for the second part of the postulate—that is, for the view that * This line of thought will be developed in a later paper. Common Sense of Lelativity. 599 the physically infinite velocity has actually been measured. The first and third parts of the postulate are scarcely dubitable. But perhaps such arguments are unconvincing ito the physical iustinct, so I proceed to considerations which should overcome any difficulty which is felt in connexion with the great importance attributed to the velocity of light wheu dealing with phenomena which appear to have nothing to do with light. These considerations are based on the fact, obvious when it is pointed out, that such velocities as distinguish practically a disturbed from a quiet system, velocities, that is to say, which are not physically infinitesimal, can only be measured by optical or electrical means. When we hear of the “ velocity of the earth relatively to the sun” or the “velocity of a @-particle relatively to its source,’ association leads us first to think of a quantity measured by the distance travelled relatively to a metre scale during the passage of the hand of a chronometer over a certain part of its tace. But a little reflection will show that this is not what we mean by these velocities: we have never held a metre rod up ayainst the sun or an electron and observed the change in relative position. It would lead us too far to inquire here what exactly we do mean by such an expression as the “ velocity of a 8-particle,” but it could be shown quite easily that that expression has no meaning what- ever, unless we assume the truth of the fundamental equations of the electron theory *. And since those equations involve the velocity of light, it is not surprising that that quantity enters when we are considering the velocity of an electron. “ The velocity of a B-particle ” is called a * velocity ” because, within a certain range of values, the number representing it is the same as the number representing a velocity measured by a scale and clock (as is shown by the Rowland experiment), but, outside the range within which the scale-and-clock measurements of velocity are applicable, ‘‘ the velocity of an electron ” is dependent for its meaning on certain theories. To inquire whether, outside this range, this velocity would agree with that determined by the scale and clock is as absurd as to inquire whether, if all triangles had four sides, all circles would be square. * Some remarks on this subject are to be found ina paper on “ The Principles of Dynamics,” Phil. Mag. xix. p. 168. SS eS a Ta roa 510 Mr. Norman Campbell on the A ee Il. The Consequences of the Principle. 6. But the chief objections which are raised against the Principle of Relativity are urged, not so much against the foundations of the Principle, as against its consequences. Two consequences seem to cause especial difficulty, and these will be considered. The first difficulty concerns the “‘ composition of velocities.” The Principle of Relativity leads to the conclusion that, if an observer on a quiet system S measures the velocity of a quiet svstem 8’ relative to him and finds it u,and if an observer on S’ finds the velocity relative to him of a third quiet system S” to be v, then the observer on S will find the velocity of S” relative to him to be . ee == =a = nie Cae U+UV =a 3 Uv 1+ 3 Ez and not, as experience with small velocities might lead us-to expect, utv. (u and v are taken in the same direction.) This conclusion seems absurd to many people. Let us inquire into the consequences of rejecting it and substituting the law w=u+v. Wemustthen, of course, reject one of the fundamental propositions (B), (C), or (D); the rejection of (A) would not help us, because this proposition is not implied in the conclusion. Now, if an objector proposed to reject (B) or (C) I should have no argument to use against him, for the experimental evidence for these propositions is just as strong and just as weak as that for the proposition w=u+v. All these propositions, as well as (A), can be tested only by com- paring the experiences of different observers, whe have been moving relatively to each other with high velocities, when they meet again ona quiet system. Now since no two human beings have ever, within historic times, moved relatively to each other with a uniform velocity of 10* cm./sec. and subse- quently compared their experiences, and since, on the other hand, we do not expect to detect divergencies from the laws of a quiet system or from the laws predicted by the Principle of Relativity until the relative velocity reaches at least 10° em./sec., the evidence for all these propositions is extremely precarious. Nor does it seem in the least likely to become less precarious: so far as I know, nobody has made the faintest suggestion as to how a relative velocity of more than 10* between two human beings might be attained in such a way that they could perform delicate measurements. The one proposition among those which are fundamental to the Principle of Relativity which there appears to be some hope ‘Common Sense of Relativity. d11 of establishing definitely is (D) : we have the source of light relative to which we are moving with a velocity of 3x 10° always available in the stars, and it is not too much to hope that some day experimental ingenuity will succeed in measuring the velocity of the light from it with an accuracy of one part in ten thousand. It seems to me incredible that anyone, who understands what he is doing, will really propose to reject definitely a proposition which he may hope to prove in the near future in favour of one for which there is never likely to be the smallest direct experimental evidence. But I think these people do not understand what they are doing: they have been confused by the most fruitful cause of confusion, the habit of using one word to denote two quite different ideas. ‘‘ Velocity’ is commonly used to mean either “mathematical velocity ” or “ physical velocity.” Mathematical velocity is defined as the ratio of a certain variabie z to a certain variable ¢. From the definition of a variable and a ratio, it follows that Gy) gt iD ates Aye RAR this is a perfectly purely logical conclusion, and to deny . would be absurd. On the other hand, “ physical velocity ” its simplest meaning is a number equal to the ratio of oe numbers—one representing the groups of metre rods that have to be placed together in order that their ends may coincide with certain points, and the other expressing the occurrence of certain events In an instrument calleda clock. From the definition nothing whatsoever can be predicted as to the relations of u, v, and w, but experiment shows us that, for all values of wu and » which can be attained practically in this way, if u =, b= thera) = a, mental proposition has become so familiar, and the association This experi- of the experimental w and v with the mathematical = and = so habitual, that people who do not think very deeply about these things have come to believe that «w means the same thing as a oie that, therefore, since it would be absurd to deny that “2 ae [= 58 that u+ov=w, There j is no more absurdity in being forced ‘to deny this assertion in the face of fresh evidence than there was in the necessity for Mill’s Central African philosopher , itis absurd to deny 512 Mr. Norman Campbell on the having to deny in the face of fresh evidence his previously undoubted proposition that ‘all men are black.” 7. But the greatest difficulties in connexion with the Principle of Relativity appear to concern certain propositions about length and time. In what follows I shall, for brevity, discuss only time: everything I say will apply, mutatis mutandis, to length. The Principle of Relativity leads to the following conclu- sion. Suppose I examine a number of clocks which, with me and my instruments, form a quiet system, and I find that they all go n times as fast as my standard clock. That is to say, for the quiet system, the “law” of these clocks is that they | iP : return to some standard state when t = —, where P is any Nee : integer. Now one of these clocks is transferred to a system moving relatively to me with a velocity v. Let us suppose that, at the moment when ¢=0, this clock is just passing me, so that z=0. Then the Principle of Relativity states that the “ law ” for the disturbed system of myself and the clock is that the clock returns to a standard position when t'= Z AS Nas cea les or when Ae ——|)=-— c s i or, since z=vi, when P=. n That is to say, the clock now agrees, not with the clocks with which it formerly agreed on the quiet system, but with one on the quiet which goes = as fast as those clocks. There is nothing new in the form of this conclusion. The crudest arguments based on the oldest theory of light lead to the conclusion that the rate of a clock as obsei ved by a certain observer must change with the relative motion of clock and observer. Tor, it will be argued, the observer does not see the clock “as it really is at the moment,” but “as it was a time T earlier, where T is the time taken for light to reach the observer.” And on these lines it is easy to show that the apparent rate of a clock moving away from the observer with a velocity v is (1 —*) times the rate of the same clocks observed at rest. It is only the magnitude of the change concerning which the two theories differ. “Yes,” says our objector, “ thatis all very well : of course the apparent rate of the clock changes with motion, but does Common Sense of Relativity. ails the real rate change?” We immediately inquire what the “real rate”? means. He is at first inclined to assert that it is the rate observed by an observer travelling with the clock, but when we inquire relative to what clock that observer is to measure the rate he becomes uneasy. He cannot compare another clock travelling with him, for if the ‘‘real rate” of one clock has changed, so has the “ real rate” of the other ; and he cannot use a clock which is not travelling with him, because he admits that he does not see such a clock “as it really is.” Pressing our inquiries, I think we shall get an answer of this nature. “If I take a pendulum clock to some place where gravity is different, the rate of the clock will change. It isa change of this nature which I call a change in the ‘real rate,’ and I want to know whether there is any change of that kind, on the theory of Relativity, when the clock 1s set in motion.” Now why does our objector ealla change of the first kind a change in the “real rate” ? The reply 1 is to be found in the history of the word “ real.” The word is intimately associated with the philosophic doctrine of realism, which holds that the most important thing that we can know about any body is not what we observe about it, but its “ real nature,” which is something that is independent of observa- tion. Now, of course, a quantity which is wholly independent of observation cannot play any part in an experimental science, but there are quantities which are independent of observation in the more limited sense that they are observed to be the same by whatever observer the observation is made. The term ‘‘real’? has come to be transferred from the philosophical conception to such quantities. The “real rate ” of the clock is said to change when it is transferred toa place where gravitation is different, because all observers agree that the rate of the clock which has been moved has under- gone an alteration relatively to that which has not been moved. Now in the conditions which we are considering the observers do not agree. If A and B, each carrying a clock with him, are moving relatively to each other, they will not agree as to the rate of either of their clocks relative to A’s standard or to B’s standard or to any other standard. The conditions which, in the case of the alteration of gravitation, gave rise to the conception of a “real rate” are not present : in this case there is no “‘ real rate,’’ and it is as absurd to ask whether it has changed as it would be to ask : question about the properties of ‘round square. However, some people, who in their eagerness to escape the reproach of being metaphysicians have adopted without inquiry the 54 Mr. Norman Campbell on the oldest and least satisfactory metaphysical doctrines, are so enamoured of the conception of “ reality ” that they refuse to give it up. Finding that the observations of different observers do not agree, tliey define a new function of those observations, such that it is the same for all observers, and proceed to call this the “‘ real rate.” This function, according — to the Principle of Relativity, is Qn’, where n’ is the rate of the clock as seen by an observer relative to whom it is iravelling with the velocity v : according to that Principle, if we substitute in that function the appropriate values for any one observer, the resulting number will always be the same. So far no overwhelming objection can be raised. The function is important in the theory, and, if care is taken to note the precise meaning now attributed to the word “ real,” there is no harm in calling it by that name. But now certain writers commit an extraordinary series of blunders. They not only inquire whether the real rate changes with the velocity, a question which, as the real rate is defined as that function which does not change with the velocity, is utterly trivial, but they actually give a negative answer. ‘They see that the expression for the real rate contains v explicitly and rush to the absurd conclusion that the real rate changes with the velocity. No wonder that they soon involve themselves in a hopeless maze of paradox. Asa matter of fact the “ crude argument” given above shows that the second definition of “ real’’ had been intro- duced before the Principle of Relativity. It had been recog- nised already that observers would not agree as to the rate of a clock: the conception of the clock “as it really is,” intro- duced in that argument, means (if it means anything) that function of the observed rate of the clock and its velocity relative to the observer which is the same for all observers. But the logical order of the argument is reversed. Instead of proving from the “ real rate” of the clock, which we do not know, the observed rate, which we do know, we should say that the observed rate of the clock is n, and that our theory of light leads to the conclusion that nf (ts - will be the same for all observers. Whether that conclusion or the conclusion reached by the Principle of Relativity is correct can only be determined by experiment, and the experiment has not yet been tried. , It is the great merit of the Principle of Relativity that it forces on our attention the true nature of the concepts of Common Sense of Relativity. 515 “real time ” and “ real space ?? which have caused such end- less confusion. If we mean by them quantities which are directly observed to be the same by all observers, there simply is no real space and. real time. If we mean by them, as apparently we do mean nowadays, functions of the directly observed quantities which are the same forall observers, then they are derivative conceptions which depend for their meaning on the acceptance of some theory as to how the directly observed quantities willvary with the motion, position, etc. of the observers. ‘‘ Real” quantities can never be the starting point of a scientific argument ; by their very nature they are not quantities which can be determined by a single observation : the term ‘‘ real” has always kept its original meaning of some property of a body which is not observed simply. All the difficulties and apparent paradoxes of the Principle of Relativity will vanish it the attention is kept rigidly fixed upon the quantities which are actually observed. If anyone thinks he discovers that that Principle predicts some experi- mental result which is incomprehensible, let him dismiss utterly from his mind the conception of reality. Let him imagine himself in the laboratory actually performing the experiment: let him consider the numbers which he will record in his note-book and the subsequent calculation which he will make. He may then find that the result is somewhat unexpected—-to meet with unexpected results is the usual end of performing experiments,—but he will not find any contra- diction or any conclusion which is not quite as simple as that which he expected. 8. There is one further point sometimes raised in con- nexion with the Principle on which a few words may be said. lt is sometimes objected that the Principle “ has no physical meaning,” that it destroys utterly the old theory of light based on an elastic ether and puts nothing in its place, that, in fact, it sacrifices the needs of the physical to the needs ot the mathematical instinct. That the statement is true there can be no doubt, but the absence of any substitute for the elastic zether theory of light may simply be due to the fact that the Principle has been developed so far chiefly by people who are primarily mathematicians. It is well toask, can any physical theory of light be produced which is consistent with the Principle ? The answer depends on what is meant by a “ physical theory.’ Hitherto the term has always meant a “ mechanical theory,” a theory of which the fundamental propositions are 516 On the Common Sense of Relativity. statements about particles moving according to the Newtonian dynamical formulz. In this sense a physical theory is im- possible if the Principle of Relativity be accepted, for the same reason that a corpuscular theory of light is impossible, if the undulatory theory of light be accepted. Newtonian dynamics and the Principle of Relativity are two theories which deal in part with the same range of facts ; they both pretend to be able to predict how the properties of observed systems will be altered by movement. If they are not logically equivalent they must be contradictory: in either case an ‘explanation’ of one in terms of the other is impossible. It can be easily shown that they are contradictory : if the Principle of Relativity is true, Newtonian dynamics must be abandoned *. I shall deal with this point rather fully in a. Jater paper ; here it will suffice to point out that Einstein has been forced in his development of the subject to deny Newtonian dynamics at an early stage. He states that the fundamental equations of his electron theory are me=eX, ete., and then puts =v, where v is the velocity of the electron relative to the instrument exerting the force eX. But, if Newtonian dynamics are true, “ is not this relative velocity, but the velocity of the electron relative to the centre of mass of the electron and the instrument. Since the mass of the electron can conceivably become infinite, the distinction, negligible in practice, is of great importance theoretically. On the other hand, if a “‘ physical theory ”’ of light means, as I think it means, a theory which draws an analogy between light propagation and the propagation of a disturbance through some mechanism, composed of rods and strings and fluids and such things, then there is no reason apparent why a physical theory of light should not be constructed which is consistent with the Principle of Relativity. But, of course, the laws according to which rods and strings and so on are supposed to act, must be changed from those predicted by Newtonian dynamics to some laws predicted bya mechanical theory consistent with the Principle. This development also is left for future discussion. * This conclusion is reached by Sommerfeld in a recent paper, Anz. d. Phys, xxxiil. p. 684, &e. (1910). Apparatus for production of Circularly Polarized Light. 517 Summary. 1-5. The assumptions made by the Principle of Relativity are stated and an attempt made to render some of them mure plausible at first sight. 6. A difficulty connected with the composition of velocities is examined and found to be due to verbal confusion. 7. The confusions introduced by the word “real” are discussed. 8. The relation between dynamics and relativity is con- sidered briefly. Leeds, November 1910, LVIII. On Apparatus forthe Production of Circularly Polarized Laght. By . Wan Dp ae a Scholar of Trinity College, Cambridge *. . \HE relative merits and dbfects of the quarter-wave plate and Fresnel’s rhomb as used in the production of circularly polarized light are well known. Jor clearness it may be well to state them here. When we are dealing with monochromatic light, the quarter-wave plate has the great advantage that as its axes are rotated the transmitted circularly polarized beam is not displaced laterally as the plate is rotated. Such a lateral motion is experienced with a Fresnel’s rhomb. On the other hand, when we are dealing with white light, the use of a quarter-wave plate is impossible since the phase-difference introduced between the com- ponents transmitted, differs considerably for different wave- lengths. In such a case the [Fresnel rhomb must be used to convert plane polarized light into circularly polarized light, and vice versa. The present investigation was undertaken with the idea of combining as far as possible the advantages of the Fresnel rhomb and the quarter-wave plate in one piece of apparatus ; the end being to produce an apparatus for which the emergent beam should be in line with the incident beam, and the various angles of ee being so arranged as to produce a phase-difference of ” for as large a range of wayve- Jength possible. 2 Let the incident vibration upon a refracting medium be represented by = t . y= oe : * Communicated by the Author, having been read before the British Association at the Meeting at Sheffield, 1910, sia eT ye a then supposing a phase-change to be produced on reflexion, the reflected amplitude may be represented by the complex quantity «+z, and the reflected vibration may be written ~ ’ y=(a+fe). where « and £ are real. This may be written 518 Mr. A. &. Oxley on an Apparatus for the y=r. oe ; git ep ont +8) : ue where 7= Va’+ 6’, and O=arc. tan 7° | Hence we see that a phase-change is produced on reflexion amounting to @. | Let Ww be the angle of incidence of a beam of plane- | polarized light incident on the surface AB of the rhomb : Ee) Gio 1), | ig. a. If y and ware respectively the corresponding angle of refraction and index of refraction, then we haye sinyr=p.siny’. | For total reflexion to take place, yy’ will be complex. Writing y= E+ 1, sin po=yp.sin (E+ 7) =p.sin €&.coshn+pm.t.cos &. sinh ». then Equating real and imaginary parts, p.&.cos &.sinhy=0, cos€=0 or sinhyn=0, whence T Ona). n=0 gives the ordinary case of refraction. Production of Circularly Polarized Light. d19 Taking == y> we obtain the following law of refraction,— SUM va COS, Stile), 0), at), sob reba eanlns GL) Also AUN ee - +. | For the vibration whose plane of polarization is in the plane of incidence *, Fresnel’s sine law becomes with the law of refraction (1), sin (yr— z — 7) Z | ear, jie licaoas Nda Wea Dig ie Tk sin (ht = + 10) ‘““b** being the amplitude of the reflected, and “a” that of the incident light. Writing b=r.e™, we have an cos (r— nt —nt) A ae “cos (Yr +e) cos Bein = a). oa cos (y= nt) Hence a ras(O ag) Meester, and ee ete, cos (w+ 71) For the component polarized perpendicular to the plane of incidence, the tangent law becomes tan (hee 3 — nt) tan (r+ > +L) or writing Caan sewn Dawen _ cot cot (yp— — nt) cot cot ht) Nie Gan: cot (r+ nz) cot (r—ne)’ and as before we find +’ =a, and Avs cot (=e) ‘ OCTET ae SCI RA Ora (; * Fresnel’s theory is assumed. i i S52 ae A RE 520 Mr. A. E. Oxley on an Apparatus for the Using equations (2) and (3) we see that the phase- difference produced at a single reflexion between the com- ponents polarized parallel and perpendicular to the plane of incidence, is given by @—@’, where | 9-6") _ sin (ar— 72) i an msi (et 70) he 2. sin?(ar—ne) ~ 2sin (—ne) sin (pr + 70) __ cos 2p. cosh 2n +4. sin 2. sinh 2n—1 re cosh 2n—cos 2f =cos 0—6'+.esin 0—@’. , Writing @-0’=A, cos 2 . cosh 2n— 1 cos A=cos 6—6'= cosh 2n—cosh 2p * °° (4) Since from (1) sinyty=p.coshn, we have, using (4), Writing 2=cos 24, Poh? — 2) — le ae eosene or a? —a{l—p?+(1+y’) .cos A}+ cos A +p?(1—cos A)=0 If the beam of plane-polarized light incident normally upon the surface of the rhomb, be vibrating in a plane inclined at 45° to a principal section of the rhomb, a phase-difference amounting to A will be produced between the components, and their amplitudes will be equal. : 1 : (Ca Now taking p= [5] (glass to air) and A= =. we find on substitution that 2? and therefore cos? 2W is real. Moreover, both values of ware real, one being about 75° 17 ce : F * A may be of the form g Lenntm @ Where 2 is any integer and m is either 1 or 0. Production of Circularly Polarized Light. 521 while the other is near the critical angle. The larger angle is chosen for reasons given below. Now let there be two rhombs placed as shown in fig. 2, Fig. 2. Cc D G which 1s a medial section. A beam of plane-polarized light enters along CD, is reflected at D, E, F,and G,and emerges along G, H, so that C, D, G, H, are collinear. There will be a retardation of one component on the other by A, and A= ie if the angle of the rhemb is 75° for ordinary glass 8 (u= rai): Hence for such a rhomb the total gain of one component on the other will be 4 x = =4ir— a and there- fore the emergent light will be circularly polarized *. Further, since C, D, G, H, are collinear, on rotating the system there will be no lateral motion of the circularly polarized beam, and no readjustment of successive pieces of apparatus is needed. The fact that the relative retardation of one component on the other has to be of the form nr+y, where y is an acute angle and n odd, in order that 4 should come out real is in accordance with the result first found by Lord Kelvin, that the phase-differenee between the components resolved in and perpendicular to a principal section (for substances at our disposal) is oblique f. In total reflexion the phase-difference introduced between the vibrations executed in and perpendicular to the plane of incidence, is not entirely independent of the wave-length, although nearly so, and it is best to choose the angle of incidence so that the dependence of A on X, or the “colour effect,” as we may call it, is as small as possible. * According to Lord Kelvin the component polarized parallel to the plane of incidence is accelerated on the perpendicular component. See Baltimore Lectures, p. 400. + Loe. cit. Rlul. Mag. 8. 6. Volu21. No. 124, April 1911. 2M tanA. 22 Mr. A. E. Oxley on an Apparatus for the Irom equation (4') we have c ee 2 ; a) 225 22 9 | ; C : ye = GOs ay). cos A =—EOs ny (“eae = 1) =. we we and differentiating psa dmically, dA _4sin’ mal ail cos Daf il dp VU cos? V2 2 sin? — p?)— pw? Oe sin’y— w?(1 4 cos 2p) | ff ot 1 : ( Blane ap—_ He cose sin? ap—p?. cos? apf” i Stem Cosa en = | : 1 ; Hence for given A ( = Hence in so far as this effect is concerned large angles of incidence are desirable. In this respect the new arrange- ment has a further advantage over the Fresnel rhomb, tor although the two reflexions in the latter are replaced by four reflexions in the former, yet the total colour effect, if we ‘choose the larger value of vw, is only half that of a Fresnel rhomb. The Colour effect is examined below. Uviol glass was chosen as the most suitable material for the Bi-rhomb (as this form of the apparatus is called) since Production of Circularly Polarized Light. 523 the loss of light by absorption’ is small and the dispersion low. Equation (5) becomes oa} —w (ta) cost soon + y'(1- -- COs 3) =0, where “= 15 es for the D, line, and x=cos 2, where wp is the angle of the rhomb. Hach of the values of yf is above the critical angle, which. is 41° 42’ for uviol glass. Both at w= 74° 382 and wy = 42° 34'°8, we get relative retardation amounting to a for each reflexion, and the larger value is chosen since then A is practically independent of variationsin dA. For fused quartz, of refractive index 1°5533 for the D, line, the larger value of is 75° 4'°5. } I£ we consider an interval comprising a range of ® equal to that between the C and F lines *, and call the increment or decrement in pw corresponding to it + dw (w<1) we can make an estimate of the variation in A for any X from the value which obtains for the D, line. We have from (6), le sue 2sin? : ee ar ye rae ‘ox in ae 2 pe? . cos? Ww sinh — pe? ary ype 1) p tan aS sin Pie A era Here he 38/, for Dy limes Ma wh = rsogsie for uviel glass. oh Substituting we find dA =+0°:052. Cal Therefore for four reflexions, the extreme phase-difference for the C—F interval considered above will be 4. dA =0°208. saa eo) OAc onl Uitte * The total interval for which.6A is calculated comprises an interval d. ahs on each side of the D, line, 2. e. an interval of approximately 1700 a U. on each side. Tea RE + Se ae SS 524 Mr. A. E. Oxley on an Apparatus for the It w=54° 37’, which is the angle of Fresnel’s rhomb (« being 1/1°51) equation (6) ae a GS and there being two reflexions the extreme phase-differenece for the C—F interval amounts to +0°406. Hence, the variation of A with pw is only half as great in the Bi- rhomb as it is in Fresnel’s rhomb, for a given wave-length. The dimensions of the Balok —Owing to the large value of y the Bi-rhomb for a given aperture is rather incon- yeniently long. If S denote the aperture, and y the angle of the rhomb, the length in the direction of the incident light when the full aperture is utilized will be sti n i \ i wee tan 2y + coty : and if A be 11 ems., and p=y=74° 38’, the length is 74 cms. This has to be doubled, and the total length of the Bi-rhomb of aperture 1+1 ems. is approximately 15 cms. Since, however, the value of dA for y= 42° 34’8 for the Cao four reflexions amounts to +3°6, it is better to sacrifice compactness for efficiency. In many experiments the length of the Bi-rhomb may be no serious objection to its use, but by the following method involving three reflexions an apparatus has been devised which combines practically all the advantages of the form just considered with compactness. ABCDE in fig. 3, shows the bi-trapezoidal form of the section. The faint lines show the trace of a beam of light through the instrument, the beam passing through symmetrically with respect to the line EHF. Thus the emergent beam, although it has suffered lateral inversion, is not displaced as a whole from the incident beam. If @ be the angle of incidence on the face AH, ¢ is the angle of reflexion from the face ED. Let ¢’ be the angle of incidence on the face BC. Clearly $’ must be greater than Production of Circularly Polarized Light. 525° the critical angle from glass to air, and since we must have from the geometry of the figure, the relation o=$47 satisfied, we must have > 65° 51’, taking @/=41° 42’ as the critical angle. Consider the component of the incident vibration which is _ polarized perpendicular to the plane of incidence. Jet d be the angle of incidence, @ the absolute phase-difference pro- duced on the component for this angle of incidence, 0 the corresponding phase-ditference for angle of incidence ¢’. Then, with the usual notation, we have ene CO Oey cosp+ nt and for the three reflexions indicated in fig. 3: e(204+6")o i eM cos gaat cosé+ni- cosd’ +b For the component polarized parallel to the plane of incidence, let 6 and 6/ correspond to @ and 6' respectively. Then as before 2548! jee p— ie) * cot p!—mn't cotp+ty-~ cot’ +n't If A’ be the relative retardation, then will cos A = Re [Sindee ing we asec) sin p+ ne sin db +n't e = ! jo sin where ree IN i ib ja and Oval his Sms From equation (8) sin @. cosh y—2z cos. sinh 777” cos A’=R eee el S< sind .cosh y+ecos d. sinh ° ’ [sind -cosh of — sens Sat sin d’ . cosh 7’ + 6cos ¢’. sinh 75 Ls (sin @ .cosh y—« cos d . sinh 7)‘ . Tae iene sh? 2 inh? (sin? d . cosh? n+ cos” d . sinh? 7) (sin d’ . cosh 7! —¢ cos ¢’ . sinh 7)? (sin? d!. cosh? 7 + cos? ¢' . sinh? 7’) * R is used to denote the real part of the function which follows it, 526. Mr. A. E. Oxley on an Apparatus for thé: lf now A’ is to be - where n is an odd integer, then the real part of the expression on the R.H.S. will be zero, This gives the following equation for ¢: (sint ¢ . cosh*.7—6 sin? d . cosh? 7. cos? @ . sinh? y+ cos’ ¢. sinh* 7) x - (sin? 6! . cosh? 7'—cos? ¢’ . sinh? 7’) + 8. sin @. cosh 7. cos # . sinh 7. sin gp’ x cosh 7’. cos ¢’. sinh 7! (cos? ¢ . sinh’ 7— sin? ¢ . cosh’ 7) = 0, Lene Coy vher SI sin ’ hone cosh 7 = nee ?, cosh 1’ = Bt fog and | ¢ « o= i ap Tk This is an equation of the sixteenth degree in sin @, 2. ¢. of the eighth degree in a. An approximate solution has been found by trial and error, and taking wy =1°5035, the value of ¢ is approximately 72° 48’, From the relation ae T $=2p-7 we get ay =e Asa test of the accuracy of these values of @ and @’, the equation (4') was used to find A’ for angles of incidence ¢ and p’. If we call the relative phase-differences 0— -6 and 6’—6' in accordance with the notation on p. 525, then from the equation we find, Oo — ie Ae 6’ — 3 =a—A2?° 42-2 which gives 2(6— —8)+(6’—8) = 2a +89° 5 Dae _ A still more accurate value of @ was found as follows. ‘Taking ¢=73° 49’, the total phase-difference for the three reflexions calculated as above is 8 es 2(0—8) + (0 — 8) = 2m + 90° 22.7 Taking OS 73° 48’, it may be shown that 2(0—8) + (0— ye Ir +89° 57/4, Production.of Circutarly Polarized Light. 527 - On a simple interpolation we obtain ¢6=73* 48"6. There- These values give a phase-difference between the com- aT | ponents on emergence equivalent to 5 for the D, line, and 7 2 : the emergent beam will he circularly polarized. Since $=73° 48'°6, the colour effect as calculated from equation (6) for a single reflexion will be nearly the same as for a single reflexion in the Bi-rhomb, for which $6=74° 38"2. There are two such reflexions, and the colour effect is 2.dA = +0102, so far as they are concerned. For the é {0 — FF : : aH he reflexion at angle of incidence ¢’, the valueof dA =+0°18. (Oy ae Hence the total colour effect is a little less than that for an ordinary Fresnel’s rhomb. For an aperture of S cms. (S is the breadth of the beam of light) the length of the Bi-trapezoid will be 2.8.tan y, where y is the acute angle of the trapezoid. Also the: greatest breadth of the trapezoid is AB= S (cot x —tan 2y). if y=75°, S=1 cm., the length in the direction of the incident light is 7°5 cms., approx. i.e., half the length of the previous form. CD is then about 1°7 ems. The Azimuth of the Incident Vibration. With g@=73° 486 the phase-difference for the three re- flexions, between the components polarized parallel and perpendicular to the plane of incidence, is exactly z f sodium light, fe being 1°5035. q Since, however, the component polarized parallel to the plane of incidence is more copiously reflected on crossing the joint than the perpendicular component is, the usual azimuth (viz. 45°) would produce elliptically polarized light, the major axis of the ellipse being parallel to CD (fig. 3). By adjusting the azimuth, the transmitted beam can be made circularly polarized. The new azimuth can be found in two ways. (1) Eeperimentally—Using a Babinet’s compensator or the Bi-rhomb (which produces cireularly polarized light, since here the beam crosses the joint noimally) we can analyse the light transmitted by the Bi-trapezoid, and adjust or ERP y VS PO-aD , =a 3 SSS a SS ~~ = at eee = 528 Mr. A. E. Oxley on an Apparatus for the the azimuth of the incident vibration with respect to the latter so that the beam of plane-polarized light atter passing through it would be circularly polarized. (2) By Calculation.—The new azimuth can be determined more accurately by calculation. Consider the layer of air forming the joint. Let @ be the angle of incidence in the Fig. 4, air Glass “p> A glass x —— -— glass (fig. 4), ¢’ the angle of refraction. The ratio of the refracted amplitudes in the air-gap is 2sing'.cosd@ sin(¢+¢’) cos (P—g') _ ite “sin (f+) "Zan greasp ~~ “*P-) the azimuth of the incident vibration being 45°. After the second refraction this ratio is cos? (p~¢’). If we make the azimuth of the incident vibration Q where 1 t QO= > 1 1 an cos” }p—are. sin («sin d) }’ © being measured from the edge AB (fig. 3), the trans- mitted amplitudes will be equal, and since their phase- difference is still = the emergent beam of light will be circularly polarized. Using the known values of ¢$ and yp, we find Or aoe rat An estimation of the ellipticity of the orbit corresponding to any value of dA=¢ (say), can be obtained as follows. Let the transmitted components be represented by the equations «= 7. cos wt, y =rsin (wt+%), The equation of the orbit is (1+) 4 ye? Ue tay. 34+ 3= I, Production of Circularly Polarized Light. 529: and defining the ellipticity (e) as the ratio of the difference of axes to major axis, we have Now taking an interval equal to that between the Cand F lines, as before, the value of € for the Bi-rhomb is 0°208, and therefore 0203) an we = ee eee) D ; 90 ae For the Bi-trapezoid and the same limits of wave-length, we find [=0° 28, 9 ie wee = -0096, while the corresponding ellipticity for the Fresnel rhomb is ees BE Ly al. YU The ellipticities are practically the same for fused quartz as for uviol glass. Table 1. shows the value of ¢ for different wave-lengths for the Bi-rhomb (2. e. for angle of incidence 74° 38/-2) and the ellipticity of the corresponding orbit as calculated from D) a ny Table 11. shows the corresponding values of A, &,¢ for the Bi-trapezoid; Table III. is for the Fresnel rhomb. HIVACB IL En: We | | | Line. NOCASU ant Grad) san ender.) é. ! 9296-0 0073 | 0417 ‘014 | TE a a 7668°5 0038 0-215 ‘0076 | 7596-0 ‘0036 0-208 ‘O07 | immedi eee ae) ke 6708-2 ‘0017 0-099 ‘0034 | iN (OL er en 58962 ‘0 0-0 0 AUMeree MM Meee ect 5350°7 = "0012.9 0-069. | — 0024. | Hl Violet ee ee: 4340-7 —-00338 | —0189 | —-0066 | 4196-0 —-0036.:| —0:208 | —-007 | 2496-0 OTST tee ONT chs Ola | | | 4 f ay a % i Ss = Sam J80 Mr, A. E. Oxley on an Apparatus for the TABLE IT. | | | | ' mes | Line. | A(A.U.). | ¢ (vad). | Z (deg.). é. | | yy) | | | | | | | | 9296-0 009 | O"d4 O19 | wed) A Sek) WA eats can 7668°5 | 0057 0-32 010. | | 75960 | 0048 0-28 0096 Ta meat 10.2 eaten neh et | -6708:2 | ‘0023 0:13 0046 | NCD Ae teats near ak rcs| 5896-2 ‘0 0-0: 4) neue [AD ore@mi eee ae ea ate t ae 53507 | —"0015 | —0-086 ---003 | - iter iolec tweeee ferent ic. ce 43407 | —-0044 | —0-25 | —005 | | | 41960 | — 0048 —0:275 | ---0096 | 9496-0 | —-0U095 | —0:54 —v19 | | | Apna: Line. | (A.U;).. | 'Z @ad.). | Z (deg ae | 92960 |- -0142 | 0-802) | aie | reed) eaAeanear Ae Ie ole 76685) 4 uy e074) |) 9 O48 ‘DL | 7596-0 | OO71 | 0-407 O14 | Wilgitred wie sick eyes 67082 | 0084 | 0195 |) tae WR oe 56962 | 0 | oO | [Tlereen ...........- S Shon 5350°7 —-0023 | —0132 | —0046 | iEiawioletia £..00....cscas. ect 43407 | —-0064 | —0367 | —013 | | 4196-0 | —0071 | —0-407 | —0l4 |: | 2496-0 | —-0142 | —0-802: |) O2anm | | | In fig. 5 the values of e given in the above tables are plotted against A, and a comparison of ordinates for given wave-lengths shows the relative efficiency of the three forms for light of that wave-length. The efficiency is unity for the-D; line. Curves I, I]; III refer to Tables 123 aame respectively. When A is > 5896 A.U., € is regarded as positive, and after passing through e=0 at A=5896, € is regarded as negative. Comparing each of the new forms with the Fresnel rhomb, it will be seen that the Bi-rhomb is about twice as efficient as the Fresnel rhomb for any given wave-length, while the Bi-trapezoid has an intermediate efficiency about one and a half times that of a Fresnel rhcmb. But the Bi-rhomb has the disadvantage of being incon- veniently long for a givenaperture. This adds to the amount of absorption which, however, is reduced to a minimum by the use of uviol glass or fused quartz. In those experi- ments where length is not an objectionable factor, this ferm may prove useful. In the Bi-trapezoid, in which three reflexions are used, the efficiency is a little higher than that Production of Circularly Polarized Light. Od of a Fresnel’s rhomb, while tlie length is not much different. Hence in these respects the qualities of the new forms com- pare favourably with those of a Fresnel rhomb, but in Fig. 5. Bi | addition each has the advantage that the emergent beam of circularly polarized light is always in the same straight line as the incident beam whatever the orientation of the polarizer. In the British Association Report for 1851, Prof. G. Stokes described a new form of elliptic analyser consisting of a plate of selenite which retarded waves of mean refrangi- bility by about a quarter wave-leneth. This, mounted in the manner described, was found to yield very accurate results, but a little experience is needed in experimenting with it as the tint produced perplexes the operator. It seems probab!e that the second form of polarizer described above would ‘prove useful in the analysis of elliptic vibrations, as it could ~be accurately set with respect to the analysing Nicol’s prism, ‘an adjustment it is difficult to make with a quarter-wave plate, for in this case the positions of the axes are not accu- rately known. In the present form of analyser the edges ot the glass (these should be accurately worked) forming the aperture take the places of the principal directions of the crystalline plate. Moreover, the tint would be absent. This 52 Mr. W. Wilson on the Effect of Temperature apparatus could be used for examining white light which had a definite vibration form. The polarizers were tested with monochromatic light, and using a quarter-wave plate as analyser, the transmitted lig ht was found to be circularly polarized. Removing the quarter- wave plate and using white light, the intensity of the field of view remained constant however the analysing Nicol’s prism and polarizer were orientated, providing the azimuth of the incident vibration with respect to the polarizer was kept con- stant and at the requisite amount. Il urther, using the Bi- trapezoid polarizer to examine an elliptically polarized beam, produced by a Fresnel’s rhomb with white light, no trace of tint could be discerned. Hach form of polarizer is mcunted in a metal cylinder which is provided with central square apertures. In conclusion I wish to tender my best thanks to Prof. W. M. Hicks and to Prof. Sir J. J. Thomson for the great interest they have taken in this work. Trinity College, Cambridge, October, 1910. Note added March 16, 1911. In the Bi-trapezoid polarizer, since the azimuth 0 of the incident vibration depends upon the refractive index, the efficiency of this form is slightly reduced, and becomes very nearly equal to that of Fresnel’s rhomb. e LIX. The Effect of Temperature on the Absorption Coefficient of Iron for y Rays. By W. Wison, M.Se.* HE tollowing experiment was made in order to determine if the absorption coefficient of iron for y rays varies with the temperature. A negative result was obtained. The arrangement of apparatus is shown in fig. 1. y¥ rays from a tube 8 containing about 30 m.g. radium bromids passed through an iron block B7 cms. in thickness. The block was contained in a mufile-furnace C, and could be heated by means of a blowpipe. The intensity of the rays after passing through the turnace and iron was measured by an electroscope HK. The distance from the furnace to the electroscope was 150 cms., and an asbestos sheet D was placed between them as shown. Under these circumstances, the heating of the furnace was found to have no direct influence on the electroscope. * Communicated by Prof. E, Rutherford, F.R.S.” on the Absorption Coefficient of Iron for y Rays. 538 - In order to prevent a redistribution of the emanation in the tube containing the radium owing to heating effects, it was enclosed in a water-bath R and shielded from the furnace by a sheet of asbestos. 3 -_-— _-_-— —_— — — - ie Experiments were made both while the iron was being heated and while it was cooling, the temperature being measured by means of a platinum-iridium thermo-couple. It was found that the readings were not so consistent while the block was being heated as while it was cooling. This is due to the fact that the temperature of the room rose con- siderably when heating was taking place, while when the heating was stopped the hot air soon dispersed, and the temperature did not vary while the iron was cooling. Results obtained in three experiments are given in Table I. AUAIBIGY Je iL i LI. TIT. Pais, a's | | | | Temp. © C.| Ionization. Temp. © C.| Tonization. | Temp. ° C. | Ionization. 630 305 || 678 | 387 570 | 3:38 Oo esGO aU OSl sit Sac e Nal ASOrWel yl S36 SO eS a meno cn kro a en meel Omori sa AG le Se HOWe Uh wn ohe | Meas TP ROTO. 336 Dee On ml nletTO.. I) S465, Ol SO eR Ais 50) ATT i 3-400) i ah estas ng Cee ee ooO neal) 348 | | | 349 | S50) ei) ses 1) B40 | | 290) SHON 307) 13-49 | . Pela a eee | | | Hos 184s 4b | } ae I ~ a = a te 534 On the Absorption Coefficient of Iron for y Rays. An experiment was made heating the furnace alone, and the ionization was the same at high and low temperatures. — It will. be seen. that the ionization in the electroscope is slightly greater at high than at low temperatures. The mean of a number of experiments showed the rate of increase of ionization to be *0020 per cent. per degree centigrade. This can be shown to be due to the change in density of the iron as follows. Let I, be the intensity of the rays falling on the block. The intensity, I, of the emergent rays is given by L=I) e-™, where / is the thickness of the block and X the coefficient of absorption. ke,” Now X is proportional to the density of the block, and therefore at any temperature @ degrees higher is given by r 1+32@> Similarly the thickness of the block at this temperature is (1+a0)/. We therefore have for small values of 20 . j= i e—Al( — 206) oly 00 = 2anrlI. where « is the coefficient of linear expansion of iron. The percentage rise in ionization per degree centigrade is therefore 200aQX/. i In an actual experiment Ig was found to be 8:11 div. per min. and I 3°55 div. per min. “. Al= log I,— log 1=°820. Taking « as ‘0000117 we obtain an increase in ionization of -00193 per cent. per degree, which is in good agreement with the value ‘0020 obtained experimentally. Thus the temperature of the iron within the limits examined affects the absorption coefficient by an amount to be expected from considerations of the change of density of the absorbing substance. That is, the absorption of y rays by iron depends only on the actual number of atoms traversed by the rays without regard to the temperature of the body which they compose. These experiments were carried out in the Physical Labo- ratories of the University of Manchester, and I wish to express my best thanks to Prof. Rutherford for the kind interest he took in them. (onyiile and Caius College Cambridge, February 1911 iX:. The Heat of Mixture of Substances and the Relative Distribution of the Molecules in the Mixture. By R. D. _ Kueeman, D.Sc., B.A., Mackinnon Student of the Royal Society, and Clerk Maxwell Student of the University of Cambridge * \ HEN two quantities of different substances are mixed and no new molecules are formed, we will assume-in this paper that the heat of mixture is the change i in the total potential energy of the matter due to the ‘chemical” attraction between the molecules. We made a similar assumption ina previous paper in the case of the internal heat of evaporation of a pure liquid, which led to results agreeing well with the facts. But even if a change in the intra-molecular potential energy of a number of molecules with a change in their state of aggregation takes place, it could only be small in comparison with the change in the potential energy due to the attraction between the molecules, since the change in the internal latent heat of evaporation of a liquid with temperature is large in comparison with the corresponding changes in the intra- ‘molecular energies of the vapour and liquid, that is, the specific heat at constant volume of the vapour and liquid. We shall deduce formule on the above assumption in this paper applying to various cases of mixture of two or more substances, using the law of attraction between molecules which was deducedt from the internal latent heat and surface-tension of liqnids. The law : attraction between a Gv/m) my)? two molecules of the same kind is (| , where oi Ile Tis the temperature of the molecules and z their distance of separation, «, denotes the distance of separation of the mole- cules in the liquid state at the critical temperature ‘l',, Sv/m denotes the sum of the square roots of oe atomic weights of the atoms of a molecule, and b(— =) is an arbitrary ’ ? 1° function of the ratios % Be AS the exact form of the ele. function ¢, cannot be deduced from latent heat or surface- tension data there is little other data relating to physical or chemical properties of substances from which any informa- tion on the exact form of gd. can be obtained. We have * Communicated by the Author. T Phil. Mag. May 1910, p. 783. 536 Dr. R. D. Kleeman on the shown, however, that @, must be a function of both z and T*, and that it probably varies very little with T or zt. Since the attraction between two molecules is equal to the product of their powers of attraction the function ¢, must consist of two factors of the same form, that is sud b Zo 2 ay (= 7 )=b4 = a), Pa . since y 3 V3 sa (GY = GY GY ot ne QP The form of the functions B, ...... C., as will be seen, depends on the form of the function ¢, (= a in the law of attraction between molecules and the relative distribution of the mole- cules in the mixture. As the form of the function @, is not yet exactly known, we are not yet able to evaluate these functions in any given case. The expression ee the value of Ly», can be obtained in several different forms, depending on the quantity in terms of which the relative distribution of the molecules in the mixture is expressed. Jor example, if the relative distribu- tion is expressed in terms of the mean distance of separation of the molecules in the mixture, 7. e. in terms of ps ( m™% ai N21 + Ng My = M, j where p; denotes the density of the mixture, Ly.», becomes | a (Sr/m,)? =F (“4+ male i )avm S/ my (S 3 Ni Tey \ Alea aE evar} {te (4 2)" where D, Ey, E., and F are functions of the ratios 05. iba RO Pe, Pe j s¥ , Te, Heat of Mivture of Substances. 5At It is possible to develop formule for the heat of mixture of liquids which do not involve any knowledge of the relative distribution of the molecules in the mixture. Let AB in the figure be a cylinder containing four pistons c, a, J, d. c CL 7) Be 0 1] Z = I. i = $ =f =4 = =~ tee? Suppose the space C between the pistons @ and 0 filled with a mixture of molecules 1 and 2 in the proportion of 7, to m9. Suppose the piston a pervious to molecules 1 but not to molecules 2, and the piston 6 pervious to molecules 2 but not to molecules 1, so that the space between the pistons v and a is filled with saturated vapour of molecules 1, and the space between the pistons 6 and d filled with saturated vapour of molecules 2. Suppose the pistons ¢ and d be moved towards the ends of the cylinder till the increase in the masses of the vapours 1 and 2 be equal to m, and ny respec- tively, and suppose that at the same time the pistons a and 0. are moved so as to remain in contact with the mixture. During the process the piston a has displaced a volume OV ny P35e, My of the mixture and pz, its density. and ¢, 1s the concentration of the mixture, where V is the volume of a gram | of the molecules 1. The quantity oe ory 2 may be PL 2 called the volumea molecule 1 occupies in the mixture ; it is a quantity which is of interest and importance and will be discussed in the next section. Similarly, the piston 6 dis- . ° Tho places a volume of the mixture equal to pss TL: Let py denote the pressure per cm.? on the piston c and pg that on d, and let the corresponding pressures on the pistons a and é be p,.' and p,! respectively. Now according to Clapeyron’s thermodynamical equation we have dV n dp OV nT dp’ L 4—Pasm ap Pe =m —p3— aL om Ee eal aye? 7s a Pde Moy dil OV n Apo dV nl dp,’ Lin, + Pats Pag, en = Fh Thee ean MM, a ) 2 where Ly, and Ln, denote the internal heats of evaporation of n arms. of molecules 1 and n, grms. of molecules 2 respectively, 542 Dr. R. D. Kleeman on the and v, the volume displaced by the piston ¢ and vy that dis- placed by the piston d. x, and v, are given by the equations yy OV Ny (m + Ps an wt) ps1, dV ny Case M,) 08="2 where p, and p; denote the densities of the saturated vapours of molecules 1 and 2 respectively. Now In,=n,/—Ln,”, where Jin,’ denotes the internal heat of evaporation of 7, erms. of molecules 1 from the mixture into a vacuum and In,’ the heat of evaporation of , grms. of the saturated vapour of molecules 1 into a vacuum. Similarly, | In, = or 7. Ln,” ‘ Now we have seen that expressions for Ln,'’ and Ln,’ are at once obtainable from the law of attraction between mole- cules. The values are 4/3 eh mis (sf) (> Vm)? or, (M, and noH Ps 4/3 MN “M, M, (> VATED 5) where tgp AL H, rane o(T ’ i, and xb mT ~ H.= o( ris)» a, and x, denoting respectively the distances of separation of the molecules in the vapours of molecules 1 and 2. Since Lr»,= Ln! + Ly,’ we have for the heat of mixture by means of the above equations the expression OV No 5 do! Ny 6V 7, moe dp, ] {P52 31, f E al a 3 "PSM SL at OV my } dp,’ f OV na dpe ac Lett E 0 | i Tae wi: f Ta P| Ue “a es ! —Hp,t9+ Apt | arr V4)? f | ; jie -— ——s —- 4 3 vB 4/3 ees es 2 ul % Hops : + Aspes } M3 > J mes Heat of Mixture of Substances. 043 It should be observed that the functions H,, Ay, Hy, A», have all the same form, namely ¢3;. The form of $3 depends on that of ¢, in the law of attraction between molecules, but this is not exactly known. There is, however, some evidence that 3 varies only slightly with the temperature and distance of the molecules*. In the absence of further information we may therefore take A,, A, H,, H, equal to the same constant for a given temperature. If the values of A, and A, are available, it is best to take the mean of A, and A, to represent the constants A,, Ay, H,, H,, as they have the same value at corresponding temperatures. Other- wise the value of this constant can be obtained from Table V. p- 796, Phil. May. May 1910 ; at a temperature of 20° C. in the case of ether it is equal to 3742. This will probably give fairly accurate results. If the density of the mixture of saturated vapours is so small that the constituent vapours obey Dalton’s law of partial pressures, we have py=p,’ and p.=p,'. The formula for the heat of mixture then becomes en mt) Ge ) {2 _ (2) 1 ( dp2 LE P3 I ral Lie Ps P3 ral ) ey UA —— 3 Me Ee = 1/0 M2 (ee /m,)?— Asp2 Mi (> V/ my)”, L 2 since At nae OV m. _ mtm OC] M, P3 OC M. ho P3 f When this is not the case there would be some difficulty in practice in obtaining the values of the quantities p, p,', 2, Po’. A special case of mixture which is of interest is a saturated solution of a salt in a liquid. To obtain a formula for the heat of mixture let the piston @ in the figure be removed and the semi-permeable piston 6 replaced by a solid one. Jet the piston ¢ be moved away from the mixture till n; grms. of liquid 1 have evaporated ; ny germs. of salt pil then be deposited. From Clapeyron’s equation we then ave Ln, —Ln,"’ a E — (my +N. ag no) ck a) P4 he” PS P2- =1| (gee eNee P4 ce P3 Po al 3 * Loe. ett. ee 2 SS =e Se ee = as a 5A Dr. R. D. Kleeman on the (ort) 9) P3 P2 ~ where is the change in yolume of the liquid and salt. The heat of mixture then 1s Be ny ((m +7) EVE dp, | ) nH, a cj ta Tan Pew Lar OY POR en OR Tr: (Sm) TAG Pa Noe =e ie si (4) Cc vm). It should be observed that the latent heat of evaporation L,’ of a grm. of liquid 1 intoa vacuum, which was expressed by i ch ry. 1 | dp Pe eee ee ee ae ) G ey aie, where p is the density of the saturated vapour and p its pressure. The value of 1, in the equation may be expressed in the form fp \%8 a Ay (£) (> ¥m,)?. 1 Another case of mixture of substances which is of interest is a saturated solution of a liquid—say of molecules 1—in a liquid of molecules 2. To obtain a formula for the heat of mixture in this case, let the piston 0 in the figure be replaced by one which is solid. Let the pistons ¢ and a be moved in the same way as before till n, grams of molecules 1 have evaporated from the mixture: the corresponding mn, grams of molecules 2 will form with a part of the mixture a saturated solution of molecules 2 in the liquid 1. Let p,;’ denote the density of this solution, and suppose the ingredients are in the proportion of m,' to n,. If x denotes the number of grams of the mixture which combine with the mn: grms. of molecules 2, we have . He , 2n,/n,+ mM, ny Ny + INo/ No + Ng ia itp. i Heat of Mixture of Substances. D45 The change in volume of the liquid phase during the evapora- tion is therefore ee 1 (Ny ++ 2)— —(t+Ng)— =v say. Ps P3 From Clapeyron’s equation we then have Rel bs oad LRN VA FOR (a eee Ln,’ — Ln, +p(2 ») pov Ey Nan lait, (A Baier as where Li. as before, is given by n, Hy Pa ae. GENS) M, \M, (Z/m)?. If the heat of mixture of a saturated solution of molecules 1 in 2 per gram of mixture be denoted by H,, and that of a saturated solution of molecules 2 in 1 per gram of mixture by H,’, we have ; 4/5 SEES H, (ny + Tg + B) = Lin,’ = ia Ay ( | (> Jim) + H t (My + x), M, M, where 2 and Ln,’ are given by the above equations. This equation expresses the relation between the quantities H, and H/. If the molecules 2 instead of 1 be allowed to evaporate the above equations apply if the suffixes 1 and 2 and the symbols H;and H;’ are interchanged, and p; is written for py ‘The last equation thus becomes Ny 4/3 ean M, 2 it) (Z4/m)? + HC + 2). H,'(y + +2) =L,,'— The elimination of H;’ from the above two equations gives an expression for the value of Hy. The value of either H; or H;’ may also be obtained by means of one of the formule given previously. It will be observed that some of the formule for the heat of mixture obtained apply whatever the changes are that take place when the substances are mixed, such as the formation of new molecules, &c. Both types of formule obtained are perfectly general, and therefore apply when one or both of the substances are in the gaseous state. It is unnecessary to develop formule for the mixture of three or more liquids, as these can now be developed without difficulty along the lines indicated. 7 a a eS SS 546 Dr. R. D. Kleeman on the Volume of Occupation of a Molecule in a Mixture. The decrease in volume of a large mass of a mixture of substances on removing from it a single molecule, may be called the volume of occupation of the molecule in the mixture. This quantity it appears has not yet been defi- nitely defined and discussed ; it seems of importance, and a consistent study of it should lead to interesting results. It is intimately connected with the relative distribution of the molecules in a mixture. Using the same notation as before, and denoting the volume of occupation of a molecule 1 by 3, we have Se We have further in the case of a i mixture of molecules 1 and 2 in the proportion of 2, to ny that | Ny Ng m+n, a, oo dee Va AN ee SOF, ova lea Whe ’ M, Ms, P3 My M, P3 where a denotes the ratio of the concentration of the mole- cules 2 to that of 1. We have seen that the decrease in volume of a mixture when a molecule 1 is removed is 4%,, therefore if Pj. denote the intrinsic pressure of a liquid, 7. e¢. the pres- sure due to the attraction between the molecules, P;,2 5, is approximately the work done on the molecule by the mixture during its removal. If the temperature of the mixture is kept constant during the process, an equivalent amount of energy in the form of beat has to be supplied to the mixture. This amount of work is equal to the internal heat of evaporation Li,’ of a molecule 1 into a vacuum, or equal to the ordinary heat of evaporation if the density of the saturated vapour is very small in comparison with that of the liquid. In the case of a mixture we have therefore at low temperatures L,/=Pj,25, and L,’=Pi.5,. From these two equations we have Ly'/Ls' =54/92= i +3 OV /6V OC} Ole q or the internal heats of evaporation of two molecules J and 2 are to one another as their volumes of occupation in the mixture. | When the density of the saturated vapour is not small in comparison with that of the liquid the value of I.,’’, the heat of evaporation of a molecule 1 of the vapour into a vacuum, is not negligible in comparison with that of I,', and we have for the ordinary heat of evaporation L,/—hL,"=P; 25, — P4235)’, where L,”=P4,9'51’, P4,2 denoting. the intrinsic Heat of Mixture of Substances. 547 pressure of the mixture of saturated vapours and $,/ the volume of occupation of a molecule 1 in the vapour. Similarly we have for the ee vot evaporation of a mole- cule 2, Ly’ — Ly!’ = P,292.—P 4,2 5,’, where L.'’= P’; 29,". If the temperature of a mixture is sontuelll: increased, a temperature will ultimately be reached when L,’—L,/’=0, inwhich case P,.3,=P'123,'. Similarly, L,’—L,’ will pass through zero for some temperature, when we have Py, %2=P%4,252. If both the expressions pass through zero ! oe Now this equation would also apply when the relative concentrations of the different molecules in the liquid mixture and the saturated vapour are the same, since, as we have already remarked, the relative distribution of the molecules should then be the same. It appears, therefore, that when the internal heats of evaporation of the molecules of a mixture pass through zero at the same temperature the relative con- centration of the different molecules in the vapour and the liquid must be the same. Further, since we then have L,’=L,” and L,’=L," the density of the vapour and liquid must be the same. A saturated solution of molecules 1 in a liquid of mole- cules 2 has the same partial vapour pressures as a saturated solution of molecules 2 in a liquid of molecules 1, since they remain in equilibrium in contact with one another. It follows S, therefore, from Clapeyron’s equation that at low tempera- tures, when the volume of the vapour of a grm. of molecules of the same kind is large in comparison with the corre- sponding volume of the mixture, the heat of evaporation ot a molecule of the same kind is the same for each mixture. Therefore, if P’ is the intrinsic pressure of one of the mixtures and V,’ and V,’ the volumes of occupation of the molecules 1 and 2, and the corresponding quantities for the other mixture are P”, V,’’, V2’, we have P’V,=P"V;,’, Van Wt Neti Ws i or the ratio of the volumes of occupation of the two kinds of molecules is the same in each mixture; further, we have PNG, py=yro the intrinsic pressure of each of the mixtures is at the same temperature we have and P’V,’=P"V,". From these equations we have inversely pr oportional to the volumes of occupation of one of the molecules. O48 Dr. R. D. Kleeman on the Test for the Formation of New Compounds in a Alieture. If two quantities of different liquids containing an equal number of molecules be mixed and no new molecules are formed, the relative distribution of the molecules in the mixture can be deduced as we have seen. The functions B,, B., Cy, and C,, in the formula for the heat of mixture given, can then at once be formed if the form of the function ¢, in the law of attraction is known. It will then be most convenient to express the relative distribution of the mole- cules in terms of that of the molecules of one kind in the mixture. We have pointed out that there is some evidence that the function ¢3 does not depend very much on the magnitude of the variables it contains. If we assume it approximately a constant an approximate formula for the heat of mixture can be obtained. Let us first find the value of Ln», on this supposition. Whena molecule 1 is removed from the mixture the work done against the attraction of the remaining mole- cules i is ; Gey uf) (S./m,)?, where m=o( Tt; p7 denotes the density of the molecules 1 in the mixture, and x, denotes their distance of separation. This is at once obtained by supposing the molecules 2 absent, when we are dealing with a liguid in which the il series are evenly distributed as in a pure liquid, and a corresponding formula applies: The work done against the attraction of the mole- cules 2 in removing the molecule 1 is obtained as follows. Suppose the molecules 2 replaced by molecules 1. The work done against the attraction of all the molecules is then 3 psn Ne BIA sets ie x oe Wiggin) (&vm)% where Wi=$5(= ae and x;y, denotes the mean distance of separation of the mole- cules. The term which is raised to the 4/3 power is equal to = we have expressed this quantity in terms of quantities Ey relating to the mixture. The work done against the molecules 1 which replaced the molecules 2 is therefore Wigan) (3 Vm)? —u( tt) Sage as is at once evident from the nature of the distribution of fhe molecules. The work done against the molecules 2 if they Heat of Mixture of Substances. 549 are again put into their place is the above expression = Vm > Vm, (2/m)" certain of the quantities ze, and Tc, occurring in W, and x. This will at once be evident if we consider that this expression is supposed to be derived by a direct application ot the law of attraction between two molecules 1 and 2, viz. ease 20 7 lh a Pa zo, ; T) : b= a, ) Sm SV my. But since wu, and W, are each of the form @3, and therefore do not depend much on the magnitude of the variables they contain, we may suppose both these quantities equal to a fonsiant. The w ork done when a molecule 1 is removed from a mixture against the attraction of the remaining mole- cules 1 and 2 is therefore 4/3 ie {Ww (a=) = 74 99 (ff) bE Vin SV my tn ff) ( au multiplied by ,and substituting xc, and To, for where ae approximately. If the density of the molecules 1 in the mixture is expressed in terms o3M, of that of the mixture we have a= Similarly the ; M, ae M 9 : work done when a molecule 2 is removed is p32 -\*8 p ie { Wf M+, ar) — Uy e yo }avms Ving + (Pe) (= VY m2)’, where W,=u,.=W,=u,= a constant approximately, and maps Ps M+ M, This constant at 20° C. is equal to 3742, and the heat of mixture of , gram of molecules 1 with ng grams of mole- p nN s > ecules 2 so that —b seat niet temperature therefore given by M,” M,’ 4/3 pat NE 37424 (7 i nie ae at Le DBI AL Teen! Tite 4/3 na +((xg aM ) - (ff) ra 2 (> s/o)? 1 2 paul \e® oy" Weert + (ae aE ) (é devin rl the molecular weights of M, and M, being expressed relative to that of hydrogen. 4 o 550 Dr. R. D. Kleeman on the As a test for the formation of new molecules in a mixture of two substances we have then that if it is found that the heat of mixture of the substances in masses proportional to their molecular weights differs considerably from that caleu- lated by the above or previous formula 2, we may conclude that new molecules are formed. The extent of the formation of new molecules in a mixture of two liquids in any given proportion will probably depend, however, on the proportion between the constituents. The various formule for the heat of mixture of liquids all labour under the unavoidable disadvantage of appearing as the difference between quantities which are usually very large in comparison with this difference, which has the effect of magnifying all errors, and thus usually preventing a good agreement with the facts being obtained. If one of the substances is in the form of a vapour the formule suffer less from this defect. This is a point to be borne in mind when the formule for the heat of mixtures are used. A simpler test than the above for the formation of new molecules in a mixture of equal numbers of different mole- cules is the following. It will be evident, from the relative distribution of the molecules in such a case (which we have discussed), that the volume of occupation is the same for each kind of molecule. It follows, therefore, that if nc new dV 6V molecules are formed we must have 50 = 3a" 1 2 involves quantities which can be easily measured. In using this formula in practice, however, we are hampered by the difficulty—which occurs also in connexion with the previous formule — of not being sure of the molecular concen- tration of the liquids used, as these may be polymerized to some extent. Water is an example: it has been shown to be polymerized from surface-tension and other considera- tions, and consists probably of a mixture of molecu'es poly- merized in different degrees. We are therefore not able to use the extensive data on the solution of substances in water in connexion with the above formula, till we possess more reliable information on the size of a water molecule. On the other hand, the data relating to other solvents is not sufficiently extensive to be of any use. But experiments having the object of testing the formula could be easily and rapidly carried out. | : The above considerations suggest a slightly different way of testing for the formation of new molecules. Since the internal heat of evaporation of the molecules 1 and 2 when This formula Tleat of Ahiature of Substances. Dai the vapour pressure is small may be written P)»..—p3 and : SMG OV cee P12.

m,)?, where A,= o,(— Ty ). 5 Ue, Cy The heat of evaporation of a molecule 1] is therefore the above expression multiplied by ee and substituting my )* ve, and Te, for certain of the quantities ve, and Tc, oecurring in Ay. But since A, has the torm @; it does not vary much with the magnitude of the variables it contains, and we may therefore as before take it approximately equal to a constant. If the values of A, and A, in the expressions for thé internal heat of evaporation of liquids 1 and 2 are available, this con- A,+Az, 2 available the constant may be obtained from some single liquid, as explained before. Ata temperature of 20° C. the heat of evaporation of a grm.-mol. of molecules 1 would be approximately equal to stant is best taken equal to . IPf these values are not 4/3 O77 4¢ 9 : om iene : 3742 Gi ) y Vm, V me calories, Vio NO) Dr. R. D. Kleeman on the and this is approximately equal to the heat of solution at that temperature. If it is fouad that the heat of solution is equal to this value, we may conclude that probably every molecule 1 replaces a molecule 2 in the mixture. Let us apply these results to the heats of solution of a number of gases in large quantities of water. The internal heat of evaporation L, of a grm.-mol. of water into a vacuum is given by Ai)" 9 af Mo)” m,) vm if the molecular weight of water is given by the chemical formula H,O. But since the water molecules are polymerized we inust multiply M, and S4/m, each by some appropriate constant which expresses tiie degree of polymerization. The formula may therefore be written Ps 4/3 ee Ta=pAd( 4) (2a7ary)?, where uw is a constant. The heat of evaporation L, of a erm.-mol. of molecules 1 from the solution is therefore 4/3 as Me L,= pal 9?) ./m, SV. From these two equations we have bay we, fms where L, may now be taken as the heat of solution. The following table contains the heats of solution per grm.-mol. of a number of gases in a large quantity of water at a temperature of 18°2 C. They were taken from Nernst’s ‘Theoretical Chemistry,’ 4th edit. p. 599. | | | Ge | Heat of Sol. L,2 vin, : a | Heat ob Sol. |. Evin, | | | per grm.-mol. Se Gas. pee mol. | “25 A! ity | NEE i430 11,370 Ci aero) =a EEE 800... 9,088) HMETCINeR | 107.810 11,730) 9) HBr ...| 19,940 ; 17,140 || CO, a 5,880 19,310 | MeO Pe TE LONE 90,650 | | | | ye a Fleat of Mixture of Substances. 5dD The third and sixth columns of the table contain the corre- sponding values of L, calculated by means of the above equation. The internal heat of evaporation of a grm. of water at 20° C. is 561°5 cal., and the value of L, therefore 561°5 x 18=10,110. It will be seen that there is a rongh agreement between the two sets of values for the gases NH,, HF, HBr, HI. It seems, therefore, that a molecule of these gases replaces a water molecule on solution. This result is, however, not quite conclusive, as it may happen that hydrates are formed and the resultant molecules occupy such positions relative to the molecules of the solute that the total change in potential energy is the same as if no new molecules were formed and each dissolved molecule replaced a molecule of the solute. The calculated heats of solution do not agree, however, with the calculated values in the case of CO,, HCl, and Cl,. It is usually assumed that CO, molecules do not change on solution in water, since the concentration obeys Henry’s law. The nature of the disagreement with CO, would then indicate that a CO, molecule is on the average further away from the surrounding water molecules than a water molecule in pure water. But it also follows (referring to what has gone before) that the volume of occupation of a CO, molecule is less than that of a water molecule. These two conclusions do not at first sight agree with one another. But it must be remembered that the volume of occupation of a molecule is the total change in the volume of the liquid when a mole- cule is removed, and it may therefore happen that when a molecule is removed from a mixture a change in the volumes of occupation of the neighbouring molecules takes place, most probably an increase, which changes the magnitude of the volume of occupation of ne removed molecule considerably. These results combined, therefore, seem to indicate that a molecule of CO, in water is surrounded by two shells of water molecules, the outside one being more dense and the inside one less dense than pure water. In the case of the other two gases Cl, and HCl, all we can say is that a dissolved molecule either does not occupy a similar position as a molecule of water or a hydrate is formed. Cambridge, Feb. 1, 191]. Phil. Mag. 8. 6. Vol. 21. No. 124, April 1911. 20 STS SS = see =. ~d Aa hare: _ _ ae ” iy LXI. On Metallic Colouring in Birds and Insects. By A. A. MicHELSON™. [Plate IV.] WE the exception of bodies which shine by their own light, the appearance of colour in natural objects is due to some modification which they impart to the light which illuminates them. In the great majority of cases this modi- fication is caused by the absorption of part of the light which falls on the object, and which, penetrating to a greater or smaller depth beneath the surface, is reflected, and finally reaches the eye. If the proportion of the various colours constituting white light which is absorbed by the illu- minated body is the same for each, the light which reaches the eye has the same composition as before, and we say the body itself is white ; but if this proportion be different, the resulting light is coloured, and the coiour of the body itself corresponds to that colour or colour combination which is least absorbed ; it is complementary to the colours which are most strongly absorbed. Thus the light from a green leaf in the sunshine, after penetrating a short distance in the substance of the leaf, is either transmitted or reflected to the eye. In its passage through the substance of the leaf it has lost a considerable part of the red light it originally contained, and the resulting combination of the remaining colours produces the effect of the complementary colour or green, as can readily be shown by analysing the light into its elementary colours bya prism, and comparing the resulting spectrum before and after the reflexion from the leaf. The same explanation holds for all the paints, dyes, and pigments in common use. These colour effects occur in such an immensely greater proportion than all others combined, that the occasional appearance of an exception is all the more striking. The rainbow and the various forms of halo are almost the only instances of prismatic colours which occur in nature. There remain only two other possible methods of producing colour. A. Interference, including Diffraction. B. “ Metallic”? Reflexion. It has been abundantly proved that the usual “ flat,” “dead,” ‘‘uniform” colouring, brilliant as this sometimes * Communicated by the Author. On Metallic Colouring in Birds and Insects. 555 can be, e.g., in birds, butterflies, and flowers, finds its simple explanation in the existence of pigment cells; so that the same cause (doubtless with many modifications) is here effective as in the great majority of cases previously considered. But the lively, variable ‘“ metallic” glitter of burnished copper or gold; the reflexion from certain aniline dyes ; the colours of certain pigeons, peacocks, humming-birds, as well as a number of butterflies, beetles, and other insects, requires another explanation. While cases under A occur occasionally in nature—for example, in the colours of thin films, in the iridescence of mother of pearl, and (as an accessory) in the colours of the rainbow and of certain halos—they are so rare and so readily distinguished from the true metallic colours that they may be most conveniently treated as exceptions after the subject of surface-colour has been considered. The designation “ metallic” at once suggests that there may be some common property of all these colours which is typified by the metals themselves. But, as is well known, the principal characteristic which distinguishes the metals from all other substances in regard to their action on light, is their extraordinary opacity. A very important consequence of such great opacity is that light is practically prevented from entering the substance at all, but is thrown back, thus giving the brilliant metallic reflexion so characteristic of silver, gold, copper, &. In fact, the distance to which light can penetrate in most metals is only a small fraction of a light-wave ; so that a wave- motion such as constitutes light, strictly speaking, cannot be propagated at all. Again, as this opacity may be different for different colours, some would be transmitted more freely than others, so that the resulting transmitted light would be coloured ; and the reflected light would be approximately complementary to the transmitted colour. For most metals the difference is not very great; so that the reflected light, except in the case of gold and copper and a few alloys, is nearly white. Inthe case of the aniline dyes, however, there is a marked difference, as is clearly shown by their absorption spectrum. In transmitted light, even a very small thickness of fuchsine shows no yellow, green, or blue, and gives as a resultant of the remaining colours a beautiful crimson. The light which it reflects, however, is just this yellow and green which it refuses to transmit, and it ac- cordingly shimmers with a metallic golden green colour, which changes when the surface is inclined, becoming full AOrZ, 556 Prof. A. A. Michelson on Metallic green, or even bluish green when the illumination is sufficiently oblique*. The chief characteristics by which “metallic” reflexion may be distinguished may be summarized as follows :— 1. The brightness of the reflected light is always a large fraction of the incident light, varying ‘from 50 per cent. ‘to nearly 100 per cent. 2. The absorption is so intense that metal films are quite opaque even when their thickness is less than a thousandth of a millimetre. 3. If the absorption varies with colour, that colour which is most copiously transmitted will be the part of the incident white light which is least reflected—so that the transmitted light is complementary to the reflected. 4, The change of colour of the reflected light with changing incidence has already been mentioned. It follows the in- variable rule that the colour alwa ays approaches the violet end of the spectrum as the incidence increases. If the colour of the normal reflexion is violet the light vanishes (changing to ultra-violet), and if the normal radiation be infra-red it passes through red, orange, and yellow as the incidence increases, While the criteria just considered are the simplest and most convenient for general observation, it is to the more rigorous results of more refined optical methods that we must look for the final test of the quality of reflexion in any given case; to determine whether or not a colour phenomenon may be due to “metallic”? reflexion or to one of the other general causes. Such optical tests are furnished by the effect of reflexion upon polarized light. The elements of the resulting elliptic vibrations may be expressed in terms of the amplitude ratio IR of the components, and of the phase difference P corre- sponding to the angle of incidence I, as in the following tables for silver and for glass. The very marked difference in the run of the numbers in these tables may be rendered still more striking by plotting the values as ordinates of the curves shown in Pl. IV. fig. 1, which gives ata glance the form of the “phase”? curve e (full line) and the * amplitude ”’ curve (dotted line) for silver, steel, graphite, selenium, flint glass, crown glass, and quartz. It is evident * The change in colour is very much more marked when the light is polarized perpendicularly to the plane of incidence. As the angle of incidence approaches the angle corresponding to the geomtaing angle ” the colour is a deep blue or even pimple Colouring in Birds and Insects. ooT | Silver. | | Glass. | i ne | i SN eee Bs eA stile he). R. pow, 00 Outen TcOD |. 45 Ra OL Jie tl un OS gaa 40 [USS a a 14 hipaa rake nn OU) 30 30° OB /S4s a BOON) S00) 25 eee ts i Ok i) al ACE Mee nOOiyaie |i |) OR.) a. 09 SOR MeOH eM MOOR Ke ais LQ.) 60° | 10 | 88 | 60° | “DO Tf ee 20 ere TOO my ker BO ma We a 2a.” | 80° OSIM MMs aa) AMS A-Data 4 PO ealivin OU. eh ADMIRE OUS- tesa ity TAO eden Werth AO | | that metals have a smoother phase curve than semi-metallic substances like graphite and se'enium, and these show less abrupt changes than do transparent substances such as glass an quartz. In tact we may take the steepness of the curve where it is steepest (better where the phase difference is 1/4) as a measure of the transparency of the substance; and theory shows that this steepness is in fact inversely proportional to the absorbing power of the substance. Starting with the formula of Cauchy 2k sin? i cost (v? cos? r+ k?) cos? a—sin* 2’ tan A= differentiating for 2 and putting in the resulting expression A=7/2, and I for the corresponding angle of incidence, we have dA sin I(tan? I+ 2) sir When (k=coeff. of absorption). On the other hand, if the phase change be the result of a surface film, and we start with the corresponding approximate formula tan A; =e tan (1+ 7) we find da, dt a (e=coeff. of ellipticity). : In this case the steepness is inversely proportional to what Jamin has termed the * ellipticity.” Se Seite alee ee So |) ri I 1 ey nt ia! 2 rn err 558 Prof Anke Michelson on Metallic In point of fact both causes are effective ; and for semi- transparent substances it is impossible to obtain results which agree even approximately with experiment by either formula. But the rigorous expression of Cauchy, which contains both k and i, is so unwieldy to be practically useless. | The difficulty may be obviated by making use of the empirical 1 relation H=EH,.+ Hz, where H= Thai? which may be trans- lated to mean that the actual “ ellipticity” is made up of two parts which combine additively ; one due to the surface film and the other resulting from absorption. If the medium under observation be transparent H,=0, hence E-=H. If it be opaque Ee is small compared with Hz, so that approximately E,= EH. For semitransparent media it will be necessary to deter- mine the absorption, k, by direct measurement, from which Ex may be calculated by means of the formula k ae sin I(tan? I+ 2)’ and E, may then be determined by HKe=H~—Ex-. In the case of substances like fuchsine and diamond green, in which the medium is almost perfectly transparent for certain colours, we may find Ee. for this colour; and if it be correct to assume that EK. does not vary with the colour, the value of H,=H—H,. may be accurately determined for the semi-transparent and the opaque portions of the spectrum. A fairly good test is that furnished by selenium. The incidence I corresponding to A=90° is nearly independent of the colour, being about 71°. The value of < calculated by the preceding formula is, very nearly, “ 2a gat ie ae ae Following are the values of = E, E;, and &*. * These last are taken from the results of Professor Wood, Phil. Mag. 1902, vol. 111. ; : . ; Colouring in Birds and Insects. 559 Selenium. d eilal j y : FR : Ex. 10Ex. If 6990 | ~— 83 043 000 (00 cote | 6410 DI "048 OOD 065 “C9 | 6075 | 18 056 U13 13 13 aps Tiat jlwes 16 062 019 19 20 5410 | 13 ‘O77 “O34 34 V8 00a ae 091 048 48 -40 4740 | 9 STALL "(68 68 53 4405 | 8 | "125 ‘082 62 ‘6l | Following are similar series of observations for fuchsine and for diamond green. Fuchsine. { d Fa E 10Ex. log nm 670 50 ‘020 00 04 640 40 025 O05 05 615 18 "059 35 88 590 @) ATL ‘GO OS) 560 q “14 ey 1:4 525 6 16 1°4 ez 500 4:5 22 9-0 15 470 4-5 22 2-0 1:0 Diamond Green. dA » i E 10K, | og 700 12 08 Oe os | 680 8 |) ied Tk 660 7 14 | 1:3 1:3 | 640 Sid 2090) sh a ee 15 620 4 525, 2°4 1:5 600 4 25 2°4 Te? 560 6 16 1°5 *65 540 25 “04 03 23 520 60 “G16 0:06 "08 500 8&0 7012 0:09 02 480 70 014 0-04 Ol 460 45 022 0:12 Zo 440 15 ‘OUT 0°57 40 ae I, _ incident light 7 I ~ transmitted hight * No account was taken of the loss by reflexion, CDT ee ee ee RE AR IRIE oe ets 560 Prof. A: A. Michelson on Metallic The quantities in the last column are proportional to k ; but the actual values of & thus deduced from observations of transmitted light are considerably less (about 1/3 of the value given), possibly because of the unevenness of the film which makes the measurement of the actual thickness (of the order of one thousandth of a millimetre) uncertain. The agreement in the last two columns of the tables, while somewhat imperfect, is still enough to show that the results are of the right order of magnitude—and if it be remembered that the properties of the specimens vary considerably with the method of preparation, it is probable that the outstanding differences may be thus accounted for. In any case the agreement is much better than it would be, had the ellipticity been attributed to absorption alone. In the aniline colours the absorption varies enormously with the colour, and we have all the gradations from metallic reflexion to almost perfect transparence combined in a single specimen. The measurement of the phase-change and the amplitude-ratio for these substances show changes in the form of the curves almost identical with those given in the preceding figure. Pl. IV. fig. 2 shows the curves obtained for fuchsine and for diamond green. It may be noted that in both these figures the “phase” curve is much more characteristic in its changes than the “amplitude” curve. These specimens are prepared by dissolving the aniline colour in hot alcohol, filtering hot, and covering a hot glass surface with the solution. The alcohol evaporates rapidly, leaving a mirror surface of thickness of the order of a thousandth of a millimetre. : A quite remarkable alteration occurs in the phase curves when the solution is diluted. The film deposited is then very much thinner than before (f:0m one-tenth to one one- hundredth of the former thickness) and, for some colours, the thickness is so small that considerable light is reflected from the surface of the glass. The resulting phase curve may then be negative, as shown in Pl. IV. fig. 3, for the colours red, orange, and yellow*. Such a result has been predicted from theoretical con- siderationst, but so far as 1 am aware, no attempt has been made to show that this depends on the colour of the incident * The lower curves show more clearly how the maximum value of & varies with the colour. + Drude, ‘ Theory of Optics,’ p. 294. Colouring in Birds and Insects. 561 light. This, however, follows, if we consider that the con- dition for such a negative phase curve is that the transition layer have an index of refraction greater than that of the second medium ; and as the refractive index for magenta is low at the blue end of the spectrum and high at the red end, the inversion of sign is strictly in accord with the theory, of which indeed it furnishes a striking confirmation. On applying the simpler general tests of metallic reflexion to the case of iridescent plumage of birds, scales of butterflies, and wing-cases of beetles, one is at once struck with the close resemblance these bear to the aniline colours, in every particular: for 1st. The intensity of the reflected light is much greater than for the “‘non-metallic” plumage, &c., in some cases approaching the value of the reflexion factor of the metals themselves. 2nd. The reflected light is always coloured, showing either a rapid change of index of refraction, or of coefficient of absorption with the wave-length or colour; and, indeed, it may perhaps be objected that these colours are far more vivid than any of the reflexion hues of the aniline dyes, or of any other case of “surface colour” hitherto observed. 3rd. In the eases which could be investigated for this relation (unfortunately rather few) the transmitted light 1s approximately complementary to that which is reflected. 4th. The change of co'our with changing incidence strictly follows the law already mentioned—the colour always changing towards the blue end of the spectrum as the incidence increases. This remarkable agreement has been pointed out by Dr. B. Walter in an admirable essay, “‘ Die Oberflichen- oder Schiller-Farben,”? and it is shown that none of the other causes of colour phenomena (in particular interference and diffraction) can be effective ; the laws which govern these last being totally different. It is, therefore, somewhat sur- prising to find that the contrary view is still held by many eminent naturalists, and it is hoped that the further evidence here presented may serve to emphasize the distinction between “metallic” or “surface” colour and the.remaining classes oD of colour (due to pigments, interference, and diffraction). In attempting to apply the more rigorous optical test of the measurement of the phase-difference and amplitude-ratios, one is met at the outset with the serious difficulty of the absence of true “ optical” surface. In fact, the materials we 562 Prof, A. A. Michelson on Metallic have to deal with (feathers, butterfly scales, beetle wing- cases) are so irregular that the quantity of “regularly” reflected light which is brought to a focus by the observing telescope is insignificant, and is often masked by the light diffusely reflected. But by the simple device of replacing the objec- tive of the collimator and of the observing telescope by low- power microscope objectives of small aperture, these difficulties are so far removed that it has been possible to obtain results which compare favourably with those obtained with the aniline films. In some of the measurements it has been found possible to deal with a single butterfly scale; and in these the irregularities of the surface were often insignificant, or of such nature that they could be taken into account. Following is a diagram showing the results of a set of measurements on a beetle having a lustre resembling burnished copper. Beside it isa duplicate of the preceding observations on a thin film of magenta (PI. IV. fig. 4). The correspondence between the two sets of curves is so remarkable that it leaves no room to doubt that in this ease the metallic coppery colour of the wing-case is due to an extremely thin film of some substance closely analogous in its optical qualities to the corresponding aniline dye*. The thickness of the magenta film was not very accurately deter- mined, but from the fact that it was deposited from a solution of 1/20 of the concentration of that which produced the cor- responding thick film (whose thickness is about 0°005 mm.), it is estimated that the thickness of the thin film is of the order of 0°00025 mm. It is, doubtless, unsafe from this to draw any more definite conclusion regarding the film of the wing-case, than to say it is probably of the same order. An attempt was made to check this estimate by the following simple device. A portion of the ellipsoidal wing-case of mean radius R was removed by passing over it very lightly a piece of the finest emery-paper fastened to a flat piece of wood. This left a clean elliptical hole of mean radius 7 showing the edges of the “metallic” film, whose width, h, could not be appreciated in a microscope with a half-inch objective. If this be estimated at less than 0°001 mm. the relation = = = pom: gives t, the thickness of the film, less than 5mm. a ten-thousandth of a millimetre. A second specimen of the same general coppery lustre, * The character of the curves for the organic film is considerably more “metallic ” than the corresponding curves for magenta. Colouring in Birds and Insects. 563 gave a set of curves (Pl. IV. fig. 5) which showed a double reversal; the phase-curve being positive for crimson and red, negative for orange and orange yellow, and positive again for the yellow, green, and blue. A series of curves fora very thin film of magenta (estimated thickness 0'00005 mm.) gave results surprisingly resembling those of the beetle. The second point of inversion being, however, in the green instead of the yellow, and the ‘ ‘metallie” character of the film being much less marked than in the beetle wing-case. The resemblance in the lower curves, showing the variation of maximum steepness with the colour, is even more striking. It can scarcely be doubted, therefore, that here again the ‘‘ metallic” colour is produced in a film whose thickness is of the order of a ten thousandth of a millimetre or less. A third example (PI. IV. fig. 6) is added, in which the correspondence is less marked, for the purpose of illustrating the general character of the curves for a case of green metallic lustre. There is in fact no aniline colour which shows an accurate correspondence, but the same magenta curves may be referred to for comparison. The beetle wing-cases furnish in many cases a fairly smooth surface, and the difficulties in obtaining the necessary measurements are far less than when working with feathers of birds or with butterfly scales. Nevertheless, as Pl. 1V. fig. 7 shows, the same general characteristics obtain in these, in both the phase-curves and the amplitude-ratios. It may be noted that the two curves do not always correspond *, but it is probable that the difference may be explained by the difficulty in obtaining accurate results with surfaces so irregular. It is worthy of note that in all of these curves (except that furnished by a red humming-bird feather) the curves are negative ; from which it is fair to conclude that the film which produces the surface colour is very thin. The total number of specimens which have been examined is perhaps not so large as it should be to draw general con- clusions, and it is clearly desirable that it be extended ; but so far the evidence for surface film, as the effective source * If we take the approximate formula /=tan 2y, it 1s at once apparent that the dotted curves in our diagrams should have ‘the highest minimum value for all the cases of oreat opacity. Thus the opacity may be in- ferred from the dotted curve for W as well as from the full curve for dA/di, and in general the indications in the two cases show a rough agreement, the steeper full curves corresponding to the lower dotted curves, and vice versa. 564 Prof. A. A. Michelson on Jletallic of the metallic colours in birds and insects, is entirely con- clusive. It is clear that in all of these curves the descriptive colour corresponds in general to that colour for which the full curve is least steep, and for which the dotted curve is highest ; and is complementary to the colour for which the full curve is steepest and the dotted curve is lowest, as we should expect; since the former corresponds to high re- flective power, while the latter is characteristic of transparent substances with but moderate reflecting power. EXCEPTIONS. Morpho alga. The measurement of the phase-difference in the light reflected from the blue-winged butterfly (Morpho alga), instead of being zero at normal incidence, had values which ranged from 0°15 to —0°15, and which were found to vary with the orientation of the specimen. There were also cor- responding changes in the general character of the phase and amplitude curves, all of which showed clearly that the whole phenomenon is considerably complicated by a structure of the scales. An examination under the microscope revealed the presence of exceedingly fine hairs (which can only be seen in reflected light) arranged without much regularity with their length parallel with that of the scale*. It was at tirst natural to attribute the blue colour to the light diffracted from these hairs ; and it is not impossible that some of the silky sheen which these butterflies exhibit, is at least in part due to these hairs, whose diameter is much less than a light-wave, and which are therefore in the same relation to the light-waves as the small particles which cause the blue colour of the sky. But the changes in colour with varying incidence, so characteristic of true ‘‘ surface colours, ” were precisely the same in this specimen, and were practically independent of the orientation ; whereas the changes with the angle of incidence, which should result on the hypothesis that the colour is due to diffraction, should follow an entirely different law. Another species of butterfly (Papilio Ulysses) was also examined and found to yield normal surface-colour curves, as * There are three varieties of scales, of different shapes. These are arranged in overlapping layers, the outer layer being quite transparent and the lower one opaque. ‘lhe middle layer is the one showiny blue by reflexion and brownish-yellow in transmitted light. Colouring in Birds and Insects. 565 shown in Pl. IV. fig. 7 (No.6). There is in this case no - such minute linear structure as in the case of Morplvo alga ; and as here the phenomenon is clearly a case of ‘‘ surtace- colour,” so it is highly probable that the same cause is effective in the case of Morpho. Many other specimens were subsequently examined, but all fell into one or other of the two classes typified by these two. Diamond Beetle. If a specimen of the beetle popularly known as the Diamond Beetle be examined with a low power under the microscope, the bright green dots on the wing-case are seen to consist of depressions from which spring brilliant and exquisitely coloured scales; the colours varying throughout the range of the spectrum (green, however, predominating). The colours exhibited by these scales are so vivid and raried, and the changes so rapid with varying incidence, that it was at once evident that the effect must be due to diffraction from regular striations, which were accordingly looked for under a ~ magnification of about 1000 diameters. There were occasionally and indications of striated structure, but so uncertain that if other indications had been less decided it might have been concluded that some other cause must have been effective. But on putting the microscope out of focus a moderately pure spectrum was observed, and by measuring the angles of incidence and diffraction of the various colours, the “ grating” space could be determined, and was found 40 be of the order of a thousandth to a oe thousandth of a m'llimetre. The specimen was next examined by reflected light* and the striations at once appeared, the count of the striations giving numbers agreeing very well with the calculated values. Frequently a single scale showed two or even three series of striations, giving corresponding srectra. in three different directions. Another important feature of these “‘ gratings? ‘ was shown in the fact that the light is all concentrated in a single spectrum, showing that the striations must have an unsymmetrical saw Sonal shape f. ~ The observation is somewhat difficuit on account of the very small working space when using high powers. + It may be noted that. the objection that the colours of birds and insects cannot be due to diffraction on account of the equalizing effects of the varying angles of incidence and diffraction, would not apply if the striations are so fine as to give practically a single spectram extending over a range of 45°, ee ae 566 On Metallic Colouring in Birds and Insects. On immersing the specimen in oil or other liquid little or no change is observed, except in those specimens in which a small communicating aperture exists in the neck (point of support) of the scale. The oil can be seen to gradually fill the interior, and simultaneously all trace of colour vanishes*. It appears, then, that the colour in this case is due to fine striations on the interior surface of the scale. Plustiotis resplendens. This is a beetle whose whole covering appears as if coated with an electrolytic deposit of metal, with a lustre resembling brass. ° Indeed, it would be difficult for even an experienced observer to distinguish between the metal and the specimen. On examination with the Babinet compensator it was found that the reflected light was circularly polarized even at normal incidence, whether the incident hight was polarized or natural. The proportion of circularly polarized light is greatest in the blue, diminishing gradually in the yellow portion of the spectrum and vanishing in the orange-yellow—for which colour the light appears to be completely depolarized. On progressing towards the red end of the spectrum traces of circular polarization in the opposite sense appear, the proportion increasing until the circular polarization is nearly complete in the extreme red. It was at first suspected that the phase difference (not always as great as one quarter, but varying between ‘15 and 25) was due to linear str ucture, as in the case of Morpho alga; but on rotating the specimen about the normal no change resulted. The effect must therefore be due to a “serew structure”? of ultra microscopic, probably of mole- cular dimensions. Such a structure would cause a separation of natural incident light into two circularly polarized pencils travelling with different speeds, and having different coefficients of absorption. Such cases have been observed in some absorbing crystals; but whereas in these the difference in absorption between the two circularly polarized pencils is quite small compared with the total absorption--here one of the two is almost totally reflected, while there is scarcely a trace of the other. If this hypothesis be correct, however, the selective ab- sorption (or reflexion) being reversed at the other end of the * Sometimes a faint indication of colour remains (usually greenish) which shows the characteristics of surface colour. It is probable that this surface colour acts conjointly with the effect of diffraction, and indeed the character of the spectrum indicates an excess of green which may be thus accounted for. Sehlémileh’s Theorem in Bessel’s Functions. 567 spectrum—then for the orange-yellow the resulting light should be compounded of these two; and the resulting light should be plane-polarized, not depolarized. The depolarization is in fact only apparent; for on using a moderately high power objective it is at once evident that there is a structure in the wing-case which causes a difference of phase between the components varying very rapidly from point to point; and the resulting plane of the plane-polarized light varies with corresponding rapidity, leaving no trace of polarization when the observation is made with a telescope. The absorption coefticient for this specimen is quite of the order of that of the metals; and the thickness of the “metallic” film is of the order of a ten thousandth of a millimetre. I take this opportunity of expressing my appreciation of the courtesy of Messrs. R. M. Strong, V. H. Shelford, W. L. W. Field, and H. B. Ward, to which I am indebted for bringing the literature of the subject to my notice, and for the specimens on which these observations are based. Ryerson Laboratory, University of Chicago. LXII. Ona Physical Interpretation of Gillett Wepre in Bessel’s Functions. By Lord Rayueicu, O.M., F.R.S* HIS theorem teaches that any function 7 (7) which is finite and continuous for real values of + between the limits r=0 and r=z7, both inclusive, may be expanded in the form 1) =a + aI (7) + 230 (27) +asJo(37)+...,- ~ (1) J, being the Bessel’s function usually so dencted; and Schlémilch’s demonstration has been reproduced with slight variations in several text-books f. So faras I have observed, it has been treated asa purely analytical development. From this point of view it presents rather an accidental appear- ance ; and I have thought that a physical interpretation, which is not without interest in itself, may help to elucidate its origin and meaning. The application that I have in mind is to the theory of * Communicated by the Author. + See, for example, Gray & Mathews’ ‘ Bessel’s Functions,’ p. 30 ; Whittaier’s ‘ Modern Analysis,’ §165. — eS — —— sae SS eS [Se SSS a 568 Lord Rayleigh on a Physical Interpretation of aerial vibrations. Let us consider the most general vibra- tions in one dimension € which are periodic in time 27 and are also symmetrical with respect to the origins of & and t. The condensation s, for example, may be expressed s=b)+b, cos Ecost+b, cos 2&cos2t+...,. . (2) where the coefficients bo, b;, &c. are arbitrary. (For simplicity it is supposed that the velocity of propagation is unity.) When t=0, (2) becomes a function of &€ only, and we write HN(E)=b, +6; cos ++ b, cos ZE- P. h eee in which F(&) may be considered to be an arbitrary func- tion of € from 0 to 7. Outside these limits F is determined by the equations R(—-@)=F(E+ 27) =F (2). . eee We now superpose an infinite number of components, analogous to (2) with the same origins of space and time, and differing from one another only in the direction of &, these directions being limited to the plane xy, and in this plane distributed uniformly. The resultant is a function of ¢ and r only, where r= Vv (a+), independent of the third coordinate z, and therefore (as is known) takes the form S=d)+a1))(”) cos t + dgJ (27) cos 2¢ + azJ9(37) cos 3t +. .., (5) reducing to (1) when ¢=0*. The expansion of a function in the series (1) is thus definitely suggested as probable in all cases and certainly possible in an immense variety. And it will be observed that no value of & greater than 7 con- tributes anything to the resultant, so long as r < 7. The relation here implied between F and fis of course identical with that used in the purely analytical investigation. If ¢ be the angle between &€ and any radius vector r to a point where the value of f is required, €=rcos¢, and the mean of all the components F(&) is expressed by > HO | oes er « O The solution of the problem of expressing F by means of f is obtained analytically with the aid of Abel’s theorem. And here again a physical, or rather geometrical, interpreta- tion throws light upon the process. * It will appear later that the a’s and 0’s are equal. Schlémilch’s Theorem in Bessel’s Functions. 569 Hquation (6) is the result of averaging F' (£) over all directions indifferently in the zy plane. “Let us abandon this restriction and. take the average when & is indifferently distributed in all directions whatever. The result now bé- comes a function only of R, the radius vector in space. If 6 be the angle between R and one direction of & &€=R cos @, and we obtain as the mean Aa "E(Recos 4) sin 6d0= 5, (EOE CO) 3 Ca) e/0 where F,/=F. This result is obtained by a direct integration of F(&) over all directions in space. It may also be arrived at indirectly from (6). In the latter f(r) represents the averaging of F(€) for all directions in a certain plane, the result being independent of the coordinate perpendicular to the plane. If we take the average again for all possible positions of this plane, we must recover (7). Now if @ be the angle between the. normal to this plane and the radius vector R, r= Rsin 6, and the mean is lH Cust) Sumi TOM ee reas (8!) 2/0 We conclude that 1 Ls f(Rsin @) sin 0dO=F\(R)—-F(0),. . 9) which may be cones ed as expressing I in terms of /. If in (6), (9) we take F(R)=cos R, we find * *"Jo(R sin 0)sin@d@?=R-'sin R. v9 } Differentiating (9), we get: FR)=(" f(R sin @) sin @ dé varkk i f'(K sin @) (1 — cos’ @) dé. i aa ar (10) Now R f° cos’ @/"(R sin 6) dd= { cos @ . d/(R sin @) Os i f(X sin @) sin @ dé. * Enc. Brit. Ayt. Wave Theory, 1888; Scientific Papers, vol. iii. p. 98. Phil. Mag. 8. 6. Vol. 21. No. 124. Apron, 2 SSPE = a =a a 2 ma — 570 Schlémilel’s Theorem in Bessel’s Functions. Accordingly zinc ay F(R) J=A+R( "(Rsin 6) d@. . (11) That f() in (1) may be arbitrary from 0 to 7 is now evident. By (3) and (6) io) “ ( ; dd{by+ 6, cos (7 cos h) +h. cos (2r cosh) + ...} 0 = by 4b (7) +b.) (27)+-..5 . . « Se where 1(7 2 | =| rea, ae |e cosmf F(E)dE.. . . (13) Further, with use of (11) @ by =f(0) + = tare) (/'@sin 0).d0, previous paperst the writer has shown that the air- equivalents} of metal foils decrease with the speed of the alpha particles entering the foils. For sheets of different metals of equal air-equivalents, the rates of decrease are * Communicated by the Author. + Amer. Journ. Sci. vol. xxvi. pp. 169-179, Sept. 1908 ; ibid. vol. xxviil. pp. 857-372, Oct. 1909; Phil. Mag. vol. xviii. p. 604, Oct. 1909. { By air-equivalent is meant fhe amount by which the range of the alpha particle is cut down by its passage through the foil. BoP 2 Be tt ae ——- D2 Mr. T. 8. Taylor on the Jonization of Different approximately proportional to the square roots of the respective atomic weights. On the contrary, the air-equiva- lents of hy drogen sheets increase while the hydrogen- equivalents of air sheets decrease with the speed of the entering alps particles, and at such a rate as to be in agreement See the square root law observed for the decrease of the air- equivalents of the metal sheets. A comparison of the Bragg ionization curves, obtained in atmospheres of air and hydrogen, when the pressure of the air was so reduced that the range of the alpha particles from polonium was the same in air as it was in hydrogen at atmospheric pressure, showed differences which are sufficient to account for the variations in the air-equivalents of the hydrogen sheets with the speed of the alpha particles. These differences between the Bragg ionization curves in air and hydrogen suggested that some such differences might be found between the ionization curves obtained in other’ gases, and it was for the purpose of making a detailed comparison of the ionization curves obtained in different gases that the present experiments were begun. Continuation of Heperiments. The apparatus used was the same as had been nse in ee previous experiments*. The sheet-iron case, enclosing the apparatus proper, was replaced by a solid iron case which could be readily exhausted. Polonium was used-as -the source of ray sand w applies) in a brass cylinder of such dimensions that the rays emerging from the cylinder fell well within the limits of the ionization chamber for all available distances of the source of rays from the ionization -ehamber. In the determination of the ionization curve in any gas, the vessel enclosing the apparatus was first evacuated and then the gas admitted very slowly till the pressure it exerted was such that the range of the alpha particles was exactly 11:1 centimetres, hich was the maximum range available with the apparatus. The Bragg ionization curve was then obtained in the usual manner by observing the deflexion of the needle of the Dolezalek electrometer in scale-divisions per second for various distances of the source of rays trom the ionization chamber. In this manner, the Bragg ioniza- tion curves were obtained in the puees aa vapours given in Table I. The curves in figs. 1 and 2, and the dotted ones in fig. 3 represent the ionization curves obtained in the above manner in the gases as indicated below the figures, * Loe. cit. Ce Gases by the Alpha Particles from Polonium. 0 73 ical: (Oe en ee be fee Moe Be a pa a) fe 3 4 S 6 tf & g 10 fl The ordinates are the deflexions in millimetres of the electrometer needle per second. ‘The abscissee are the distances in centimetres of the polonium from the ionization chamber. Curves I, II, and III were obtained when the maximum range of the alpha particle was exactly 11:1 centimetres in hydrogen, air, and methyl iodide, respectively. _ Wig: 2. 7: /F le /0 heat g : Be ai Ee, ae fa i FM Ee SN Sa We ESS CMT aN ene 3 A 5 Seay 8 9 10 TE The ordinates are the deflexions in millimetres of the electrometer- needle per second. ‘The abscissz are the distances in centimetres of the polonium from the ionization chamber, Curves IJ, I, and HI were ok- tained whenthe maximum range of the alphaparticle was exactly 11:1 centi- metres in methane, ethyl chloride, and carbon disulphide, respectively. BOT SSeey a EE SR OO OO ee ee ee 2: 574. Mr. T. 8. Taylor on the Ionization of Different respectively. The dotted portion of each curve in figs. 1 and 2 is assumed to be the form it would take were it possible to move the polonium entirely up to the ionization chamber. At any rate, such assumed portions of the curves can differ but little from the actual curves. It is to be noted, that the ionization curves shown in figs. 1 and 2 are plotted differently from the regular Bragg ionization curve in that the values of ionization are taken as ordinates and distances of the source of rays from the chamber as abscissee, instead of vice versa as is usually done. Although the curves in figs. 1, 2, and 3 represent some differences from one another in regard to the relative amounts of ionization for corresponding distances of the source of rays from the ionization chamber, all of them are of the same general form. From a re-determination of the velocity of the alpha particle at different points in its path, and the assumption that the ionization produced at any point in the path of the particle is proportional to the energy consumed, Geiger* has shown that the ionization I at any point in the path is given by the relation G [= G—ay? ; where c and r are constants and a is the distance from the source of rays. By comparing this theoretical ionization curve with the experimental curve obtained in hydrogen for a pencil of rays, Geiger found the two to agree very closely. This theoretical curve has been compared with the experi- mental curves obtained in each of the gases and vapours given in Table I., and a very close agreement between theoretical and experimental curves was found for each gas. To make this comparison, it was necessary to determine the constants r and ¢ for each gas. For the value of r, Geiger used the average range of the alpha particles in the pencil of rays. Since the maximum range of the alpha particles in the cone of rays used in the present experiments was always 11:1 centimetres, the average range of the alpha particles in this cone of rays emerging trom the cylinder containing the polonium was slightly less than 11:1 centi- metres. Consequently 10°8 centimetres was taken as the value of the average range of the alpha particle, that is 10°8 centimetres is supposed to represent the average dis- tance the alpha particles travelled in each gas before losing * Proc. Royal Society, Seriés A, vol. Ixxxiii, no. A 365, p. 506. Gases by the Alpha Particles from Polonium. BS their power of producing ions. In order to determine c for any one gas, the ionization (ordinate of the ionization curve, figs. 1, 2, and 3) and the corresponding distance x of the source of rays from the ionizacion chamber (abscissa of curve) were substituted in the equation C Fig. 3. t t \ i 1 \ 1 \ fixe O / 2 3 4 G 6 7 § 9 (OMT The full line curves I, II, and III are the theoretical ionization curves for nitrogen, sulphur dioxide, and ether, respectively as obtained hy substituting the corresponding values of ¢ given in column 2, Table L, in the equation c = (Fay te? where r=10°8. The dotted curves I, U, and III are the experimental ionization curves for nitrogen, sulphur dioxide, and ether, respectively, and are plotted similarly to the curves in figs, 1 and 2. and the equation solved for c. Separate values of ¢ were thus obtained for various distances of the source of rays from the ionization chamber between x=0 and 9°5 centi- metres, and the mean value of these separate determinations found for each gas. The mean values of ¢ as found in the Se eee SAE OE BE =o er Sy eT 1 os A — eta ae OS. == a= 076 =Mr. T. 8. Taylor on the Ionization of Different above manner for all the gases and vapours used are recorded in column 2, Table I. TABLE I, | Ratio of the total | Relative c Area under | 2 4:0 of area! Gas or areaunder| experimental a ave’) ionization in the energy or theoretical | curve as mea- poeeeweny gas to that in air.| required Wallnidk livided ‘ed with | mental curve ate ie pour. | curve divided) — sure wit hey 0 produce | by 7:33. planimeter. : Taylor. | Bragg. | an ion. air ee 124 | 980. 7 | 1-00 fies cease 10-00 966 96 0-99 1:00 101 CE Sse 14°73 1301 88 1:33 1:33 0°75 GE dieser 12°65 1156 91 tats 0°85 C,H,Cl 14:05 1251 89 129 1:32 O77 OSs tecaen: 15°60 1355 87 ‘38 1:37 073 jee Pa Dont a 4, eed Bees BIG ae ete an Pee ele 2 Se SE DASA era) ans 14-64 1249 85 ze ae 1:00 Nea 15°81 1206 87 0-96 0-96 104 COs esse: 15-01 1262 84 101 1:08 0°99 OR Sine ramen ae 16°72 1415 85 1:13 1:09 0°88 CHE OR: 19-42 1702 88 1:56 1-3. 0-74 BAN Fass are 13°27 1182 89 ae 1:00 ShO ee einer ioe 15°30 1223 80 1:03 0:97 13(O axe 17°70 1530 86 1:29 O77 1g e) area 18°32 1527 83 1:29 0-77 Ge (Cee Bec ioe ae geese) 1-00 {18 6 Cir 17°68 1535 87 1:29 O77 The full line curves I, II, and III in fig. 3 represent the theoretical curves for nitrogen, sulphur dioxide, and ether, respectively, as obtained by using the values of c as recorded in column 2, Table I. for the respective gases. The dotted curves are the corresponding experimental curves and, as can be seen, agree very well with the theoretical curves. The agreement between the theoretical and the experimental curves for the other gases was equally as good as it is for those given in fig. 3. In some cases the agreement was much closer. This agreement between theoretical and ex- perimental curves confirms the assumption that the energy consumed is proportional to the ionization produced. The ionization at any point of the path of the particle being given by the relation is We ae ety ee (n—a)bs Gases by the Alpha Particles from Poloninm. 344 the total area under this theoretical curve is a measure of the total ionization produced by the alpha particle in the gas. If A; represents the area under the theoretical curve, then °p , wr) A= | Lae aN ee e 0 mG (r—a)'8 — 3/2 e(r)23 = 7-33 ¢ (* being equal to 10°8 centimetres). Hence c is 3/22 of the area under the theoretical curve when the average range of the alpha particle is 10°8 centi- metres in any gas whatever. The values of ¢ recorded in column 2 of Table I. are then 3/22 of the area under the theoretical ionization curves in the respective gases. ' The areas under the ionization curves being proportional to the energies consumed in the production of ions in the respective gases, the value of c in any one gas depends upon the total ionization produced in the gas, and consequently upon the energy required to produce an ion in the gas. Then the ratio of the area under the experimental curve to ¢ should be a constant. By dividing the areas under the experimental curves as measured with a planimeter and recorded in column 3, Table I., by the values of ¢ for the corresponding gases, the values recorded in column 4 were obtained and, as can be seen, are approximately constant. The arens under the ionization curves being the measures of the relative ionizations produced in the gases, the ratios of the total ionization produced in the gases to that produced in air were determined by finding the ratio under each curve to the area under the corresponding comparison air curve. After the determination of the ionization curve in each gas the ionization curve was always obtained in air to be used as a basisof comparison. The ratios of the ionizations produced in the different gases to that produced in air are recorded in, column 5 of Table I. Bragg*, by a less direct process, determined the ratio of the total ionizations in gases to that in air and his valnes are recorded in column 6. There isa fairly good agreement between the values as found by Bragg and those found by a more direct process of measurement of the area enclosed by the axes of references and the ionization curve for each gas. Since the energy of the alpha particle is entirely consumed before it ceases to produce ions, the energy required to pro- duce an ion in any given substance will vary inversely as the ratio of the total ionization in the substance to the total ) ") * Bragg, Phil, Mag. vol. xiii. pp. 833-857, March 1907. 978 Lonization of Gases by Alpha Particles from Polonium. ionization in air if the energy required to produce an ion in air is always taken as the basis of comparison. The values of column 5 of the Table are the ratios of the total ionizations produced in the gases as compared with the total ionization produced in air. Consequently the reciprocals of these ratios are the relative amounts of energy required to produce an jon in the substance as compared with the energy required to produce an ion in air. The values recorded in column 7 are these reciprocals of the values in column 5, and hence are the relative amounts of energy required to produce an ion in the gases as compared with that required to produce anion in air. These values indicate a considerable variation of the energy required to produce an ion. The heavier and more complex molecules are apparently more readily ionized than the lighter and less complex ones. This is probably due to the electrons in the heavier and more complex mole- cules being in a less stable arrangement than they are in the lighter and less complex molecules, and hence more readily drawn out. In conclusion [I wish to express my thanks to Prefessor Bumstead for his valuable suggestions in connexion with the work and for loaning me the apparatus. I am also indebted to Professor Bcltwood for furnishing me the preparation of polonium. Results. 1. The ionization curve obtained in various gases and vapours with polonium as the source of rays is of the general form c a (r—a yi? ii where I is the ionization ; ¢ is a constant for any one gas depending upon the total ionization produced, and conse- quently upon the energy required to produce an ion in the given gas; 7 is the average range of the alpha particles in the cone of rays; and «2 is the distance from the source of ruys. . The agreement between the theoretical and the experi- mental curves confirms the assumption made in previous papers by the writer* and by Geiger, that the ionization produced by the alpha particle is proportional to the energy consumed. 3. The values of the ratio of the total ionization produced by the alpha particle in different gases to the total ionization * Loe, cit. + Loe: ct. A New Form of Earth Inductor. 579 produced in air as found by Bragg have been confirmed by a more direct process. | 4. The energy of the alpha particle consumed in the pro- duction of an ion depends upon the nature of the molecule ionized. It apparently requires less energy to produce an ion in the gases or vapours which have heavy or relatively complex molecules than it does in those gases of lighter or less complex molecules. | Laboratory of Physics, University of Iinois, Urbana, Illinois, January 28, 1911. LXIV. A new Form of Earth Inductor. By Jamus HE. Ives, Ph.d)., Associate Professor of Physics, and 8. J. Mavcu.y, of the University of Cincinnati *. HE instrument described in this paper is the result of a is desire of the authors to construct an Earth Inductor which could be used to measure the vertical component of the Earth’s magnetic field directly, without the use of the magnetometer and the ordinary rotating Earth Inductor. In the form described, it was only used to measure the vertical component, but it could easily be adapted to determine, directly, all of the elements of the Earth’s field. The essential parts of the device were a square of brass tubing supported in a horizontal plane upon a wooden frame, €,¢,¢3C, in the perspective drawing of fle. 1; a sliding conductor, s;s,, connecting two sides of the square, which Fig. 1. could be moved parallel to itself ; and an insulated wire enclosed within the tubing. In the figure, the brass tubing is shown in solid, and the wooden frame in dotted lines. The tubing had outer and inner diameters of 1:12 and ‘97 em. respectively. The square was 100 cms. long on a side, from centre to centre of the tubing. It was cut in two places, shown at a and 6 in fig. 1, on opposite sides. At a the gap was 1:2 cm. long, and atb-3 cm. These gaps were filled in * Communicated by the Authors. 580 Prof. J. E. Ives and Mr. 8. J. Mauchly on with closely fitting, split, hard rubber tubes, having shoulders on them to keep the two parts of the brass tube the proper ih distance apart. At 6, two binding posts were soldered to the ii ends of the brass tubes. The tubes forming the square were tt carefully soldered together at the corners, ¢, C2, ¢3,¢ Lhe i insulated copper wire within the tubing had a diameter of 0°1007 cm. The ends of this wire were brought out at a. The sliding conductor, s; s,, was attached securely to the lower side of a wooden bar, and consisted of a stout brass wire, ‘318 cm. in diameter, soldered at its ends to square brass plates, 3°25 cm. on a side, to which were attached half-cylinders of brass , 3°20 em. long, engaging with the brass tubing. It could be moved parallel to itself through any desired distance, between wooden stops attached to the wooden bar ab. At a was a closely-fitting split brass sleeve, about 3 cms. long, sliding upon the brass tube, which could be slid over the gap at a, forming a conducting bridge between the two tubes ¢,a and c.a, when desired. The binding posts at 6 were connected to a two-coil Du Bois-Rubens armoured galvanometer made by Siemens and Halske, having a resistance, with the two coils in parallel, of about 2°5 ohms, a period of about 4 seconds, and a sensi- tiveness, as adjusted for this experiment, of about 2 x 107~° ampere for one millimetre at a distance of one metre. This galyanometer is described in the Zeitschrift ftir tists un mentenkunde, Jahrg. 1900, p. 65. The arrangement of the circuits is shown in fig. 2, where $1 So is the slider ; G, the galvanometer ; Rt;, a resistance in the tube circuit to give a convenient deflexion ; G1 Cy Cs C4, the square of brass tubing ; B, battery in the wire circuit ; K. , key =. —— = a. SSS SS SS ee a — Saree 99 29 A,, ammeter 4 a eed R,, a rheostat tu adjust the current to a suitable 29 39 Sis) value. The experiment consisted in varying the flux, due io the earth’s field, through the rectangle s; s. c3 cy by moving the slider s, 59, al then calibrating the galvanometer by making or breaking a current in dies wire circuit, and noting the deflexion pr Sadeieed in the tube circuit: The procedure was as follows :—With the tule circuit open at a,and the wire cireult open at K, the slider was moved through a suitable distance between stops, and the a New Form of Larth Inductor. d81 hallistie deflexion, 6,, on the galvanometer noted. The actual distance moved through was 15:l em. The de- flexion was proportional to the flux cut, which was equal to Z, the vertical component of the Earth’s field, multiplied by the area, A, described by the slider. Fig. 2. _ To determine to what change of flux this deflexion of the galvanometer corresponded, the slider s, s. was removed, the tube circuit closed at a, and the ballistic deflexion & of the galvanometer observed when a current of two amperes was made or broken in the wire cirewt. The value of the current in the wire circuit was given by the ammeter A;, which had been previously calibrated, and by means of which its value could be determined to within one per cent. If the self- inductance, I., of the square of brass tubing is known, the 582 A new Form of Earth Inductor. flux, ®, producing this deflexion can be calculated, for &=LI, where I is the current made or broken in the wire circuit. This follows from the fact that since there is no magnetic field within a metal tube due to a current flowing uniformly along it, the Mutual Inductance of the two circuits is equal to the Self Inductance of the square of tubing. We then have: One fea SG Se : 5 * Rtg eat ee ] Sk (1) L can be calculated approximately by taking the mean radius, p, of the tube, and assuming that its walls are vanishingly thin. In this case, the current lies entirely on the surface, and L=8i [log + —0-774 |, -) . er where / is the length of one side of the square *. L can be determined to a greater degree of accuracy by substituting for p, in formula (2), the geometric mean distance, a, of the cross-section of the tube. a is given by loge a=log. a (a,?—a,?)? 2 ay 4 a—a.* where a, and a, are the outer and inner radii of the tube f. The mean radius of the tube was ‘523 cm., and the inductance of the square circuit of brass tubing, 100 em. ona side, trom (2), was therefore equal, approximately, to 3587 em. The geometric mean distance of the cross-section of the tube by (3) was °536 cm. Using this instead of the mean radius in (2) we get, to a greater degree of approximation, L=3568 cm. The mean throw of the galvanometer when the slider was moved through 15:1 ems. was 7:06 ems. The mean throw when a current of 2 amperes was made or broken in the wire circuit was 6°19 ems. We therefore have yee 106 3 3968 x2 6°19 1510 An indirect determination of Z, made in the same place and under the same conditions with a magnetometer and Edelmann Earth Inductor, gave °548 c.G.s. unit for the vaiue of Z. Our inductor was only used to determine Z, but it could easily be adapted to determine the horizontal intensity, H, =()°539 c.a.s. unit, * See Fleming, ‘Electric Wave Telegraphy,’ pp. 98-100. + Maxwell, ‘ Electricity and Magnetism,’ vol. 1, § 692; Rosa & Cohen, Bull. Bur. Standards, 5, 1908, pp. 50-52. Notices respecting New Books. 583 and the magnetic declination by supporting it in a vertical plane, in such a manner that it could be rotated about a vertical axis, as shown in fig. 3. Its plane could be rotated until no deflexion of the galvanometer was obtained when moving the slider, This would give the direction of H. It could then be turned at right angles to this position, and the magnitude of H determined in the same manner as has already been described for Z. Our inductor was 100 cms. on a side. It could, however, without any serious loss of efficiency be reduced in size to 50 cms. on a side or less. An advantage of this instrument, when suitably constructed, would be that 1t would be universal, as it could be used to determine all the elements of the Earth’s field. The method is also very direct, involving only the determination of a current, and the calculation of a self-inductance. University of Cincinnati, June 1910. LAY. Notices respecting New Books. Physics. By C. Rrpore Mann and G. Ransom Twiss. Revised Edition. Chicago and New York: Scott Foresman & Company. Pp. 424. With Ulustrations. ips say that this book will be received with very different feelings by different readers is only to say that it possesses individuality. It is an attempt to replace the usual elementary course of physics by one in which the problems are more likely to be significant to the average high-school boy or girl just beginning the study of physics. Instead of the “Physics which every Physicist must know” they give what they consider to be the ‘“ Physics every child should know.” The authors are to be commended on the TS an Jr ethameie eave ee = & EET = a See Se 29 TMT ela 584 Notices respecting New Books. choice of their material and in general upon the way they present it. At least this is so if it be clearly understood that the book is to be treated as an introductory account intended to rouse the interest of the pupil in scientific things. At the same time lovers of the logical presentation of ideas will continue to wonder whether it is really necessary to sacrifice so much in order to rouse the pupils’ interest. Our own opinion is that the va media is best. The pupils’ interest must, of course, be aroused; but unless, at the same time, an endeavour be made to logically develop the subject, they go without one of the main advantages of a scientific training. Regarded as an introduction to physics, however, the work of the authors has been well done. Athough terms such as force, energy, and work are introduced with the slightest definition only, yet we do not think the pupil will have anything serious to unlearn at a later date. The book covers the ground of mechanics, heat, light, sound, electricity, and magnetism, the illustrative cases being selected from the technical side (electric bells, telephones, &e.). We observe only one mistake. Radium has not the heaviest atom known. This honour is possessed by Uranium, with Thorium as a good second. Crystailine Structure and Chemical Constitution. By Dr. A. E. H. Turron, /.R.S. London: Macmillan & Co. 1910. THIS is an account of the work of the author during the last twenty years in connexion with the properties of certain crystals. The detailed accounts of this work are only to be found in scattered journals and proceedings of scientific societies, the result being that it is perhaps not so well known as it ought to be. At any rate, by bringing the various parts of it together into a short monograph, the author has succeeded in presenting a picture of the researches on which he has expended so much time and ingenuity, which will go far to bring him the greater credit he deserves. There are few cases which form a better illustration of the enormous amount of scientific advance which can be effected by the continuous and thoroughgoing application of an investi- gation to the examination of one small department of knowledge. Dr. Tutton’s work has consisted in the measurement, with the highest possible degree of precision, of some of the chief physical properties of selected groups of crystals, with the object of ascer- taining, without any of the uncertainty existing at the time when he began his researches, whether.any small differences exist in these properties and whether these differences present any cor- relation with the position of the metallic bases of the crystal in Mendelejef’s series. For the resulis obtained reference must be made to this book itself, where a sufficiently full summary is given of them and of the apparatus by which the results were obtained. But it may be said here that Dr. Tutton has succeeded in his aim to set free from uncertainty some of the disputed questions of erystallography.. The book is written by an enthusiast; and if sometimes his en- thusiasm drives him into digressions which do not seem to belong to the main theme, we do not doubt that this small defect will be forgiven him. MICHELSON. “Copper Beetle (%o. 165) Magenta, Thin Film (T=.05Te) Fuehsine Diamond Green. Fie. 5. “Copper Beetle Ibo.2 Noage nla J hinFiln T= 0.01 Jo Nagenta, Thick Film Magenta, Thin Film VIN GMMEY OR Phil. Mag. Ser. 6, Vol. 21, Pl. IV. Fie. 6. 60" 70" 607 Eg rr tor 5" al Y | Crimson Sreen Beetle Diamond Sceen (Thin Fitm) Green Peacock Feather Hummins Bird It Green Ham. Bi Blue Wins Butterfly (He 6) st rd POR; LONDON, EDINBURGH, ann DUBLIN | PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCIENCE. [SIXTH SERIES.] Ry tf WAY 1911. LXVI. On the Mscharge from an Electrified Point. By A. M. Tynpau, M.Sc. (Bristol), B.Sc. (Lond.), Lecturer in Physics in the University of Bristol”. [Plate V.] NOME experiments, chiefly on (1) the pressure of the S Hlectric Wind, and (2) the field in the neighbourhood of a discharging point{, led Prof. A. P. Chattock and the author to adopt certain theories as to the nature of point dis- charge which at the time were capable of explaining most of the results obtained. The author has since extended the experi- ments, and has obtained results which are to a certain extent at variance with these theories. It seems desirable, therefore, to review the situation in the light of the more recent experimental facts. The theories of point discharge originally suggested may be briefly summarized thus :— Suppose thata point P is gradually charged with electricity; the field near its surface is at first unable to do more than clear away the few initially present ions in its neighbourhood as fast as they are formed. But as soon as it is strong enough to impart to the positives among them sufficient energy to enable these to ionize fresh molecules in their turn, * Communicated by the Author. + Chattock and Tyndall, Phil. Mag. [6] xix. p. 449 (1910). { Chattock and Tyndall, Phil. Mag. [6] xx. p. 277 (1910). » Phil. Mag. 8.62 Vol. 21 No. $25. May L911. 2 Q) 1 586 Mr. A. M. Tyndall on the ordinary discharge with glow sets in. For both signs of discharge the supply of positive ions is kept up by ionization due to negative ions, these having been produced by pre- viously formed positive ions and so on. Both signs of ion ib have, therefore, to be able to ionize as each produces the | other. Since positive ions require a stronger field for this | than negative, it is always the field required by the positive ions which has to occur at the point P. Suppose now that ions of opposite sign to the charge on P are supplied to it from an external point N in its neighbour- hood. When FP is negative the field necessary for glow at its surface is unaltered. When P is positive, the supply of negative ions, which is in other cases kept up by ionization near P by positive ions, is now continuously supplied from without. Glow therefore appears at P when the field near it has reached that in which negative ions will ionize. This field (f—) is less than that (f+) in which positive ions will ionize. Experiments on the pressure of the electric wind further suggested that the ions streaming away from a glowing point | are not at first fully formed, and travel a distance from the | point varying from 4 centimetres in the case of negative ions in pure hydrogen to 3 or 4 millimetres in air, before | ee ar they completely cluster. Now unclustered ions ionize in a lower field than ions fully grown, so that, as N is made to approach P, the field for glow at the latter begins to drop when the distance between N and P is decreased to values less than this critical distance. | Thus when P is positive and N very close to it, P is sub- 1) | jected to a corpuscular bombardment and its field sinks to il 3S See ee bs ee _ Hi that (f.) necessary for corpuscles to ionize. ia Lastly, when the current is small, ions of one sign only alt, traverse the greater part of the distance between point and Al plate, except in negative discharge in pure hydrogen. | In the light of the experiments described below it appears Hit that certain parts of this theory must now be abandoned. i | Thus it will be shown that except in the case of negative ions al in certain oxygen-free gases the growing ion theory presents ai very serious difficulties, and that effects originally attributed al to unclustered ions are better explained on the view that, Vii accompanying the process of ionization by positive ions, there is an emission of rapidly moving neutral bodies which | have a range of a centimetre or two at atmospheric pressure. | The statement that in air for small currents, ions of one | sign only are present between the glow and the point and the plate is probably also not correct. It seems likely that Discharge from an Electrified Point. 587 at all currents there is a back discharge from the plate increasing with current and becoming very intensified at particular points on the plate under certain special conditions. . | EXPERIMENTAL RESULTS. The experimental results may be divided into two groups : (1) Experiments on the strength of field at the surface of a point ; and (2) Experiments on the pressure of the Electric Wind. (1) Experiments on the strength of field at the surface of a point. Most of the original theories were apparently verified by some experiments, in which the field at a hemispherically ended point P was determined in terms of the mechanical pull on its surface. If » is the radius of the point, f the field at the centre of its surface, and P the mechanical pull due to the lines of force which start from the end of P, then JE j= — x constant. The value of the constant is 2°83 for a positive point and about 3:07 for a negative point. The particular value of f at which under given conditions glow first appears at P is referred to below as /9. The supply of ‘‘external ions”? was obtained by spraying ions on to the point P from a point N in its neighbourhood. N was a point of smaller radius than P,so that N discharged more readily than P. The tendency of N to start first was further increased by causing P to project through a plate, the plane of which was only a few millimetres behind P. P and N were both horizontal and in the same vertical plane, and they were so arranged that the vertical component x and the horizontal component y of the distance between them could be varied at will. When P was positive and external ions were supplied to it from N placed opposite to it, and at a distance y greater than about 2 cms., fp was reduced to about 0°6 of the value it had when N was replaced by a plate. On the above theories this field when N was present was f—,and when N was absent f+. f— was about 0°8 f+. As y was decreased below 2 ems. fy rapidly decreased, and apparently the curve of /) and y cut the axis of 7) at a positive 2Q 2 588 Mr. A. M. Tyndall on the value of fo, assumed to be the field f£in which corpuscles ionize. f, was about 0°14 f+. The author has extended the work on these smaller values of y, and has found that a variation in the radius of a positively charged P, which causes a marked variation in the value of the field at P, does not appreciably affect the value — of the critical distance (referred to in what follows as yg) at which the drop in field sets in. Thus in Curves 1 (PL. V.) the values of f) are plotted as ordinates with the corresponding values of y as abscissee. Two points were experimented upon, one of radius 0°062 cm. (crosses) and the other of radius 0°018 ecm. (circles); the vertical scales of the curves are arbitrary, and are adjusted to make the two curves identical at large values of y. As a result it appears that the two curves are then coincident throughout ; this points to the conclusion that the cause of the drop in fp does not lie with P but with N. On the other hand, except when the two points are a millimetre or two apart, the values of these fields are independent of the size of N, provided that the latter is discharging vigorously before P starts. Under these conditions yo, therefore, has a value which is independent of the sizes of P and N; itis about 2 centimetres. Similar experiments have been carried out with a negative P. These were not previously tried because P and N always sparked at distances under several centimetres. Assuming, however, that the field just when the spark passes is the true glow field fo. it has since been found that, in this case also, Jo drops rapidly as y decreases. This is shown in Curves 2, where the upper curve is the f)-y curve and the lower the current-y curve. P was of radius (026 cm. and N wasa very fine wire. Whether it is permissib!e or not to take the field just when the spark passes as the true ionizing field, the drop in field at short distances is shown to be real by the current-y curve, which is similar in form to ihat obtained when P is positive and is giving true point discharge; this drop in current at short distances implies a falling field. Also by removing the plate bebind P, true point discharge was obtained for smaller values of y, and the same drop in fo was observed. The observed currents were in this case too great, because although the lines of force ending on the sides of the point do not contribute to the pull, an appreciable amount of current is supplied tv them when the plate has been removed. The reason why the two points sparked when P was backed Discharge from an Electrified Point. 589 by a plate can be explained as follows: N is giving positive ions to P at the ordinary high values of field at N. P starts discharging: this means that a sudden supply of negative lons enters the high field at N, the value of the field “there being considerably greater than that necessar y to cause the arriving ions to ionize. When the flow from P to N is con- centrated by a plate behind P, this may mean a very sudden increase in’ current from N to P which may possibly result ina spark. Throughout the work with a negative P the point was very unsteady, no doubt owing to this tendency to spark, and attempts to find the effect of varying r have given exceedingly irregular results. With each point investigated, however, a similar drop in fy has been observed for small values of y, and it appears not unlikely that, as in the case of a positive P, y) is also independent of the size of P. For negative discharge y, 1s about 1°5 cms. Sometimes the discharge takes the “streamer” form, in which there is a continuous glow stretching across between the two points even when they are a centimetre or two apart. The field at P is in this case greatly reduced, especially when P is large. An isolated case of this effect, which has since been frequently observed, was given in a previous paper (loc. cit.). When P is positive, sparks also pass between P and N at short distances, but it has been found that the value of y at which this occurs becomes smaller as the radius of N is decreased. In fact the results obtained when N was a piece of the finest platinum wire cut obliquely with scissors show that the values of the fields f, given in the previous work are not correct. These were obtained by extrapolating the Jo-y curve to y=0, N being throughout a sewing-needle. Lhe dotted curve in fig. 1 shows similar results which were obtained when P was of radius ‘062 cm. and N ‘039 em. But using the above very fine N the values shown in full lines in Curves 1 were obtained, and it appears not unlikely from these that f and y become zero almost simultaneously. The same applies to the /o-y curve for a negative P with a very fine N in Curves 2. The reason for this sparking when P is positive is also clear. When y is quite small the positive ions from P at first find a field at N which is larger than necessary to give ionization there, so that a sudden increase in current results. The phenomenon i is, therefore, similar to that occurring at a negative P, except that at a positive P it only occurs at ‘small values of y. For a very sharp point, the lines of force spread rapidly and the distance y must be deoreased, so that the field reonnaeew oe we a 590 Mr. A. M. Tyndall on the along the path of the discharge may be uniform enough to cause the sudden current to take the spark form. Some interesting results were obtained when N was made larger than P. P was a positively charged point 0°0115 cm. in radius. N was 0-039 cm. in radius, and was backed by a plate which was adjusted at such a distance that though N was larger than P, it started to glow either before or after P according to the distance y between them. In Curves 3 the full line shows the glow-fields at P plotted with y under these conditions. The dotted curve was obtained with N a piece of the finest platinum wire, and hence always first to start discharging. In the region BC of the full curve N started glowing first; near A, P started first, and between A and B there was a transition stage in which, as far as could be ascertained by eye, the two points started together. Curves 4a give a few of the field-current curves obtained with the large N point at varying values of y. The circle on each of the upper curves gives the current at which, as far as one could tell by eye, the point N started to glow. It is probable that the currents thus circled are too great because N could not be viewed in darkness owing to the proximity of a glowing P. It will be seen that in the BC region the current-field curves are similar to those obtained in previous work with a fine N point (Phil. Mag. xx. 1910). In the region AB, where P either started first or simultaneously with N, it will be seen that there was no sudden drop in field at P when N started. It would appear, therefore, that though N was glowing, the negative ions supplied by it to P were not numerous enough to bring the field at P down to that in which negative ions will ionize. As the distance y was in- creased, the proportion of current supplied by N increased and the field at P decreased until the conditions were re- versed and N supplied practically all the current. The field at P was then that in which negative ions will ionize. These experiments also throw light upon the effect which was obtained when N was large but not so largeas P. It was found in this case that at very short distances the field at P tended to become constant, as is shown in the dotted line of Curves 1 discussed above. Now at these dis- tances, which were only slightly greater than those at which sparks occurred, N seemed to discharge almost simultaneously with P. Consequently the number of ions supplied to P was small, and hence the field at P was higher that that when N was-a fine point discharging vigorously. - | Discharge from an Electrified Point. aoe When P was negative the presence of N, whether dis- charging or not, made practically no difference to the field at P, as shown in the field-current curves 45. This is also what one would expect on the above theories, if N at short distances is only supplying a small proportion of the current between the points. At greater distances, whatever the size of N,a large supply of positive ions from it to P has no effect on the field at P because that must still be the field in which positive ions ionize. Lastly, an interesting’ effect was obtained when P was made a white hot platinum loop. When P was positively charged fy was greatly reduced, being from 40 to 70 per cent. lower than that when P was cold and other conditions re- mained the same. The greatest reduction occurred when y was small. A much smaller reduction of from 5 to 10 per cent. occurred when P was negative. This is what might be expected from the well-known fact that the mobility of a negative lon increases much more rapidly with temperature than that of a positive ion. ‘The fields, therefore, in which these ionize will be correspondingly lower, and the decrease in fo will be most marked when N is supplying negative ions. (2) Experiments on the pressure of the Electric Wind. It has been shown by Chattock* for various gases, and later by Chattock and the author (loc. cit.), for mixtures of hydrogen and oxygen, that if a point is placed opposite a plate at a distance z from it, the average pressure on the plate p for a given current density is a linear function of z when the values of the latter are of the order of a few centi- metres. If, however, these values of p and <¢ are plotted and the curve is produced to p=0, it cuts the axis of z at some positive value, referred to in what follows as 2p. For negative discharge in “pure” hydrogen 2 was as much as 4 centimetres, but it rapidly decreased to 3 or 4 millimetres on the addition of small percentages of oxygen or air. For positive discharge under the same conditions it was 3 or 4 millimetres throughout. In air ¢) appeared to have about the same value, 3 or 4 millimetres, for both signs of discharge. It was suggested that 2) was the average distance that an ion travelled before clustering. Support for this view was obtained from some experiments of Franck t when discharge * Chattock, Phil. Mag. [51 xlviii. p, 401 (1899). } J. Franck, dn. d. Phys. [4] Bd. xxi. p. 984. 592 Mr. A. M. Tyndall on the occurred in air from the surface of a fine wire, his value of 7 mms. instead of 3-4 mms. being attributed at the time to the fact that the average field within 7 mms. of a straight wire was greater than the average field within 3 mms. of a fine point. : The possible effects of field and current upon 2) seemed interesting, and the wind pressure apparatus was set up again by the author, to investigate the question. The apparatus was that previously used for the work on hydrogen, the experiments now, however, being confined to dry dust-free air. Values of p and z were obtained for various values of current densities C. If it be assumed that in a distance dz ions of one sign only are present, then the specific velocity v of the ions may be calculated from the expression The typical result is given in Curves 5 for a current of 12-2 microamperes, with the values of p as ordinates and those of z as abscissze. It will be seen that the curve is made up of three parts, of which the centre part AB is straight ; it is from this part AB that the values of v in past work have been calculated. Omitting for the present the ends OA and BC from the discussion, it is clear that if the above expression holds the slope of AB should be proportional to C. The author has found, however, that this is not the case. Curves 6 show the relation between oe and C for positive and negative discharge, C being measured in microamperes per sq. centi- metre. . The fall curves are the experimental curves and the dotted lines are calculated from the above expressions, assuming that v has constant values 1°32 and 1°84 cms. per sec. per volt cm. for positive and negative discharge respectively. Jt will be seen that as the current increases ap falls short dz of the value necessary for proportionality with C; conse- quently the calculated values of the velocities of the ions increase. The readings were all very irregular, and the points given on the curves are actually means of a number of readings differing from one another sometimes by nearly 20 per cent. Discharge from an Lleetrified Point. 593 The author found that these irregularities were due, at any rate in part, to the presence of ozone in the discharge-tube, because (1) they increased with the lens th of the experiment and diminished again with a long wait, and (2) they were reduced by placing a tray of powdered manganese dioxide in the discharge vessel, to decompose the ozone which was formed during discharge. Hffects due to ozone could not be completely eliminated in this way, because it is generated in the path of the discharge itself, but the oxide no doubt decreased the amount of it which was present at any given moment. The effect on 2 of altering the size of the point and the current was then determined. Three points were used, all of platinum: two, A and B, had hemispherical ends and were of diameter 0:078 cm. and 0:0043 cm. respectively. The third, ©, was a piece of the finest platinum wire cut obliquely with scissors. Now the field at A was nearly four times as great as that at B*; that at C probably many times greater still, The values of z) obtained were, however, nearly the same for all. Thus the values of z) for A were 0°43 cm. in positive dis- charge, and 0:40 cm. in negative discharge. The corre- sponding values for B were 0-44 and 0-41, and those for C were (32 and 0°36. Those values were also independent of the current, as will be seen from the following table, if experimental discrepancies are allowed for. (m es peres Zot. | Cores per sq. cm.). 0:05 “45 “40 0-14 “47 "38 0-14 38 "28 0-31 43 49 0-31 "42 “40 0-69 "45 "35 kG "43 43 “44 “40 * Obtained from the expression f:0'45= constant, Phil. Mag. xx. p. 270 (1910). Be SS eS Fs OSE ESS 594 Mr. A. M. Tyndall on the Discussion oF RESULTS. It would seem at first sight that the drop in field at short distances when P is negative may be explained, as when P is positive, by the theory that at these distances P enters the unclus‘ered region of the discharging N. That is to say, Zand y) would receive a common explanation which would be that they are indications of a region near a discharging point in which some of the ions are unclustered. As a detector of this region the field method would be more sensitive than the wind pressure method because the presence of a very few unclustered ions has a much greater effect on the value of the ionizing field than on the value of the wind pressure. There are, however, great difficulties in the way of the acceptance of this theory, and of these the following are the chief :— It is not easy to explain on the above view why 2» is constant for different points. Thus one may imagine that an ion tends to cluster when its velocity has decreased to some critical value ; if the velocity of an ion close to the point is proportional to the field, this implies that growth occurs when an ion enters a certain critical field. Now assuming that the inverse square law of field holds near a point, the values of the fields at say 4 millimetres from the two points A and B may be calculated. Their ratio is about 65 to 1. It is of course probable that the inverse square law does not hold as far from these points as this, but the difference between the two fields is too great to be accounted for in this way. Moreover, the following deductions on the growing ion view do not fit the experimental facts. When no external ions are supplied to P, glow discharge starts when the positive ions in its neighbourhood ionize. These having just been produced by negative ions are new and consequently unclustered. On this view then, when discharge has once started, the field at such a point is that in which unclustered positive ions ionize. Now when P is negative and external ions are supplied from N at very short distances, the positive ions which start the discharge are again unclustered. The fields in the two cases should there- fore be the same. It is found experimentally, however, that the field at a point opposite a plate is at least six times greater than that at the same point supplied with external ions at small values of 7. Again, assume that of dee two fields the one at a point es —o ee Ss Discharge from an Electrified Point. 599 opposite a plate is the true ionizing field for unclustered positives, and that the effects of external ions at short dis- tances must be explained in some other way. Then when N supplies negative ions from a distance greater than y,, the field at P is f—. One would expect, however (though not with certainty), that 7/— would be greater than the ionizing field for unclustered positives, whereas experiment shows it to be considerably less. Suppose, on the other hand, that the positive ion at birth is already of molecular magnitude, and that the negative ion is the only one to go through a clustering stage. This would explain the effects at a positive P for all values of y. In fact, in the previous paper (Phil. Mag. xx. 1910) the possibility ofa positive clustering ion was ignored from lack of experimental! data. The theory, however, would not explain (1) why the field at a negative P decreases as N is brought within yo, and (2) why there is a 2) wind effect in positive discharge. On these grounds the author considers that, at any rate for air, the above growing ion theory must be abandoned. If in air the negative ion is at birth a corpuscle, it must immediately cluster by taking on an oxygen molecule or molecules. Further evidence for this view is given below. With regard to possible alternatives the following considera- tions present themselves :— 1. A 2 effect will be present if there is a large amount of ionization occurring within a few millimetres of a discharging point. Such ionization may occur if the point is emitting radiations with powerful ionizing properties, the resultant positive and negative ions giving mutually neutralizing wind pressures. On this view, it is not clear why the field at P should fall when brought close to a discharging N, because although ions are being produced close to P by the radiations from N, the field at P before it can discharge must be such as to make these ions produce more. It is true that radiations falling on the surface of a negatively charged P may give _ photoelectric effects, and a supply of negative ions may thus be obtained at a low field, but to fit the facts there must be similar effects at P when it is positively charged. Also it is probable that any such photoelectric currents will be negligibly small compared with the currents used in the above experiments. 2. The author tentatively suggests the following hypothesis as one which seems to fit most of the experimental facts. It is well known that electrically neutral bodies—possibly doublets formed by the union of positive and negative ions— are present in discharge-tubes at low pressures ; evidence —_ 596 Mr. A. M. Tyndall on the has also been obtained by Lonsdale* for their existence at atmospheric pressure. It has further been suggested by Sir J. J. Thomson + that the emission of these bodies accom- panies the process of ionization, so that it isnot unreasonable to suppose that such are emitted from the glow region near a i point. If their initial velocity is high, they may travel some | distance from the point before their kinetic energy falls to that of the molecules surrounding them. Now suppose that they are emitted from N and that the distance y, isa measure of their range: then they will have no effect on a point P when it is placed at a distance y greater than yj. But when P is brought within the distance yo, doublets possessing considerable energy will strike its surface. By this bombardment energy will be communicated to P in an amount which will increase as y is made smaller, and this supply of energy may be available to aid the process of ionization at P,so that the nearer that P approaches N, the smaller will be the field necessary to start the discharge at P. The supply of doublets, however, must be kept up by N, hence the size of P will not affect the value of y. Also the range of the doublets will be independent of N if it is assunied that the initial velocity of expulsion from N is constant. Some of the doublets in the distance yp will either lonize the gas or will themselves break up, so that z) receives an explanation on the lines suggested in (1). It is consistent with this view that z) is independent of the size of the dis- charging point, and is practically the same for positive and negative discharge in air. Now ¢ is much less than yo. It is, however, really the value of z from 0 to A in Curve 4 that should be compared with yo: this value is about 0°8 cm. in air. Also it is pos- sible that z, may be too low because the wind pressure method may not be sensitive enough to detect the small number of ions occurring at greater distances. In negative discharge in very pure hydrogen 2 is many times greater than in air, but for various reasons, detailed below, the growing ion view gives in this particular case a better explanation. It seems necessary to suppose that these doublets are pro- duced only during the ionization of molecules by means of positive ions. If, for instance, doublets were produced during . es a ee iT ionization by negatives they would take the place of the F ‘| positives in being a fresh source of negative ions, and the . if i * Lonsdale, Phil. Mag. B xx. p. 464 (1910). i + Thomson, Phil. Mag. [6] xvii. p. 821 (1909). Discharge from an Electrified Point. aO7 field, therefore, at a point would be that in which negative ions will ionize, and this does not seem to be the case. This doublet theory is thus consistent with all the experi- mental facts above stated. Changes of Wind Pressure with Current. There is still the fact to be explained that in the wind- dp pressure work the values of — per unit current decrease dz with increasing values of current. Le : The fact that -- is constant for values of z included in the centre portion of Curve 5 implies that the carriers of the current are not generated in the main body of the gas, but travel a distance which does not differ very appreciably from the whole distance z between the point and the plate. The observed fall in 2 per unit current might occur if the current is carried by two kinds of ions, clustered and unclustered, both travelling from point to plate and varying in relative numbers with the current. Although, as shown below, such a view is possible for negative discharge in hydrogen containing traces of oxygen, it does not seem probable in air”. The simplest explanation is that the current between the point and the plate is not wholly carried by ions of one sign only, but that there is a certain amount of back discharve trom the plate which at very small currents is negligible in amount, but which increases as the current and the field at the plate increase. That such a back discharge can exist in air, even at low eurrents, is shown by the part BC of Curve 5. In this region there is a rapid fall in a . Ata current of 1 micro- ampere the value of z at which the curve leaves the straioht is about 6°0 centimetres, and at 10 microamperes it is about * It is true that in a recent paper (Phil. Mae. Feb. 1911) Sir J. J. Thomson has shown that positive ions O, and O, are present in the positive rays at low pressures. It is, therefore, conceivable that these and similar clusters with negative charges may be present at atmospheric pressure 1n varying amounts depending on the valne of the current flowing from the point; if they carry an appreciable percentage of that current, the observed changes in a may thus be produced. ; ae ) 598 Mr. A. M. Tyndall on the if 2°8 centimetres. This sudden change in of was always . accompanied by a speck of light on the plate; this is no i | doubt the source of a back discharge which effectually re- i duces the value of p. The etfect was observed in both positive i and negative discharge. Such a phenomenon will occur, if iv; at any region on the plate the back discharge accidentally increases ; the lines ot force from the point will then con- verge towards that region and concentrate the current there, thus tending to increase the back discharge still further, and so intensify the concentration of the lines. In the earlier work on hydrogen when, in negative discharge, this back discharge was known to be present, very marked effects of this kind were observed (see Phil. Mag. xix. p. 455). This instability apparently sets in when the field at the plate reaches some critical value, since the necessary distance between point and plate decreases as the current increases. For values of z less than that from O to B,it may be assumed that the back discharge is general over the surface of the plate, but increases with increasing field and current. Discharge in Hydrogen. Franck* has suggested that gases may be arranged in the following order according to the magnitude of the affinity which their molecules possess for negative electrons :— chlorine, nitric oxide, oxygen, hydrogen, nitrogen, argon, and helium. According to him chlorine and oxygen mole- cules, for instance, have a strong affinity for negative ions, but the gases at the other end of the series have comparatively little. Thus he has shown that when argon and nitrogen have had all electropositive impurities such as oxygen and chlorine eliminated trom them, the velocities of the negative ions rise to very high values, even at atmospheric pressure. For instance, in pure argon the velocity of a negative ion was 206°3 ems. per volt em., and in pure nitrogen 80-145 cms.; but traces of oxygen reduced these to normal values 1-70 and 1°84 respectively. The velocity of the positive ion was normal throughout ; thus in pure nitrogen it was 1:27 and in impure nitrogen 1°30. Previous to this Prof. A. P. Chattock and the author (Phil. Mag. xix. 1910) found effects in the wind-pressure work in hydrogen which may be similarly explained. These experi- ments, however, were complicated by the presence of an unknown amount of back discharge from the plate, so that, in negative discharge in pure hydrogen, the direction of the * Verh. d. D. Phys. Gesell. xii. pp. 291 & 613 (1910). Discharge from an Electrified Point. 599 electric wind was sometimes even reversed. It is, therefore, impossible to say what the actual velocity of the negative ions was, but in the light of Franck’s work it is almost certain to have been high. It was found that the wind pressure rapidly rose with the addition of slight traces of air, that is to say, the calculated velocity of the negative ions rapidly fell, as was found by Franck in argon and nitrogen. On Franck’s view this is due to the tendency possessed by the negative ions, when oxygen is present, to attach themselves to mole- cules and become clustere:l; in oxygen-free inert gases they remain for a longer time in a corpuscular state. This is supported by the results obtained for 2 in pure and impure hydrogen. In pure hydrogen for negative dis- charge, 2) was about 4 centimetres, but with the addition of 2 ver cent. of oxygen it fell toaboutO-3cem. If it is assumed that 4 centimetres is the distance that negative ions travel in pure hydrogen before clustering, one would expect a rapid decrease in this distance when small percentages of oxygen are added ; the negative ion in pure hydrogen would thus eluster within a distance depending on the amount of oxygen present. The growing ion view as an explanation of 2) may, therefore, be retained for the particular case of hydrogen, either pure or containing not more than a few traces of oxygen. It is not so easy to explain the form of the p-z curves for different percentages of oxygen. If for negative discharge in hydrogen, curves of the type of curve 5 are taken, it is found that, in general, whatever the percentage of oxygen present the part AB is straight but that its slope for a given current rapidly increases with the addition of oxygen. Thus at ‘045 per cent. oxygen z was found to be 2:4 centimetres, but the slope of the curve was constant and about 1/6th of that in ordinary impure hydrogen. ‘The fact that at a given percentage = is constant, implies that outside the distance 2 no further clustering takes place, and yet the ions appear to move much faster than in impure hydrogen. At the time the facts were explained by the theory, that in pure hydrogen there was a considerable amount of back discharge from the positive plate, which gradually ceased with the addition of oxygen. Now the slightest reversal of the electric wind in pure hydrogen shows that back discharge was un- doubtedly present ; but if the negative ions are corpuscular in nature they will not contribute any appreciable wind, and the amount of back discharge which is necessary to give the observed reversal need, therefore, only be very slight. WW 600 Mr. A. M. Tyndall on the i ? i A bright glow was also observed on the plate in pure ii | hydrogen. This might be explained either on the view that a the plate was bombarded by corpuscles or that the plate was a source of back discharge : a corpuscular bombardment may i of course itself be a cause of back discharge. At any rate, it the glow was connected with the above phenomena in that it gradually disappeared as oxygen was introduced. The following argument reconciles the results with Franck’s theory :— In some work on the combination produced by point dis- charge in hydrogen containing traces of oxygen*, evidence was obtained for the theory that, at low percentages of oxygen much of the oxygen was concentrated in films at the electrode surfaces. Owing tothe suggested affinity between oxygen molecules aud negative electricity, the density of this film at the cathode will be far greater than at the anode, and will increase as the percentage of oxygen in the gas increases. Now in negative discharge the negative ions which take part in the discharge will be produced at the point, and if they are ejected from the point surface they will pass through this concentrated layer of oxygen. A certain number of them will immediately take on oxygen and others will escape uninfluenced. The percentage of the clustered to the unclustered will depend on the density of the film, that is to say, on the percentage of oxygen present. Once free of this film they will continue their path without further change, since it may be shown that for the percentages of oxygen considered the chances of further collisions between ions and oxygen molecules in the main body of the gas are very remote. Since the clustered ions are wind producing and the un- clustered are comparatively not, the value of ~ will increase with increasing percentages of oxygen. Also, since 2, for these clustered ions is on this view very small, the resultant Z, for the whole discharge will also decrease with increasing percentages. To explain the residual 2 of about 3 millimetres in very impure hydrogen, when all the ions are clustered, some such theory as the doublet theory suggested above is necessary. The unclustered ion is, therefore, a special case for negative discharge in oxygen-free gases only. Franck’s theory thus offers a very satisfactory explanation of all the wind-pressure results obtained in hydrogen con- taining traces of oxygen. * Chattock and Tyndall, Phil. Mag. [6] xvi. p. 24 (1908). Discharge from an Electrified Point. 601 Changes of Pressure accompanying Point Discharge in Hydrogen. | This view, that the negative ions in pure hydrogen are corpuscular in nature, also throws light on the net changes of pressure which result in closed vessels from point discharge in hydrogen. If there is any oxygen present water is formed, and there is in consequence a small decrease in pressure. It was found, however, that even if oxygen is very carefully ex- cluded, a very small contraction still occurs (loc. ct. Phil. Mag. 1908). This effect was attributed to. an absorption of ions at the surfaces of the metal electrodes. With a copper plate this contraction amounted to 2 atoms per ion—the ex- pression “ per ion “ibeing defined as “ per hydrogen atom set free in a water voltameter placed in series with the discharge vessel.”” At the time it was generally supposed that an ion was a charged molecular cluster. From the results of the work of Wellisch and others, on the mobilities of the ions, there is now however considerable evidence for the theory that an ion is either molecular or atomic in size at atmospheric pressure. Now,if in hydrogen the negative ion is a corpuscle, there can only be absorption of gas at one electrode, the plate in positive discharge and the point in negative dis- charge. The two atoms absorbed, “ per ion,” must thus be carried in the gas by a single positive hydrogen ion. ‘This is, therefore, evidence for the view that the positive ion in pure hydrogen at atmospheric pressure is acharged molecule. But if there was an evolution of gas from the cathode and. an absorption of gas at the anode, as has been found by Skinner * to occur in glow-discharge in hydrogen at low pressures, the observed fall in pressure might have been merely a small difference effect. The author rejects this view for the following reason. The point used was a piece of the finest platinum wire and therefore of very small surface. By making it the cathode it could be easily de- nuded of surface gases by discharge—at any rate temporarily. The same electrodes were used for months without change, and for the greater part of that time the point was the cathode. If the Skinner effect were responsible for the result, one would expect that after a long negative discharge from the point, the contractions per coulomb in following dis- charges would differ widely according to whether the discharge was. positive or negative—that is to say, according to whether the plate or the denuded point was the cathode. * Skinner, Phil. Mag. xii. p. 481; Phys. Rev. xxi. pp. 1 & 169. Phil. Mag. 8. 6. Vol. 21. No. 125. May 1911. 2k 602 Mr. A. M. Tyndall on the It was found, however, that the contractions on the average were the same in both cases. It seems, therefore, that in the purest possible hydrogen, with point and plate no longer “fresh,” emissions from the cathode are only corpuscular. in nature. Again it may be argued that this contraction was due to residual impurity in the gas. The reasons for not accepting this view are set forth in the above paper. There is no doubt that, in negative discharge in pure hydrogen, a little back discharge occurs. If the ions in this are generated in the gas at the surface of the plate, the above conclusions will not be affected. But if they are emitted from the metal itself, this is not the case. However, it is shown above that it is very probable that the amount of back discharge was quite small, so that its effect on the result may be neglected. The theory of Franck will also explain the amount of combination which is produced in negative discharge between hydrogen and oxygen, when the latter is present in small quantit ies. In the above paper it was shown that when oxygen is present in hydrogen, combination occurs between them, and as far as one could tell between them only, even when nitr ogen is present in large excess. The results receive a simple explanation on the view that to make oxygen and hydrogen combine, they must be suitably presented to one another together with an electric charge. The efficiency of negative discharge is greater because the affinity of negative ions for oxygen is great. Since their affinity for nitrogen is, according to Franck, even less than that for hydrogen, the tendency to unite nitrogen and hydrogen will be small, at any rate with small currents. One may assume that having caused a certain amount of combination, a negative ion may either be retained by the products or in some way may be rendered no longer efficient as a combining agent. Other corpuscles will attract oxygen, but may not be ‘suitably presented to hydrogen before the time of entry into a molecular life. Now if, as suggested, some of this oxygen in negative dis- charge is concentrated as a film on the point, then, as the percentage of oxygen increases, the density of this film increases. At first the chances of free corpuscles producing combination in the neighbourhood of the point will greatly increase with this increasing densitv, and the combination per coulomb will rapidly increase. With higher percentages of oxygen, however, oxygen will pre edominate over hydrogen in the surface film, and the free corpuscles will be decreased in Discharge from an Electrified Point. 603 number by clustering with oxygen before hydrogen can be suitably encountered. The curve of combination per coulomb and percentage of oxygen will thus pass through a maximum value; this was found to occur experimentally, the combina- tion per coulomb being greatest at 0°008 per cent. oxygen, and being then three times greater than at 14 per cent. oxygen. There was apparently a much smaller maximum in positive discharge ; this also is reasonable because the main oxygen layer is then at the negative plate, where the number of free corpuscles is very much less, if not nil. SUMMARY. 1. Further measurements of the field at the surface of a discharging point have been made, and it is suggested that some of the observed effects, originally attributed to the action of unclustered ions, are due to an emission of uncharged doublets during ionization by positive ions. 2. When extrapolated, the curve of wind pressure on a plate () and the distance between point and plate (z) cuts the axis of z at a positive value zp. 2 is inde- pendent of the field at the point surface and of the current. It is in general about 4 millimetres in length, but is much greater in negative discharge in pure hydrogen. The effects in air are explained by the doublet theory, and in hydrogen by the view that the negative ions in that gas are, in the main, cor- puscular in nature. 3. The apparent velocities of the ions in air, as measured by the wind-pressure method, become greater as the current between point and plate increases. This is explained by the presence of a back discharge from the plate, which prevents the method from being appled with accuracy except when the current is small. 4. Evidence is adduced in support of the view that the positive ion in pure hydrogen is a charged molecule. ). The amount of combination which point discharge pro- duces in hydrogen containing different percentages of oxygen and nitrogen, and the form of the wind- pressure curves in these mixtures are discussed and explanations are offered. Pe Wile BOR 4 |, eS LXVIL. A New Method of Measuring the Luminosity of the Spectrum. By Frank Auuen, W.A., Ph.D., Professor of Physics, University of Manitoba, Winnipeg ™ \ J] HEN a ray of light entering the eye is periodically interrupted by a rotating sectored disk, a sensation of flickering is prodneed until the interruptions reach a certain critical frequency at which the impressions of the separate flashes of light become fused into one continuous sensation. This peculiarity of vision, in one form or another, was observed and commented on by philosophers in ancient times, but was first quantitatively investigated by the Chevalier D’Arcy, who in 1765 measured the least time a revolving glowing coal required to trace an apparently con- tinuous circle of light. In more recent times this subject has been studied by numerous investigators, and many phenomena of interest and importance elucidated. It was discovered by Ferry T, and subsequently, in another manner, by Porter}, that the duration of the sensation of syachareodelned brightness of a flash of light, at the critical frequency of interruption, depended only on the luminosity of the light and not in any way on the colour. The duration of the impression was found by both investigators to be inversely proportional to the logarithm of the luminosity of the light. In the course of some investigations on colour vision in which constant use was made of the measurement of the critical frequency of flicker, it became desirable to measure the luminosity of the spectrum in some direct manner. As apparatus for the more commonly used methods was not available, a method based on the above principle of Ferry and Porter was devised which is believed tu possess some new features. The arrangement of apparatus is shown in fig. 1. Light from an acetylene flame (A), after concentration by a lens (B), passed through an open sector of the disk (D), which was rotated by an electric motor, then through two Nicol prisms (Hand F) arranged with their principal sections horizontal, thence through the spectrometer (G), and was finally viewed in a Hilger eyepiece (H) in which all the light of the spectrum, except a narrow central band of any desired colour, was cut off by means of adjustable shutters. In the path of the light at Ca small mirror was set so as * Communicated by the Author. + I. S. Ferry, Am. Journ. Sci. vol. xliv., 1892. 1 T. C. Porter, Proc. Roy. Soc. vol. lxiii., 1898; vol. Ixx., 1902. Method of Measuring the Luminosity of the Spectrum. 605 to reflect white light to a similar mirror mounted on the eyepiece H, which reflected it down through a hole in the eyepiece upon the polished sloping top of a steel pointer, Bigs ls S c . Pol Anal ee ol : "aati Sega aay 4 Murror [re with which the instrument was provided instead of cross- hairs, which finally reflected the light to the eye. The observer, therefore, looking into the eyepiece could see a small patch of white light, and immediately above it a patch of suitable size of any desired colour of the spectrum. The white light could be reduced in intensity to any desired amount by changing the inclination of the mirrors at Cor H, while the intensity of the spectrum was controlled by rotating the polarizer (H). When the disk (D) rotated both patches of light flickered necessarily at the same rate. The spectrometer used was of the Hilger automatic type, which gave a dispersion slightly in excess of twelve degrees. In measuring the luminosity of the spectrum the method of procedure was, first, to lower the intensity of the patch of white light until it was of the same luminosity as a patch of violet (A=°414 w) of undiminished brightness, the principal sections of the nicols being parallel, as near the end of the spectrum as it was possible to make exact measurements upon. The intensities of the white and violet lights were considered equal when the critical frequency of flicker was the same for both. The white light now became the standard of comparison, and was maintained continuously at this low intensity through all the observations. Each selected part of the spectrum was in turn brought into view and reduced to the luminosity of the white patch by rotating the polarizer an amount depending on the brightness of the part of the spectrum under observation. In every case the same critical i 606 Prof. F. Allen on a New Method of | frequency of flicker of both white and coloured lights was ie | taken to mean equality of brightness. The brighter the hi spectral light the greater was the rotation of the polarizer required, since a smaller portion of light was sufficient in that case to equal the luminosity of the standard of comparison. 1} The luminosity of each part of the spectrum is inversely pro- portional to the intensity of the portion of light passing I through the nicols, which is proportional to the square of the cosine of the angle between the principal sections of the ie | prisms. i Observations were made upon nineteen portions of the ie spectrum. These are given in Table I. The results are I | shown graphically in the luminosity curve in fig. 2. i | TABLE I. i | Angle Fi | between 5 1 Reduced to i " planes of eee Coscia a maxe — 1.00: Remarks. i nicols=a. 414 w g° 1 1 0-16 |Standard of comparison. 442 go Se) 1:02 0:16 460 30°? ‘671 1-49 0:24 "480 63° 206 4:85 0-78 “491 (02 117 8°55 1:42 “500 76° 45! 052 19 05 3°10 DiS 83° 70148 67°34 Nalet "000 84° 54! ‘0079 126°53 20°6 564 87° ‘0033 365°4 59°4 593 87° 40' ‘0016 603°5 98-0 Oe) Li hab (tie Gas, 5 5 EA (615) (100) Not observed. ‘601 87° 40’ 0016 603°5 98 621 Sioa 0023 4348 70°6 648 86° 0048 205°5 33°4 663 84° 70109 91°57 15°23 ‘677 noe ‘0364 27°47 4-47 “695 73° 24' ‘0816 1225 1:99 yf 2) | D5° 24’ 322 Spl eo 7 YFO , A oa : ae | 3 ey oy ; as ae | Equal to standard of comparison. In plotting the curve the maximum ordinate is given an arbitrary value of 100, and the others are reduced proportionately. At least three independent observations were made on each colour, and these always agreed with each other very closely. In thirteen of the nineteen cases i the settings of the polarizer differed from the mean value by oy | less than one per cent. ; in the remaining six the differences Wh. were greater. i The ordinates of the curve do not represent absolute lumi- re nosities of the spectrum, but only values relative to the Measuring the Luminosity of the Spectrum. 697 chosen standard, which was ultimately the violet colour of wave-length -414. Since, however, this standard was of LUMINOSITY AQue very low intensity, it need not be considered except in the ends of the spectrum where the luminosity is small. The accuracy of the measurements is influenced, especially in the extreme red and violet, by stray white light, which was unavoidably present in ‘sutficient amount to influence the measurements of the brightness of the spectrum in those feebly luminous regions. The method as originally devised was to adjust the speed of the sectored disk until the critical frequency of flicker of the standard violet was reached. The disk was then to be maintained constantly at this rate of rotation, and each colour in succession reduced in intensity by rotating the polarizing prism until critical frequency was reached. The luminosity could then be determined as before. By this method the auxiliary white standard could be dispensed with, and the luminosity of the spectrum obtained directly. It was found, however, impossible to maintain the motor ata sufficiently constant ‘speed, and the method modified as described in this paper was substituted. I ee | if 608 J LAVIII. On the Comparison of Two Self-Inductions. By Professor A. ANDERSON™*. FYXWO conductors connected in parallel, whose resistances are P and Q, coefficients of self-induction L and N, and coefficient of mutual induction M, are equivalent to a : : P single conductor of resistance 2 2 P+Q induction ine in cases where the current is , and coefficient of self- not oscillatory. It follows that a system consisting of a coil of resistance P and coefficient of self-induction L connected in parallel with a non-inductive resistance 8 has an inductance equal to pen q (P +S)?" It is thus possible, by shunting a coil with a non-inductive resistance, to reduce its effective self-inductance, and to make the latter equal to that of another coil whose coefficient of self-induction is less. A very easy and, possibly, useful method of comparing the coefficients of self-induction of two coils is readily deduced ; and, though it involves a double balance, there is little more experimental difficulty in it than in the measurement of a resistance. Referring to fig. 1, the coils A and B, of which A has the higher self-induction, are placed in the two arms of icra, a Wheatstone bridge, B in series with a variable resistance R, and A shunted by a variable shunt of resistance 8. The * Communicated by the Author. On the Comparison of Two Self-Inductions. 609 resistance CE is equal to the resistance ED. If the resistance in the arm DF is equal to. that in Ci there will be no per- manent current in the galvanometer, and if the self-induction in DF is equal to that in CF, there will be no transient current when the battery key is put down after the galva- nometer key. The method consists in varying § and R till both these conditions are fulfilled. Denoting the resistances of A and B by X and Y, and their coefficients of self-induction by L and N, we have, then, L XO X$ y= (1+3), ONS area rr It is possible that Y may be greater than = yt and, in that case, the resistance R should be in CF in series with the system consisting of AandS. Any inconvenience of moving the variable resistance R from DE to CE will be avoided by having variable resistances in both arms. The following experiment will illustrate the ease with which the adjustment can be made. Resistance of A=109°3 ohms. Resistance of B= 14:5 ohms. BR in ohms, Sinohms, | Wi Ree 0 16°7 Right. 10 315 Richt. 20 50 Right. 30 75 Right. 40 109 Right. 50 157 Right. 60 233 Right. 70 371 Right, but small. 80 697 Left. 72 414 Right, one division. 73 434 No kick. 74 465 Left, one division. N the coefficient of self-induction of B is therefore equal to 4342 L ii \ Goamiggay? ye _ * Very careful measurements by Mr. W. G. Griffith of the self- inductances of these coils, in which Lord Rayleigh’s method was used, gave, for A, 0'236 henry, and, for B, 0:15] henry. 610 On the Comparison of Two Self-Inductions. When the kick is large it is not necessary to have an accurate balance for permanent currents; a rough balance will suffice. But when the transient currents are nearly zero, the permanent balance must be as good as possible. In the above experiment an ordinary mirror galvanometer of the Thomson type was used. The following is, perhaps, the easiest way of applying the principle of the method :— SN S} The self-inductions, Land N, of the coils whose resistances are R andS are to be compared. R, and 8, are non-inductive resistances equal, respectively, to R and 8. There is thus a balance for steady currents. The Q’s are equal non-inductive shunts which are varied till there is no transient current. We have then Na eee . Thus N must not be greater LS R(R+Q) a than “55 Otherwise, the shunts must be applied to S and §,. University College, Galway. March 138, 1911. a rae t Prroler h ye YAS, eh LXIX. Notes on the Electrification of the Air near the Zambest Falls. By W. A. Dovucuas Rupes, M.A., Professor of Physics, University College, Bloemfontein”. HE electrification of the atmosphere is always very marked near a waterfall, and it seemed of sufficient interest to take a series of observations in the neighbourhood of what is probably the largest fall in the world. It is hardly necessary to describe the structure of the fall, but it may be noted that it possesses the peculiar feature of falling over the side of the gorge rather than over one end. The Zambesi just above the fall is nearly a mile and a half wide, but at the fall it narrows to about 1900 yards. The width of the chasm is about 350 ft., the depth 420 ft. The river below the fall pursues a course at right angles to its original direction, and a very energetic churning of the water ensues, an enormous cloud of spray being formed, which may be seen for many miles. The actual height of the cloud above the gorge varies considerably, the maximum height noted by the observer being about 600 ft. in the early morning, falling to about one-third of this height at midday, increasing again towards the evening. The instrument used in taking observations was an electroscope of the Exner type attached to a telescopic stand, so that its height above the ground could be varied, a wire tipped with radium serving as the collector. Obser- vations were taken at a fixed station and also at different distances from the falls. Owing to the peculiar formation of the river-bed just above the fall, it is possible to take observations at the very edge of the fall itself, and also at various points in the river above the fall, as well as from the side of the gorge opposite ; but the results obtained in the immediate neigh- bourhood are not of much value, as the charges obtained were so great that the instrument used was incapable of measuring them. A wire poorly insulated, and stretched halfway across the gorge close to the bridge, about half-a- mile from the fall, became so strongly charged that sparks 2 or 3 mm. in length were easily obtained from it. Observations at the Fixed Station. These consisted in taking the potential gradient at frequent intervals. * Communicated by the Author. 612 Prof, W. A. Douglas Rudge on the The potential varied enormously but to some extent uni- formly with the time, and changed sign during the morning. Observations were begun before sunrise 6.30 to 7, and were continued at intervalsuptol0 p.m. The position chosen was such that the cloud from the falls intervened between the observer and the rising sun. In the early morning at a distance of about one mile from the fall the charge was negative. The amount of electrification depended upon the joint effect of the sun and the cloud, a large increase being seen when the rays of the sun were able to break through the cloud. The maximum was reached when the position of the sun allowed it to shine above the cloud. During the whole period of observations, the sky was quite free from ordinary clouds. The maximum electrification occurred at about 8 a.M., though no quite precise time can be stated, for, as might be expected, the wind had a very appreciable influence. From 8 o’clock the potential fell and reached a minimum value about 10.30 at the station where the obser- vations were made, but when the electroscope was carried nearer to the falls, the potential rose to a negative value ; and on taking the instrument some distance further from the falls, a slight positive electrification was obtained. From this period on to 2 pM. the electrification was nearly always zero, but afterwards a positive value was developed. As the country round the falls is thickly wooded, obser- vations could only be taken satisfactorily where free from trees. The fixed station was at a height of about 100 ft. above the falls, and no high trees intervened, or, rather, the line drawn from the station to the falls’ cloud passed well over the intervening trees. At 6 P.M. the positive electri- fication was well-marked, and it increased in value up to 7 o'clock, after which it fluctuated, falling once or twice to zero and changing sign. At 9.50 it was positive. In the evening when the charge was positive at the fixed station, it was still as strongly negative near the falls ; the sign changed at about 2000 paces from the falls. Observations were taken in a canoe at points on the river above the falls. At a distance of a mile above, the value had fallen to zero, at which it remained for a considerable distance. At 74 miles above, a landing was made upon a rock in the rapids, and here the electrification was decidedly positive but comparatively small in amount—100 volts per metre. The potential falls very rapidly above the falls. Livingstone Island is perched at the very edge of the fall, and is about 200 yards in length measured in the direction ee ee a ee Electrification of the Air near the Zambesi Falls. 613 Tape I. Potential Gradient at Fixed Station. Height of Potential PD Time. Electroscope Charge. | difference ae epee above ground. in volts. P Nie GES YAM. cea ee oe. 160 cms. — (i) 47 TEU BM assole ote; 55 — 75 47 EC iSetacnscuss “F - 175 110 TUS RC ne it — 200 125 1230 eee pnaere 3 — 250 156 PEO conc Dacedges A — 375 237 PETAR Ah cos) x evace's - -- 500 314 20) 2 eae is — 250 156 PESO O eeue cena es! 5 510 317 SMR SW achcrter ess is — 525 330 BOON vc cisscisele 23, . - 27 171 SL) OE ee 5s — 175 110 NOON Ie Wises see — 0 0 Y PROM REM ccs ticn lh. ” + 25 16 DEE NM Men Grate visaan/ " a 25 16 PE er gseeis ses: %» a 25 16 SrOUM Mn renuc.. 00. 2 + 40 25 BO 0 ee eee %9 aR 50 3l AL) 0h ae eae %9 a 50 ol SOMO el ts areicis 2 0 % ate 100 62°5 6.0 : SMa i a 120 (i Ove as as 5 =f 0? 0 OMS eke s facie: id + 150 93 (C205). nee ‘ + 150 93 TROON Rees ces o4 ” + 125 78 Uae sane %9 a 140 88 SRO BTR apis chs cs HS 5° 100 62°5 LCSD eee 3 a5 100 62:5 Between 10.30 and 2 the charge was very small. of the river. At the end nearer to the falls, the electroscope was at once charged to the maximum, while at the further end it had a value of only 600 to 800 volts per metr>. A set of observations was next taken at different distances from the falls, starting from the fixed station at a time when the electrification at that place was ata minimum. Distances were measured by pacing, and detours had to be made in order to obtain spaces tree from trees ; therefore the results are only approximate. The total distance between the fixed point and the falls was about 2,500 paces, and observations were made at intervals of 100 paces, with the result: shown in the table. After 1000 paces had been made, the charge was too great to be measured directly by the instrument, so Hig 614 Hlectrification of the Air near the Zambesi Falls. that the comparative values had to be obtained by noting the time required by the leaves to reach the maximum. At 1000 paces the time was 50 secs. OO : D0) bas 5, 1200 >> » 30, 57 OO) 6 %5 20s, At this point the indication became rather erratic owing to the spray being carried by the wind towards the instrument. When the spray was encountered on the confines of the “Rain Forest,” the electroscope was charged to a maximum in a second or two, in some cases in even less, so that the leaves were continuously moving up and down. {Unrerea OE Potential Gradient at Different Distances from Fall. | : . Height of Potential Sao erce ome walls | leet tostae Charge. Gradient. | es ia | above ground. Volts per metre. +5) 0 ee ee “rae ate 20 cms. | _ 5,000 OOO eee is AN lors 20 — % TAO Oe Senne 205; = 3,500 Ope eek Satta | OMe = xs SILO site eee tee uar ame | 50. 3; _ 2,400 HEOOW He: earn HO = = 1,700 LOO nee f as se. 8 a oe BOs, = 1.390 SOO Pere eros hn deta | SONG, | = 1,000 1510.0) Sisnee | eee aera 50. ,, | = Hs AO OM Ae ee SE 8 50: 45 — a 15) 0) eae Ses a em ee 190.5 = 500 Lb OOO Soe sie conn fede: <3. | 16002 = 425 NON | 160. habe 300 CUO can eoree es | GOMES | = 250 OOO ee Teenie sete LGVee ss = 200 PAU UIO seine Bs bere | NGO 1s; = 100 aI O ES eee teeta ahaa | GO = 79 EO arenes eter G0 | -- 75 POU Re Aon sic eat | LGOsss = 50 DAO): sais Asad dare LG 8 | — 50 DROS R Ae stan: GABh. da: | LOOM ye. 0 0 * Values uncertain. At the bottom of the gorge, but screened from the falls by a bend, the electroscope diverged to a maximum in: 15 seconds, but if the spray got carried by the wind towards the instrument, it was at once charged toa maximum. At Livingstone, eight miles from the falls, the potential had fallen to a low value, and always, during the short time spent there, was positive. On the way to Livingstone some Photographs of Vibration Curves. 615 observations were made from the moving train by projecting a radium-tipped wire connected with the electr oscope out of the carriage window. When the steam from the engine assed over the carriage, a very large positive charge was shown by the electroscope. The potential gradient appeared to be of the order of more than 1000 volts per metre, due to the steam, as when that was blown away from the carriage, the potential fell to practically zero. On the southward journey, observations were taken at most of the stopping-places, including the summit of the Matoppo Hills. The electrification was always positive, no abnormal values being observed. At the edge of the falls the potential gradient if it increased at the same rate would have been enormous, but no doubt in the cloud itself there would be some conduction going on (certainly convection). From the rate at which the elec- troscope leaves diverged to a maximum, the charge at, the falls would be peony BY e times as great as that at a distance of 1100 paces, viz.: 25,000 volts per metre. The curve shows the rate up to within 500 paces, This of course is one day’s observation; no doubt the potential gradient would vary from time to time. ae ey LXNX. Photographs of Vibration Curves. a 7 By CaN eeANtAN: | Vi aA [Plate VI] rin Aaa of vibration-curves forwarded with this note possess certain features of interest which seem to justify their publication. Hxperimental work on vibration- curves relating to the sonometer, violin, and pianoforte that has been published in recent issues of this Journal, con- siderably interested me and induced me to undertake some work in the same direction. The photographs (figs. 1to 9, PI. VI.) were obtained, working with an apparatus a description of which has already been published elsewhere f. The idea of the construction of this apparatus was sue- gested to me in 1908 by the problem of the motion of the bridge of the vioiin. I recognized that the bridge is subject to a normal forcing of double the frequency of the oscillation, * Communicated by the Author. + See ‘ Nature,’ Dec. 9, 1909, on “ The Maintenance of Forced Oscilla- tions of a New Type,” and ‘The Journal of the Indian Mathematical Club,’ October 1909, 616 Mr. C. V. Raman on of the string, and might, under suitable circumstances, be expected to oscillate with the double frequency. ‘To verify this point, I did, at that time, think only of direct aural observation on a specially constructed model. This idea was worked out by me immediately and with success. To hear a note of the double frequency it was essential that all sounding parts that would emit the fundamental should be abolished. In other words, the model sonometer (for so it was) had only, so to speak, a very much magnified bridge, .e., only a sounding-board normal to the wire, instead of, as usual, one parallel “to it. This was arranged without difficulty. The sounding-board was fixed in a rigid frame; one end of the stretched wire was attached to the frame, and the other end normally to the centre of the sounding-board. It was verified by comparison with a sonometer of the ordinary type that the note emitted by the instrument had double the frequency of the vibrations of the wire, in whatever way the latter was set in vibration. | | A Vi term thunderbolt is given in common use both to the rare phenomenon of ball lightning and to meteoric stones. In the latter case it only has meaning, in so far as their luminous path resembles lightning or that they cause great atmospheric disturbance, and it is here used to describe the former. The sing eularity of ball hghtning lies in the com- plete isolation of a gaseous sphere having no envelope, yet within which there is energy stored by previous electrical action. This is in the end liberated with explosive violence. From the scattered records of its appearance the following facts may be regarded as established. It is observed as a luminous blue ball, occurring after lightning flashes of great intensity, and either falling slowly from clouds or moving * Communicated by the Author. Prof. W. M. Thornton on Thunderbolts. 631 horizontally some feet above the earth’s surface. It is seen more often at sea than on land, and both vertical and horizontal movements are recorded in each case. One of the most interesting records of its appearance is given in an account of a storm at sea in Hakluyt’s Voyages by Pedro Fernandez de Quiros, three falling in one day. Ball lightning appears to move under gravitative action on a mass somewhat denser than air, or horizontally in a feeble air-current or electric field of force. It has been observed to follow the course of a conductor such as a water-main, and in most cases to burst on reaching water. It has also been seen to burst in mid-air. That it has some elastic cohesion is shown by its spherical shape and by its rebounding from the earth—in one case at least—atter falling vertically. The features of its end are significant ; the ball simply ceases to be and an explosion wave travels outwards from the spot. In all cases its dis- appearance is followed by a strong smell of ozone. There are records of its curious selective behaviour in the neighbourhood of conductors. Thus a fireball came down a chimney, approached a person in the room (who slowly avoided it), retired up an old flue papered over, breaking throu;zh the paper, and finally burst with great violence on reaching the chimney-top, doing considerable damage. It may be inferred from this that its undoubted immense energy is not in the form of any surface charge which would have had many opportunities of dissipation in such a journey. From the circumstances of its origin, it is clear that there ean be nothing present in it but the gases of the atmosphere. That their molecular condition is abnormal is shown by the light which permeates the whole, and the only possible infer- ence from this is that there is atomic rearrangement proceeding actively within the mass. This blue colour is characteristic of a state of air in which there is proceeding intense electric dissociation, as for example in the immediate neighbourhood of a highly charged needle-point. The chief product of molecular change ‘under electric stress in air is ozone. This is shown by the ‘fact that ata charged point ozone is given off, freely at the negative, and to a much less extent at the positive pole. Nitric oxide is not produced in this case, and it appears to be necessary to have streams of sparks to give rise to the formation of nitrogen compounds in air. The absence of nitrogen compounds i is shown by the action of the electric atte from charged points on paper dipped in a solution in alcohol of tetramethyl p. p. diamido-diphenyl-methane, which in the presence of ozone turns violet-blue and with nitrogen 632 Prof. W. M. Thornton on Thunderbolts. compounds yellow*. Inthe electric wind no yellow coloration is to be seen. Ifa stream of ozone produced electrically in a Siemens tube from oxygen is passed over a metal plate attached to an electroscope charged with positive electrifi- cation, the leaves collapse, and the rate of decay is pro- i portional to the speed at which the gas is passed through the ! ozonizer. The discharge is accelerated by the influence of a negatively charged plate, showing clearly that the fresh ozone carries a negative charge. This suggests an explanation of the origin of the energy in ball lightning. On the occurrence of a flash of lightning from a charged cloud there is an immediate readjustment of the surface electrical conditions, and in certain cases there is the so-called return flash, closely following the first, caused by such a readjustment of the distribution of charge on cloud and earth. If at any projecting part of a negatively charged cloud the stress is nearly but not quite suthicient for a second flash, there will be for a time ionization on a great scale with the formation of ozone which, when sufficiently local in production, gathers into a ball, is repelled and falls. The volume produced depends on the energy immediately available. The process is of the same nature as the point discharge ; but whereas under the stress possible in a laboratory the space in which the glow occurs has a radius of about half a millimetre, under the colossal stress in thunder-clouds it may quite well occur simultaneously ina space a yard in diameter, that of the largest fireball known. At the mast-heads and yards of ships at sea in tropical thunderstorms a blue light is frequently seen—St. Elmo’s fire —a toot or more in radius. All records agree that a thunderbolt is somewhat heavier than air. Nitrogen is lighter than air, and no allotropic form of it is known, though oxides of nitrogen are produced under the influence of streams of electric sparks. Oxygen is slightly heavier than air, ozone is nearly 70 per cent. heavier. The gravitating force on a sphere of ozone a metre diameter in air is 430 grammes—nearly a pound weight. Such a sphere would descend at a rate quick enough to be called a fall. On one of half this size the force would be 54 erammes. For such quantities not to fall but to travel horizontally there must be electrostatic repulsion from the earth requiring, since ozone carries a negative charge, a similar charge on its surface, which is known to generally exist. Itis improbable that the cloud and earth ‘below it * Fischer & Braemer, Ber. vol. xxxviii. No. 3, p. 2633 (1905), “ On the Production of Ozone by Ultra-violet Light.” Prof. W. M. Thornton on Thunderbolts. 633 should both be negatively charged at the same instant. It would be the normal thing for the ball to approach the earth with considerable velocity, as is recorded in well authenticated eases. The fact that sometimes it turns off parallel to the earth’s surface indicates that if, at the moment of the last discharge, the earth was locally positive, its sign has changed by reason of the electrical discharge, and is as usual negative. The reason why the gas gathers intoa sphere is that since the energy of ozone is for a given mass greater than that of oxygen, whilst the volume is less, the foree between molecules of oxygen and ozone is an attraction, which decreases in the aggregate as recombination proceeds until it has the same value as the repulsion due to the usual molecular bombardment in gases. The temperature rising on account of the heat set free equilibrium would be quickly reached, the cooling of the ball by radiation and its motion through the air giving it stability. On reaching water, for which ozone has a strong affinity, or anything which causes ozone to decompose with great rapidity, the ball explodes. The most conclusive evidence for any suggested constituent is whether this contains energy in such a torm that it can be quickly liberated. It is well known that ozone reverts to oxygen, and it remains to see whether the energy liberated by this change is sufficient to account for the effects observed on its sudden occurrence. In the conversion of a gramme of oxygen into ozone 29°6 kilogramme-degree-centigrade units of heat are ab- sorbed. A sphere of 50 cm. diameter contains 62°5 litres. For the complete change of oxygen at 1°45 grammes per litre to this volume of ozone at 2°14 grammes per litre, there would be required 2615 of the above heat units. A clearer view of what this means is obtained hy expressing it in mechanical units. Hach kilogramme-degree unit is equivalent to 3094 foot-pounds; the total energy of transition is therefore 8 million foot-pounds. It is unlikely, since there is recombination proceeding by diffusion within the ball as it falls, that the whole mass could be pure ozone on reaching the earth, but the dissipation of one tenth of the above energy explosively in the tenth of a second is at the rate of 15,000 horse-power. The energy of the explosion-wave is well accounted for by this. There is in addition the sudden expansion when ozone is changed into oxygen, in this case 20 litres for a sphere of 50 cm. diameter of ozone. Phil. Mag. 8. 6. Vole 21; No. 125. May 1911. 21 634 Dr. W. Wilson on the Discharge of The facts which may then be stated in favour of thunder- bolts consisting mostly of ozone in active recombination are :— 1. Ozone is said to be observed on their dissipation. 2. The gas of which they are composed is heavier than air. Ozone is the only gas denser than air produced in quantity under electric stress in air, as distinct from streaming spark-discharge. 3. On reaching the earth thunderbolts are frequently deflected and travel horizontally as if repelled. The earth’s surface and ozone are both in general negatively charged. 4. The energy liberated on the transition of ozone to oxygen in the volume of a fireball is sufficient to account for the explosive violence with which it bursts. 5. The blue colour usually observed with it is associated with the sparkless electrical discharge in air which causes the production of ozone. It is also observed when oxygen and hydrogen combine explosively; when nitrogen is present the colour of the explosion flame is yellow. These considerations lead one to suggest that the principal though not perhaps the only constituent of thunderbolts is an aggregation of ozone and partially dissociated oxygen, thrown off from a negatively charged cloud by an electric surge after a heavy lightning discharge. LXXIV. The Discharge of Positive Electricity from Hot Bodies. By Witu1am Witson, Ph.D., Assistant Lecturer in Physics, University of London, King’s College*. | a recent paper on the positive electrification due to heated aluminium phosphate (Phil. Mag. Oct. 1910), A. E. Garrett has described the effect of the presence of water in the salt in temporarily increasing the positive electrification produced. As I have observed a similar phenomenon, I propose to publish a preliminary report of research which is still in progress on the discharge of elec- tricity from hot bodies. Experimental Arrangement.—The platinum wire, p (fig. 1), formed one arm of a Wheatstone bridge arrangement, and was 13 cm. in length and 0:2 mm. in diameter. The adjacent arm contained a known adjustable resistance, R, of thick eureka wire (immersed in paraffin oil), and a shorter * Communicated by the Author. Positive Electricity from Hot Bodies. 635 piece, ¢ (3 cm. in length), of platinum wire exactly similar to p. This shorter piece of wire had also leads like those Hie. of p, and served to compensate for their resistance. The wire was of pure platinum supplied by Messrs. Johnson & Matthey. The other arms of the bridge consisted of two resistances, A A, each of 5000 ohms. A battery, b, of 10 to 12 accumulators supplied the current, which could be regulated by means of a variable resistance, 7. Before a measurement of the leak from the platinum was carried out, the adjustable resistance, I, was given asuitable value. The current was then started and the resistance, 7, reduced till the galvanometer, g, indicated no current. In this way the wire, p, was given a definite temperature, namely that at which the resistance of a definite portion of it was equal to R. The arrangement thus served to heat the wire, p, and to determine its temperature. In fig. 2 (p. 636) is shown the way in which the platinum wires, pand c, wereconnected to the thick copper wires, ww, conveying the current, and the arrangement for measuring the “thermionic” current. ‘lhe wire, p, was connected to thick platinum leads fused through the ends of glass tubes, L L. The wires, ww, dipped into mercury which covered the platinum leads. The whole was fixed by means of a tightly fitting indiarubber stopper, g, in a test-tube shaped vessel provided with side tubes, ¢¢, by means of which dry air could be supplied to it. Surrounding the platinum wire was an aluminium cylinder, D, 2 cm. in diameter, supported at the end of a straight copper wire which was insulated from the rest of the vessel by means of sulphur, 8, and the other end of which dipped into mercury contained in a cup, M, in a paraffin block. One terminal of a delicate moving coil- galvanometer, G (1 scale-division = 6°32 x 10-" ampere) was connected to the negative pole of a battery, B, of 100 volts 3 2T 2 636 Dr. W. Wilson on the Discharge of whose positive pole was connected through w to p. When p had been raised to the desired temperature in the manner ie 2. SSSss;s;sc SG DB_[SAQAQ|\|[/Fr SSS already described, the other terminal of the galvanometer was connected with the mercury in M and the ensuing current from p to D measured. The experiments were con- ducted in air, at atmospheric pressure, which had been freed from dust and moisture by passing through tubes containing cotton-wool, calcium chloride, caustic potash, and phosphorus pentoxide. Piminution of Positive Leak from Hot Platinum not due to Heating only.—It is a well-known fact that the positive leak from hot platinum diminishes with continued heating. My experiments point to the conclusion that this phenomenon is not due to the temperature alone, but is rather a consequence of the discharge of positive electricity. The foilowing obser- vations illustrate this:— aie In the initial observation the current was represented by 186 scale-divisions, the potential-difference between the wire, p, and the cylinder, D (fig. 2), being 100 volts. The tempe- rature of the wire was about 1100°C. After this measurement the cylinder, D, was completely insulated by disconnecting at M, the wire being maintained at exactly the same tempe- rature as before. At the end of 10 minutes the connexion Positive Electricity from Hot Bodies. 637 at M was re-established and the current again measured. It was now found to be 176 scale-divisions—that is to say, not very much smaller than the initial leak. The potential- difference of 100 volts was maintained for 10 minutes (with the same temperature as before) and the current noted at the end of this period. This time it was found to be only 111 scale-divisions. The discharge from the hot wire was thus reduced to a much greater extent during the second than during the first period. Finally the cy iinder was again insulated for a further period of 10 minutes, the temperature of the wire not being allowed to vary. When the current was again measured it was found to be 111 scale-divisions. These observations show that the diminution in the positive current from the platinum is not an ordinary fatigue effect, but is due to the decrease in the quantity of ionizable matter which the platinum contains. Influence of Water on the Positive Leak from Platinum.— One of the most remarkable phenomena in connexion with the emission of positive electricity by hot platinum is the increase in the discharge caused by the presence of water. I observed this effect by accident during experiments on potassium sulphate. This salt when heated on platinum wire apparently discharged large quantities of positive electricity. It was put on the wire by dipping the latter in a concentrated aquevus solution of the salt. At a temperature of 1100° C. and with a potential-difference of 100 volts the leak was initially far beyond the range of measurement of the gal- vanometer. This leak decayed with great rapidity at first and more slowly and somewhat irregularly afterwards. The CURRENT /N GALVANOMETER SCALE DIVISIONS. eae eee eee : | (0) 2 4 6 8 10 = Atinures way in which the current varies with the time is shown in fig. 3. Asit was found that the same effect could be produced 638 Dr. W. Wilson on the Discharge of by simply dipping the platinum wire in distilled water, the discharge in the case of potassium sulphate was due mainly, if not entirely, to water. The similar effect observed by A. HE. Garrett, J. c., in his experiments on hot aluminium phosphate is possibly an increased activity of the platinum on which he heated his salt, and occasioned by the com- paratively large quantity of water which the salt contains. To obtain this increase in the positive discharge from platinum, it is not necessary that there should be water vapour present in the atmosphere surrounding the wire during the measurement of the discharge. ‘This is shown by the following experiment :—The platinum wire was placed in an atmosphere saturated with water vapour and heated for some minutes by a current, the temperature being raised as far as was possible without risking fusion of the wire. The latter, after being allowed to cool in this atmosphere, was then placed in the measuring apparatus (fig. 2), the air in which was perfectly dry, and with the usual potential- difference and temperature the leak was found to be greatly increased. It would appear therefore that water causes or accelerates the production in the platinum of something which can emerge at sufficiently high temperatures in the form of positive ions. In ali cases the activity induced by water decreased with great rapidity, which indicates that it is confined to the outer portion of the wire. It should be mentioned that a much greater increase in activity was pro- duced by actually dipping the wire into water and introducing the wet wire directly into the measuring apparatus. Another experiment, which showed the effect of water ina very striking way, consisted in dipping the wire into a solution of calcium nitrate. It was then heated in the measuring apparatus (fig. 2}, till the salt was reduced to calcium oxide and the discharge was reduced to small dimensions. So long as the wire with its coating of oxide remained in the dr atmosphere of the apparatus, the leak continued to be almost inappreciable. A 5 minutes’ exposure of the coated wire to the air of the room, however, was sufficient to induce a leak of 50 to 60 scale-divisions. Again the leak decayed with great rapidity. ! } Nature of the Ions in the Induced Discharge.—The carriers of the positive electricity in the case of the normal leak from hot platinum consist probably to a large extent of carbon monoxide. The value of < obtained by Richardson for the positive ions from platinum, and Horton’s spectroscopic work in the case of aluminium phosphate heated on platinum Positive Electricity from Hot Bodies. 639 (Proc. Roy. Soc. Dec. 1910), support this view. Further, the value of = found by Garrett, J. c., for some of the ions emitted when aluminium phosphate is heated on platinum is consistent with the view that hydrogen ions are also emitted from hot platinum. Now if we suppose—as I think we are justified in doing— that even the purest platinum contains traces of carbon, the water effect can easily be explained. When the platinum is heated in the presence of water vapour, the water will be decomposed with the formation of carbon monoxide and the liberation of hydrogen. This may occur even at tempe- ratures considerably below that at which the positive leak begins to be appreciable. These gases will naturally diffuse into the platinum and re-emerge under suitable conditions in the form of positive ions. There is also the further possi- bility that water may accelerate catalytically the production of carbon monoxide when platinum is heated in air. Positive Leak from Hot Aluminium Phosphate.-—Both Horton and Garrett heated the aluminium phosphate on platinum. The salt was made into a paste with water, and moreover contains in any case a large amount of water. As this water will affect, temporarily at any rate, the activity of the platinum, the question naturally arises: What part of the discharge from the hot phosphate are we entitled to ascribe to the salt itself ? Hven when it has been heated so long that the effect due to the presence of water has died down, it is still con- ceivable that the salt may facilitate in some way the escape of ions from the platinum. I failed to observe any positive discharge from aluminium phosphate when the latter was heated on a Nernst filament, the temperature of which was. very much higher than that employed in the experiments with a platinum wire. The experiment was carried out both ona Nernst filament provided with the usual heating arrange- ment and on one not so provided. In the former case the galvanometer showed an appreciable deflexion (20 to 30 scale- divisions) whale the heater was in operation. This deflexion diminished to zero immediately the heater was cut out. It was therefore due to the platinum wire of the heater. While the Nernst filament was glowing no deflexion of the galvano- meter could be observed. The experiment was done under conditions as closely resembling those in which a platinum wire was used as possible. It seems quite likely that aluminium phosphate, at any rate when prepared from aluminium acetate, may contain traces of carbon, and therefore some part of the positive 640 Mr. G. H. Livens on the Initial Accelerated discharge observed when the salt is heated on platinum may have its origin in this carbon. The experiments described above, however, suggest that the platinum plays an important role in the ionization of the carbon monoxide and other pro- ducts formed by heating platinum or platinum coated with aluminium phosphate. Summary.—(a) The activity of platinum is not reduced by continued heating merely, but only under conditions which admit of a positive discharge from the metal. The loss of activity is therefore due to the diminution of the quantity of matter—carbon, carbon monoxide, or whatever it may be— which emerges from the platinum in the form of positive ions at sufficiently high temperatures. (b) The activity is greatly increased by heating the platinum in the presence of water. This effect is possibly due to the production of carbon monoxide and hydrogen in the platinum or at its surface, and can be observed even when the platinum is heated in a dry atmosphere, provided it has been previously heated and allowed to cool in an atmosphere saturated with aqueous vapour. (c) There is apparently no positive leak (or only a very small one) when aluminium phosphate is heated on a Nernst filament, and therefore the leak observed when the salt is heated on platinum is either mainly a leak from the platinum itself, or the latter plays an important réle in its production. Further research on the subject is being carried out, of which a full account will be published later, Wheatstone Laboratory, University of London, King’s College. February 1911. LXSXV. The Initial Accelerated Motion of a Perfectly Con- ducting Electrified Sphere. By G. H. Livens, B.A., Lecturer in Mathematics, Sheffield University *. Sue ae papers have recently been published dealing with this subject ; those particularly under review here are by G. W. Walker (Proc. R. 8. vol. Ixxviil. and Phil, Trans. 1910). In the present paper the same subject is dealt with in a manner similar to that given by Walker, but the complica- tions of considering any material mass that the sphere may possess are entirely avoided, the subject being discussed purely from the electromagnetic standpoint. The use of * Communicated by the Author. Motion of a Perfectly Conducting Electrified Sphere. 641 spherical polar coordinates is also adopted, following a very kind suggestion from Dr. Bromwich. The general method consists in imparting to the sphere, in a manner which will hereafter appear, a uniform acceleration and deducing the initial field purely from geometrical con- siderations. The effective force on the sphere is then calcu- lated, and the coefficient of the acceleration in the expression for this force is taken as representing the electromagnetic mass of the sphere. I. When the sphere starts from rest. The sphere is perfectly conducting of radius a, with a total charge e, and the acceleration is s, applied in a direction which is taken as the polar axis of the coordinates, the centre of the sphere coinciding initially with the origin. The acceleration is considered so small that the displace- ment of the sphere in the time taken by radiation to travel 2 across the sphere is small compared with the radius ; —, is small compared with a. ¢ We have obviously only to deal with a case of symmetry about the polar axis. The Maxwell equations for the field outside the sphere can then be written Be | ON. ano os Y)=(- y? Op’ rsin 6 Or. (X, Y, Z; a, 8, y) being the usual components of the electriciand magnetic vectors along the polar directions, and w=cos@ and w=rysin@. This leads as usual to the equation for vp, dip Ore he aoa ed? — dr 1 ye Op? ? of which the known general solution is pern( BBY ffl gy Me The solution being restricted to the only necessary case of expanding waves. We now attempt to find the field outside the sphere at the end of a time ¢, to the first order in the acceleration. The AY small displacement of the centre of the sphere is = =. and the equation to its surface is si r=atEcos 0. 642 Mr. G. H. Livens on the Initial Accelerated We try a solution involving the first order harmonics only. We add on to the initial field the first harmonic solution of the general equations, and attempt to satisfy the boundary conditions. We take 8) Daas Bs, sin 9 AS Of’ +f), Y=—,- ( y? 9 of +f th sin @ oN rea Gd Mugs J is now interpreted as a function of (et—7r4-a). As Prof. Love points out, there are two conditions to be satisfied, one at the front of the advancing wave boundary, which started out from the sphere at the initial instant, and the other one at the surface of the sphere itself. The con- ditions at the wave front can easily be seen to be PX — Dn NG = orr NG iead: | 977 == ce the initial field (Xo, Yo, Zo, a, Go, Yo) existing undisturbed outside this boundary. These give f/(0)=0, f(0)=0. The condition at the surface of the sphere is that the tangential electrodynamic force is zero. In the ease under discussion this is the same as that the tangential electric force should be zero, the magnetic part of the former, con- taining the product of two small quantities, being zero to the first order. This leads to the condition that y— sin OX _ a 3 account being taken of the fact that the centre of the sphere is not at the origin of the coordinates ; this is equivalent to a’f''(ct) + af'(ct) +f (ct) —e€=0. We now use #=ct and AES, and then the equation for yous : | af" (a) +af (a) +f(¢)—Az?=0. The solution subject to the wave-boundary conditions is 4A Aa? ad ais a == ie Cy LN mace V3 Ie + Ax? baer! 2 Aax. Motion of a Perfectly Conducting Electrified Sphere. 643 With this form of f (x being now interpreted generally as ct—r-+a) we have fulfilled all the conditions, satisfied by the field, to the first order. The field is, therefore, completely determined under the restrictions imposed. The density of the charge on the sphere is given to the first order by Amo=X at r=at+écosé e Hes ng ft many (Ct) ae J (ct) —eF) ; Ga or considering the ae satisfied by /f, ON ny Anro= ae cos @. We now find the resultant force on the sphere, which is obviously along the polar axes. The component of the electrodynamic force in this direction is P'= X cos 0—Y sin @, and at the surface of the sphere this reduces to puck 2h COS ) ; pe (5 ee Thus the total force on the sphere in the direction of motion is of 2a 5 . r= ( \ P’cad’ sin 6 dé dpi 2) _7 2 Zep” aa cc is the effective electromagnetic force on the sphere. ow (IN (1-e"# cost °)- ane 2a sin 8 3S and since 1 se A= 5 a we have 2 e's us aa s Wel 2a. 644 Mr. C. H. Livens on the Initial Accelerated The electromagnetic mass of the sphere, defined as the ratio P/s, is therefore a ct 4 (1-6 eos ses 2 @° @- 2a Gig 3 ac 2a i 3 ~ 3 ae? V3 ee) The value of m is initially zero, but rapidly approaches the 2 ordinarily assigned value : = The sphere therefore starts off without offering any electromagnetic inertia to its motion initially. The force and mass, however, both differ from zero at any finite time, however small, after the initial instant. This fact is explained quite easily from general principles. The system to be moved is specified by a certain state in the zether around the sphere. Now the ether at any place is not affected by any motion of the sphere until after the time that radiation, leaving the sphere at the initial instant of its disturbance, takes to reach that place ; and it cannot, therefore, offer any reaction to the motion of the sphere until after that time. Other conclusions, similar to those deduced by Walker, can be deduced ; the only distinction being that none of the results here given involve the “ material”’ mass of the sphere. The production of a small uniform acceleration causes a readjustment of the charge distribution. The readjustment of the charge, however, involves an oscillation which sends out a damped periodic wave train into the ether. ‘The oscil- lations and wave-motion are, however, soon damped out of the system, and a sort of steady state is reached in which 2se e 4Ano=-; -;cos6 i OF DQ p2 P= D a) Sie 3 ac Il. The sphere accelerated from a uniform motion with velocity v. The velocity is supposed to have been uniform for an indefinite time before the initial instant considered. Now, according to Larmor*, if we refer the phenomena to a set of axes moving with a velocity v, the fundamental equations of the theory, the Maxwell equations referred to moving axes, assume exactly the same form as they had originally referred to fixed axes. * See ‘Aither and Matter,’ pp. 173-176. Motion of a Perfectly Conducting Electrified Sphere. 645 In fact, if the motion of the sphere is along the axis of «, and if (2, ¥; 2; ¢;) be the coordinates of space and time in the moving axes, connected with (a,¥,z,t) those in the fixed axes by the relations then Maxwell’s equations referred to moving axes assume the form d An Gane D5 h,) == (Oil (a, Dy. C1) ‘1 iy. ee dare? ace by, ¢) = Curl, Chis J15 hy), aly where (7; #1 43 a 0: &) are related to the actual vectors in the field by the equations ; 2 ae meas UC UE ts Ci On h,)=e (< v J Aare?’ h+ un): (ay, b;, c1:) =e (€—2a, b+ 4arvh, c—Arrvg). Thus if the values (1, 91,413 a 0, ¢,) given as functions of (2, ¥, 21 t;) express the course of change of the ethereal vectors of any electrical system referred to the axes (2, 7 2 t)) at rest in the ether, then NIK € (e#f g—- ms pst m= b), e (e2a, b+4arvh, c—Anvq) expressed by the same functions of the variables oes ab —i,/ oy a elu, 2 eat moe fe will represent the course of change of the ethereal vectors (f,9,h; a, b,c) of a correlated system of moving charges referred to axes (a’, y’, 2’) moving through the ether with uniform translatory velocity (v, 0,0). Moreover, in this correlation between the courses of change, in the two systems, elements of charge occupy corresponding positions in the two systems and are of equal strengths. However, electro- dynamic forces per unit charge are not the same in the two a systems. They are, however, related in an obvious manner. 646 Mr. G. H. Livens on the Initial Accelerated In fact, if the force per unit charge on any element executing any motion in the system at rest be determined as (P, Q, R)), then the force per unit charge on the element executing the corresponding motion relative to the moving axes is deter- mined by (P, Q, R), where (P,, Q: It,) = e (e? Je, Q, R). Again, the accelerations of any point determined as (s1, $y’, 5;''), when referred to the axes at rest in the ether, become when they refer to the point executing a corresponding motion in the correlated system (s, s’, s’), where SAS Oy =e? (ES, €25\ E25 |): *) if the point is always near the origin of the coordinates. Thus if in the case of a system referred to fixed axes we have determined the equations of linear motion, considered merely in its electrodynamic aspects, in the form (P; Q1 Ry) = Cy 81, my‘ 81, "51", m, m,'m,'" being determined as the electromagnetic masses for the system in motion, as a whole, along the three axes of coordinates ; then the equations of motion for the same system executing a corresponding motion referred to axes in _uniform motion along the z-axes will be (P, Q, R) = (mye??s, mye 8)’, m1" s,"’). The electromagnetic masses for the new system, which has the additional motion with velocity v along the w, axis, are therefore (m, m’', m") = (mye??, m, €2, me). From these preliminary remarks it will be at once seen that the solution which we have already obtained for the initial accelerated motion of a charged perfectly conducting sphere can be transferred to the case for the sphere initially in uniform motion with any velocity v, if the vectors are inter- preted properly. One other condition has to be satisfied. The equation of the sphere has to remain of the form wty?+22=a?, and to ensure this we must accept the Lorentz contraction hypothesis, which I propose to do, and have already tacitly done in the previous general discussion. In the solution transferred to the case of initial uniform motion the general electrodynamic vectors (1 91 A1)(a@ 6; ¢,) determined in terms Motion of a Perfectly Conducting Electrified Sphere. 647 of the actual field vectors by relations already given, cor- respond to the values already obtained for the electric and magnetic vectors, when these are expressed as functions of the coordinates (a,' y,! z,' t’) referred to the moving axes. Thus if (X, ¥;Z,; 2,71) are the components of the electrodynamic vectors referred to spherical polar coordinates moving with the cartesian axes, but whose polar axis is in the direction of the applied acceleration, which need not necessarily be the direction of the uniform motion, then the field referred to moving coordinates is determined by relations similar to those already obtained. i e 2 cos @ ay 1 1 a sin @ Noi ee ee ma 13) SIMO, Wann Cilia ae Ge viata)’. where AA,a@? -" sin y,,/3 i ET 2 wa + Aye?—2A,ax,, es and M=ch—y+a, A= 528° The density of the charge on the sphere is given by t e 27'"' cos 6 470, = — — ae 5 a a and the force on the sphere in the direction of the accelera- tion is Brcus: uae ct, V3 9 os, ose V/ 3ct P=555 = 24 gos J 5 tO gi VON 3 ac? 2a 3 ac? V3 2a Whence we deduce that the electromagnetic masses of the sphere are 2 9 ct Re 2 saad = Qh / re é are ct a 2 ~ _; ‘ ate E a a ces at/3 Tae a sin V det, i 2a 3 ac V3 5) 3.ac They are all three initially zero, but tend rapidly in an oscillating manner to the values usually obtained from quasi- stationary principles a 648 Prof. C. G. Barkla on the Conclusions can be drawn similar to those indicated in the previous case considered. The introduction of acceleration into the motion of the electrified sphere, which was previously uniform, results in a small disturbance of the initial uniform distribution. The disturbance and rearrangement of the charge give rise to its oscillation which sends out the damped wave-train into the ether, but the oscillation and ether dis- turbance soon die away, and the system settles down into a steady state of motion with a uniform acceleration. The electrical reaction to the starting of the accelerated motion is initially zero, but rapidly assumes its steady value. The case of a rigidly charged dielectric sphere possesses some additional characteristics, and I shall reserve the dis- cussion of it for a future paper. : Sheffield, February 1911. LXXVL Note on the Energy of Scattered X-radiation. By CHARLES G. BARKLA*. N 1904 the writer made an experimental determination of the energy of Rontgen radiation scattered by light elements (Phil. Mag. May 1904), and applied the result to calculate the number of scattering electrons in a known quantity of matter, on the theory of scattering given by Sir J.J. Thomsont. With the data for - and e of an electron at that time available, the number of electrons in a cubic. centimetre of air under normal conditions of pressure and temperature was found to be 6x10”, or between 100 and 206 per molecule of air. Using the more recently deter- mined values of =, e,and n (the number of molecules per cubic centimetre of gas) t, the calculation gives the number of scattering electrons per atom as about half the atomic weight of the element. In a recent paper, however (Proc. Roy. Soc. A. lxxxy. pp. 29-44), Mr. Crowther has determined the energy of the scattered radiation from experiments on aluminium. He * Communicated by the Author. y~ The theory was first given, together with the expression for the energy of the scattered radiation, in the First Edition of ‘ Conduction of Electricity through Gases.’ Hi: < =173X 10 eis: (Bucherer) ; e=1:55x 10-29 em.u. (Ruther- ford & Geiger); n=2°8xX1C"* (Rutherford). Energy of Scattered X-radiation. 649 estimates the energy of radiation from a given mass to be six times that given by the writer, and concludes that the number of scattering electrons per atom is three times the atomic weight. Now it was early shown by the writer* that elements of low atomic weight, up to and including sulphur, scatter to the same extent mass for mass, and that the scattering is independent of the penetrating power of the Roéntgen radia- tion used. Mr. Crowther later verified both these results fT. We should therefore expect to find the same amount of energy scattered in the two cases from equal masses of air and aluminium. The experimental values are evidently in conflict, and it becomes a matter of interest to examine them, not merely for the sake of any evidence the result may afford as to atomic constitution, but in order to explain certain phenomena of absorption. From the results of experiments by the writer 1t was con- eluded that a layer of atmospheric air of 1 centimetre thick- ness scatters about ‘00024 of the energy of Rontgen radiation passing through it. Thus if —dI, represents the diminution due to scattering in the intensity of a beam during transmission through a layer of air of thickness dz, then dI,=—:00024I dx. Calling the quantity ‘00024 the co- s e Ss a e efficient of scattering s, we get — = 2 approximately, where p is the density of air or any substance of low atomic weight. Mr. Crowther’s value in the case of aluminium is 1°18. We will briefly consider these results in the light of other experiments. (1) As the radiation considered is scattered radiation, it involves a corresponding diminution in the intensity of the . ry ° ° ° e ¥ . primary beam. ‘That is if the intensity of a beam proceeding in a given direction be expressed by the equation [=Ipe-*, » cannot be less than s, or the total diminution of intensity of the beam proceeding in the original direction of propagation can- not be less than the loss due to scattering alone}. Yet the total * Phil. Mag. June 1903, pp. 685-698: May 104, pp. 543-560; June 1906, pp. 812-828. t+ Phil. Mag. Nov. 1907, pp. 653-675. (Hydrogen is a possible though not certain exception to the first Jaw.) | For want of a better term A will throughout be termed the absorp- tion coefficient, though some of the energy is merely scattered and some is re-emitted in a different form. Correctly it is the rate of diminution with distance of the primary beam, as a primary beam, or it is the rate of diminution of intensity of an infinitely narrow pencil of radiation during transmission through matter. In experiments on absorption care has to be taken to get the primary beam after transmission practically free from the scattered and re-scattered radiations, as well as from the fluorescent X-radiation and the corpuscular radiation, or at any rate to arrange for the effects to be small and to correet for them. Phil. Mog Seca VolemaNcwieos way loli, 20 650 Prof. C. G. Barkla on the : : Xr mass absorption coefficient ( ¢ tion of quite ordinary penetrating power, such as certainl obeys the laws of scattering, was shown by Barkla and Sadler* to be only about °41. Tt is difficult to ee this with ) in carbon of a certain X-radia- Mr. Crowther’s estimate of scattering, 2. e. — * =1:18. (2) Much more penetrating beams of homogeneous X- radiation have since been found by the writer, and the mass- absorption-coefticient for these in carbon has been found as low as about °25, or about 1 of Mr. Crowther’s scattering coefficient. Thus either for these rays the scattering is much less than for the more absorbable rays, or there is a con- siderable discrepancy between these results. Neither theory nor experiment indicates that for these very penetrating rays there is any diminution in scattering ; both in fact indicate that the scattered radiation carries away the same fraction of the energy of primary radiations differing widely in penetrating power. (3) In aluminium itself, the mass-absorption-coefficient rX : : : (*) of certain penetrating rays is much less than 1:18. The lowest value experimentally found is about °6 for a fluor- escent X-radiation characteristic of cerium. (4) The simple laws of absorption found by Barkla and Sadler point to the conclusion that the mass scattering coefficient is of the order of magnitude of 2. The absorption of Rontgen radiation in a substance A bears an approximately constant ratio to the absorption in a substance B, through long ranges of penetrating power. The limits to this law are that the radiation used must not extend in penetrating power beyond that of any radiation characteristic of either A or B, and must not be near one of these characteristic radiations. on its:more penetrating side. ‘This proportionality is evidently true only when the total absorption is great compared with the portion of it due to scattering, for as has been pointed out, the scattering is independent of the pene- trating power of the radiation as well as the particular light element which is producing the scattering. There is thus a constant term in the variable mass- absorption- -coefficient r : 2 S —. Jt then we subtract a constant quantity - due to scattering from this, we expect the law: to. hold even for small Xr seat, cape a: | values of —, for there is no obvious reason why the law should be departed from just when scattering becomes an appreciable * Phil. Mag. May 1909, pp. 739-760, Energy of Scattered X-radiation. 651 fraction of the whole absorption. Now the quantity « which the writer has found from the equation | (5) ba NTA Sat G = —wv PZAl | suffixes denoting the absorbing substance | varies in different eases from ‘16 to °25. It is very probable that with care the absorption coefficients will be determined with greater accuracy, and consequently « brought within narrower limits. The results are, however, sufficiently accurate to give the order of magnitude. This agrees very well with the value directly determined by the writer. It should also be pointed out that a value for a = constant os : Ms perenne (i.e. *) as high as 1:18 would involve complete violation in place of otherwise close agreement with simple laws of absorption. (5) The penetrating power of the fluorescent radiations from various elements shows no tendency with increasing atomic weight of the radiator to approximate to the ‘emu suggested by a scattering coefficient (*) as large as 1°18. On the other hand, there is such an approximation in the : é ideas Py absorption in carbon to an inferior limit for Cc) equal to something of the order of °2. There thus appears abundant, consistent, and apparently conclusive evidence that the intensity of Rontgen radiation scattered by light elements is of the order of magnitude found directly by the writer in early experiments, and that Mr. Crowther'’s result is several times too great. The theory of scattering as given by Sir J. J. Thomson leads to the conclusion that the number of scatterin g¢ electrons per atom is about half the atomic weight in the case of light atoms. * This applies to atomic weights not greater than 32, with the possible exception of hydrogen. Accurate results have not been obtained for the intensity of radiation scattered from heavier elements owing to the difficulty in many cases of getting rid of the fluorescent X- radiations superposed on the scattered radiation. Barkla & Sadler estimated the intensity of the radiation scattered from silver to be about 6 times that from an equal mass of the light elements. ‘There is also indirect, but by no means conclusive evidence that still heavier atoms scatter to a greater extent. Measurements might easily be made in a number of cases with a fair degree of accuracy. ‘The subject is worth further investigation. 2U 2 ee 652 Miss [tuth Pirret and Mr. F. Soddy on the Evidence is also given of a limit to the penetrating power of Rontgen radiation. Unless the laws of scattering some- where break down, the lowest possibie value for : is about "2. This has been approached in the case of absorption by carbon. -LXXVII. Vhe Ratio between Uranium and Radium in Minerals. 11. By Ruta Pirrer, B.Sc, and FREDERICK poppy, MAY Eas." | ie a previous paper on this subject (Phil. Mag. 1910 [6] xx. p. 345), a short account was given of the determi- nation of the ratio of radium to uranium in Ceylon thorianite and a specimen of Portuguese autunite. The preliminary results went to confirm those of Mlle. Gleditsch (Compt. Rend. 1909, exlviit. p. 1451; exlix. p. 267) in that the ratio in autunite was found to be considerably lower than in pitchblende ; but the results with thorianite were not equally conclusive. Only one specimen of thorianite was compared with the old pitchblende standards prepared some years ago. Further investigations with several different specimens of thorianite, pitchblende, and autunite were therefore carried out on the same lines. Uranium Analysis. —The methods of estimating the uranium in thorianite and autunite have already been described. In the case of the pitchblendes the mineral was first dissolved in nitric acid, the solution diluted, filtered, evaporated to dryness, the residue dissolved in hydrochloric acid, treated with sulphuretted hydrogen and filtered. ‘The filtrate, after heating and oxidizing, was poured into a mixture of ammonium hydrate, sulphide, and carbonate, corked, and left over night. The filtrate from this precipitate was heated, acidified by nitric acid, and the uranium precipitated by microcosmic salt and sodium thiosulphate in presence of acetic acid. The precipitate was ignited in a porcelain crucible and weighed in the form of a green compound of constant composition. It was then converted, by means of a few drops of strong nitric acid, into uranium pyrophosphate and weighed again after ignition at a dull red heat (Brearley’s ‘Analytical Chemistry of Uranium,’ p. 7). In some cases Patera’s Method (Fre- senius, ‘ Quantitative Analysis,’ vol. 11. p. 310) was employed, or a modification of it in which, instead of the uranium being weighed as sodium uranate, it was, after separation by * Communicated by the Authors, 693 means of sodium carbonate, estimated as pyrophosphate as before. Table J. contains the results of the estimation of uranium in the various minerals. The thorianites were all specimens of Ceylon thorianite. “ThI, Thi«, Thlb” were from the same sample of the variety richest in uranium and con- taining a small residne (3 per cent.) insoluble in acid. “Th X,”? was a poor specimen of the mineral, and contained nearly 24 per cent. of insoluble material. ‘Th CCC” and “Th HE” were specimens of mixtures of the two varieties (which differ chiefly in their respective high or low percentage of uranium). ‘The pitchblende “‘ PI’ was a specimen from Joachimsthal which contained comparatively little uranium and proved troublesome in the analysis and unsatisfactory in the results. The other (“J.P.A ’ and “J.P.B ”) was a picked Ratio between Uranium and Radium in Minerals. AB Eas Percentage of Uranium. | | hb) (2) (3) (4) (5) Mean. | _ (Thi, Thia, ThTd...| 20:804@ | 20-1le | 19-86c¢ | 19:32¢ | 19-80c 20:06 | S | 20°56 6 | 20:28d | 19:99 d | 19°74d | 20:'15d } 4 | § "E.|) TON .Cn UC ce aoe 949a@ | 1071d | 10-40d 10-00 BE 4 GSO |S. cosas (eee | gue Ne ob os | OO Cs Sona ace ease 13-02a@ | 1404@ | 138°44a 13°61 =I Uth B 13°20 6 | 14146 | 13°82 6 \ 2 EST eee ig se Le DI Gerdes ACO ae WA aa ei A 4 3 179361 17-955 |foce | oe | eee 17-62 28 Jen CARE ett AG Riana a 33°35¢ | 28:06c¢ | 30°81 d | 34:99 a | 31:20e \ 31:86 ers : Sad) | 26.2210) Maen SVE alle hanes : S78 ldP.A, JP.B ......... 62:37 a | 60°73 | 6177¢ | 61344 61-2 5 61:396 | 59-986 | ...... 61-01 I yt = GAP casos swe stent 1230G ay. German East African 1 70°31 A \ ShbobOe el Mododod sul apeoge. lb saodacc rai 3 © a pitchblende. 28) (LUIS BREE ae 20°04 | 19°90¢ | 1954d | 21:56d | 19954 |} 99.19 ge 19°65 6 |} ~ & =) A dace Abe aon ee 47°82a | 47°85a } 48°44 ow < 49°30 6 TEM TSC See A a RPC RS SE TNT BCE ¢ | PAUNDAGILO sy cases de seeds 20°45 a | 21'l6a 21:98 | | 90°85 b Deine b } Gieveiaevere sy) ijay) -@ mio eelae Vilicnesatelerele ~ oe * The other autunites referred to in Tables IJ. and III. were used in connexion with other work (Le Radiwm, 1910, vii. p. 295), and details of the uranium analysis need not be given. a denotes that the uranium was weighed as the green compound before con- version into pyrophosphate. OT ” ,° ‘ as uranium pyrophosphate. Cc ” ” ” ” as UeOx d ” ” ” 9 as UO,. 8 ae sodium uranate. 654 Miss Ruth Pirret and Mr. F. Soddy on the specimen of Joachimsthal pitchblende containing a much larger proportion of uranium. “ G.H.A.P.” was a German Hast-African pitchblende. The autunites were all specimens of Portuguese autunite. A specimen of the new mineral pulbarite was obtained from Mr. Simpson of the West Australian Government Survey. It is described in the ‘Australian Mining Standard,’ 7/9/10 (Chem. News, 1910, cli. p. 283). Method of Estimation of Radium.—The radium was esti- mated in the solution of the mineral by the emanation method. The apparatus used was at first (series A) that described by one of us (Phil. Mag, 1909 [6] xvii. p. 846). Later a new apparatus was employed with a microscope of about four times less magnifying power. With this instrument the leaf was charged positively instead of negatively, and was not kept charged during the period of three hours pre- ceding the measurements (series B and CC). A period of about three months elapsed between series Band U. In all twenty-seven radium preparations were used including the six old pitchblende standards already described (loc. cit.), representing twelve different minerals. These were pre- pared by dissolving quantities of minerals which could be accurately weighed, and taking a known fraction by weight. of the solution. Table IL. shows the results. Column (1) gives the desig- nation of the preparation, those bracketed being of the same specimen of the same mineral. Column (2) gives the number of milligrams of uranium in the preparation. Columns 3, 4, 5 give the number of divisions of leak of the electro- scope-leaf per minute in the three respective sets of observations, and 6, 7, 8 the same per milligram of uranium present. The old standards J. to VI. had existed for nearly four years, and comparing the results in Table II. with the deter- minations previously published (loc. cet.) made at the time of preparation, the variations among the standards are rather larger than they were initially, but do not show any certain influence of ave. “RI” and “RIL” were made up on June 16, 1910, from a solution of radium bromide obtained from Professor Rutherford, and described by him as con- taining 1570x10-” gram of radium. ‘The Rutherford- Boltwood ratio is assumed in stating the quantity of uranium in these (3-4 x 1077 gram of radium per gram of uranium ). Ratio between Uranium and Radium in Minerals. DABiE ET: 690 Portuguese Autunites. Specimen. Pilbarite .. eutee eorsscoe sereoese escse eoece ewessece Uranium Qmilligrms). | Leak (divisions per 1 min.). eercoe eercoe on eee eocsen | Leak per 1 min. per 1 milli- gram Uranium. e@oeees ecsccen eorcece e@oecce eseccoe ececee eecoee oerece €eceee eoscoe 67°44 eescee eecroe eo-cee eeecee ecoree eereee eet eee eo cee [12°54] 14:17 14°43 14-42 eoeeed eorcee CG. [12:98] 13°53 14 (Hl Discussion of Results —In interpreting the results of Table II. have not all equal weight. B and C series. it must be remembered that the determinations The specimens on which the A series of measurements was performed contained as a rule too small quantities of radium to give the best results with the less sensitive electroscope and method employed in the The measurements of Table II. enclosed in square brackets may be at once eliminated from further con- sideration on this account. In addition many of the solutions had been prepared a considerable time, and may have undergone changes by keeping. herecen the measurements in the B and C series, and it is evident from the Table that the sensitiveness of the instrument had somewhat increased in the interval, before with the old instrument. Three months elapsed as has been observed i Hy i 656 Miss Ruth Pirret and Mr. F. Soddy on the The first thing is to reduce the three series of measurements i given in the last three columns of Table Il. to a common i standard. This has been done by taking first the mean valne ‘ in each of the three series for each of the minerals, except : the autunites, and then taking the mean of these means. The 4 result is, in divisions per milligram of uranium :— i) t A 68-00, B37 9, C 14:24. i ie } Hence to reduce all the determinations to the same standard , as the C series, the A series must be multiplied by the factor i 0:212, and the B series by 1:033. This has been done in ; Table IIT. | i TaBLeE III. a Divisions per milligram of Uranium | (reduced values). t | | | | / i | | A (cor.) | B(cor.). | C. Means. | See MGs ere CSE AN LAST Lo | | i So | Tea a Sian I hae \ r Battles Sean hpeen So 13-66 1423 || | é Sire Ol ee ee Teak esrecsec 1418 14:14 ioe : eee Aes Ca Ree |) SENS oe 1382 Meee monn poe | | Ske ede pane Rane 13662 | AS53419 | } | EG ae eeme 12 64 1282 | 1271 12:72 5) UE Pea eee ASO YA. Pis5ee | i pees Wie ||| ee: 13-59 ie |} em | ee a ee EEE EE Es | G.E.A.P Le a610 16 At AGG 16-22 | | aces a a Se = B Spe Riba ae eve A gl ee WS aN HL 8 d. A | TE en de 15°20 1462 i t oe coi | 14:89 14°89 1462 | a eee ee eee 14-88 14-41 14-60 ‘ BB BOs eA EN eee ee 14°47 i Bb, sb eeceicslipay 4m 13°68 A DR KD) -.2)-cccl, Gyles ae 13°51 13:55 ‘| mec | er 13°56 13:46 BBE foe o2 ct sceall/ |) eae a ee 13°14 AG OS SRR 6°20 Cedi ES. \paciaebe \ a 0 Th Re ce 6°05 2 Oe i ere | PAPA eecces wen .c5 2 9°80 SUC SUE is Walla ier BR es 65 | 7a Ct ae L715 LN oth 0 eS 6°12 BS gene's Ja\ve a Weck Sk AE an ART He AR i bg Ge 3°33 | CSO Weer ing dks dan Wosski-s 10-22 |J Pilbarite s+. il oma BEA RR 8:00 Ratio between Uranium and Radium in Jhinerals. 657 Dealing first with the pitchblendes, the mean results for the old standards are :— A 13°83, B i374, Ole a which gives a mean result for this mineral of 13°85. The mean of the three determinations for the mineral “J.P. A ” and “J.P.B” is 13°77. So that from these two results the mean value 13°8 may be taken with considerable confidence as representing Joachimsthal pitchblende. The specimen PI, giving a mean result of 12-72, must be rejected, for, as will be seen from Table I., its uranium analysis is far from satisfactory. Indeed this mineral proved most trouble- some to analyse, the uranium content being low and the proportion of foreign constituents high. ‘The mean of the four determinations with Rutherford’s radium standard is 14:9; so that, if this is taken as the primary standard, the ratio of radium to uranium in Joachimsthal pitchblende is 3°15(x10-‘). The original value given by Rutherford and Boltwood was 3°8, which was lowered subsequently to 3-4 owing to an error in the uranium analysis (Boltwood, Am, Journ. Sci. 1911, xxv. p. 296). The radium solution pro- vided by Professor Rutherford was part of the original employed by these investigators; so that, assuming the solution has not changed since its preparation, our results indicate that the corrected value is still somewhat high. The value we have arrived at, 3°15( x 10"), is in good agreement with the following results of Mlle. Gleditsch, obtained with Mme. Curie’s standards of radium (Mme. Curie, Radioactivité, i. p. 441):—St. Joachimsthal pitchblende 3°21, Norwegian Cléveite 3°23, Broggerite 3°22, Portuguese chalcolite 3°24. Dealing now with the thorianites, the Table shows that although the first specimen investigated, “ThI,” as recorded in the last paper, gives an undoubtedly higher value than Joachimsthal pitchblende, it is alone of those examined in this respect. Five different uranium analyses and six esti- mations of the radium in three solutions of this mineral give the mean result 14:6, which is about 6 per cent. higher than that for Joachimsthal pitchblende. Even if the highest uranium result and the lowest radium result are compared, the value arrived at would still be as high as the mean for Joachimsthal pitchblende. But the other thorianites examined give an entirely different result. The mean five determi- nations on the three minerals is 13°47. Omitting “Th E,” for which only one determination has so far been done, the mean is 13°55. Hence the later results have not confirmed 6598 Ratio between Uranium and Radium in Minerals. these of Mlle. Gleditsch that the ratio for thorianite is 20 per cent. higher than for pitchblende. It is interesting to notice that investigations published subsequently to Mlle. Gleditsch’s work (Marckwald, Ber. Chem. Ges. 1910, xh. p. 3420; Soddy, Trans. Chem. Soc. 1911)) xGime p. 72) show that in separating the radium she must also have separated the mesothorium quantitatively from the thorianite, though this does not account for her results. We are inclined to ascribe the undoubtedly high value obtained for “ Th I” to contamination with radium before it came into our hands. All the other specimens were samples of large quantities purchased direct from the importers, but * Th I” was obtained from a retail dealer who handles radium preparations and sells spinthariscopes. We attach no im- portance to the high result for the single specimen of German East-African pitchblende, as the same doubt arises. It came to us through Mr. Russell from Prof. Marckwald’s laboratory, where chemical work on radium has long been carried on. With regard to the autunites, all of which came from various mines in Portugal, the values range from that of “AD,” which has 74 per cent. of the equilibrium amount, to that of “AS,” which has only 24 per cent. and is the lowest vet recorded, that described by Mr. Russell (Nature, Aug. 25, 1910) from Autun, France, having 27 per cent. It is inter- esting to note that both these specimens “ AD” and “AS” were from the same property {compare Ann. Reports, Chem. Soc. 1910, vii. p. 264). The pilbarite is also low (64 per cent.), but the mineral, described as probably a hydrous pseudomorph of an anhydrous parent mineral, is evidently much altered. The main point at issue, that possibly the life-period of ionium is sufficiently extended to cause the equilibrium ratio of radium to uranium to be less in a geologically recent mineral like Joachimsthal pitchblende than in an ancient mineral like thorianite, although no doubt still an open one, certainly receives no support from these measurements. Physical Chemical Laboratory, University of Glasgow. March 22nd, 1911. fy 2659 - J LXAXVIIT. An Apparent Softening bE RM on Mane abe mission through Matter. By CHARLES A. SADLER, D.Sc., Oliver Lodge Feilow, and ae I. STEVEN, 1. AN ee SCa, Lecturer in Physics, U niversity of Liverpool*. LARGE proportion of the rays emitted by the anti- cathode of an ordinary X-ray tube oe of necessity be absorbed in their passage through the glass walls, and consequently the nature of the emergent radiation is some- what modified. A recognition of this fact has led various investigators to place a thin aluminium window in the walls of the tube, and to examine the nature of the radiation pro- ceeding through it. In particular, Kaye found that, with such a bulb, the amount of the radiation emitted for a given potential difference varied with the nature of the anticathode used, and also gave data which indicated that in some cases a fairly homogeneous beam was emitted. Subsequently it was pointed out that these beams were largely composed of the homogeneous radiation characteristic of the particular metal used as anticathede, but at the same time there was also present a certain amount of scattered radiation. If a piece of glass, 1 mm. thick, were placed in the path of such a beam, it would practically absorb the whole of the homogeneous radiation together with the softer constituents of the scattered, the remainder being similar in character to the radiation from an ordinary bulb. Should the characteristic radiation, however, be very easily absorbed, and of no great intensity, a comparatively thin layer of glass or other sub- stance would suffice to ent out this portion, but the beam would still contain components which are much softer than those present under ordinary circumstances. If these conditions can be experimentally realized, it is obvious that the issuing beam approximates much more closely to the scattered radiation as it leaves the anticathode itself. Now, as the characteristic radiation of aluminium is known to be very soft and of feeble intensity compared with the scattered radiation, it seemed desirable to investigate the nature of the radiation proceeding from a bulb having a thin aluminium window and fitted with an aluminium anticathode. The scope of the inquiry was limited to a measurement of the penetrating power and heterogeneity of the rays emitted at different stages of exhaustion, but here an ee difficulty arose. When the beam had reached a certain penetrating power, it appeared to become softer on cutting * Communicated by the Authors. The expenses of this research have been partially defrayed by a Government erant through the Royal Society. + Phil. Trans. A. ccix, pp, 128-161. 660 Dr. Sadler and Mr. Steven on an Apparent Softening off part of the radiation by sheets of different substances. As the explanation of this softening was not manifest, it was decided to make it the subject of further study. ies di , Lead Sera The arrangement of the apparatus employed is shown in fig. 1, drawn to scale. The bulb of the usual spherical form had sealed into it three side-tubes all in one plane, two of which were at opposite ends of a diameter, while the third was along a radius perpendicular to the others. One of these tubes had a fixed aluminium cathode C, curved so as to con- centrate the cathode rays on the anticathode B placed at the centre of the bulb. A thin sheet of aluminium (‘00367 em.) was fixed between two similar brass disks. One of these was soldered to a tube which slid on the glass tube facing the anticathode. The radiation under examination was limited to two beams, which passed through two holes (‘4 cm. diameter) in the brass disk situated symmetrically with respect to the axis of the tube, the distance between their centres being 1°5 cm. A lead screen prevented any stray radiation from the bulb from entering the two electroscopes (of the usual Wilson type), E,, E,, which were used for testing the radiation. In the lead screen were two holes (‘5 cm. diameter) corre- sponding exactly to the two holes in the brass plate. The of Réntgen Rays in Transmission through Matter. 661 radiation entered the two electroscopes by precisely similar holes. A guide was placed at S, so that absorbers could be placed and replaced in exactly the same position as required. Between the bulb and the pump a tube containing charcoal was connected, and by surrounding this with liquid air the exhaustion was facilitated. When the liquid air had been in position for some time, the bulb reached a steady state. The nature of the radiation from the bulb was then tested by placing sheets of different substances at S, and the amount by which the rays were absorbed could be deduced from the readings of the two electroscopes. When the radiation was cut down by sheets of aluminium and the absorbability of the remainder tested by a thin sheet of aluminium (:V0305), the beam appeared very heterogeneous as shown by the following table :— TaBLe I. | Previous per cent. absorption | Subsequent per cent. absorption | by Aluminium sheets. by Aluminium test-piece. : 0 | 34°3 87-4 | 26-9 56°53 23 75'S | 17°38 | 80°6 19) 90 0 | 10-4 When, however, the beam was cut down by sheets of copper and the remainder still tested with the same aluminium sheet, it now appeared fairly homogeneous, as shown in Table IT. | TasBLe II. | | Previous per cent. absorption | Subsequent per cent.absorption. | by Copper sheets. . by Aluminium test-piece. 0 | 316 48°3 | Ale | 73 | 30°0 | | In general, it was found that when the beam was cut down by a substance which, under the stimulus of a suitable Réntgen radiation, emits a characteristic homogeneous radiation con- siderably in excess of that which it scatters, e.g. Ni, Pe, &e,, 662 Dr. Sadler and Mr. Steven on an Apparent Softening results similar to that for copper were obtained. On the other hand, if cut down by substances of low atomic weight, which chiefly scatter incident radiation, e. g. C, Al, &e., or by substances of higher atomic weight, e. g. Au, in whic the characteristic radiation is not excited except by very penetrating beams, the results were similar to those given in ‘lable I. It. might be supposed that these effects were due to secondary radiation superposed on the primary beam, but direct experiment showed that when the beam was cut deme by copper or iron at 8, the amount of secondary radiation entering either of the electroscopes EH, or EH, would not account for 1 per cent. of the observed ionization. The use of the charcoal cooled by liquid air as an aid to exhaustion had however its disadvantages. After the bulb had been running for a considerable time the discharge tended to become intermittent, and the readings obtained were very irregular. The charcoal was therefore dispensed with, and the pump alone depended on for exhaustion. Under these new conditions, the discharge at any stage appeared much more regular, and there was the added advantage of being able to investigate the rays at an earlier stage in the exhaustion. By continuing the pumping, higher stages of exhaustion could be reached, and the process, although slower, proved more reliable. The beam was now tested by a thin sheet of aluminium (00305 cm.), and the absorption was found both before and after cutting down the beam by iron (‘00124 cm.). Witha moderately penetrating beam the absorption by the aluminium was considerably greater after transmission through the iron than before transmission, and this apparent softening was ereater as the initial primary beam became still harder. This is clearly shown in the following table :— TasueE IIT. | . | | Per cent. absorp- Per cent. absorp- Hobe “Ady -00305) tion by Al (‘00305)) Per cent. absorbed | Per cent. increase a es after cutting down) by the iron sheet | in absorption by | beam primary beam | (00124 em.). the aluminium. | ; by iron. ! 48°38 Sar 7 65 14-2 | 45°5 B72 67 25°8 | 30°4 46-0 63°8 51-3 2it 39°6 54:5 85-0 | of Rintgen Rays in Transmission through Matter. 663 Similar results were found when the beam was cut down by nickel and copper, aluminium still being used as test-substance. When the beam was cut down by aluminium and tested by aluminium, the absorption decreased slightly at first, but more rapidly as successive sheets were placed in the primary beam. TABLE IV. aaa | | Per cent. previously absorbed | Per cent. subsequently absorbed by Al sheets. by Al (00305). 0 | 27°2 307 | 268 Ogio 20a 88-0 | 12°8 | If the beam was eut down by aluminium and tested by nickel, the same apparent softening occurred. TABLE V. Per cent. previously absorbed | Absorption by a Nickel sheet by Al sheets. (00095 cm.). 0 | 45°7 30 55°4 0 | 47-9 As this softening had not been observed when the charcoal tube cooled by liquid air was used, it was again resorted to in order to bring the bulb to the same stage of hardness. The observations showed that the phenomena were similar but not so pronounced. Discussion of Results. When a primary beam passes through a thin sheet of an element, there is a loss of energy (measured by the decrease in the ionization it is able to produce in a given volume of air) which may be due to :— : A scattering of a portion of the incident energy ; . A transformation of energy into the production of. 3 homogeneous radiation characteristic of the leicna” . A production of a corpuscular radiation accompanying both the scattered and the homogeneous radiation. 664 Dr. Sadler and Mr. Steven on an Apparent Softening Jt has been shown®* that this characteristic radiation is only produced by a more penetrating radiation, and the increase in absorption accompanying its production is a maximum for an exciting beam only slightly more penetrating. In Table VI.* are given the absorption coefficients in different substances of the characteristic radiations of the elements Cr: Ag. From columns II. and III. it will be seen that these radiations are in increasing order of penetrating power when tested by elements in which they excite little or no characteristic radiation; and from column IV. that the absorption coefficient in iron is a maximum for nickel radiation. Tasie VI. Mass Absorption Coefficients (A/p). | ABSORBER, | | RapIaTor. | | C. Al. Fe. Ni. Cn,)4/\) sane Calis at th 153 | 1560 | 1088 | 129 | 143° |) a7ome 1 Ste nee alee! 10-1 885)|. (661) | 888 | ob aie Cane ene 796°) TVG) 6n2 | 67:2 | vos eee In nae 658 | 5017) Blt |. 563 | *bI-80 | eaeees WAC ches enacts Sa D2 Maret) 26S 0) | 62:7.) D3s00) aaa Ze es ALG 30th) 265 555 | “50 Eee ee 9:49) 20584 | 166. | 176. |) eae (gSon Geena. 2-04 189) 11638 | 1413 | 1493" aia Pee Haken eros 4] 25 A 2279) ONES 27-1 We should expect that if we placed a sheet of iron in our heterogeneous primary beam, it would specially absorb the constituents of about the penetrating power of nickel radia- tion, and more penetrating constituents to a lesser extent. The portion of the beam too soft to excite iron radiation would be absorbed only to a limited extent. For example, it will be seen that a constituent of the hardness of chromium radiation, of which the absorption coefficient in aluminium is 136, would be absorbed to a less extent by iron than that of the radiation from selenium, the absorption coefficient of which in aluminium is only 18°9. On the whole, then, the beam after passing through iron would be richer propor- tionately in these softer rays than before, but these latter are much more easily absorbed by aluminium than those specially absorbed by iron, and therefore we would expect the beam to be softer to aluminium after. passing through iron than before. * Barkla & Sadler, Phil. Mag. May 1909, pp. 739-76 of Réntgen Rays in Transmission through Matter. 665 This explanation of the phenomena is borne ont by the following experiments. \ Fig. 2. An electroscope E; (fig. 2) was placed with its aperture parallel to the primary beam. A guide 8’ was arranged so that different substances could be placed in the path of the beam entering E,. The secondary radiations excited in these could be measured by E3. The ionization in electroscope H, serving as a standard. Strips of Ti, Cr, Fe, Cu, Zn were used as radiators. The amounts by which the ionization in E; was diminished when sheets of iron and aluminium respectively (absorbing the same percentage of the primary) were put at 8, were noted. These diminutions in the secondary radiation indicated which constituents of the primary were cut off by the iron and aluminium respectively. The results are tabulated below. TasieE VII. Per cent. by Per cent. by A/o for | which Al in pri-| which Fe in pri- \/p for Secondary| secondary | mary (cutting off| mary (cutting off | secondary | Radiator. | radiation | 63 percent.) cuts |62°8 percent.) cuts| radiation | in Al. down secondary} down secondary in Iron. | radiation. radiation. | Tae sees 230 68:1 542 173 0) ee 136 63°7 62 104 MO, Sssaaeee 88°5 53°5 69°6 66:2 Chee, 47-7 45:3 es: 268 AiWertesscaeiss 39°4 42:3 46:2 221 ~ Phil. Mag. 8.6. Vol. 21. No. 125. May 1911, 2X 666 Dr. Sadler and Mr. Steven on an Apparent Softening It will be seen from column IV. that the iron specially cuts down the secondary radiation from iron as predicted, 2. e., the iron selectively absorbs those constituents of the primary capable of exciting radiations in itself. It will be seen also that the softer radiations, too soft to excite iron radiation but capable of producing considerable radiation — from Ti and Cr, are cut down by the iron to a lesser extent than the average. These latter, on the other hand, are cut down more readily by the aluminium than those which specially excite on iron (see column III.). It is interesting in this connexion to show the subsequent absorption by aluminium of the primary beam after being cut down by iron and aluminium respectively to the same extent (63 per cent.). Of the initial beam aluminium ‘00305 absorbs 28:9 °/, 9 rh 9 cut down by Fe or) 9? 93 41-4 = ” ” ” ye Al ” ” 9 222% The softening of the primary beam when cut down by various substances and tested by a thin sheet of a substance other than aluminium, has also been observed. The following two examples indicate, however, that the occurrence of the effect depends on certain relations existing between the radiations characteristic of the absorber and of the test substance. | I. Cutting down the primary beam 60°3 per cent. by iron: Per cent. absorbed by test substance: Per cent. of Before cutting After cutting secondary from Test Substance. down by Iron. down by Iron. test substance cut down by Iron. IN rey eke 49 56°6 D4°9 te eRe ene 60 2°6 65 Il. Cutting down the primary beam 50:7 per cent. by nickel : Per cent. absorbed by test substance: Per cent. of Before cutting After cutting secondary from Test Substance. down by Nickel. down by Nickel.” test substance cut down by Nickel BN inte hie ATT 46-2 62°7 A ee a 60°3 67:9 52 From what has been said in a previous portion of the of Iéntgen Rays in Transmission through Matter. 667 paper, it will be gathered that a primary beam, after passing through a thin sheet of i iron, is deficient in those constituents which are especially capable of exciting iron radiation. If, then, the issuing beam is absorbed by a further sheet of iron of the same thickness, we should expect the second sheet to absorb much less. This is well shown in the first example above, for the iron sheet which previously absorbed 60 per cent. now only absorbs 52°6 per cent. A similar line of reasoning explains why the beam is apparently harder to nickel after passing through nickel. An examination of Table VI. shows clearly that nickel absorbs those constituents which especially excite the charac- teristic radiation of iron (¢. g. those of the hardness of nickel and copper radiations) to a much less extent than it absorbs more penetrating radiations (e. g. those of the hardness of Zn, As, Se, &ec.). But Table VIL. column IV. clearly shows that the beam after passing through iron is richer in those con- stituents which can be specially absorbed by nickel, for these are cut down to a lesser extent than the average. As a further test of this point, the secondary radiations from iron and nickel respectively were measured before and after cutting down the primary beam by iron. A reference to the last column of the first example shows that while the whole beam is reduced by 60 per cent., the constituents specially capable of exciting secondary homogeneous radia- tion in nickel are only reduced by 54°9 per cent. ; it also well illustrates the fact that iron selectively absorbs those con- stituents specially exciting radiation in itself—this absorption 65 per cent. being above the average. The results given in example ie equally confirm these views. The phenomena so far described can be readily duplicated with a beam composed of suitable proportions of various homo- geneous radiations. Jor instance, a beam composed of the homogeneous radiations from iron, nickel, zinc, and arsenic each equally contributing to the ionization produced, shows the following properties when tested by the sheets used in these experiments :-— A sheet of nickel absorbs of the composite beam 63°53 per cent., but after transmission, the absorption by a similar sheet of nickel falls to 52 per cent.; on the other hand, if the beam had previously passed through i iron (cutting off 76°2 per cent.). the subsequent absorption by the same sheet of nickel rises to 78°6 per cent. A sheet of aluminium. absorbing 33:7 per cent. of the 2X 2 i disint ane all enines SS SS Se meng teh ether 668 Softening of Leéntgen Rays in Transmission through Matter. composite beam, absorbs 38°6 per cent. after cutting down by iron, and 38-2 per cent. after cutting down by nickel. Other experiments have been carried out on similar lines, using anticathodes of different metals. In general, the phe- nomena observed are of the same kind as those already described. There are one or two outstanding features which are being further investigated, These include :—1. An ap- parent considerable softening of a primary beam from certain anticathodes when tested by aluminium after having been cut down by aluminium. 2. A variation in the components of primary beams of the same average hardness produced under different conditions. | In conclusion, we wish to point out the importance of using as a test-substance, when comparing the penetrating powers of different beams, one in which the fraction of the total absorption due to the emission of secondary characteristic radiation is small, e.g. Al, C, &e. Attention may also be directed to the use of the dis- tinguishing properties of the characteristic radiations of . various elements as affording an effective means for the analysis of heterogeneous beams. Summary. An apparent softening of a heterogeneous primary beam in the process of transmission through matter has been observed. This effect is shown to be connected with the selective absorption by a substance of those constituents of the beam, which can readily excite its characteristic homogeneous radiation. Confirmation of this view has been obtained by an analysis of the primary beam. We wish to place on record our appreciation of the kindly interest Professor Wilberforce has shown and the encourage- ment we have received from him throughout the course of these experiments. George Holt Physics Laboratory, University of Liverpool. March 27th, 1911. [ 669 J XL iL he Scattering of aand 8 Particles by Matter and the Structure of the Atom. By Professor HE. RurHeRrorD, F.R.S., University of Manchester *. § 1. PT is well known that the 2 and B particles suffer deflexions from their rectilinear paths by encounters with atoms of matter. This scattering is far more marked for the 8 than for the « particle on account of the much smaller momentum and energy of the former particle. There seems to be no doubt that such swiftly moving par- ticles pass through the atoms in their path, and that the deflexions observed are due to the strong electric field traversed within the atomic system. It has” generally been supposed that the scattering of a pencil of @ or 8 rays in passing through a thin plate of matter is the result of a multitude of small scatterings by the atoms of matter traversed. The observations, however, of Geiger and Marsden f on the scattering of @ rays tiiieate nee some of the @ particles must suffer a deflexion of more than a right angle at a single encounter. They found, for example, that email fraction of the incident « par bicles, about 1 in 20,000, were turned through an average angle of 90° in passing through a layer of gold-foil about -00004 cm. thick, which was equivalent i in SoD EO en of the « particle to 1-6 milli- metres of air. Geiger f showed later that the most probable angle of deflexion for a pall of « particles traversing a gold- foil of this thickness was about 0°87. A simple calculation based on the theory of probability shows that the chance of an a particle being deflected through 90° is vanishingly small. In addition, it will be seen later that the distribution of the « particles for various angles of large deflexion does not follow the probability law to be expected if such large deflexions are made up of a large number of small deviations. It seems reasonable to suppose that the deflexion through a large angle is due to a single atomic encounter, for the chance of a second encounter of a kind to produce a large deflexion must in most cases be exceedingly small. A simple calculation shows that the atom must be a seat of an intense electric field in order to produce such a large deflexion ata single encounter. Recently Sir J. J. Thomson § has put forward a theory to * Communicated by the Author. A brief account of this paper was communicated to the Manchester Literary and Philosophical Society in February, 1911. + Proc. Roy. Soc. Ixxxii. p. 495 (1909). t Proc. Roy. Soc. Ixxxiii. p. 492 (1910). § Camb. Lit. & Phil. Soc. xv. pt. 5 (1910). 670 Prof. EK. Rutherford on the explain the scattering of electrified particles in passing through small thicknesses of matter. The atom is supposed to consist of a number N of negatively charged corpuscles, accompanied by an equal quantity of positive electricity uniformly dis- tributed throughout a sphere. The deflexion of a negatively electrified particle i in passing through the atom is ascribed to two eauses—(1) the repulsion of the corpuscles distributed through the atom, and (2) the attraction of the positive electricity in the atom. ‘The deflexion of the particle in passing through the atom is supposed to be small, while the average deflexion after a large number m of encounters was taken as 1/m.6, where @ is the average deflexion due toa single atom. It was shown that the number N of the electrons within the atom could be deduced from observations of the scattering of electrified particles. The accuracy of this theory of compound scattering was examined experimentally by Crowther* in a later paper. His results apparently confirmed the main conclusions of the theory, and he deduced, on the assumption that the positive electricity was continuous, that the number of electrons in an atom was about three times its atomic weight. The theory of Sir J. J. Thomson is based on the assumption that the scattering due to a single atomic encounter is small, and the particular structure assumed for the atom does ae admit of a very large deflexion of an @ particle in traversing a single atom, ‘nnless it be supposed that the diameter of the sphere of positive electricity is minute compared with the diameter of the sphere of influence of the atom. Since the a and £ particles traverse the atom, it should be possible from a close study of the nature of the deflexion to form some idea of the constitution of the atom to produce the effects observed. In fact, the scattering of high-speed charged particles by the atoms of matter is one of the most promising methods of attack of this problem. The develop- ment of the scintillation method of counting single « particles affords unusual advantages of investigation, -and the researches of H. Geiger by this method have already added much to our Eewledee of the scattering of a rays by matter. § 2. We shall first examine theoretically the single en- counters f with an atom of simple structure, which is able to * Crowther, Proc. Roy. Soc. lxxxiv. p. 226 (1910). + The deviation of a “particle throughout a considerable angle from an encounter with a single atom will in this paper be called “ single” scattering. The deviation of a particle resulting from a multitude of small deviations will be termed ‘ ‘compound ” scattering. ' ‘ , Scattering of a and.8 Particles by Matter. 671 produce large deflexions of an e particle, and then compare the deductions from the theory with the experimental data available. Consider an atom which contains a charge +Ne at its centre surrounded by a sphere of electrification containing a charge = Ne supposed uniformly distributed throughout a sphere of radius R. e is the fundamental unit of charge, which in this paper is taken as 4765x107! Es. unit. We shall suppose that for distances less than 107” cm. the central charge and also the charge on the a particle may be sup- posed to be concentrated at a point. It will be shown that the main deductions from the theory are independent of whether the central charge is supposed to be positive or negative. For convenience, the sign will be assumed to be positive. The question of the stability of the atom proposed need not be considered at this stage, for this will obviously depend upon the minute structure of the atom, and on the motion of the constituent charged parts. In order to form some idea of the forces required to deflect an « particle through a large angle, consider an atom containing a positive charge Ne at its centre, and surrounded by a distribution of negative electricity Ne uniformly dis- ° tributed within a sphere of radius R. The electric force X and the potential V at a distance r from the centre of an atom for a point inside the atom, are given by Bs Ie r —— Ne € _ =) 18 Eee V=Ne(_ OR sm Suppose an « particle of mass m and velocity u and charge I shot directly towards the centre of the atom. - It will be brought to rest at a distance 6 from the centre given by 1 3 b? SUNT oN. ce teens gmu"= Nek (; aprons an) It will be seen that 6 is an important quantity in later calculations. Assuming that the central charge is 100e, it can be calculated that the value of 6 for an a particle of velocity 2°09 x 10° cms. per second is about 3:4 x 10° em. In this calculation 6 is supposed to be very small compared with R. Since R is supposed to be of the order of the radius of the atom, viz. 10-8 em., it is obvious that the a particle before being turned back penetrates so close to ————————I 672 Prof. E. Rutherford on the the central charge, that the field due to the uniform dis- tribution of negative electricity may be neglected. In general, a simple calculation shows that for all deflexions greater than a degree, we may without sensible error suppose the deflexion due to the field of the central charge alone. Possibie single deviations due to the negative electricity, if distributed in the form of corpuscles, are not taken into account at this stage of the theory. It will be shown later that its effect is in general small compared with that due to the central field. Consider the passage of a positive electrified particle close to the centre of an atom. Supposing that the velocity of the particle is not appreciably changed by its passage through the atom, the path of the particle under the influence of a repulsive force varying inversely as the square of the distance will be an hyperbola with the centre of the atom S as the external focus. Suppose the particle to enter the atom in the direction PO (fig. 1), and that the direction of motion Fig. 1. Pp’ on escaping the atom is OP’. OP and OP!’ make equal angles with the line SA, where A is the apse of the hyperbola. p=SN=perpendicular distance from centre on direction of initial motion of particle. Scattering of a and.8 Particles by Matter. 673 Let angle POA=8@. Let V=velocity of particle on entering the atom, v its velocity at A, then from consideration of angular momentum DN 0. 0. Irom conservation of energy Nek tmV?7=tmv?— A? b y= V? (1- si) Since the eccentricity is sec 0, SA=S0+0A=pcosec 6(1+ cos @) =p cot 0/2, p?=SA(SA—b) =p cot 0/2(p cot 0/2 —4), ae otie: The angle of deviation ¢ of the particle is 7—20 and cot g/2 =P OMe eet sa a? ta? CL) This gives the angle of deviation of the particle in terms of b, and the perpendicular distance of the direction of projection from the centre of the atom. For illustration, the angle of deviation ¢@ for different values of p/b are shown in the following table :— ibe eo) 10 5 Oe 1 ees Gun yc 15 GS Deedee DSO) Sn Oma bagae 1)2° § 3. Probability of single deflexion through any angle. Suppose a pencil of electrified particles to fall normally on a thin screen of matter of thickness ¢. With the exception of the few particles which are scattered through a large angle, the particles are supposed to pass nearly normally through the plate with only a small change of velocity. Let n=number of atoms in unit volume of material. Then the number of collisions of the particle with the atom of radius R is wR?nt in the thickness ¢. * A simple consideration shows that the deflexion is unaltered if the forces are attractive instead of repulsive. 674 Prof. E. Rutherford on the The probabilty m of entering an atom within a distance p of its centre is given by n= mpnt. Chance dm of striking within radii p and p+dp is given b yi dm=2rpnt .dp = nb? cot d/2 cosec? ¢/2 dg, . (2) since cot 6/2 =2p/6. The value of dm gives the fraction of the total number of particles which are deviated between the angles @ and o+dd¢. The fraction p of the total number of particles which are deflected through an angle greater than @¢ is given by p = znil? cot? /2. -« | ) The fraction p which is deflected between the angles 9, and o, is given by p = ntl? (cot? $ — cot? ees It is convenient to express the equation (2) in another form for comparison with experiment. In the case of the a rays, the number of scintillations appearing on a constant area of a zinc sulphide screen are counted for different angles with the direction of incidence of the particles. Let » =distance from point of incidence of a rays on scattering material, then if Q be the total number of particles falling on the scattering material, the number y of « particles falling on unit area which are deflected through an angle } is given by be Qdm __ nth’. Q . cosec* g/2 (5 I~ Ia sin &.db 1672 am Qf Since p= et, we see from this equation that the number of « particles (scintillations) per unit area of zine sulphide screen at a given distance 7 from the point of ~] Or Scattering of 2 and B Particles by Matter. 6 incidence of the rays is proportional to (1) cosect 6/2 or 1/p* if d be small ; (2) thickness of scattering material ¢ provided this is | small ; (3) magnitude of central charge Ne ; (4) and is inversely proportional to (mu?)?, or to the fourth power of the velocity if m be constant. In these calculations, it is assumed that the « particles scattered through a large angle suffer only one large deflexion. For this to hold, it is essential that the thickness of the scattering material should be so small that the chance of a second encounter involving another large deflexion is very small. If, for example, the probabiiity of a single deflexion @ in passing through a thickness ¢ is 1/1000, the probability of two successive deflexions each of value ¢ is 1/10°, and is negligibly small. The angular distribution of the « particles scattered from a thin metal sheet affords one of the simplest methods of testing the general correctness of this theory of single scattering. This has been done recently for « rays by Dr. Geiger *, who found that the distribution for particles deflected between 30° and 150° from a thin gold-foil was in substantial agreement with the theory. A more detailed account of these and other experiments to test the validity of the theory will be published later. § 4. Alteration of velocity in an atomic encounter. It has so far been assumed that an & or 8 particle does not suffer an appreciable change of velocity as the result of a single atomic encounter resulting in a large deflexion of the particle. The effect of such an encounter in altering the velocity of the particle can be calculated on certain assump- tions. It is supposed that only two systems are involved, viz., the swiftly moving particle and the atom which it traverses supposed initially at rest. It is supposed that the principle of conservation of momentum and of energy apples, and that there is no appreciable loss of energy or momentum by radiation. * Manch. Lit. & Phil. Soc. 1910. 676 Prof. E. Rutherford on the Let m be mass of the particle, v, = velocity of approach, Vy = velocity of recession, M = mass of atom, V = velocity communicated to atom as result of encounter. Let OA (fig. 2) represent in magnitude and direction the momentum mv, of the entering particle, and OB the momentum of the receding Fig. 2. particle which has been turned through an angele AOB=¢. Then BA represents in magnitude and direction the momentum MV of the recoiling atom. »B (MV )? = (mv)? + (sve)? —2m?v,v, cos d. (1) By the conservation of energy MN 2a ee ces a2) Suppose M/m=K and v.=pv,, where A DNS << she From (1) and (2), (K+1)p?—2p cos = K—1, _ cosh if [oe ot or P= ka Kem / K?—sin? d. Consider the case of an @ particle of atomic weight 4, deflected through an angle of 90° by an encounter with an atom of gold of atomic weight 197. Since K=49 nearly, gan aero or the velocity of the particle is reduced only about 2 per cent. by the encounter. In the case of alummium K = 27/4 and for 6= 907 = 7010, j It is seen that the reduction of velocity of the « particle becomes marked on this theory for encounters with the lighter atoms. Since the range of an a@ particle in air or other matter is approximately proportional to the enbe of the velocity, it follows that an a particle of range 7 cms. has its range reduced to 4°5 cms. after incurring a single ws): a —— - Scattering of a and 8 Particles by Matter. O77 deviation of 90° in traversing an aluminium atom. This is of a magnitude to be easily detected experimentally. Since the value of K is very large for an encounter of a 8 particle with an atom, the reduction of velocity on this formula is very small. Some very interesting cases of the theory arise in con- sidering the changes of velocity and the distribution of scattered particles when the a@ particle encounters a light atom, for example a hydrogen or helium atom, A discussion of these and similar cases is reserved until the question has been examined experimentally. § 5. Comparison of single and compound scattering. Before comparing the results of theory with experiment, it is desirable to consider the relative importance of single and compound scattering in determining the distribution of the scattered particles. Since the atom is supposed to consist of a central charge surrounded by a uniform distribution of the opposite sign through a sphere of radius R, the chance of encounters with the atom involving small deflexions is very great compared with the chance of a single large deflexion. This question of compound scattering has been examined by Sir J. J. Thomson in the paper previously discussed (§ 1). In the notation of this paper, the average deflexion ¢, due to the field of the sphere of positive electricity of radius R and quantity Ne was found by him to be $ Me Nek ey er) ee a The average deflexion ¢; due to the N negative corpuscles supposed distributed uniformly thr oughout the sphere was found to be Hoveliy” Te ail 5 mu? ht 2 d2 = The mean deflexion due to both positive and negative electricity was taken as (p+ bo) Ina similar way, it is not difficult to calculate the average deflexion due to the atom with a central charge discussed in this paper. Since the radial electrie field X at any distanee » from the 678 Prof. E. Rutherford on the centre 1s given by y R? it is not difficult to show that the deflexion (supposed small) of an electrified particle due to this field is given by X=Ne(3 a) where p is the perpendicular from the centre on the path of the particle and 6 has the same value as before. It is seen that the value of @ increases with diminution of p and becomes great for small values of ¢. Since we have already seen that the deflexions become very large for a particle passing near the centre of the atom, it is obviously not correct to find the average value by assuming @ is small. Taking R of the order 10-%em., the value of p for a large deflexion is for a and 8 particles of the order 10~™ em. Since the chance of an encounter involving a large defiexion is small compared with the chance of small deflexions, a simple consideration shows that the average small deflexion is practically unaltered if the large deflexions are omitted. This is equivalent to integrating over that part of the cross section of the atom where the deflexions are small and neglecting the small central area. It can in this way be simply shown that the average small deflexion is given by 3c 6b Pie: This value of ¢, for the atom with a concentrated central charge is three times the magnitude of the average deflexion for the same value of Ne in the type of atom examined by Sir J. J. Thomson. Combining the deflexions due te the electric field and to the corpuscles, the average deflexion is 3 OS eae 15°4\1/2 (p+ 2")? or gp (554+ “KY It will be seen later that the value of N is nearly proportional to the atomic weight, and is about 100 for gold. The effect due to scattering of the individual corpuscles expressed by the second term of the equation is consequently small for heavy atoms compared with that due to the distributed electric field. Scattering of « and B Particles by Matter. 679 Neglecting the second term, the average deflexion per atom is — We are now in a position to consider the relative effects on the distribution of particles due to single and to compound scattering. Following J. J. Thomson’s argument, the average deflexion 6, after passing through a thickness ¢ of matter is proportional to the square root of the number of encounters and is given by Onl eee nt ea Ci an VaR SER ee age J nt, where n as before is equal to the number of atoms per unit volume. The probability », for compound scattering that the defiexion of the particle is greater than ¢ is equal to pa ies 3 Qar Jons ntly 2— — - Consequently d GA Next suppose that single scattering alone is operative. We have seen (§ 3) that the probability py, of a deflexion greater than ¢ is given by AY L b? nt log py. P= a .n.t cot 7/2. By comparing these two equations pz log pp = —'181¢? cot */2, @ is sufficiently small that tan 6/2=/2, 2 log py= — "712. Ii we suppose p.=‘d, then p,;="24. Le iO lly p= 0004. It is evident from this comparison, that the probability for any given deflexion is always greater for single than for compound scattering. The difference is especially marked when only a small fraction of the particles are scattered through any given angle. It follows from this result that the distribution of particles due to encounters with the atoms is for small thicknesses mainly governed by single scattering. No doubt compound scattering produces some efteet in equalizing the distribution of the scattered particles ; but its effect becomes relatively smaller, the smaller the fraction of the particles scattered through a given angle. 680 Prof. E. Rutherford on the § 6. Comparison of Theory with Experiments. On the present theory, the value of the central charge Ne is an important constant, and it is desirable to determine its value for different atoms. This can be most simply done by determining the small fraction of « or 8 particles of Known velocity falling on a thin metal screen, which are scattered between @ and ¢+dd where ¢ is the angle of deflexion. The influence of compound scattering should be small when this fraction is small. Experiments in these directions are in progress, but it is desirable at this stage to discuss in the light of the present theory the data already published on seattering of a and 8 particles. The following points will be discussed :— (a) The “diffuse reflexion” of a@ particles, 7. e. the scattering of « particles through large angles (Geiger and Marsden). (b) The variation of diffuse reflexion with atomic weight of the radiator (Geiger and Marsden). (c) The average scattering of a pencil of « rays trans- mitted through a thin metal plate (Geiger). (d) The experiments of Crowther on the scattering of 8 rays of different velocities by various metals. (a) In the paper of Geiger and Marsden (loc. cit.) on the diffuse reflexion of @ particles falling on various substances it was shown that about 1/8000 of the « particles from radium C falling on a thick plate of platinum are scattered back in the direction of the incidence. This fraction is deduced on the assumption that the « particles are uniformly scattered in all directions, the observations being made for a deflexion of about 90°. The form of experiment is not very suited for accurate calculation, but from the data available it can be shown that the scattering observed is about that to be expected on the theory if the atom of platinum has a central charge of about 100 e. (}) In their experiments on this subject, Geiger and Marsden gave the relative number of a particles diffusely reflected from thick layers of different metals, under similar conditions. ‘The numbers obtained by them are given in the table below, where < represents the relative number of scattered particles, measured by the number of scintillations per minute on a zinc sulphide screen. Scattering of a and B Particles by Matter. 681 | Metal. Atomic weight. Z. | 2) A8/2- Pendie es | 207 | 62 | 208 | Go ee erecta Go). | Baca Platinum ...... Kane sai 63 Low {ps2 i eee | 119 34 26 Silver.i css os | 108 oy, | 241 WOopper- occ. | 64 145 225 LUNG) ORAS | 56 10-2 | 250 Aluminium ...! 27 | 3:4 | 243 Average 233 | On the theory of single scattering, the fraction of the total number of « particles scattered through any given angle in passing through a thickness ¢ is proportional to n.A%é, assuming that the central charge is proportional to the atomic weight A. In the present case, the thickness of matter from which the scattered @ particles are able to emerge and affect the zine sulphide screen depends on the metal. Since Bragg has shown that the stopping power of an atom for an a particle is proportional to the square root of its atomic weight, the value of nt for different elements is proportional to 1/ V/A. In this case ¢ represents the greatest depth from which the scattered a particles emerge. The number z of @ particles scattered back from a thick layer is consequently proportional to A°*” or z/A®? should be a constant. To compare this deduction with experiment, the relative values of the latter quotient are given in the last column. Considering the difficulty of the experiments, the agreement between theory and experiment is reasonably good * The single large scattering of « particles will obviously affect to some extent the shape of the Bragg ionization curve for a pencil of arays. This effect of large scattering should be marked when the a rays have traversed screens of metals of high atomic weight, but should be small for atoms of light atomic weight. (c) Geiger made a careful determination of the scattering of a@ par ticles passing through thin metal foils, by the scintillation method, and deduced the most probable angle * The effect of change of velocity in an atomic encounter is neglected in this caloultions Phil. Mag. 8 Ss wOs Vol. at: No. 1 Seay May T9LL. 2 ‘ 682 Prof. E. Rutherford on the through which the « particles are deflected in passing through known thicknesses of different kinds of matter. A narrow pencil of homogeneous a rays was used as a source. After passing through the scattering foil, the total number of a particles deflected through different angles was directly measured. The angle for which the number of scattered particles was a maximum was taken as the most probable angle. The variation of the most probable angle with thickness of matter was determined, but calculation from these data is somewhat complicated by the variation’ of velocity of the « particles in their passage through the scattering material. A consideration of the curve of distribu- tion of the e particles given in the paper (loc. cit. p. 496) shows that the angle through which half the particles are scattered is about 20 per cent greater than the most probable angle. We have already seen that compound scattering may become important when about half the particles are scattered through a given angle, and it is difficult to disentangle in such cases the relative effects due to the two kinds of scattering. An approximate estimate can be made in the following way :— From (§ 5) the relation between the probabilities p, and p, for compound and single scattering respectively is given by po log pyp=—'721. The probability g of the combined effects may as a first approximation be taken as q= (prt pe)”. If ¢="5, it follows that pi= 2 and jpo—"46. We have seen that the probability p, of a single deflexion greater than ¢ Is given by Pa= a WD COL -w/ 2. Since in the experiments considered ¢ is comparatively small p WV Po ils 2NeH NV wnt Gord aaieed? =. Geiger found that the most probable angle of scattering of the # rays in passing through a thickness of gold equivalent In stopping power to about ‘76 cm. of air was 1° 40’. The angle @ through which half the e particles are turned thus corresponds to 2° nearly. p=" OOOL cm. 5 n= 6°0T X 10” - w (average value) =1°8 x 10°. yi eO ahs eeesemmniiss (== 476 oe A) ae Scattering of « and 8 Particles by Matter. 683 Taking the probability of single scattering ='46 and substituting the above values in the formula, the value of N for gold comes out to be 97. | For a thickness of gold equivalent in stopping power to 2°12 ems. of air, Geiger found the most probable angle to be 3°40’. In this case ¢=00047, 6=4°4, and average u= 1:7 x 10°, and N comes out to be 114. Geiger showed tbat the most probable angle of deflexion for an atom was nearly proportional to its atomic weight. It consequently follows that the value of N for different atoms should be nearly proportional to their atomic weights, at any rate for atomic weights between gold and aluminium. Since the atomic weight of platinum is nearly equal to that of gold, it follows from these considerations that the magnitude of the diffuse reflexion of « particles through more than 90° from gold and the magnitude of the average small angle scattering of a pencil of rays in passing through gold- foil are both explained on the hypothesis of single scattering by supposing the atom of gold has a central charge of about Oe. (d) Experiments of Crowther on scattering of 8 rays.— We shall now consider how far the experimental results of Crowther on scattering of @ particles of different velocities by various materials can be explained on the general theory of single scattering. On this theory, the fraction of 8B particles p turned through an angle greater than ¢ is given by p= ic” .t. 0? cot? b/2. In most of Crowther’s experiments ¢ is sufficiently small that tan ¢@/2 may be put equal to $/2 without much error. Consequently Guam G. Oe) il yg De On the theory of compound scattering, we have already seen that the chance p, that the deflexion of the particles is greater than ¢ is given by Oar? "log py= ———n.t. 0. p / Si 1 64 Since in the experiments of Crowther the thickness ¢ of matter was determined for which p,=1/2, @* =" 967 n t b?. For a probability of 1/2, the theories of single and compound Be Wy 2 | 684 Prof. E..Rutherford on the scattering are thus identical in general form, but differ by a numerical constant. It is thus clear that the main relations on the theory of compound scattering of Sir J. J. Thomson, which were verified experimentally by Crowther, hold equally well on the theory of single scattering. For example, if t, be the thickness for which half the particles are scattered through an angle ¢, Crowther showed that d| Wt,» and also ae Vtm were constants for a given material when ¢@ was fixed. These relations hold also on the theory of single scattering. Notwithstanding this apparent similarity in form, the two theories are fundamentaily different. In one case, the effects observed are due to eumulative effects of small deflexions, while in the other the large deflexions are supposed to result from a single encounter. The distribution of scattered particles is entirely different on the two theories when the probability of deflexion greater than dis small. | We have already seen that the distribution of scattered a particles at various angles has been found by Geiger to be in substantial agreement with the theory of single scattering, but cannot be explained on the theory of compound scat- tering alone. Since there is every reason to believe that the laws of scattering of a and @ particles are very similar, the law of distribution of scattered 8 particles should be the same as for @ particles for small thicknesses of matter. Since the value of mu?/E for the 6 particles is in most cases much smaller than the corresponding value for the « par- ticles, the chance of large single deflexions for 8 particles in passing through a given thickness of matter is much greater than for « particles. Since on the theory of single scattering the fraction of the number of particles which are deflected through a given angle is proportional to kt, where ¢ is the thickness supposed small and & a constant, the number of particles which are undeflected through this angle is propor- tional to 1—kt. From considerations based on the theory of compound scattering, Sir J. J. Thomson deduced that the probability of deflexion less than ¢ is proportional to 1—e-#/ where pw is a constant for any given value of ¢. The correctness of this latter formula was tested by Crowther by measuring electrically the fraction I/I) of the scattered 3 particles which passed through a circular opening sub- tending an angle of 36° with the scattering material. If ie 1 — Catl. the value of I should decrease very slowly at first with Scattering of a and 8 Particles by Matter. 685 increase of ¢. Crowther, using aluminium as scattering material, states that the variation of I/I) was in good accord with this theory for small values of t. On the other hand, if single scattering be present, as it undoubtedly is for & rays, the curve showing the relation between I/I, and ¢ should be nearly linear in the initial stages. The experiments of Madsen * on scattering of @ rays, although not made with quite so small a thickness of aluminium as that used by Crowther, certainly support such a conclusion. Considering the importance of the point at issue, further experiments on this question are desirable. From the table given by Crowther of the value ¢/ “t,, for different elements for @ rays of velocity 2°68 x10" cms. per second, the values of the central charge Ne can be calculated on the theory of single scattering. It is supposed, as in the case of the a rays, that for the given value of g@/ “t, the fraction of the 8 particles deflected by single aoe through an angle greater than @ is 46 instead Ob 5D: The values of N calculated from Crowther’s data are given below. Atomic -— Element. | Soieue ¢/M tm. N. Miiminione ees): 27 4-95 22 Coppeuyin. wes: ast eee. 63:2 10:0 42 SITET HAA Bee eM als ate ON 108 15°4 78 a Giana ie en eee, 194 29:0 138 | It will be remembered that the values of N for gold deduced from scattering of the « rays were in two calcula- tions 97 and 114. These numbers are somewhat smaller than the values given above for platinum (viz. 138), whose atomic weight is not very different from gold. Taking into account the uncertainties involved in the calculation from the experimental data, the agreement is sufficiently close to indicate that the same general laws of scattering hold for the a and § particles, notwithstanding the wide differences in the relative velocity and mass of these particles. As in the case of the « rays, the value of N should be most simply determined for any given element by measuring * Phil, Mag. xviii. p. 909 (1909). 686 Prof. E. Rutherford on the the small fraction of the incident 8 particles scattered through a large angle. In this way, possible errors due to small scattering will be avoided. 3 The scattering data for the @ rays, as well as for the a rays, indicate that the central charge in an atom is approximately proportional to its atomic weight. This falls in with the experimental deductions of Schmidt*. In his theory of absorption of 8 rays, he supposed that in traversing a thin sheet of matter, a small fraction « of the particles are stopped, and a small fraction § are reflected or scattered back in the direction of incidence. From comparison of the absorption curves of different elements, he deduced that the value of the constant 8 for different elements is propor- tional to nA? where n is the number of atoms per unit volume and A the atomic weight of the element. This is exactly the relation to be expected on the theory of single scattering if the central charge on an atom is proportional to its atomic weight. § 7. General Considerations. In comparing the theory outlined in this paper with the experimental results, it has been supposed that the atom consists of a central charge supposed concentrated at a point, and that the large single deflexions of the « and @ particles are mainly due to their passage through the strong central field. The effect of the equal and opposite compensating charge supposed distributed uniformly throughout a sphere has been neglected. Some of the evidence in support of these assumptions will now be briefly considered. For con- creteness, consider the passage of a high speed « particle through an atom having a positive central charge Ne, and surrounded by a compensating charge of N electrons. Remembering that the mass, momentum, and kinetic energy of the a particle are very large compared with the corre- sponding values for an electron in rapid motion, it does not seem possible from dynamic considerations that an a particle can be deflected through a large angle by a close approach to an electron, even if the latter be in rapid motion and constrained by strong electrical forces. It seems reasonable to suppose that the chance of single deflexions through a large angle due to this cause, if not zero, must be exceedingly small compared with that due to the central charge. It is of interest to examine how far the experimental evidence throws light on the question of the extent of the * Annal. d. Phys. ty. 23. p. 671 (1907). Scaltering of « and 8 Particles by Matter. 687 distribution of the central charge. Suppose, for example, the central charge to be composed of N unit charges dis- tributed over such a volume that the large single deflexions are mainly due to the constituent charges and not to the external field produced by the distribution. It has been shown (§ 3) that the fraction of the « particles scattered through a large angle is proportional to (NeE)*, where Ne is the central charge concentrated at a point and E the charge on the deflected particle. If, however, this charge is dis- tributed in single units, the fraction of the a particles scattered through a given angle is proportional to Ne? instead of N’e?. In this calculation, the influence of mass of the constituent particle has been neglected, and account has only been taken of its electric field. Since it has been shown that the value of the central point charge for gold must be about 100, the value of the distributed char ge required to produce the same proportion of single deflexions through a large angle should be at least 10,000. Under these conditions the mass of the constituent particle would be small compared with that of the a particle, and the difficulty arises of the production of large single deflexions at all. In addition, with such a large distributed charge, the effect of compound scattering is relatively more important than that of single scattering. or example, the probable small angle of de- flexion of a pencil of « particles passing through a thin gold foil would be much greater than that experimentally observed by Geiger (§ d-c). The large and small angle scattering could not then be explained by the assumption of a central charge of the same value. Considering the evidence as a whole, it seems simplest to suppose that the atom contains a central charge distributed through a very small volume, and that the large single deflexions are due to the central charge as a whole, and not to its constituents. At the same time, the experimental evidence is not precise enough to negative the possibility that a small fraction of the positive charge may be carried by satellites extending some distance from the centre. Evidence on this point could be obtained by examining whether the same central charge is required to explain the large single deflexions of # and @ particles ; for the 2 particle must approach much closer to the centre of the atom than the 8 particle of average speed to sutter the same large deflexion. The gener: al data available indicate that the value of this central “charge for different atoms is approximately propor- tional to their atomic w eights, at any rate for atoms heavier than aluminium. It will be of ereat interest to examine 688 Scattering of a and 8 Particles by Matter. experimentally whether such a simple relation holds also for the lighter atoms. In cases where the mass of the deflecting atom (for example, hydrogen, helium, lithium) is not very different from that of the « particle, the general theory of single scattering will require modification, for it is necessary to take into account the movements of the atom itself (see § 4). It is of interest to note that Nagaoka * has mathematically considered the properties of a ‘‘Saturnian’”’ atom which he supposed to consist of a central attracting mass surrounded by rings of rotating electrons. He showed that such a system was stable if the attractive force was large. From the point of view considered in this paper,.the chance of large deflexion would practically be unaltered, whether the atom is considered to be a disk or a sphere. It may be remarked that the approximate value found for the central charge of the atom of gold (100e) is about that to be expected if the atom of gold consisted of 49 atoms of helium, each carrying a charge 2e. This may be only a coincidence, but it is certainly suggestive in view of the expulsion of helium atoms carrying two unit charges from radioactive matter. The deductions from the theory so far considered are independent of the sign of the central charge, and it has not so far been found possible to obtain definite evidence to determine whether it be positive or negative. It may be possible to settle the question of sign by consideration of the difference of the laws of absorption of the @ particle to be expected on the two hypotheses, for the effect of radiation in reducing the velocity of the @ particle should be far more marked with a positive than with a negative centre. If the central charge be positive, it is easily seen that a positively charged mass if released from the centre of a heavy atom, would acquire a great velocity in moving through the electric field. It may be possible in this way to account for the high velocity of expulsion of « particles without supposing that they are initially in rapid motion within the atom. | Further consideration of the application of this theory to these and other questions will be reserved for a later paper, when the main deductions of the theory have been tested experimentally. Experiments in this direction are already in progress by Geiger and Marsden. University of Manchester, April 1911. * Nagaoka, Phil. Mag. vii. p. 445 (1904). | 689 J LXXX. On Extremely Long Waves, emitted by the Quartz Mercury Lamp. By H. RuBens and O. von BAanyer”. N advance in the spectrum towards the side of the long waves is extremely difficult, while using pure hearin radiators. If the source of heat does not possess selective qualities, the intensity of radiation in the long-waved spectrum diminishes with the 4th power of the wave-length. It is true this intensity of radiation grows in proportion to the temperature of the source; but in a much higher degree (with the 4th power of the absolute temperatur re) the total energy of the radiant body augments, from which the desired part of radiation must ie “peal out by certain processes. An increase of the temperature of the source of light will, therefore, in many cases scarcely involve an ad- vantage for the present purpose. In the long-waved spectrum the Welsbach mantle has proved the most advantageous source of heat of purely thermoradiant character, because of its very favourable selective qualities. But even here, no rays of much greater wave-length than 108 have been obtained. This paper gives a description of experiments undertaken with a view to enlarge the knowledge of the infra-red spectral region, by employ ment of sources of light, from which the radiation is emitted by incandescent gases. As far as pure radiation of temperature is concerned, such light- sources are selective in the highest degree. Besides, the possibility of an existing long-waved infra-red luminescence radiation must here be considered. Our arrangement of apparatus is identical with that used recently by R. W. Wood and one of us for the isolation of long-waved rays{. It is founded on the use of quartz-lenses, which, because of the extreme difference of the index of refraction for heat-rays on both sides of the region of absorp- tion in quartz (1°50 to 2°14), can be so adjusted as to con- centrate the emitted long-waved radiation on a _ given diaphragm, while the ordinary heat-waves are dispersed. Our method is further founded on the selective absorption of quartz and on the effect of certain central screens. or all details concerning the apparatus and the method reference must be made to the above-cited paper. The first sources of light we now used were strong leyden- jar sparks between electrodes of zinc, cadmium, aluminium, iron, platinum, and bismuth ; the spar ks were produced by a * Communicated by the Authors. + H. Rubens & R. W. Wood, Phil. Mag. Feb, JO11. 690 Profs. H. Rubens and O. von Baeyer on Extremely 40 em. inductor, using alternating current for the inner coil. We have, however, in no case succeeded in obtaining a per- ceptible radiation in the observed long-waved spectral region. As little suecess was gained when we used the electric are with carbon electrodes, or with Bremer carbons and carbons with iron-salt filling, if the investigation was limited to the electric are itself. It is true, in both these last-named cases our micro-radiometer always showed small irregular deviations, which undoubtedly were due to long-waved radiation ; but it is not improbable that this radiation is emitted by ‘solid particles in the electricare. The observed effects were neither regular enough nor sufficiently strong to allow of a closer Inv vestigation. A comparativ ely very strong long-waved radiation was, however, obtained with the quar tz-mer cury lamp, especially at higher consumption ofenergy *. Witha current of 4 amperes on 100 volts, the are being ‘about 80 mm. long, a deflexion of more than 50 mm. appeared in our micro-radiometer. When the lamp had burned some time this deviation proved so constant that it could easily be measured down to fractions of a per cent. A few preliminary experiments showed us that the observed long-waved radiation of the mercury-lamp must possess a composition essentially different from that of the Welsbach mantle, the mean wave-length of which had, under the same conditions, amounted to 108. We found, for instance e, that l | | | Sense mm. 6 eer cent. | Per cent. | Per cent. | Per cent. | Quarioigscnce ce |41-7 12-1 25-4 51:8 58°9 Amorphous quartz .|_ 2°00 12°5 24-2 — 60:0 HUG OLIGO® es. s are ee |). (Og 53 19:4 39°5 42°2 Rock-salt ............ ie ae 0-5 57 16°5 22°95 Swlpines Li sss. | 210 0 | 36 11-7 167 Diamond |y.2.4.s22>-1.0el 26 45°3 64°5 = — [Selatan aaa fosz | 68 | 129 24-2 Et Waving... 23... fees. o026 0°055 16°6 38'8 51°5 550 Glasiies We ak eesiosens | 0-18 rai | | 90 21 259 12/920 1S ee 3 03 57:0 | 72°3 | 825 85°5 Hard rubber ......... | 0-40 39°0 Sto 7) 98s 65°3 ANINOET Sino 35 2he tee bel | 1:28 11-2 16-4 32°72 34°8 veers. Sect. | 1-80 07 29 | 100 ee. Black Paper .2..-..-.| 0-11 33°95 BOs |. Seo 79°0 Black Cardboard ...| 0°38 21 fy 1) ees 36-7 Céllaloid uA. | 026 | 162 P6276 \9) 387 43:5 peer (| 0-019 | 555 =| G03 627 ATELY — o-ceeeeecnee {| 0 038 | 83-0 | 88:4 89:8 * A mercury-amalgam lamp, containing 20 per cent. bismuth and 20 per cent. lead on 60 per cent. mercury gave nearly the same results. Long Waves enutted by the Quartz Mercury Lamp. 691 a 14°66 mm. thick layer of quartz transmitted 46°6 per cent. of the isolated radiation, when the mercury-lamp served as sources of light; and only 21°7 per cent. when the Welsbach mantle was adopted as radiation-source. This table shows, for a large number of substances, the transmission of long-waved radiation isolated by means of quartz-lenses. Both sources of light were in use, D, being the transmission observed with the Welsbach mantle, D, with the mercury-lamp as radiator. We have, moreover (under D;), exhibited, for the same substances, the transmission of the radiation of the mercury-lamp filtered by a 2:0 mm. thick layer of amorphous quartz. It could be assumed from the beginning that the observed radiation cf the mercury- lamp consisted of two parts, one of which emanated from the hot quartz-walls of the tube, the other from the mercury vapour itself. For the separation of the latter part we at first deemed a filter of melted quartz most suitable Later on, we found that a ray-filter of black cardboard proved still more efficient for the isolation of the long-waved radiation emerging from the mercury vapour. In ‘the last column of our table (D,) we have therefore exhibited the results of the measurements on transmission, obtained after substituting a filter of black cardboard, 0°38 mm. thick, for the amorphous quartz. By reference to the table it wil immediately be seen that for all substances the values D,, D,, D3, and D, form an ascending series. So far as substances are concerned whose region of absorption is known to be situated at shorter wave- lengths (as quariz, fluorite, rock-salt, and sylvine), this course indicates an increase of the mean wave-lengths of the corresponding. radiations. We must therefore attribute a greater mean wave-length to this radiation of the mercury- lamp than to that emitted by the Welsbach mantle; we must further assume a greater mean wave-length for the radiation of the mercury-lamp filtered by black cardboard, than for that which passed through amorphous quartz. This assump- tion is in a still higher “degree justified by the behaviour of black paper and black cardboard, because in such media, in which the principal loss of energy is due to diffuse dissipation, the transmission must strongly 1 increase with growing wave- length. The rise of the mean wave-length, which the radia- tion of the mercury-lamp shows after the introduction of the radiation-filters employed, is according to our opinion due to the fact that the short-waved radiation of the quartz-tube (which is nearly of the same quality with that of the Welsbach mantle) is much more strongly absorbed by these filters than 692 Profs. H. Rubens and O. von Baeyer on Extremely the evidently much longer-waved radiation of the mercury vapour. The extremely high transmission of quartz is of particular interest for these kinds of radiation. On caleu- lating the coefficient of absorption 100 D,- from the transmission for the 41:7 mm. thick quartz plate, cut perpendicularly to the axis (d being the thickness of the plate in mm,, D,/ the transmission, corrected on account of the loss by reflexion), we obtain for the here investigated radiations the following values of g,:— Gi=0°044 ; q2=0°026 ; g3=0°0089 5 q=0'0057. It is evident that the rays of the mercury-lamp filtered by black cardboard must penetrate about 8 times as thick a quartz-layer as the rays emerging from the Welsbach mantle, before being attenuated to the same fraction of their primary intensity. Similar circumstances prevail with amorphous quartz, but here the absorption-power for the four investigated radiations is about 20 times as great as at the natural modification. Glass and mica seem, like fluorite, rock-salt, and sylvine, to belong to the substances whose main region of absorption lies among the wave-lengths below 100. The high trans- mission of paraffin, hard rubber, and amber, well-known as good isolators, is not surprising; neither is the small absorp- tion of the elements diamond and selenium. Water shows a far smaller absorption power for the radia- tion emitted by the mercury-lamp (particularly after its fiitration through quartz or black cardboard) than for the rays emanating from the Welsbach mantle. The reflexion from the water surfaces can also not be considerable, as the values of the coefficient of absorption calculated from both layers of different thickness agree satisfactorily without con- sideration of the reflecting power. This would not be so in case of a considerable loss by reflexion. We may, therefore, assume that in these spectral regions water still possesses a refractive index of small magnitude, lying far closer to the value observed in the visible spectrum than to the square- root of the dielectric constant for slow vibrations. As our measurements of absorption cannot give quan- titative determinations as to the average wave-length of the investigated radiations, we have attempted to measure the wave-length by aid of the previously employed inter- ferometer*. The interference-curves obtained with the * H. Rubens & H,. Hollnagel, Phil. Mag. [6] xix. p. 761 (1910). = vlog nat. Long Waves emitted by the Quartz Mercury Lamp. 693 quartz-mercury lamp—omitting the radiation filter—showed very irreeular character. Nevertheless, it was evident that the main element of the investigated radiation was supplied by a radiation of about the same mean wave-length as that resulting from the Welsbach mantle with this arrangement. But as soon as a 15 mm. thick layer of quartz was oon the aspect was changed. The first minimum, which had been observed for unfiltered radiation at a thickness of the air-film of about 5 divisions of the drum* (26%), now did not appear before a thickness of the layer of air of 8 divi- sions (42 4). If the thickness of the inserted layer of quartz was increased to 42 mm., the first minimum appeared only at a distance of the interferometer plates of about 13 divisions (684). At the same time the interference curve showed a much smoother course. The originally observed irregular maxima and minima had nearly quite disappeared ; “and, besides the mentioned minimum at 13 divisions, in some series of observations a faintly marked maximum appeared in pear ie a | f a eens 3 Aes) LS ee EE Bas. 0 70 40 3O 40 50 the further course of the curve. Such an interferometer- curve is exlibited in the accompanying figure (curve a). Curve (5) of the same figure was observed in the same w ay with insertion of the 2 mm. thick plate of amorphous quartz; curve (c) with insertion of the black cardboard (0-4 mm. thick). Curve (c) shows the wave-character most distinetly. Here the minimum lies at 15 divisions (784) and the * One division of the drum to 5:25 4. 694 Long Waves emitted by the Quartz Mercury Lamp. maximum at 30 is more conspicuous than in the other curves. But even in this curve an accurate determination of these points is very difficult. The assumption is certainly justi- fiable, that the radiation filtered through black cardboard contains a greater amount of this long-waved part than it does after purification by the quartz-filter. We had already deduced this fact from the results of the absorption-table. It is still an open question, whether this long-waved radiation consists of several nearly homogeneous kinds of rays of different wave-length—as would be expected upon the assumption of a luminiferous radiation of mercury vapour —or whether it is a continuous radiation, covering a larger spectral region, such as thermo-radiators mostly possess. The results of the interferometer measurements are unable to warrant us in settling this question. But we can safely deduce from our observations, that a large part of this radia- tion emerging from the quartz-mercury lamp possesses a mean wave-length of about 30 x 2 x 5°23 w=314 w or nearly 4 mm. With a view to confirming our supposition, that this extremely long-waved radiation originates in the mercury vapour itself and not in the hot quartz-tube of the lamp, we cite the following reflection:—-As the intensity of the radia- tion from a black body diminishes with the 4th power in the region of great waye-lengths, amorphous quartz (which at A=100 w behaves nearly like a black body) might at the threefold wave-length send forth at most the 8lst part of the energy it emits at 100». But at the relatively low temperature of the quartz-mantle such a feeble radiation would not be discernible. We could, moreover, show by experiment that the observed long-waved radiation came from the mercury vapour itself. The intensity of radiation was measured shortly before and after the break of the current of the lamp. On introduction of the cardboard-filter the observed intensity of radiation fell, immediately after the interruption of the current, to about 30) per cent. of the initial value, and then slowly diminished more and more. ‘The same experiment without cardboard-filter only produced a decrease of radiation of about 30 per cent. after the inter- ruption of the lamp-current. We have, lastly, investigated the radiation we obtained by substituting for the quartz- mercury lamp a piece of amorphous quartz, heated by a Bunsen-flame. ‘This radiation proved to be even of a some- what smaller wave-length than that emitted by the Welsbach mantle under equal conditions. Less than two per cent. of it ren ey through black cardboard, and only ten per cent. through 2 mm. of amorphous quartz. Geological Society, 695 That the observed long-waved radiation is not emitted by the quartz walls is, therefore, an established fact, and it is highly probable that it originates in the luininous mercury vapour * But the question is not solved, whether we are dealing with a radiation of temperature or of luminosity. According to measurements of Messrs. Kiich and Retschinskyf, the mercury vapour of the quartz-mercury lamp possesses a temperature which amounts to many thousand degrees. In this case, the observation of such long-waved pure temperature radiation is not impossible, if the radiating mercury vapour possesses strongly defined selective absorption j in that spectral region. The main result of this investigation is the fact that heat rays of a wave-length of about 03 mm. may be extracted from the radiation of the mercury-lamp in sufficient force to permit an investigation of their qualities. The infra- red spectrum thereby sustains another enlargement of 14 octaves. LXXXI. Proceedings of Learned Societies. GEOLOGICAL SOCIETY [Continued from p. 32. ] December 21st, 1910.—Prof. W. W. Watts, Sce.D., M.Sc., F.RB.S., President, in the Chair. Y [ ‘HE following communication was read :— ‘The Keuper Maris around Charnwood Forest.’ By Thomas Owen Bosworth, B.A., B.Se., F.G.S. The area under Bm icvatior comprises some 300 square miles, including the towns of Leicester, Loughborough, Coalville, and Hinckley. As has been shown by Prof. Watts, the Charnian roclis project through a mantle of Triassic deposits which once com- pletely covered them. In numerous quarry-sections the relation of the Keuper to the pre-Cambrian rocks is well exposed. The quarries generally have becn opened in the summits of the more or less completely buried hills. A quarry is so worked that its outline follows the contour of the buried hill: consequently, the section presents but a dwarfed impression of the irregularity of the rock-surface. Nevertheless, considerable undulations are observed, * Tt is, moreover, not qnite out of the range of possibility that this long-waved radiation could consist of relatively very short Hertzian waves, which are produced by electric oscillations in small mercury drops. But it seems improbable that any condensation of mercury vapour would take place in the path of the current, ¢ e. in the hottest part of the tube. t Kuch & Retschinsky, Ann. d. Phys. xxii. p. 595 (1907). Fl i 696 Geoloyical Society. and wherever there are any sections at right angles to the contours, the rock-slopes are seen to be remarkably steep. Contoured maps have been prepared, showing the features of some of these covered peaks. On the buried slopes, and in the gullies, are screes and breccias : and bands of stones and grit are present in the adjacent beds of marl. All these stones, in every case, are derived only from the rock immediately at hand. They never resemble pebbles, but often are fretted into irregular shapes. Where exposed to the present climate, the Charnian igneous rocks are deeply weathered and disintegrated. But the same rocks beneath the Keuper are fresh right up to the top, as also are the rock-fragments in the marls. The Keuper marls le in a catenary manner across the gullies, and probably across the large valleys also; for they dip away steeply in all directions around each buried peak. There has been almost no post-Lriassic movement in Charnwood. Nevertheless, the beds must have been originally laid down horizontally, for they are in no way peculiar, and contain the normal seams of shallow-water sediment. All the points of contact of any one bed with the Charnian rocks lie on one horizontal plane. The inclination of the strata must, therefore, be due to subsequent sagging. The Upper Keuper deposits accumulated in a desert basin, of which parts were dry and parts were occupied by ever-shifting salt-lakes and pools. In these waters the red marls were laid down. The red marls are of several different types, and are usually well-bedded. The principal ingredients are a certain aluminous mineral in very small particles and a much smaller proportion of very fine quartz-sand. There is generally 20 or 30 per cent. of dolomite present, in the form of minute rhombs. The grey bands include various kinds of rock. Each band usually contains one or more seams of well-bedded sandstone or quartzose dolomite, and may safely be relied upon to indicate the bedding. The irregularities are due to irregularity in the bleaching above and below these porous seams. The abundant heavy minerals are garnet, zircon, tourmaline, staurolite, rutile, magnetite. These are found in every sediment— marls, sandstones, grits, breccias, etc. The grains are intensely worn. The quartz-grains are sometimes evidently wind-worn. The sand in the grey bands is coarser and more abundant than that in the red marls. Hach grey band marks the introduction of coarser sediment into the basin. The false bedding is mainly from the south-west. The bands are of wide extent and are due to inflows of fresh water from the surrounding hills, which from time to time spread themselves far and wide over the dry portions of the desert, and were often completely desiccated before reaching any. pre-existing pool. Where these waters evaporated the quartzose- dolomite seams were formed, bearing ripple-marks and salt- pseudomorphs. The ripples indicate prevalent south-westerly winds. Phil. Mag. Ser. 6, Vol. 21, PI. V. CurvVEs 5. pias. OS , bo Ls 20 as sc Osa eo lo ¥ ue eae 4nes cm Z: Te x7 Vi Zo (=) i if a a v (=) 6 ie aH Gi 2 pA val r) ia | = ‘Gi ( Tr ecacan~bs Per Sy om 4) TYNDALT. Curves 1. Cunves 2. Phil. Mag. Ser. 6, Vol. 21, Pl. V. Curves 5. Curves 3, e+ (res Oe Curves 4. 20 a> Jo JS Cm Curves 6. c (Gol / | 4 { Uy UY Z (H 2 (F)) Ze i Ye Ze (-) V, MEZA 7 Aes Za L ——_—$—$___——_}— fo he 2:0 ( Mmtrcan-ps per Syom -) wake the re j £ u \ ‘ * bd = 7 7 ba . «igen poe i » ¥ a = i ig ’ a ; 1 Fs { 4 =) < oe by ed Les es : oe hia > et i em L | : tite oe - yal a hae on : \ : : : , £ 4 ( é F : th roe Ve id aN i 7 var an he _ _ Bd ns cu 4, wl a) SVE me eon a ‘3 =f Fee “> - ‘ } 25 Ine lee apn a <7 , -- dl > Re: . Pcs aes PS Part me S ¥ . ae i see f j tw - . if 7 or She : = a guee ; bat a Wf j 2 a ae : eee Q es Se € vA - + * \ ‘., ~ iJ ; ‘ " ; ! JF ® . ; . a : 2 i ne | ‘ . : | fi ra Tey Pane : ' | ; \ t q f : , Ai ie ir, j ) if i iy td ; ' } St ae ee } ‘ , , t ' Peto — | i ; a ! a . | 1 { \ i ; = ‘ ‘ ak | #g he ‘ ; F H ' | RAMAN. Phil. Mag. Ser. 6, Vol. 21, Pl. VI. Fic. 1. Fie. 2. ; Fig. 3. “Fa. 6. Fie. 8. Frc. 7. | | | RAMAN. . Phil. Mag. Ser. 6, Vol. 21, Pl. VII. Bren i: Enlarged photograph showing diffraction pattern. Ries: Photograph of black photometric disk. 0 mus 0 pill pla ale — ) =~ 4 a 3 ik j bik ,, SP THE LONDON, EDINBURGH, ann DUBLIN PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCIENCE. [SIXTH SERIES.] SUN NOW Ts LXXXII. On the Motion of Solid Bodies Acme Viscous Liquid. By Lord Rayuzien, OW, F.R.S.* Sa | hi problem of the uniform and infinitely slow motion of a sphere, or cylinder, through an un- limited mass of incompressible viscous liquid otherwise at rest was fully treated by Stokes in his celebrated memoir on Pendulumsf. The two cases mentioned stand in sharp contrast. In the first a relative steady motion of the fluid is easily determined, satisfying all the conditions both at the surface of the sphere and at infinity; and the force required to propel the sphere is found to be finite, being given by the formula (126) soll Oar elds Via mh in coe wee th) where yp is the viscosity, a the radius, and V the velocity of the sphere. On the other hand in dhe case of the cy linder, moving transversely, no such steady motion is possible. I we suppose the cylinder originaliy at rest to be started and afterwards maintained in uniform motion, finite effects are propagated to ever greater and yreater fidacinaees and the motion of the fluid approaches no limit, Stokes shows that more and more of the fluid tends to accompany the travelling cylinder, which thus experiences a continually decreasing resistance. * Communicated by the Author. + Camb. Phil. Trans. ix. 1850; Math. & Phys. Papers, vol. iii. p. 1. Phil. Mag. 8. 6. Vol. 21. No. 126, June 1911. 22, 698 Lord Rayleigh on the Motion of § 2. In attempting to go further, one of the first questions to suggest itself is whether similar conclusions are applicable to bodies of other forms. The consideration of this subject is often facilitated by use of the well-known analogy between the motion of a viscous fluid, when the square of the motion is neglected, and the displacements of an elastic solid. Suppose that in the latter case the solid is bounded by two closed surfaces, one of which completely envelopes the other. Whatever displacements (a, y, 8) be imposed at these two surfaces, there must be a corresponding configuration of equilibrium, satisfying certain differential equations. If the solid be incompressible, the otherwise arbitrary boundary displacements must be chosen subject to this condition. The same conclusion applies in two dimensions, where the bounding surfaces reduce to cylinders with parallel generating lines. For our present purpose we may suppose that at the outer surface the displacements are zero. The contrast between the three-dimensional and two- dimensional cases arises when the outer surface is made to pass off to infinity. In the former case, where the inner surface is supposed to be limited in all directions, the dis- placements there imposed diminish, on receding from it, in such a manner that when the outer surface is removed to a sufficient distance no further sensible change occurs. In the two-dimensional case the inner surface extends to infinity, and the displacement affects sensibly points however distant, provided the outer surface be still further and sufficiently removed. The nature of the distinction may be illustrated by a simple example relating to the conduction of heat through a uniform medium. If the temperature v be unity on the surface of the sphere r=a, and vanish when r=, the steady state is expressed by When @ is made infinite, v assumes the limiting form a/r. In the corresponding problem for coaxal cylinders of radu a and b we have log b—logr ie = Tog b—log a But here there is no limiting form when 6 is made infinite. However great 7 may be, v is small when 6 exceeds 7 by Solid Bodies through Viscous Liquid. 699 only a little; but when } is great enough v may acquire any value up to unity. And since the distinction depends upon what occurs at infinity, it may evidently be extended on the one side to oval surfaces of any shape, and on the other to cylinders with any form of cross-section. In the analogy already referred to there is correspondence between the displacements (a, 6, y) in the first case and the velocities (uw, v, w) which express the motion of the viscous liquid in the second. There is also another analogy which is sometimes useful when the motion of the viscous liquid takes place in two dimensions. The stream-function (fr) for this motion satisfies the same differential equation as does the transverse displacement (w’) of a plane elastic plate. Anda surface on which the fluid remains at rest (p=0, dy /dn=0) corresponds to a curve along which the elastic plate is clamped. In the light of these analogies we may conclude that, pro- vided the square of the motion is neglected absolutely, there exists always a unique steady motion of Jiquid past a solid obstacle of any form limited in al] directions, which satisties the necessary conditions both at the surface of the obstacle and at infinity, and further that the force required to hold the solid is finite. But if the obstacle be an infinite cylinder of any cross-section, no such steady motion is possible, and the force required to hold the cylinder in position continually diminishes as the motion continues. § 3. For further developments the simplest case is that of a material plane, coinciding with the coordinate plane «=0 and moving parallei to y ina fluid originally at rest. The component velocities wu, ware then zero ; and the third velocity v satisfies (even though its square be not neglected) the general equation dv d*v dt Dai Aree e ° = < ° : (4) in which v, equal to w/p, represents the kinematic viscosity. In § 7 of his memoir Stokes considers periodic oscillations of the plane. Thus in (4) if v be proportional to e”’, we have on the positive side vom Ae emt V (inv) | ) : i ; ‘ (5) When x=0, (5) must coincide with the velocity (V) of the plane. Jf this be Ve", we have A=V,,; so that in real 222 700 Lord Rayleigh on the Motion of quantities v= Ve 74 cos {ni—z,/(n]2v)}-. 2 a corresponds with V=V.,cosnt... 2) for the plane itself. In order to find the tangential force (—T3) exercised upon the plane, we have from (5) when «=U (=) aly eG), 0 di and T;= —p(dv/dz)o=pV,e"/ (inv) =py/(knv) . (1412) Vem =p/(am) (V+ 5}, _ . 2 er giving the force per unit area due to the reaction of the fluid upon one side. ‘‘The force expressed by the first of these terms tends to diminish the amplitude of the oscillations of the plane. The force expressed by the second has the same effect as increasing the inertia of the plane.” It will be observed that if V, be given, the force diminishes without limit with n. In note B Stokes resumes the problem of § 7: instead of the motion of the plane being periodic, he supposes that the plane and fluid are initially at rest, and that the plane is then (¢=0) moved with a constant velocity V. This problem depends upon one of Fourier’s solutions which is easily verified *. We have TOS Ni —22/4yt ao gap: 2V aes ‘ Crait 0 ON oe iz ° a ° . (195) For the reaction on the plane we require only the value of dv/dx when s=(. And dv Vv =) Se >> ]]]5.,.-. . 2 aul) a/ (avt) (i Stokes continues f ‘‘ now suppose the plane to be moved * Compare Kelvin, Ed. Trans. 1862; Thomson & Tait, Appendix D. + I have made some small changes of notation. Solid Bodies through Viscous Liquid. TOL in any manner, so that its velocity at the end of the time ¢ is V(t). We may evidently obtain the result in this case by writing V'(r)d7 for V, and t—7 for ¢ in [12], and integrating with respect to7. We thus get 1 ( V(r) dr _ 2) =- ee a ay, dx]) 2 Gi) as / (¢—T) AV (Ga) Se Wie) w/e (13) 399 and since T;=—pdv/diy, these formule solve the problem of finding the reaction in the general case. There is another method by which the present problem may be treated, and a comparison leads to a transformation which we shall find useful further on. Starting from the periodic solution (8), we may generalize it by Fourier’s theorem. Thus Vee (|) = -{ Vre/ (in/v) dn 5 ° ° (14) corresponds to vo=| Vie Lit: he aw ca “poo GES) 0 where V, is an arbitrary function of n. Comparing (13) and (14), we see that 1 EVE Gpyaia: A SRG Tt is easy to verify (16). If we substitute on the right for V(r) from (15), we get fo a) ( ve etn: dn= “0 Le cram ‘s9 nVre™ dn: aca mVne "an; and taking first the integration with respect to 7, See Ae (" edt — ene an=a/ (7 ) ‘ eint v— oV/ t—T) e/0 a/ ty 2 whence (16) follows at once. ; As a particular case of (13), let us suppose that the fluid is at rest and that the plane starts at #=0 with a velocity which is uniformly accelerated for a time 7, and afterwards remains 702 Lord Rayleigh on the Motion of constant. Thus from —x to 0, V(r)=0; from 0 to 7%, V(7)=hr; trom 7, to ¢, where t > 7, V(7)=hr,. Thus (0 < t < 71) (=) =- 1 (" hdr 2h and (é > 7) GON Wala foul er eth dla? es 2h, (i) Week /(i-7) iC i (18) Expressions (17), (18), taken negatively and multiplied by #, give the force per unit area required to propel the plane against the fluid forces acting upon one side. The force increases until ¢=7,, that is so long as the acceleration continues. Afterwards it gradually diminishes to zero. For the differential coefficient of ./t—,/(t—7;) is negative when ¢>7,; and when ¢ is great, Jt—/t—1)=4t-3 ultimately. § 4. In like manner we may treat any problem in which the motion of the material plane is prescribed. A more difficult question arises when it is the forces propelling the plane that are given. Suppose, for example, that an infinitely thin vertical lamina of superficial density o begins to fall from rest under the action of gravity when ¢=0, the fluid being also initially at rest. By (13) the equation of motion may be written 0) AGH), /i—7) 4 / Ga IV, 20 (8 Wr)dr dt a ea os Uae (19) the fluid being now supposed to act on both sides of the lamina. By an ingenious application of Abel’s theorem Boegio has succeeded in integrating equations which include (19)*. The theorem is as follows:—If W(t) be defined by yo= [Sor ot (20) ae =mid(i)—g(0)t. . - (20) * Boggio, Rend. d. Accad. d. Lincet, vol. xvi. pp. 618, 730 (1907) ; also Basset, Quart. Journ. of Mathematics, No. 164, 1910, trom which I first became acquainted with Boggio’s work. then olid Bodies through Viscous Liquid. 703 For by (20), if (¢—7)?=y, VE yig=2h gy) ay so that ab(r)dr : oe Coe V(t—22) - Oats ati) dy vt =e p'(t—17) rdr=7, b(t)—4(0)}, 0 where 7?= 27+ y’. Now, if t/ be any time between 0 and t, we have, asin (19), ; 2pv! HONE GG air a ee ont |, /(@—T) Multiplying this by (¢—2¢’)~?dt' and integrating between 0 and ¢t, we get tw! / pve t I te / t t! ee i (i dt jf TOS =o ( d - Q2) 0 OSS on J, Gay ye (Ca). v0 (¢—t’) In (22) the first integral is the same as the integral in (19). By Abel’s theorem the double integral in (22) is equal to mV (t), since V(0)=0- Thus ( V(r) dr ae 1 20 b= 7) If we now eliminate the integral between (19) and (23), we obtain simply WG) ony tae (8) dV 4p*y Aree Ry eM TE A UR a Veo a eES) as the differential equation governing the motion of the lamina. This is a linear equation of the first order. Since V 704 Lord Rayleigh on the Motion of vanishes with t, the integral may be written ApPVN (oy, gee Zt hae i é (ea ee 2 ‘(2 a ae ae Jape CE Mees (22 in which ¢’=¢.4p’v/o?._ When ¢, or @’, is great, I ee ee : 26 é da= 5 ( — 57+): or are (26) e Jt! so that SV e/a 1 es if A ge a tt ae oe ae Ultimately, when ¢ is very great, v=" \/(2)). 4 i oe § 5. The problem of the sphere moving with arbitrary velocity through a viscous fluid is of course more difficult than the corresponding problem of the plane lamina, but it has been satisfactorily solved by Boussinesq * and by Basset +. The easiest road to the result is by the application of Fourier’s theorem to the periodic solution investigated by Stokes. Ié the velocity of the sphere at time ¢ be V=V,e, a the radius, M’ the mass of the liquid displaced by the sphere, and s=,/(n/2v),v being as before the kinematic viscosity, Stokes finds as the total force at time ¢ ee I ea? Lh F= MVin4 (5 +a) aac ips Ee gint (29) Thus, if ys) Veetdn . .. en 0 Pe |S ivy os NI if int = zs F=—M A Vine {¢ 2 a+ = (14 ai dn, s e se e (31) * C. R. t. 100. p. 985 (1885) ; Theorre Analytique de la Chaleur, t. ii. Paris, 19038. f Phil. Trans. 1888; Hydrodynamics, I. ch, xxii. 1888. Solid Bodies through Viscous Liquid. 705 OF the four integrals in (31), phe Aiea ee { in Vn et dn=1V': 0 Oya ear & the fourth= Th Vea — 2a | Coe 2a Also the second and third together give ° oS) ne) CE INGED) TORE) \ V,, nz e™ dn, 4a ‘ and this is the only part which could present any difficulty. We have, however, already considered this integral in con- nexion with the motion of a plane and its value is expressed by (16). Thus :. Pilea. Sy ae Ni (aaa, (86) F=—M'4 5 Tuo nk ea ee The first term depends upon the inertia of the fluid, and is the same as would be obtained by ordinary hydrodynamics when v=0. If there is no acceleration at the moment, this term vanishes. If, further, there has been no acceleration for a Jong time, the third term also vanishes, and we obtain the result appropriate to a uniform motion K=— —_,— = — 6rrapvV = — 6rrpaV, asin (1). The general result (32) is that of Boussinesq and Basset. As an example of (32), we may suppose (as formerly for the plane) that V(t)=0 from —x to 0; V(t)=At from 0 to TG) =—hrewhen ¢ > 7. Lhenift< 7, P= — 1M’ - and when t> Ta F=—/)M' Ex ed vt— V( ava so) CO) When ¢ is very great (34) reduces to its first term. The more difficult problem of a sphere falling under the influence of gravity has been solved by Boggio (loc. ct.). In the case where the liquid and sphere are initially at rest, the solution is comparatively simple; but the analytical form vt | b] oh bees 2a es comm? (33) 706 Lord Rayleigh on the Motion of of the functions is found to depend upon the ratio of densities of the sphere and liquid. This may be rather unexpected ; but I am unable to follow Mr. Basset in regarding it as an objection to the usual approximate equations of viscous motion. § 6. We will now endeavour to apply a similar method to Stokes’s solution for a cylinder oscillating transversely in a viscous fluid. If the radius be a and the velocity V be expressed by V=V,e’, Stokes finds for the force R=—M’in Vine™ (k—ik’). . . 2 een In (35) M! is the mass of the fluid displaced; £ and k’ are certain functions of m, where m=4av(n/v), which are tabulated in § 37. The cylinder is much less amenable to mathematical treatment than the sphere, and we shall limit ourselves to the case where, all being initially at rest, the cylinder is started with unit velocity which is afterwards steadily maintained. The velocity V of the cylinder, which is to be zero when ¢ is negative and unity when ¢ is positive, may be expressed by [a o} e sin nt dn, . J vas T n in which the second term may be regarded as the real part of iL 2) ent Tt 0 nN Ges. |. nn We shall see further below, and may anticipate from Stokes’s result relating to uniform motion of the cylinder, that the first term of (36) contributes nothing to F; so that we may take MW! aay h= { em (k—ih')dn, J0 corresponding to (37). Discarding the imaginary part, we get, corresponding to (36), F=—— | (Acosnt+hk'sinnt)dn, . . (88) Since k, k’ are known functions of m, or (a and y being given) of n, (38) may be calculated by quadratures for any prescribed value of ¢. It appears from the tables that &, k’ are positive throughout. Solid Bodies through Viscous Liquid. 707 When m=0, & and £’ are infinite and continually diminish as m increases, until when m=o,k=1,k’=0. For small values of m the limiting forms for 4, k’ are wD | if Awe 470 pure y : m?(log m)?’ Ke m log m’” (ao) from which it appears that if we make n vanish in (35), while V, is given, F comes to zero. We now seek the limiting form when ¢ is very great. The integrand in (38) is then rapidly oscillatory, and ulti- mately the integral comes to depend sensibly upon that part of the range where nis very small. And for this part we may use the approximate forms (39). Consider, for example, the first integral in (38), from which we may omit the constant part of k. We have a) , oF wi” cosntdn Admv{™ cos(4va~*t.a)de keos nt dn= aa 9 i ——\9 = 5 A i en aa yc Te A 4}, m*(log m) Oe a x (log x) (40) Writing 4vt/a?=t’, we have to consider S > { “cost de Ne uae eat) 0 a (log #;? ° Tn this integral the integrand is positive from «= 0 to v®=7/2t’, negative from 7/2t! to 3a/2t!,and so on. Tor the first part of the range if we omit the cosine, af 20) ; 7 { dx (fe i Atco Piaeadooa.: | (oewe milco@aia and since the cosine is less than unity, this is an over estimate. When 1’ is very great, log (2¢!/7r) may be identified with logt', and to this order of approximation it appears that (41) may be represented by (42). Thus if quadratures be applied to (41), dividing the first quadrant into three parts, we have TT cos — Be ie ome 1 Rae om | if er | ——= Ss ST aa 0s -— | —— — = Son Ce ee en wea Ge | me Be! log |; logp— log — | loge Slog == T We Suet T ieee Poe of which the second and third terms may ultimately be neglected in comparison with the first. For example, the coefficient of cos (37/12) is equal to ; St. 6é! Dp ; g2— log ee ea: lo 3 = — SS eee eee nr ee 2 — . a Sno x _ a A a 708 Lord Rayleigh on the Motion of Proceeding in this way we see that the cosine factor may properly be identified with unity, and that the value of the integral for the first quadrant may be equated to 1/logt’. And for a similar reason the quadrants after the first con- tribute nothing of this order of magnitude. Accordingly we may take {eos nt d= 2 6 ee! es 0 GAOe by) For the other part of (38), we get in like manner i k’ sin nt dn= — Py ( sinta#’.dz 0 6 vlog x fe 0) . a] i) ee ed. aa (44) 2' log (¢'/2') a 20 In the denominator of (44) it appears that ultimately we may replace log (¢//2') by log?’ simply. Thus a Any | si = > —, ... . (4% { k' sin nt dn aioet? (45) so that the two integrals (43), (45) are equal. We conclude that when ¢ is great enough, Sv! 8yM! Han a log t! Say, a log (Avi/a?) > ae Ce But a better discussion of these integrals is certainly a desideratum. § 7. Whatever interest the solution of the approximate equations may possess, we must never forget that the con- ditions under which they are applicable are very restricted, and as far as possible from being observed in many practical problems. Dynamical similarity in viscous motion requires that Va/v be unchanged, a being the linear dimension. Thus the general form for the resistance to the uniform motion of a sphere will be F=pwWaytValy), .. . ie where f is an unknown function. In Stokes’s solution (1) 7 is constant, and its validity requires that Va/v be small*. When V is rather large, experiment shows that F is nearly proportional to V’. In this case v disappears. ‘‘ The second * Phil. Mag. xxxvi. p. 354 (1893) ; Scientific Papers, iv. p. 87. - . Solid Bodies through Viscous Liquid. 709 power of the velocity and independence of viscosity are thus inseparably connected” *. | The general investigation for the sphere moving in any manner (in a straight line) shows that the departure from Stokes’s law when the velocity is not very smail must be due to the operation of the neglected terms involving the squares of the velocities ; but the manner in which these act has not yet been traced. Observation shows that an essential feature in rapid fluid motion past an obstacle is the formation of a wake in the rear of the obstacle; but of this the solutions of the approximate equations give no hint. Hydrodynamical solutions involving surfaces of discon- tinuity of the kind investigated by Helmholtz and Kirchhoff provide indeed for a wake, but here again there are difficulties. Behind a blade immersed transversely in a stream a region of “dead water” isindicated. The conditions of steady motion are thus satisfied ; but,as Helmholtz himself pointed out, the motion thus defined is unstable. Practically the dead and live water are continually mixing; and if there be viscosity, the layer of transition rapidly assumes a finite width inde- pendently of the instability. One important consequence is the development of a suction on the hind surface of the lamina which contributes in no insignificant degree to the total resistance. The amount of the suction does not appear to depend much on the degree of viscosity. When the latter is small, the dragging action of the live upon the dead water extends to a greater distance behind. § 8. If the blade, supposed infinitely thin, be moved edge- ways through the fluid, the case becomes one of ‘“skin- friction.”” Towards determining the law of resistance Mr. Lanchester has put forward an argument} which, even if not rigorous, at any rate throws an interesting light upon the question. Applied to the case of two dimensions in order to find the resistance F per unit length of blade, it is some- what as follows. Considering two systems for which the velocity V of the blade is different, let n be the proportional width of corresponding strata of velocity. The momentum communicated to the wake per unit length of travel is as nV, and therefore on the whole as nV? per unit of time. Thus F variesas nV®. Again, having regard to the law of viscosity and considering the strata contiguous to the blade, we see that I’ varies as V/n. Hence, nV? varies as V/n, or V varies as n~*, from which it follows that F varies as V3. If this * Phil. Mag. xxxiv. p. 59 (1892) ; Scientific Papers, ili. p. 576. t Aerodynamics, London, 1907, § 35. 710 = Motion of Solid Bodies through Viscous Liquid. be admitted, the general law of dynamical similarity requires that for the whole resistance F=cpovtli?V?, . . . eee where J is the length, & the width of the blade, and ¢ a constant. Mr. Lanchester gives this in the form Vjo=ev? A? V2, .. 2 ee where A is the area of the lamina, agreeing with (48) if {and 6 maintain a constant ratio. The difficulty in the way of accepting the above argument as rigorous is that complete similarity cannot be secured so longas 0 is constant as has been supposed. If, as is necessary to this end, we take 6 proportional to n,it is bV/n, or V (and not V/n), which varies as nV*, or bV*. The conclusion is then simply that bV must be constant (v being given). This is merely the usual condition of dynamical similarity, and no conclusion as to the law of velocity follows. But a closer consideration will show, I think, that there is a substantial foundation for the idea at the basis of Lan- chester’s argument. If we suppose that the viscosity is so small that the layer of fluid atfected by the passage of the blade is very small compared with the width (6) of the latter, it will appear that the communication of motion at any stage takes place much as if the blade formed part of an infinite plane moving asa whole. We know that if sucha plane starts from rest with a velocity V afterwards uniformly maintained, the force acting upon it at time ¢ is per unit of area, see (12), Vw). .. .. aa The supposition now to be made is that we may apply this formula to the element of width dy, taking t equal te y/V, where y is the distance of the element from the leading edge. Thus F=Ip(v[ar)?V# S 4 dy=2lp(v/m)?Vibt, =. (51) which agrees with (48) if we take in the latter c=2/./7. The formula (51) would seem to be justified when y is small enough, as representing a possible state of things ; and, as will be seen, it affords an absolutely definite value for the resistance. There is no difficulty in extending it under similar restrictions to a lamina of any shape. If 8, Velocity of the Ions of Alkaly Salt Vapours in Flames. 711 no longer constant, is the width of the lamina in the direction of motion at level z, we have Tt will be seen that the result is not expressible in terms of the area of the lamina. In (49) ¢ is not constant, unless the Jamina remains always similar in shape. The fundamental condition as to the smallness of v would seem to be realised in numerous practical cases; but any one who has looked over the side of a steamer will know that the motion is not usually of the kind supposed in the theory. It would appear that the theoretical motion is subject to in- stabilities which prevent the motion from maintaining its simply stratified character. The resistance is then doubtless more nearly as the square of the velocity and independent of the value of v. When in the case of bodies moving through air or water we express V,a,and v in a consistent system of units, we find that in al ordinary cases v[/Vai is so very small a quantity that it is reasonable to ‘identity f(v[Va) with f(0). The in- fluence of linear scale meget the character of the motion then disappears. This seems to be the explanation of a difficulty raised by Mr. Lanchester (loc. cit. § 56). ro ne A 3 ‘ae LXXXIII. The Velocity of the Lons of Alkali Salt Vapours foe nonmessn sonperok st. A. WILSON, Butts. pel seve Oe McGill University, Montreal”. T was shown by the writer in 1899 that, in flames, all the alkali metals give positive ions which have equal velocities due to an electric field. This result has been con- firmed by Marx and Moreau. The value of the velocity is about 70 cms. per sec. for one volt per cm. The fact that the maximum quantity of electricity which can be carried by a definite amount of any alkali salt vapour is equal to that required to electrolyse the same amount in a solution}, shows that the product Ne has the same value in salt vapours as in solutions. Here N is the number of nositive ions formed from one gram molecule of the salt when completely ionized and e the charge carried by each ion. Since in solutions each atom of the alkali metal forms * Communicated by the Author. r Phil. Trans. A. cexxxvii. (1899). { H. A. Wilson, Phil. Trans. A. cexcvi. (1901). (2 Prof. H. A. Wilson on the Velocity of the one monovalent positive ion, the result just mentioned makes it probable that the same thing happens in the vapours. Prof. O. W. Richardson* has recently measured the ratio of the charge e to the mass m for the positive ions of vapours of the sulphates of all the alkali metals, and finds it equal to ~ the value which obtains in solutions. This makes it very probable that the positive ions are metal atoms. The fact that the haloid salt vapours give ions having the same velocities as the oxysalts in flames, shows that ail salts of any one metal give ions identical in nature. The equality of the velocities of a lithium ion and a cesium ion is difficult to explain on the view that they are simply single atoms, for we should expect-the velocity to depend on the atomic weight. The main object of this paper is to point a way out of this difficulty. In my experiments two electrodes were placed one above the other in a Bunsen flame, and the current between them was measured. If the upper electrode was positively charged and a bead of salt was placed just below it, it was tound that the current was not appreciably increased by the salt unless the potential difference between the electrodes was greater than about 100 volts. This was taken to mean that 100 volts was just enough to make the positive ions move down the flame. The potential gradient in the flame is nearly uniform except near the electrodes, so that the current density (7) is given by i=en(vy+v2), where m is the number of ions of either sign per ¢.c., v; and v, the velocities of the positive and negative ions. Jf the gas is moving upwards with velocity uw and X denotes the electric intensity, then we have vy=hX—u Ve=hh,X +u, so that the current is equal to en X(k, + k,), and is independent of u. When the upper part of the flame is filled with salt vapour n will be much larger in that part than in the rest of the flame, so that for a given current X will be proportionally smalier. ‘This diminution of X, however, does not lead to an appreciable increase in the current, when the upper elec- trode is positive, because nearly all the resistance to the passage of the current is close to the negative electrode, where the greater part of the fall of potential takes place. * Phil. Mag. Dec. 1910. Tons of Alkalt Salt Vapours in Flames. 713 Thus if X' and n’ refer to the upper part of the flame containing the salt, we have {= Xa el + hy) = Nr +h), which is independent of w, so that it does not appear at first sight why the salt should increase the current with any potential difference. If, however, X is big enough, the metal ions will move down against the upward stream of gas and will be deposited on the negative electrode. The alkali metal will consequently accumulate at the lower electrode, and since it is strongly ionized it will diminish the resistance there and so increase the current. In the case of sodium salts this accumulation ean be easily observed by the appearance of sodium light near the lower electrode*. It appears, therefore, that the increase in the current is not as was originally supposed, due merely to the current carried by positive *ons coming down, for before the salt is put in there are aircady present far more than enough ions to carry the current allowed by the resistance at the negative electrode. Suppose that an alkali metal atom in the flame is ionized for a fraction f of the time. Then its velocity due to an electric field will be f4,X instead of &,X. If, then, Xo denotes the least value of X for which the metal accumulates at the lower electrode, we have F(aX.—n) =G—f or u f= we for during the fraction (L—/) of the time the atom will be carried upwards with velocity U. The quantity which was determined experimentally was therefore not k, as was supposed, but fk, Now the con- ductivity imparted to a flame by equal numbers of molecules of different alkali metal salts increases rapidly with the atomic weight of the metal. This shows that a ceesium atom is ionized for a much larger fraction of the time than a lithium atom. Hence, since 5 both give the same value for 7k, it follows that Ay for lithium muh be really much ereater than *, for cxsium. In hot air at about 1000° C. the 7’s will be much smallei * EH, A. Wilson, Proc. R. I. 1909. Piul. Mag. 8. 6. Vol. 21. No. 126. June 1911. 3 A 714 Prof. H. A. Wilson on the Velocity of the than in a Bunsen flame, so that the values found for fh, should be much less, as was found to be the case*. The relative values of the fraction 7/ for different salts can be deduced from the conductivities which they impart to the flame. In the determination of 7k, the concentration of the metallic atoms which move down the flame is extremely small, not enongh to appreciably colour the flame except in the case of sodium. ‘The equilibrium between the atoms and negative electrons will, therefore, be determined by the equation | g=B(N—n)=anm, 2) eee where g is the number of positive ions produced per c.c. per sec. by ionization of the metal atoms, N the total number of metal atoms present per c.c., n the number of metal atoms which are ions per c.c., m the number of nega- tive electrons per c.c., and 8 and @ are constants. m will be large compared with n because the ions of the flame will be much more numerous than the ions from the metal vapour. Now f=n/N, so that equation (1) becomes B(L—f) am’ for another salt we have in the same way B'(1—f') =«'mf’ fa!_f 1-f yi = Fre rae |e © ° e ° ° 2 ap fiat @) When the conductivities imparted to the flame by different salts were compared, the salts were present in comparatively large concentrations, and the number of ions due to the salts was large compared with the number due to the flame gases. In this case, therefore, and for-another salt having the same moleculir concentration g=f'(N—n’). Here 7 will be small compared with N, so that approximately =g.--::.. 2 Hence Then (2) and (3) give by mM * H. A. Wilson, Phil. Trans, A. cexxxvil. (1899). Tons of Alkali Salt Vapours in Flames. 713 It will be observed that in this equation the /’s apply to very small concentrations, while the q’s are for a comparatively large concentration. Sir J. J. Thomson has given the theory* of the relation between the potential difference (V) between two electrodes in a flame and the current (2). He obtains the equation ie av? 1 (i an 6 ak NO drretgy/ a, When the electrodes are near together, as was the case in the measurements of the currents with different salts, the first term can be neglected, so that a3 V =F ap Sn a a Ame’ kyzki 242 If 2’ denotes the current obtained with another salt and the same V, then y=) (5 =(7) ai The experiments show that /k,=/'h,’. Hence, using (4), ane a eae fy (>) = G7) G) or L GCE se ee Here 7 and 7’ are the currents given by the same P.D: with two different salt vapours of equal concentration. ‘The ratio i/t' ought therefore to be independent of the concentration. When the concentration of the salt is not too large this is true, for then the current is proportional to a power of the concentration, which is about one-half for all salts. The following tablet gives the currents observed when a yo normal solution of the chloride was sprayed into the flame ri HOondtichicn of Electricity ieouee Gases,’ 2nd ed. ine Lhe Electrical Conductivity and Luminosity of Flames containing Vaporized Salts,” Smithells, Dawson, and Wilson: Phil. Trans. A. eexli, (1899). | DA, 2 716 Prof. H. A. Wilson on the Velocity of the using a P.D. of 5°60 volts. With this P.D., the first term in Sir J. J. Thomson’s equation can be safely neglected. Metal. Current. Gecm <6. 123 Rubidium 47). 41°4. Potassionr siz.) ) 2a moddum: 't) 4.4). a0 iar eee ES In order to use these values of the currents to calculate the 7’s we require another relation. If the positive ions consist of single atoms, then their velocities ought to be approximately inversely proportional to the square roots of their atomic weights (M), consequently / ought to be pro- portional to ,/M. Instead of (5) we can write Af lea) where A isa constant. Putting f=B /M this becomes __ AB ~~ (1—BM2)3° Two values of 2 and M then suffice to determine A and B. Using the values for ceesium and sodium gives B=0-08594 and sA = Oil. With these values of A and B we get the values of f given in the second column of the following table :— I= a | | | Metal. | a ye ky. Be ein, eee ee | 0-99 099 | al 71 Rueiamn. | 079 096 89 73 IPotasswmn: ioe. | 0°54 091 130 oh eScdtun tL een. i) ae aoe 170 170 | Lea inianee eee | 0-23 | 0-21 305 333 | The column headed f’ contains the values of f required by the observed currents. The differences between f and f’ are not very great, except in the case of potassium. It seems, therefore, that the assumption that f varies as VM is roughly true. The column headed f, contains the values of the velocities of the positive ions got by using the numbers for f, and that headed £,/ those corresponding to the numbers Tons of Alkale Salt Vapours in Flames. riely7 under f’. The value of fk, was taken to be 70 cms. per sec. The value of &, can be calculated roughly on the kinetic theory of gases, for in a flame at about 2000° (. the free path (A) vi an atom is probably about 10-4 cm.* The well known formula k,=eA/mV_ gives k,=300 ems. per sec. for an atom of hydrogen, or 120 for an atom of lithium. This is about one-third the value of k, as estimated above, which is as near as could be expected. I think, therefore, that the evidence provided by the measurements made with the object of finding the velocities of the positive ions in flames is not inconsistent with the view that these ions are single atoms of the alkali metal. Measurements on the negative ions in flames have also been made by the writer and otherst. The negative ions appear to be free electrons, so that their deposition on the positive electrode cannot be supposed to cause an increase in the current, as in the case of the positive ions at the negative electrode. It seems, therefore, that the supposed determinations of the velocity of the negative ions in flames, by finding the least P.D. required to make them move against or across the stream of gas, are based on a fallacy. When the salt is put in near the negative electrode the large re- sistance there is diminished, so that the current ought to be increased, whether the P.D. is big enough to make the negative ions move against the stream or not. The following table contains the currents observed taken from my paper f. Current. Pes (Volts). (Without salt.) (With salt.) 0 —-3 —13 0°25 —2 —10 0-5 0 — 7 0:75 +2 0 10 +3 yo 15 +3 +20 2:0 +3 +80 30 +3 +33 * Sir J. J. Thomson, * Conduction of Electricity through Gases, 2nd ed. + Marx, Moreau, and E. Gold. } Phil. Trans, A. cexxxvit. p. 517 (1899). 718 Prof. H. A. Wilson on the The increase in the current between one and two volts with salt below the upper (negative) electrode was supposed to show that above one volt the negative ions from the salt ‘moved down the flame. The increase in the current below one volt is however quite marked. It was then supposed that the flame without salt was not strongly ionized, but now it is known that the small current without salt is due to the great resistance close to the negative electrode, and that the ions present are sufficient to carry a current of probably many amperes. Such experiments, therefore, do not give any information with regard to the velocity of the negative ions. Under these circumstances it is necessary to fall back on indirect evidence. Measurements of the effect of a magnetic field on the conductivity of a Bunsen flame made by the writer* indicated that the velocity of the negative ions was about 9000 cms. per sec., which is about the value to be expected for negative electrons. We may therefore conclude that the positive ions of alkali salts in flames are probably single atoms of the metal, and that the negative ions are electrons. In a recent paper Mr. Lusby f finds 290 cms. per see. for the velocity of the positive ions of salt vapoursin flames. In his experiments the electrodes were only 3 cms. apart, so that in the absence of salt the uniform gradient was not present because the negative drop extends more than 2 cms. from the lower electrode. On putting in the salt near the upper electrede he observed a very small uniform gradient which is evidently due to the high conductivity of the salt vapour. To calculate the velocity of the ions correctly the value of the uniform gradient just below the salt vapour is required, and this should be equal to the gradient in the absence of salt since the current is unchanged at the critical potential. I think therefore that Mr. Lusby’s result is too high. t = a LXXXIV. The Number of Electrons in the Atom. By Prof. H. A. Wusson, FBS. #50.8.C., McGill Uninere Montreal t. A CCORDING to Sir J. J. Thomson’s theory § atoms may be regarded as spheres of positive electricity containing negative electrons which can move about freely inside the positive charge. The total negative charge on the electrons * Proc. Roy. Soc. A., vol. Ixxxii. + Proc. Camb. Phil. Soc. vol. xvi. Pt. 1, 1911. 1 Comiunicated by the Author. - § ‘The Corpuscular Theory of Matter,’ 1907. © Number of Electrons in the Atom. 719 is equal to the positive charge on the sphere in a neutral atom. The object of the present paper is to show how to obtain an approximate solution of the problem of the distribution of n electrons in a positive sphere and how to deduce the number of electrons in any atom from the atomic weights of the elements. Consider an electron having a charge e¢ inside a sphere of positive electricity of uniform density of charge p per c¢.c. Close to the electron the electric field is of strength = where r is the distance from the electron, so that 47e tubes of electric force come out of the electron, if the number of tubes per sq. cm. is taken to be equal to the field strength. Con- sider one of these tubes of force and let ds be an element of its length and « its cross-section at ds. The charge in the length ds is pads, so that d Lah age (Ka) =4arpa, where F is the electric force along ds. Integrating along the tube this gives F\2,—Fa=4ap | ads, where Fe, denotes the value of Fa at the surface of the electron. This shows that as we go along the tube Fe diminishes and when Fie = Arrp\z ds it will be zero and the tube will end. Now F,=e/a?, where a is the radius of the electron, and a,=a?/e, so that Fy2,;=1, hence Amp\ a ds from the surface of the electron to the end of the tube is equal to unity. Thus the volume of each tube is fe and the volume of all the 47e tubes is therefore e/p. Thus the tubes of force starting from the electron occupy a volume e/p, and this is true in any case whether other electrons are near or not. Also, since every tube of force must end on positive electricity, it is clear that the volume e/p can only contain the one electron from which the tubes start. Thus when any number of electrons are present each one will be surrounded by its own field which will occupy the volume e/p. The positive charge in the volume e/p is equal to e, so that if the sphere has a positive charge equal to the total negative charge on the n electrons in it, it will be divided up into x equal volumes each containing one electron. The energy in an element of a tube of force is equal to 720 Prof. H. A. Wilson on the F’a ds/87, and if the tube is slightly distorted the volume of each element and the value of Fa remain unchanged, so that the change in the energy in the element will be due to the change in F. The energy will be a minimum when the tube is in equilibrium, so that F will be as small as possible and therefore aas large as possible. This means that the tubes tend to become as shortas possible, their volumes remaining constant. The effect of this will evidently be to make the field round each electron tend to become as nearly spherical as possible with the electron in the middle. Consequently, to determine approximately the distribution of the n electrons in the positive sphere, it is sufficient to find how the sphere can be divided up into n equal volumes all as nearly spherical as possible and to put an electron at the centre of each of the n volumes. When n is large it is easy to see that this requires the electrons to be arranged like the centres of the shot in a pile of shot. Thus with thirteen electrons we should expect to have one in the middle and twelve arranged round it all at the same distance from it. It is easy to see from considerations of symmetry that the electrons will arrange themselves on nearly spherical surfaces concentric with the surface of the positive sphere. The condition that the fields of the electrons shall be as nearly spherical as possible evidently requires the distances between the successive surfaces to be all equal. The fields of the electrons on the surface of a sphere will form a layer the eube of the thickness of which will be approximately equal to the volume of the field of one electron. According to Sir J. J. Thomson’s theory each element in a series of similar elements, such as fluorine, chlorine, bromine, iodine, is derived from the one before it in the series by the addition of a spherical layer of electrons together with the necessary amount of positive electricity to keep the atom neutral. Let 2,, 2, 3, &e., denote the numbers of electrons in the atoms of a series of similar elements and let Aj, Ay, As, &e. denote their atomic weights. Then, if we assume that the number of electrons in an atom is proportional to its atomic weight, we can write BA,;=7,, BA,=n., &c. where £8 is a constant. Let 71, 2, &c. denote the radii of the positive spheres and let v=e/p be the volume of the field round each electron. Then we have AL Lag 5 7 12 nf UA 3 Pies =m 1 1¥ = BvA m41* - Number of Electrons in the Atom. 721 Hence 4or 3 Bv where C is a constant which should be the same for all series of similar elements. u . ® ub i ) (7'm4+1—1m) Se nN nO, Also ('m+1— 1m)? = UV approximately so that Anr Toe According to the theory therefore we ought to be able to find the number of electrons per atom from the atomic weights. (op) qn Bis CUBE Foor oF Aromic WEIGHT NATN NUN xe . ORDER IN SERIES. Ss e © e In the figure the values of A® for series of similar elements are plotted against the order of the elements in the series. 722 Dr. R. W. Boyle on the Behaviour of For some series a constant has been added to the values of A® to prevent the different lines falling too close together. It will be seen that the values of A® for each series fall nearly on straight lines and that the different lines are nearly parallel. This shows that Ah .,—A5=C is nearly constant, as was to be expected from the theory. Tke mean value of C is about O°S1. Hence we get 8=8, so that the number of electrons per atom comes out about 8 times the atomic weight in all cases. This estimate agrees as well as could be expected with recent estimates depending on the scattering of radiation by different elements. : Since n=BA we have Aa\3 3 3 ea = Nmt+1— Mins By means of this equation it is easy to calculate the number of electrons in successive spherical layers. If we take ny =8 we get the following values of mm :— Mm Tite n nee om 8 e iene Ree 8 ii H 2% Z cece corce 47 6 Li —i DEAN ie ak 142 18 Na=2Za Lato 320 40) Ka eee» See OOD 7d Rb=8d CER ee 1020 128 Cs =a The last column contains the atomic weights of the alkali metals, which do not ditfer very much from the values of nj8. Since the calculations made are only approximate, the agreement is as good as could be expected. LXXXV. The Behaviour of Radium Emanation at Low Temperatures. By R. W. Boyiz, M.Se., Ph.D., 1851 Ealubition Science Scholar, McGill University*. rEXHE researches of Rutherford ft and of Gray and Ramsayt have shown that at temperatures from —127° C. to 104° C. the emanation of radium has definite and constant values of vapour pressure corresponding to every tempera- ture. At temperatures below —127° C. the only knowledge * Communicated by Prof. E. Rutherford, F.R.S. + Phil. Mag. [6] xvii. p. 723 (1909). { Journ. Chem. Soc. xev. p. 1078 (1909). Radium Emanaton at Low Temperatures. 723 we have concerning the process of volatilization of condensed emanation is that given by the flow method of experiment originally devised by Rutherford and Soddy *. This method is best adapted, and so far has been used, for experiments with small quantities of emanation. Under the circum- stances of its use condensation of the emanation can only result in a very sparse distribution of emanation moiecules over a considerable area of cooled surface, so that the con- densed “layer” will be of much less than molecular thickness. In these cases it is probable that the phenomenon is entirely one of surface adhesion or occlusion. The object of the present paper is to describe briefly some experiments which were performed to seek further informa- tion on the process of volatilization at low temperatures. The conditions in the experiments were quite different from those in the flow method. The emanation was contained in sealed glass tubes which were as free as possible from all other gases; condensation and volatilization were confined to the point where the minute volume of condensed emanation was situated; and no current of air or other gas was required. Apparatus and Method of Experiment. The tubes containing the emanation were of the shape ABC shown in fig. 1 (p 724). The wall of the part AB was 2-5 mm. thick, and of the part BC 1mm. thick. The bore of the tube was usually about 2°55 mm. The end of the tube at A was closed by a very thin sheet of mica, which was secured to the wall by a special kind of marine glue. The thickness of the mica was equivalent to 1:9 cm. of air in its stoppage of «-particles ; nevertheless the sheet was strong enough to support the full atmospheric pressure over the opening, and thus maintain a vacuum inside the tube. For an experiment the glass tube was first evacuated to a charcoal vacuum. The required amount of purified emanation received from Prof. Rutherford was then introduced, and the tube was sealed at C. Four hours after admitting the emanation the amount present was determined by the y-ray method. The tube was then fitted to a small ionization vessel MNO in the manner shown in the diagram. The vessel and fittings were made air-tight so that the pressure of air in the ionization chamber could be adjusted to any value. With such an arrangement the ionization was usually very intense, and it was often very difficult to obtain saturation. * Phil. Mag. [6] v. p. 561 (1903). (24 Dr. R. W. Boyle on the Behaviour of The method of performing an experiment was to condense the emanation at the extreme end © of the containing tube TO BATTERY AND GALVANOMETER. FF by immersing this end in a bath of pentane cooled with liquid air. Condensation was maintained for four hours, during which time the active deposit about the tube practi- eally all decayed, and the ionization gradually decreased to a small constant value. The double right-angled bend in the tube at B prevented the «-rays from the condensed emanation and its active deposit at C having any ionizing effect in the vessel above. After keeping the emanation condensed for four hours the temperature was allowed to rise slowly, and continuous ob- servations of ionization were taken. Under these conditions, whenever any of the condensed emanation at C volatilized, the emanation vapour quickly distributed itself throughout the tube, and the portion of it going to the upper part AB Radium Emanation-at Low Temperatures. V2 marked its presence there by causing an increase of ionization. In this way the ionization measurements indicated roughly the amount of emanation in the upper part of the tube cor- responding to the various temperatures, and thus gave a means of tellowing the progress of the volatilization. So) The free end of the glass tube which contained the emanation was dipped into a very small glass bulb containing just enough pentane to cover the end of the tube and the junctions of the thermo-couples which were used for determining the temperatures. Surrounding this small bulb was an outer bath of pentane, which was itself surrounded by a bath of liquid air contained in a 6-inch, silvered, Dewar cylinder. The Dewar cylinder was kept filled to the top with liquid air until it was desired to allow the temperature to rise. The liquid air was then allowed to evaporate slowly, and the gradual lowering of the level caused a slow variation of temperature at the point where the emanation was condensed. The temperature usually rose at a rate of 05 C. per minute. Without the double bath arrangement it was found that the rate of temperature rise was not sufficiently uniform. Some trial experiments with moderate quantities of emana- tion showed, after the baths were removed, the presence of a bright point of light at the extreme end of the glass tube, and a uniform fluorescence over the rest. ‘This bright spot was due to the active matter which had been deposited by the condensed emanation. Its concentration at this one point showed that the emanation had condensed not over any considerable area but at the very tip of the tube. Conse- quently the junctions of the thermo-junction were placed in the inner pentane bath exactly at the tip of the glass tube which contained the emanation. The thermo-couple was a double, copper-constantan element of number 30 double-cotton covered wires. ‘The warmer junctions were maintained at the temperature of melting ice. Tt has already been mentioned that there was a difficulty in obtaining saturation in the ionization chamber. Since the ionization-teuperature curves afterwards shown could not be used to determine the actual amount of emanation volatilized at a given temperature, complete saturation was not essential; nevertheless, in all experiments saturation was approximately attained. For this purpose the ionization vessels employed were made very small. The one mostly used was a brass cylinder, 1 cm. in diameter and 5 cms. long, fitted with the usual central electrode and ebonite insulation. Since the quantities of emanation employed in different The condensing arrangement finally used is shown in fig. 1. 726 Dr. R. W. Boyle on the Behaviour of experiments varied over a wide range, it was necessary in measuring the ionizations to use instruments varying widely in sensitiveness. With the larger amounts of emanation a Kelvin astatic galvanometer, of which the greatest sensitive- ness was i scale-division (millimetre) for 1:2 x 10-!° ampere, could be used ; with the smaller quantities a gold-leaf elec- troscope sufficed. When using the galvanometer and large quantities of emanation, it was necessary to reduce the pressure in the ionization chamber in order to obtain saturation. But the pressure could not be too far reduced, for a diminution of the pressure caused a decrease in the current, and it was desirable to work with a fairly large deflexion of the galvanometer- needle. (Increased voltage sometimes helped to this end.) In general, approximate saturation with satisfactor y deflexion was obtained by manipulating the pressure, the voltage, and the position of the control magnet of the galvanometer. The maximum deflexions in different experiments varied from 200 to 400 divisions at a scale-distance of 1°41 metres. When using the gold-leaf electroscope the only possible adjustment to secure saturation was to lower the pressure of air in the ionization chamber. Some possible causes of error which may enter into the ie should be mentioned. The emanation produces ases—mostly carbon dioxide—by its action on the marine Bite with which the mica sheet was secured to the containing tube. If the volume of the tube were ver y small, as in the case of a capillary, these gases would have an effect in re- tarding by diffusion the passage of the volatilized emanation from the lower to the upper part of the tube. Again, with capillaries, another error due to viscosity would ‘affect the readings in the same direction. ‘The frictional resistance of the gas in the capillary would retard the passage of the volatilized emanation from the lower to the upper part of the tube. Some trial experiments showed the necessity of avoiding these troubles. The best conditions of experiment were (1) to use well cleaned tubes of not too small a bore, and therefore of not too small a cubical capacity ; (2) to perform the experiment as soon as possible after the admission of purified emanation into the tube ; (3) to obtain as nearly as possible saturation of ionization. From a number of experiments in which these conditions were fulfilled the curves shown in fig. 2 are given as samples. Radium Emanation qt Low Temperatures. IONIZATION CURRENT... i4 12 = 9 -/40 -/80 -470 -/60 (50 -/40 TEMPERATURE, _-/80 -/70 -/60 ~/59 ~{40 150 = _ - a — ee — ee ee ee pS 7 =a = aS SS rT ec Tee: SSS Sa pete ce mal 728 Dr. R. W. Boyle on the Behaviour of The ordinates represent ionization current and the abscissee the corresponding temperatures. The three curves, A, B, and C, represent three widely different quantities of emanation, which correspondingly required three measuring devices differing widely in sensitiveness. The scales of ordinates are therefore very unequal: scale of © > scale of B > scale of A. In Table I. are given the experimental details concerning the curves shown in fig. 2. The “ Partial Pressures of Hmanation” are calvulated—assuming Boyle’s law to hold approximately—from the fact that the equilibrium amount of emanation of 1 gm.of radium hasa volume of 0°6 cub.mm. at N.T.P. These pressures are included merely to give an idea of the extreme tenuities of the emanation before or after condensation. (The abbreviation “ m.r.e”’ means the amount of emanation in radioactive equilibrium with the stated number of milligrams of radium element.) Tasue I. (1) Q (2) ; a) ocala ae ee (5) a uantity oO ol. 0 artial Pressure e =. Curve. | Emanation.| ‘Tube. of Emanation. Measuring Device. ap 100 mm. He. AL vedise: 78 m.r.e 03 cc. | 6X10 mm. Hg./Galvanometer 4 pressure in ioni- zation chamber. oc Only a few mm. Bo... 0°30, 10 ,, jL5x10 Y Electroscope Hg. pressure in ‘ionization cham- | ber. ans 760 Gumi) ae blocks OO, Dis ss 2x10 », |Electroscope , pressure in ioni- | catiee chamber. The general form of the curve consists of three parts. First there is an initial, flat, or nearly flat, portion ; then a steep portion rising from the temperature axis; and then a bend towards the temperature axis, after which the curve continues to rise but much less markedly. The last portion extends much further than is shown in the diagram. As an example the following table shows the readings in an experi- ment where the galvanometer was used. Radium Emanation at Low Temperatures. 729 asa oe Deflexion before condensation ............... 25°4 em. Deflexion 4 hours after applying liquid air 0°5 ,, On allowing the temperature to rise :— Temperature. Deflexion. | —177 0°38 cm. —172 08 — 1645 0:8 | —160°5 Telos | —154:5 2°55 | —142°5 7:65 —1315 115 | —1185 13°6 | —116°5 14:6 | ‘Followed by along period of very slow rise of ionization. The form of the curve can be very simply explained on a basis of a normal behaviour of the emanation. Consider, first, curve A which corresponds to 78 m.r.e., and in which a galvanometer was used to measure the ionization. The galvanometer being a very insensitive in- strument required an enormous ionization to affect it. Apparently there was no appreciable rise of ionization, and therefore no appreciable volatilization of emanation until the temperature approached —163°C. To this is due the initial, flat, portion ot the curve. About —163° the emanation began to volatilize in larger quantity, the vaporized emana- tion distributed itself throughout the tube, and RaA and Ra C began to grow. As the temperature increased all three products—the emanation, Ra A, and RaC—increased more rapidly in the upper part of the tube. The continuously increasing number of e-rays sent out by these products caused a corresponding increase of ionization in the chamber above, thus giving the steep, rising, portion of the curve. After the emanation has all volatilized the rate of inerease of emanation vapour in the top part of the tube must fall off very greatly. On account of the low temperature at the bottom of the tube, the density of the emanation is greater at this part than in the upper part, although all the emanation has volatilized. But as the bottom temperature continues to rise slowly the density here decreases, and this causes a slow transference of emanation molecules from the lower to the Phil. Mag, 8. 6. Vol. 21. Wo; 126. Jame 1911. 3B 730 Dr. R. W. Boyle on the Behaviour of upper part of the tube. This slow increase of emanation in the upper part of the tube, with the consequent growth of Ra A and RaC, causes the ionization to increase, but much less rapidly than before, and thus gives the final, long- continued, portion of the curve. Within the range of temperature covered by the initial, flat, portion of the curve, the emanation was certainly vola- tilizing, though not in sufficient quantity to affect the galva- nometer. But a more sensitive instrument should be affected within this range, and this is shown to be the case by curve B. This curve corresponds to a quantity of 0°30 m.r.e. of emanation and an electroscope with a pressure of a few mm. Hg in the ionization chamber for measuring the ioniza- tion. Instead of the flat portion extending from the lowest condensing temperature to —163° C., as in the case of eurve A, it extends here only to —171° C., after which the curve rises, and then bends towards the temperature axis, as explained. The measuring device in this case was so much more sensitive than the galvanometer, that it could detect the changes in the vapour phase of the emanation at temperatures as low as —171° C., whereas the galvanometer could only do this as low as —163° C. A still more sensitive device, viz. an electroscope with atmospheric pressure in the ionization chamber, could detect the changes in the vapour phase at a lower temperature still. This is the case of curve C, where 0:01 m.r.e. was employed. From the above it follows that if it were feasible to condense a very large quantity of emanation, and employ at the same time only one measuring instrument possessing the required ranges of sensibility, we could obtain a single ionization curve of the form already shown. But this curve would rise immediately from the lowest temperature of condensation, and would cover a wide range of temperature before bending towards the axis of temperature. In other words, the con- densed emanation would begin to volatilize at the lowest temperature of condensation, and would continue to volatilize through a wide range of temperature until the emanation was entirely free from the condensing surface. In the experiments we cannot be sure that the amount of emanation volatilized at any temperature was the exact amount required to saturate the space of the containing tube at that temperature, and from the curves given we cannot calculate the vapour pressures. The experiments were only qualitative, and the complications introduced by the rate of rise of temperature, the growth of Ra A and Ra C with their different ranges of «-particles, effectively prevent our utilizing Radium Emanation at Low Temperatures. ene the ionization measurements for quantitative calculations. The slight ionizations at the beginning of each of the curves shown were due to the y and some @ rays from the radio- active products of the condensed emanation. The experiments show that a vapour phase corresponding to condensed radium emanation can easily be traced to a temperature as low as —180° C. Gray and Ramsay*, as the result of an experiment in which the opacity of the condensed emanation was the test of solidity, state that the emanation solidifies at —71° C., the vapour pressure then being 500mm. Hg. Under the infinitesimal partial pressures and low temperatures in the present experiments the state of the condensed emanation is not known, but whether it exists as solid, or as liquid, or as an adsorbed layer, we should expect on a basis of behaviour like ordinary gases under familiar conditions : (1) that at any temperature a vapour phase of the emana- tion would exist ; (2) that volatilization from the condensed to the vapour phase would set in as soon as the temperature com- menced to rise ; and (3) that volatilization would proceed gradually, becoming more and more rapid as the temperature increased. The experiments described bear out these expectations, and thus far the behaviour of the emanation may be said to be normal. It can be seen that the temperature of final volatilization from the condensing surface will depend on the quantity of emanation, and therefore it cannot be said, unless particular conditions are stated, that the emanation when condensed will volatilize at any particular temperature. The experiments cannot tell us whether the phenomenon of volatilization under these conditions is affected by surface adsorption or adhesion. Such matters will be settled when it can be shown definitely that in equilibrium with these infinitesimal volumes of condensed emanation there is, or is not, at any fixed, low, temperature an invariable value of vapour pressure. Already we have from Russ and Makowert a few incidental observations which suggest that at the liquid air temperature the amount of emanation vapour in equi- librium with condensed emanation is not a fixed quantity but depends on the amount of emanation condensed. Following the work already described in this paper, the moa cule t Le Radium, vi. 1909, p. 182; Proc. Roy. Soc. A, Ixxxii. p. 205. a2 732 Behaviour of Radium Emanation at Low Temperatures. writer made a number of experiments with the object of deter- mining quantitatively the vapour pressures at temperatures upwards from —180°C. The experiments were not successful, but they may be briefly referred to. The emanation was condensed in a manner somewhat similar to the one described on p. 725, in the bottom of a narrow glass tube connected to the side of a larger vessel which terminated at the top in a series of small bulbs. The whole apparatus was exhausted to a charcoal vacuum before the emanation was introduced. After condensing the emanation and securing temperature conditions as steady as possible, the condensing tube was opened to the larger vessel for half an hour in order to ensure a constant dis- tribution of the emanation. The connexion was then closed, and the emanation distributed in the larger volume was compressed over mercury into the topmost small bulb. This bulb was then sealed off, the mercury was lowered to its original position, and the same process was repeated at another temperature. After a set of experiments, the ema- nation contents of the different bulbs were measured, and from them could be calculated the vapour pressures corre- sponding to the different temperatures employed. The result of these experiments supported the conclusions arrived at from the former experiments, and showed that the pressures were of such small orders as the figures of column 4, Table I., would suggest. But the numerical results were very irregular, and they could not be even approximately repeated under the same conditions. There was great dif- ficulty in maintaining constant temperatures ; but the chief cause of the failure was due to the action of the emanation in producing, in the course of an experiment, appreciable quantities of carbon dioxide and other gases from the im- purities introduced into the apparatus by the mereury and the stop-cocks. ‘The emanation behaved as if it con- densed along with these gases, thereby becoming entrapped and not being able to escape until the gases escaped also. The experience showed that in this type of experiment special apparatus will have to be used to prevent any foreign gas entering the condensing chamber. It is hoped that the experiments may be taken up again in the near future. The writer is greatly indebted to Prof. Rutherford for the loan of apparatus and supplies of emanation, and also for his helpful suggestions and advice throughout the course of the experiments. ares LXXXVI. The Longitudinal and Transverse Mass of an Electron. By W.F. G. Swann, D.Sc. AR.CS., As- sistant Lecturer in Physics at the University of Sheffield *. i his paper on ‘ Recent Theories of Electricity” (Phil. Mag. February 1911) Prof. L. T. More refers in a note, page 214, to the difficulty of realizing the existence of a transverse mass for an electron when the velocity is zero, in view of the fact that, as he remarks, transverse mass Is defined as mass due to a change in direction only. I think that the following method of deducing the expressions for the masses, while of course it rests on the same funda- mental bases as those hitherto employed in former investi- gations, has the advantage or bringing out more clearly the real meaning of the masses, and further it does not involve the consideration of a curvilinear motion at all f. We first define force as equal to the rate of increase of momentum produced by it in the direction in which it acts. Let us find the component er accelerations A, u,v produced by the unit forces in the three coordinate directions X, Y, Z y at the instant when the electron A is moving in any direction OA with velocity p, g, r. Let U, V, W be the components of the momentum of the electron expressed as functions of p,q, ”. G —---—- K According to our definitions of the unit forces we have ol VOW ee 2 til) La ede Se Ou) Op» (dt Op : : with similar expressions involving — and a : so that NO re i esa Me Ci), ace ae * Communicated by the Author. - + The point raised by Professor More may also be met, by observing that the expression deduced for the transverse mass by the method adopted by Abraham (see ‘Ions, Electrons, and Corpuscules,’ by Abraham & Langevin) is independent ot the radius of curvature of the curve which the electron is snpposed to describe, so that it holds for an infinite radius of curvature, z.e. for a rectilinear motion. 734 Dr. W. F. G. Swann on the Longitudinal Now, confining ourselves to the case in which the electron is moving along the axis of X with velocity v, we have (rhe Gri Gre r=0 These expressions are of course the values of the so-called longitudinal and transverse masses, the last two being the two transverse masses, which are of course equal. Now although when p=v, g =0, r=0, V is zero, it does not follow that ati is also zero. Again, though each of the quantities U, V, W is zero when p=q=r=0, it does not follow that the derivatives are zero also. Of course from symmetry, when p=qg=r=0, all three masses are the same. Let us now proceed to the deduction of the expressions for the masses: to do this it is necessary to find the general expression for the momentum of an electron moving along any line. Take axes of &,, ¢, not coincident with those of x,y, 2, and let the electron move along the axis of & with velocity w. Let a, 8,y be the magnetic vector due to the motion; then the kinetic energy per unit volume is ae 2 2 2 The resultant momentum of the electron per unit volume is Ol Aapeew eos . Oy So aa("aa tf Sat 750) Since if f, 9, h are the components of the etherial displace- ment a=0, BP=-—A4rhe, y= 4790, therefore coe oe 2 OY sate 0, an —Arh, aes Arg, and ols 4, ca An(h? +9*)o = 47 P%o, where P is the component of the etherial displacement resolved perpendicular to the line of motion of the electron. The total momentum is Arr \(\ P2w dédn dé, the integral being taken throughout all space. The rest of the analysis depends on the shape and nature of the electron. If we take the and Transverse Mass of an Electron. 139 ellipsoidal electron of Lorentz the value of our integral is (see Lorentz’s ‘ Electrons,’ p. 211) Av Dwr e2 1 @\ 724 3° ac ¢? i e being the charge in electrostatic units, a the semi-major axis of the ellipsoid, and c the velocity of light. Returning to the axes of X, Y, Z, and putting #?=p?+q?+7?, we at once obtain for the components U, V, W of the momentum resolved along the axes | se ee r+et+r we Bae ce a? an ee ( P+etr —4 ee mam aie 1c" PEG eT Wa,o{1--o*" ' T. Differentiating these expressions with regard to p, q, and r respectively, and afterwards putting p=v, g=0, r=0, we obtain for the three masses the expressions usually given f, the last two being of course identical. Wy) -(°*) == tl ae ORR 0.624 Bae ye (P=) OV 26? y\ 73 Lily = ea D=v = TD) 1-4) 3 Oy F (oj -=\\) Jac C r= ft (a 2¢? (1 ae m3 =| —— }),-,»>=>— |1-s 5 Ol yes eae A T=0 If, instead of the ellipsoidal electron, we take the conducting spherical electron we of course obtain the well known expressions corresponding to that case. * Lorentz uses the “ Rational unit” of charge, which results in an expression slightly different from the above. + In obtaining this expression the field of the electron at each point in space is taken as the field corresponding to the steady motion of the electron, All methods of determining the electromagnetic masses involve this assumption. It is a very legitimate assumption for the purpose in hand, because, as is easily shown, practically the whole momentum of the field of the electron is contained within a space of the same order of size as the electron itself, and consequently the field in this region follows the motion of the electron practically instantaneously. It may be noted that it is easy to show that even if the electron were not small, the assumption would be justified for the case of the motion of an electron starting from rest.-" °° ) howe LAXXVIL. Vhe Oscillations of Chains and their Relation to Bessel und Neumann Functions. By Joun R. Arrey, M.A., BSc., late Scholar of St. John’s College, Cambridge*. HE oscillations of chains afford interesting examples of the practical applications of Bessel functions to physical problems, These functions, in fact, first presented themselves in connexion with the problem of the small oscillations of a uniform chain suspended by one end—Bernouilli’s problem. The times of vibration in this case depend upon the roots of the equation J,(<)=0. The more general function of the same kind but of higher order, viz. Jn(<), appears in the ex- pression for the time of vibration of a chain whose line- density varies as the nth power of the distance from the free end. When a uniform chain is loaded at the free end—a more general case than Bernouilli’s,—the complete solution includes both kinds of Bessel functions, viz. J,(z) and Y,(z), and their differential coefficients. The Y,(z) functions are sometimes called Neumann functionst. The following ex- periments were carried out for the purpose of comparing the observed periods of oscillation of certain “ chains” with those of ‘“‘ideal chains” calculated from the expressions giving the periods in terms of these functions. (A) Oscillations of a uniform chain. The periodic times t of the small “normal” oscillations of a uniform chain of length J, suspended by one extremity and hanging under the action of gravity, are determined by the equation t= (47/p)(U/9)?, where p is a root of the equation Jo(z)=0. The equation J,(z)=0 bas an infinite number of real positive roots corre- sponding to the different modes of vibration of the chain. The first root py=2°405 gives the period when the whole of the chain lies on the same side of its original vertical position; the second root p,=5'520 gives the period when the chain has one node; the third root p3=8°654 gives the period when the chain has two nodes and so on. In order to compare the calculated results with those obtained by experiment, the times of oscillation of a long chain were observed. A bicycle chain was employed so that the vibrations might be restricted as far as possible to one vertical plane. The observation of the periods presented no * Communicated by the Author. _ + Gray and Mathews, ‘Treatise on Bessel Functions,’ p. 14. The Oscillations of Chains. 737 difficulty when the chain was vibrating in the first and second modes, but when the chain had three or four nodes, only a limited number of vibrations were executed without assistance, and it was necessary in these cases to maintain the motion by gentle pressure of the hand near the top of the chain. The error thus introduced is however quite small. Twenty sets of 100 vibrations each were recorded for each mode of oscillation of the chain. The time was measured by means of an accurate stop-watch. This experiment, which is quite easily performed, is perhaps the simplest example of a physical problem involving the use of Bessel functions. Value of g at Morley, Yorks=981°4 cms./sec.? Length of chain=219°9 cms. Time of vibration | Time of vibration Mode of vibration. in secs. in secs. Calculated. Observed. First (no node)......... | 2°473 2470 Second (one node) ... 1-077 1:075 Third (two nodes) ... ‘687 685 Fourth (three nodes) . D04 504 Fifth (four nodes) ... "398 397 (B) Oscillations of a heterogeneous chain whose line-density varies as the nth power of the distance from the free end. This extension of Bernouilli’s problem is due to Prof. Sir Geo. Greenhill. The form of this chain, when executing its principal oscillations, is given by y=Ax ?),(2b,/2) sin (pet +k), where 4c’?= 9, 46°=p? (n+1) ; xis measured upwards from the free end, and y is measured horizontally. The fact that the upper end is fixed imposes the condition that J,,(26 /1)=0. If p be one of the roots of this equation, / the length of the chain, and 7 the time of vibration, it is easily shown that ceaCimia) (ee b/glte, so. oD ee a oe a ie ee a ee ———— 738 Mr. J. R. Airey on the To realize the conditions of the problem practically, a number of “blinds” were constructed, each consisting of about fifty or sixty wood rods, with square cross-section, sides one cm. long and fixed -25 cm. from one another. The uppermost rod was generally about 40 cms. long. ‘The shape of the “ blind” was determined by the curves y= +ca”, where 7 had the values 4, 2, 3, 1, &e., the first curve being a parabola, the fourth a triangle, &e. The rods were held in position by means of a string passing tightly into vertical saw cuts at their ends. The whole arrangement was then suspended from two loops on the string above the uppermost od. The following observations were made of the times of vibration in the different modes, ten sets of 100 vibrations each being recorded for each mode. For comparison, the values calculated from equation (1) have been added. The roots of the equation J,(z)=O0 are easily found from the formula given by Prof McMahon. Mode of Sigal pe of Time of vibration | Time of vibration apo a ind secs. SECS. in cms. Calculated. Observed. 1 l 1071 1618 1°629 2 2 “809 817 3 539 "539 4 “404 402 1 2 101°6 1547 1:552 2 z “799 ‘798 3 “a9 "536 4 ‘407 404 1 3 87:9 1-425 1437 2 4 ‘747 "745 3 507 “504 4 384 “oie i 1 107°4 1:535 1-538 2 838 ‘835 3 "578 571 4 441 437 5 B57 B59 1 | 5 86-4 1342 1352 2 4 758 “758 3 | 530 BIA 4 ‘406 404 | Oscillations of Chains. 139 (C) Oscillations of a uniform chain loaded at the fi ee end. The Bessel functions Jo(z), Jy(z), and J.(z), and the Neumann functions Yo(z), Y,(z), and Y,(z) appear in the expression for the times of vibration of a loaded chain*., If the load attached to the lowest point of the chain be n times the mass of the chain, the periods of oscillation in the different modes can be found from the roots of the equation M2) eee) Ne) (2) Jo(Az) — edo(z)—2d(z) ~— Je(2)’ where z= (4r/ts)(nl/q)?, N=[(n+1)/n]?, and 7;=time of complete vibration in the sth mode. A hbicycle-chain about 150 cms. long was suspended by one extremity and a load was attached to the other. Through two openings in the lower end of the lowest link of the chain, a steel rod was passed which supported a number of perforated iron disks about 4 cm. thick. The radii of the disks varied from 1 cm. to4 ems. The load could by this means be made any multiple or submultiple of the mass of the chain. Load equal to or greater than the mass of the chain. Mode of ees nen of | Time oes Time ohne | vibration. of n. cms. Calculated. Observed. 1 1 219-9 2'812 2'809 9 ‘730 “725 i if 171-4 2°484 2472 2 645 638 iI 2 148°8 2°370 2°361 9 ‘475 "472 1 PWG 148'8 2-410 2410 9 *396 32D 10 1488 2-495 2°425 9 ‘937 ‘236 * Routh, ‘ Advanced Rigid Dynamics,’ 1905, p. 400. 740 Mr. J. R. Airey on the Load less than mass of chain. | Mode of | Value | Length of Time of Time of vibration. of w. chain, vibration. vibration. clus, Calculated. Observed. Mite ae alee 1488 2-297 2-274 2 5) 639 ‘638 1 bhi ade dos 4 2:255 2-246 - 2 715 “15 - 1 do. 2211 | 2-200 3 172 ‘766 3 -428 | “426 ree: al | _ 1 eave an 2-128 2-114 - 8 "856 | 850 3 506 | 500 | 1 1 do. 2-092 2-076 | 2 15 -883 ‘877 3 | 537 5381 | 1 1 do. 2.072 2-058 ~ 24 890 ‘882 3 557 552 4 386 “382 (D) The general expression for the roots of the equation Yo(Az) oe 2Y0(e) —2Y1(2) Ne Vee) Jo(Az) a zd (2) —2d1(<) ¥ Jo (z) : where A is greater than unity, can be suetnedl by following the method adopted by Prof. McMahon * in finding the roots of Bessel and other related functions. Subtract log 2—y from each sidef. Then No(Az) _ No(z) es TO | N,,(z)= Y2(2) — (log 2—y)Jn(z). where 7 2 Substitute the semiconvergent series for Jo(Az), No(Az), Ke. in equation (3) and put Q)(Az)=Risin 0, Az)= Kh cos... ee O72) =s sin 7, eee) =S cosgt>* ie * McMahon, Annals of Mathematics, 1895. + Jahnke u. Emde, Funktionentafeln, 1907. Oscillations of Chains. | 741 We find, after simplification, that tan (eae - + 0)= tan ea a eh); TT a A2—-— +0=2- He aediae ar 4. or (A—l)z=nr+n-—0. [n=1, Jeo Writing y for =. we get from (4) i Roos OG oLe De PAS ae Bin and making use of Gregory’s series, 1 100 343536 ese BGs bes e e (6) Similarly Eon ed) 47520 2 TG: Ge w Gan Ee ee ee (yi) Sinee nT 1 cee ee a we find, on substituting the values of 7 and @, and writing 8 NIT rA—1L? Ne 15A4+1 54042 + 100 237600 — 34336 ae SrA(M—1)z BARB (A—1)z8 BNP 8P(A—1) 2° for an equation of the form cae eee ae where | ey ae Pe 1) 0 2h eeeoy an) and (425)°— 1073 _— 5120\(A—1) - a eit = By Lagrange’s theorem Dp g—-p . r—dpgt 2p" c=Bt+o+ athe + - ms eee UE B Close approximations to the earlier roots are not given by 742 Prof. J. EH. Ives on an Approximate Theory of an this series. In these cases, closer values are obtained by interpolation from tables of the Jo, Jj, Yo and Y, functions. The following table gives the roots of equation (2) for different values of \ ; the earlier roots were found by inter- polation, the higher roots from the general expression given above. Table of roots of equation (2). Jo(Az) _ Jo(z) ; === Se Yoke) WING) | | Value of X. i First root. | Second. Third, | Hourthe | Le egsica 6380 | 65264 | 129587 | 194116 ln As ny, | 4510 33536 | 66145 | 98-936 NV 15 2-920 14581 | 98268 | “Bema V0 «ie. (ot S147 | 15441 |" eens ISP Mit 1-905 Gi8b2) 1) 12870 19-055 WO ah) lsat os | 8-882 | - 1307s 220 al ons 3659. | 6-594 9-626 3-0 0-813 2017, 7 3-416 4-906 40 |) 0604 Tee, 2352 3-328 5-0 0-482 1122, i egoe 4) aoe LXXXVIII. An Approaimate Theory of an Elastic String vibrating, in its fundamental mode, in a Viscous Medium. By James H. Ives, Ph.D., Associate Professor of Physics in the University of Cincinnati *. a the theory of the elastic string, usually given, the effect of the internal and external friction is assumed to be so small that it can be neglected. In certain cases, however, this is not permissible. For instance, if the string vibrates in a viscous medium the external friction can no longer be disregarded. Since the displacement and velocity of any point on the string vary from point to point along it, the system is really one having an infinite number of degrees of treedom and is difficult totreat. An approximate solution may, however, be obtained by making use of its mean velocity, and regarding it as a system having only one degree of freedom. To do this, we must know the transverse displacement, g;, of the * Communicated by the Author, Elastie String vibrating in a Viscous Medium. 745 string at any point 2 asa function of z We are, I think, justified in assuming that, for its fundamental vibration, 4 ondh Ve = PLS aie, where g is its displacement at its middle point and / is its length, since we know that this is true for an undamped string, and observation does not show any sensible variation from this form of displacement when the str ing is vibrating in a viscous medium. The velocity, v,, of any point, will then be given by LG tz = usin, where v is the velocity at the middle point. The mean velocity will therefore be equal to v, and the equivalent F 2 momentum of the whole string to 7, Mle, where M is its mass. In the same way, the mean frictzonal force will be given ) by —Rv, where R is the force which would be necessary to T overcome the internal and external friction of the whole string if every point of it were moving with unit velocity. The normal pressure on the string, at any point, tending to e . a bring it back to its position of equilibrium, is equal to — 9 where 7 is the tension to which the string is subjected, and p the radius of curvature of the string at that point. Tor small curvature, a dee Tae ane == ie p igi Therefore, for any point, #, on the string, the normal pressure is given by TIT" TL Se aeEe g sin va The total force on the string tending to restore it to its position of oar is equal to ae (ae TL 4 207 _ - gsin — de = —="" 9. nf e r=0 The equation of motion of the string, considered as a 744 Elastic String vibrating in a Viscous Medium. ‘system having only one degree of freedom, is then, for its fundamental vibration, given by 2 d(= Mv) 2 2arT aes or ae or d?q Bop nig ee A Its period will be given by pe z a On R? Ml 42M? If R is small compared with M, this reduces to T= oa Ue Writing M=ml, where m is the mass of unit length, we have Ne T= aia / ss o which is the well-known formula for the period of an undamped string. The motion of the middle point of the string will be given by 20 T where g,, is its maximum displacement. 2M . ° . a The relaxation-time is equal to — La The string will cease to vibrate when Mr is a= ona [ME == Lie Nir This is the ecratecal frictional resistance which will make ita motion non-oscillatory. That the motion can be made non- oscillatory can easily be shown by immersing a string made of indiarubber, which vibrates freely in air, in glycerine. In such a viscous medium, when displaced from its positien of equilibrium and then released, it no longer oscillates about this position, but simply returns to it. University of Cincinnati, January 1911. pea ie 2M = 9 ,€ COS qaeee yes 1 LXXXIX. Some Problems in the Theory of Probability. By H. Bateman, Lecturer in Mathematics at Bryn Mawr College, Pennsylvania ™. 12 re a note at the end of a paper by Prof. Rutherford and Dr. Geiger f, I gave a method of finding the ehance that exactly x a-particles should strike a screen in a given interval of time ¢, when the average number 2 of a-particles which strike the screen in an interval of length ¢ isalready known. If the source of a-particles is kept constant and the value of w is determined from a very large number of observations, the chance in question is found to be OR Ra Pe an nme AO nan This law of probability is not new but it is not very well known, and has sometimes been used in a slightly different form. In view of the recent interesting applications of the formula, it may be useful to add a few references to my former note. In arecent article by R. Greiner, “‘ Ueber das Fehlersystem der Kollektivmasslehre,’ Zeitschrift fiir Mathematik und Physik, vol. lvii. (1909) p. 150, it is stated that the formula is due to Poisson and is known as the law of probability for rare events. Greiner refers to a treatise by Borkiewitsch, ““ Ueber das Gesetz der kleinen Zahlen,” and considers the question of the correlation of errors when the law is valid. The formula is usually obtained by a limiting process. In J. W. Mellor’s ‘Higher Mathematics for Students of Chemistry and Physics,’ 3rd edition, p. 495, the theorem is stated in the following form :— “Tf p denotes the very small probability that an event will happen on asingle trial, the probability, P, that it will happen y times in a very great number, 7, of trials is P= ee “Thus if x grains of wheat are scattered haphazard over a surface s units of area, the probability that a units of area will contain 7 grains ef wheat is met \ a7 an (an)" RC [7 * Communicated by the Author. + Phil. Mag. October 1910. Phal, Mag. 8. 6. Vol. 21, No. 126. June 1911. 30 746 Mr. H. Bateman on some Problems The particular case in which r=0 is well known in the Kinetic Theory of Gases. It was shown in fact by Clausius* that the chance that a single molecule, moving in a swarm of molecules at rest, will traverse a distance x without collision, is ie P=e77" z where / denotes the mean free path or the probable length of the free path which the molecule can describe without a collision. The average number of collisions which occur when a molecule ee a path of length x may be taken to be equal to =, and so by applying the general furmula we find 1 p] that the chance that the molecule experiences 7 collisions while describing a path of length x is da ,n " |) Oe ae n} (7 ) The general formula may also be used in the Kinetic Theory of Gases in quite a different way, as M. von Smoluchowski has shown in an interesting paper published in 1904f. Imagine a certain volume ina mass of gas to be geome- trically but not mechanically bounded, and let the number of molecules which would be erbrnal in this volume ina uniform distribution be vy. In consequence of the molecular motion the number will sometimes be greater, sometimes less than this mean value. The chance that exactly n molecules are present in the volume at a given time is pre” n! The relative momentary deviation 6 from the mean value v being defined by the equation ii ——V = V Smoluchowski determines the mean value of all these momentary positive and negative deviations. Assuming * Phil. Mag. [4] xvii. p. 81 (1859). + “Uber Unregelmassigkeiten in der Verteilung yon Gasmolekiilen und deren Einfluss auf Entropie und Zustandsgleichung.” Boltzmann Festschrift, p. 626. in the Theory of Probability. 747 that the Boyle-Gay Lussac law holds, he finds that whee v is large the mean value is given by the equation ~ 2 oO 9 Vr but that when r is not very large x aS c= ”) where & denotes the largest integer which is not greater than v. If Boyle’s law does not hold, then for large values of y where ( is the true compressibility and @) the compressibility derived from Boyle’s law. These formule have been applied by The Svedberg* to the study of colloidal solutions. He finds that in great dilution the distribution of particles corresponds very exactly to the theory and that Boyle’s law is practically exact for dilute solutions. 2. Having indicated some of the known applications of the formula, we now proceed to a few developments which may perhaps be useful in the future. Consider first the case of a number of particles which carry either a positive or negative unit charge. Ifthe average number of these particles which are present within the given volume is vy, what is the chance thai at any given time the volume contains a total charge r on account of the presence of particles of these types ? This problem is analogous to one considered by Whetham in an electrical theory of coagulation, ‘Theory of Solution,’ p- 396. In Whetham’s problem, however, the electric charges are supposed to be all of one sign, and the probability” is calculated from a different point of view with the result that Poisson’s law i e-” is replaced by the simpler law (Ay)” where A is a constant. If we suppose that positive and negative charges are equally likely to be present, then the chance that a group of “Eine neue Methode zur Prifung der Giiltigkeit des Boyle- Gp eetaoschen Gesetz fiir kolloide Lésungen.” Zeztschr. fiir Phys. Chemie, Bd. Ixxii. Heft 5, p, 547 Sets aC 2 748 Mr. H. Bateman on some Problems n particles has a total charge r is zero if n—r is odd, and equal to n! aly mens Gr , 2” ays if n—ris even. The chance is in fact the coefficient of ¢” in t 1 \% Now we have seen that if vis the probable number of particles in the given volume, the chance that at a given time there are exactly n particles is vit =i, n! Hence the chance that the volume contains a total charge equal to r units is the coefficient of ¢” in the expansion o yn (t Ly? ae Nee 5) es Baa 2t that is in the expansion of the function Be (pase e2 ( =) i Now if we use the notation employed by Basset *, we may write ye = PL). -. Sas OS = = [SE — = Sa = = ie eo Hence the chance of getting a total charge of 7 units is represented by See ——sS ev tw). 2) ce je, BO or (3) The probable value of r is clearly zero, but we may Bnd the probable value of r? by summing the series fi a a Sa ~ Se $$ — ——— See oad S i AG ¢*=—D = = ee nas * © Wydromechanics,’ vol. 11. eS SS Se ee a ee ee Se | eer in the Theory of Probability. 749 To do this we differentiate equation (2), this gives Therefore Vv Neat cS d 9 («- 7) é2 (44 3) = ee L(). Differentiating again and putting t=1, we get ye = Brel (yy). Hence the probable value of 7? is v. A somewhat similar result is obtained in Rayleigh’s ‘Sound,’ vol.i. p. 36, where it is shown that if n unit vectors whose signs can be either positive or negative, are combined so as to give a resultant of magnitude r, then the probable value of r? is n. 3. To find the most probable value of r we take the recurrence formula Ts) Ina) = 22 1,00). Since I,(v) is always positive, we have Ne = eae ees Oy Also ie) — >) Leelee Leal Oi and @ yan. Consequently, TO! iG bs Jess) L,Y) Ai (ve } These inequalities show that if a 1,0) ~ then hiv) = ho); while ips es) we have To) aL yes Tai) : faa cow, IL) and so ING S LPC, 750 Mr. H. Bateman on some Problems We may conclude from these inequalities that I, I, I;, 9D form a decreasing set of quantities ; we have finally to find whether LZ). Now it follows at once from the equations * 1 T Loy) = -| cosh (v cos })d¢, 0 IG = - (" sinh (v cos ¢) cos ddd, t that 3 Lv) <1); for clearly | sinh (v cos¢) cos d< cosh (v cos } for all values of 6. Hence r=0 is the most probable value of 7. 4. Itis also of some interest to find the value of v for which the chance e*L(v) is a maximum when 7 is given. To do this we have to solve the equation I’) ac I,(v) iM / Putting X,= 2 ) , we easily find from the recurrence formule nl) i% == gee = ee y a ao yes La = i that * Cf. Whittaker’s ‘ Analysis,’ p. 307. in the Theory of Probability. ToL Hence if X,=1, X,-1<1, we have n—ly, i! Vv n 1+ - V Le Tea ee 0; v vy v>n(r—1). On the other hand, if we assume x J, Xnii> IL aa \e-5- V column headed “g” varies slightly both because of obser- vational errors and because of Brownian movements*. Under the column F are recorded the various observed values of the times of rise through 10 divisions of the scale in the eyepiece. A star after an observation in this column signifies that the drop was moving with gravity instead of against it. The procedure was in general to start with the drop either altogether neutral (so that it fell when the field was on with the same speed as when the field was off) or having one single positive charge, and then to throw on positive charges until its speed came to the 6:0 second value, then to make it neutral again with the aid of radium and to begin over. It will be seen from Table I. that in 4 cases out of 44 we caught negatives, although it would appear from the arrange- ment shown in fig. 2 that we could catch only positives. These negatives are doubtless due to secondary X rays which radiate in all directions from the air molecules when these are subjected to the primary X ray radiation. The smallness of the number of negatives so caught shows conclusively that the greater part of the rtonization of a gas by X rays is due to the direct action of the primary rays. Towards the end of Table I. is an interesting series of eatches. The drop was as at the beginning of this series charged with 2 negatives which produced a speed in the direction of gravity of 6°5 seconds. It caught in succession six single positives before the field was thrown off. The corresponding times were 6°5*, 10*, 20*, 100, 15°5, 8-0, 6:0. The mean time during which the X rays had to be on in order to produce a “ catch ”’ was in these experiments about six seconds, though in some instances it was as muchas a minute. The majority of the times recorded in column F were actually measured with a stop-watch as recorded, but since there could be no possibility of mistaking the 100 second speed it was observed only four or five times. It will be seen from Table I. that owt of 44 catches of ions pro- duced by very hard X rays there is not a single double. Table II. is even more convincing than is Table I. This drop was held under observation for about 4 hours and 100 different catches were observed, every one of which was a single. The rays were somewhat softer than those used in obtaining Table I., and corresponded to a spark distance of about 2 inches. * Millikan and Fletcher, Phys. Zeit. xii. pp. 161-3 (1911). Phil. Mag. 8. 6. Vol. 21. No. 126. June 1911. 3D 4762 ‘Prof. R.A. Millikantand Mir Ey Pletcher on the (UAB TB lee Plate Distance 1°6 cm. Distance of Fall -0941 em. Volts 1970. Temperature 22°5 C. Radius of Drop :00007 en. No. of No. of No. of No. of g- Re charges |chargeson|| g. F. charges | charges on on drop. |ion canght,| on drop. jion caught. i2-6 Noe one Ces | Fe. | Ge re 120 | 255| 1 P Liste hy | | 260] 1 2 aes fy BEE) A sy DAxX | Rex a ne as ee 4 i i Wee 129 | 66| 2 BP 1P | 2635.) (eae ; 5 6:3 | sone 12-4*| 0 250| 1 P ae 12-4%| 0 1? 127 | 66| 2 P | 25:0 | ane Lee |. 68| 2a lee 12-4*| 0 | ope) fe ; , o5:0 | 1 ae i 123 66a ee Oa | Goi 9 te 12-6 | . 0551 4 P 48] 1 P 65/2 P 1 eam 63 | 2 P 1 P | 99 |265| 1 P 19-5 W5-S nan oe, ae tye 124%) 0 ee 250| 1 P ° 122 | 124%] 0 6-0) oe 1 I (98 10953) 6P ; : 130 | 0641) 2.P | 250| 12 een |; 67 | oaee 55 1 2 64| 2 P 1 P 95-0: |, eae ies 63| 2 ae 265° Fhe G45 (ap P Le 01 1 1e ie 12-4%| 0 123 | 12-4*| 0 Hie 25:0 | 1 oP : _ 2501 1 P he 124%] ee 65 Ae sep 25-0) |! ae ae OF Ot ua shee ic ee 95-5 |, ae ie Poe lly Dat | 66 | "a Se | 120 | 124%) 0 124%) 0 250, lr a P| ies 250| 1 Paes 67| 2 P es) 124*| 0 38| 3 P | | 95:0 |) alae ie ).- | >, >) ei |e 6612-2 + oe Se ik 1 ‘ae Be Vn Bue 124 | 66] 2 P Question of Valency in, Gaseous Ionization. PIED DO be be Dro BOVEY © oe lores) oo © 2K * * DS WW Oo bo Or bhIch oO (ep) 12:0 kK OK oe 2 SATS He NOR © akc ho eH Cow bo *K Ses ao ho — 12-1 bo — bo CDS He DP ATbO TALS? SES? CO TaBLE I]. (continued). No. of charges on drop. ihe 5 suid i 3b OE? 20 PB 0 ie Ee 0 eB Pa coer foN 0 ie PEGA a aNi ) iP a ie Zit 0 Bak Zine ie Ns Ni 0 IP 7 ei ale fh N 0 TP 24,2 0 Ip B ys AE 0 ive 2p 2 Pie 0 aly aye a © No. of charges on|| 4g. ion caught. i oa ere bet et et =a a) feet et pe 12:0 13:0 nop na) Ing) aollnc/inoll> Alpe) molnoline hg kg kg welaciiae) ro kg F. 12-4% 25-0 68 4-0 12:4% 25°0 2K So So so Cao Ge 3K bo — ty DAW BA i) bo C2 Or Oey Se Orly aS BRO KO 7K LD or SOKO K * No. of charges on drop. —— oi >) QNrS NO NE NEO HO NRO rtd hb — Boi NrOrF bore Wwe Fe Wy Nehlnef lel refine) tee} inef Ine) = Ineltne) Ine} ac) = lel Ine! MH Onm nolige) Zz ge} lao) Ineltae| 1 Ih 1 ee a a a neline| et = Ard 763 No. of charges on ion caught. 12 Pp P P P P iP ror A 100 catches, all singles. 764 Prof. R. A. Millikan and Mr. H. Fletcher on the § 3. Lonization by Secondary X rays. Table III. represents observations taken on the ionizing effect of secondary X rays under conditions as nearly as possible like those described by Townsend when he found that the positives produced by secondaries were chiefly doubles. Fig. 3 shows the arrangement of the apparatus. The plate D was covered by a thin coat of oil and only the left-hand portion of it could ke hit by the primary X rays. Despite the fact that the secondary rays from the plate D traverse the air on all sides of the drop, we succeeded in catching chiefly positives by holding the drop always within 2 or 3 millimetres of the plate C. We had hoped to clear up the apparent contradictions in preceding work on this subject by finding that homogeneous secondary X rays do in fact produce double positives, but it will be seen from Table III. that out of 84 catches there were but 3 which could possibly correspond to multiples of any kind. Of these 3 catches the two followed by an interrogation mark unfortunately happened when Millikan, who was observing at this time, had glanced away to read the stop-watch, and hence may have corresponded to two or three separate catches following in rapid succession. The third catch marked ‘‘ good” appeared to the observer (Fletcher) to cor- respond to a change in speed which happened all at once rather than in two successive steps. Nevertheless the changes were happening here, when the field was on, at about one or two second intervals, instead of six second intervals as in the work recorded in Tables I. and IJ. It is not at all per- missible, therefore, to draw the conclusion that this catch corresponded to the advent of a double ion, rather than to the nearly simultaneous advent of two separate single ions. The legitimate conclusion which can be drawn from this table is that if doubles were formed at all by the secondary rays here used, their numbers were certainly exceedingly small in comparison with the number of singles formed. Question of Valency in’ Gaseous Ionization. 765 TABLE ITT. | Plate Distance 1°6. Distance of Fall -0744 cm. Volts 1750. Temperature 24° C. Radius of Drop :000095 cm. No. of No. of No, of No. of g: F, charges |charges on] g. F. |charges on | charges on on drop. | ion caught. on drop. | ion caught. -3 A ee 68 | 72! 0 oe BAL Sp 1P 18% iL ie ie bal. yy 2 bal. ee ke SN 74 | 18% ie bla 9:0 3. aB “fe rep tine (?) 50 | 4 P : 18* ike 19* ice bak, 2 Pp : 5 pa ah ee ; A 9:0 ip 9-0 3 ie ee DD 2: ean 82 18% ap ; 3) fey ee) TP Gath ba By as ep i 9-0 Bia de | % | Hi OX | 0 me eh PB Wee Piet ame re 90 | 3 P Le Bale ur 5 9-0 2° Pp ie fee ck oh P HU ees 1N oa 2 Pp IP OU en 1P A. 1 4 IP bal. de fen Top SN ne Male ip bal Ge iP 5-0 AP idee 1P 92 Bye A a) 4B IP a F S 1P a : 1P 17* iP OO ea | bal, Zine i Z 9 50 | 4 P ZOE) Gm) ero Wy eM ab 7:0* 0 LP 70 | 18% ie ee 18-0* {GE cle | bal. pay AN | bal. 2 a «| 18% Wer IN 9-2 3. \P Le an | 69 | bal. | 2 P ae 4-9 4 P | 9-0 3 P bal. 29 Pp ta | 9-1 2 Pp IP | 7:3%| 0 ae 50 3 2?P tea 18* uae pal) 2 Pp IP | GON ge IP | | t This speed marked balanced was actually measured tvo or three times anc corresponded to a time of about 48 seconds against g. Se a Sp SS 88 eee ee eee ee Rr _ ——————————e ~~ ee S eS 766 ~=Prof. R. A. Millikan and Mr. H. Fletcher on the TaBveE IIL. (continued). No. of No. of No. of No. of g F. charges j|charges on || g. F. charges | charges on on drop. | ion caught. on drop. |ion caught. 7:0* 0 6:9* 0 1BE 9) | lege.a || Wee 17* 1 Pe bal. Sy 1 1P bal ep Bs 9p a 70| 9:7 Sep i= 5:8 4 P ee S33 Aa 6:9* 0 70x | 0 ve 16% 1 Pee 18* lose 1P 9:0 BP IP icon Ge eee 1P 55 4 P ! bal. ye IP bal. Dae 17:0* 1 ie 10-0 3 P 1P bal.=40| 2 P [45 9-0 3 1P bal. a IN 5:5 4 2 L7* Wyn de 1P bal. Dee iP 18* Tee de 1P 10:0 33, 12 bal. 2a IP 9:0 ay “Ae bal. Bat aa 1P Te 9-0 Sua IP 18* vee LP 55) 47 P bal. Pale! & bal. yg © 1B) gee | wor AS 9:0 3 pl ieee bal. Ty 12 1P 10:0 one IP bal. Dh» TP IP 58 4 Pp 1P 8'8 ey 1B 34 Ry PD bal. Je 2P 10:0 3 P LF The above tables do not represent one half of the observations which we have made upon ionization produced by either pri- mary or secondary X rays of varying hardness, and Table III. contains the best evidence which we have been able to obtain at allin support of the hypothesis that doubles are even in rare instances formed. Despite the fact that we began this investigation with the firm belief in the correctness of this hypothesis, we are forced to the conclusion that. we have now no reason whatever for assuming that the act of ionization of air by X rays ever consists in the detachment of more than one elementary charge from a neutral molecule. Question of Valency in Gaseous Ionization. 767 § 4. Lonizing effect of B and y rays of Radium. Although there was no evidence from other quarters that ionizing agents other than X rays ever produce doubles in air, we have made observations similar to the above on the ionizing action of the 8 and y rays of radium. No attempt was made to separate the effects of the @ and y rays, but we estimate from rough measurements that about three-fourths of the ions formed in these experiments were due to the & rays. A sample of the resulis obtained with the use of very small drops is shown in Table IV. Lvery one of the 20 TABLE [Y. Plate Distance 1°6 em. Distance of Fall -0741 em. Volts 1368. Temperature 25°C. Radius of Drop :00003Y2 cm. : eae See ah | Noxok |» No. of No. of No. of ie lee RES charges chargeson|, g. FR. | charges | charges on on drop. | ion caught. on drop. | ion caught. ey 40% is eee) wi Oe je a 40%* | 0) Ree Poe ay bee Gl iN eh ee ee 30°6 41-4 306 | 40* 0) : 40* 0 CO). LUN ee aes tae ih Ve aes 33°0 41-7 40-6 | 40* 0 ie: 40* 0 one oo || 1 pee t F GO oN HS 44-0 40% 0 | 40* 0 ‘ ek oe |) pees FE 58) 1 P is ae | 40% |. 0 A0* 0 coi Pe me eae eae ire 45°6 AQ* 0 40* 0 | 45-0 5-5 1 ene I P | 46:0 6°4 I de IP 40% 0 40% 0 sei Pie be ee 60 | P ee 30 , 40% 0 Age | ‘ 59 | LN eee 59 | 1 P Hu 40% 0 | 40% 0 ‘ or oN FN AG ORIR GON eerie en hss 40* 0 1 N | ¥ PES aan ee | 58 iN 3 | 20 catches, all singles. sox | 0 | | ont (bal ene Le | | LD ) 768 Prof. R. A. Millikay and Mr. H. Fletcher on the TABLE V. Piate Distance 1‘6em. Distance of Fall -0741 cm. Volts 3200. Temperature 25° C. Radius of Drop 000093 em. No. of No. of No. of No. of g: 1h charges |chargeson| 4g. 1s charges | charges on on drop. | ion caught. on drop. | ion caught. 7.0 || S120 Ga ep 10 Ul ten ; 7T0*| 0 Wp 66 eum I N 120 LN 100 Lom 70*| 0 aie 68 | 2 Nee 120 1 N 100 t WN TO*| 0 1 P 79%| 0 ‘ P 100 i os) i = 120 Lae Las 6:7 2 aN 7:0* 0 ae 200 iy ee 1 100 i oe tor 66 1 68 2 ne 200 ee) 100 NG 64/1) 99 Pp 1 P 64 Po IN 200 Dic GaN 100 1 N T2*| 0 2 8 Gey | ean tN 120 ie 100 Leu 72*| 0 LN A eatery ||, Stas 1 N 120 tN 100 Lares 65 | 2 N , 2 64) | oi 1 N 120 LN 1 N 64 Di VIN 1 P 90 ga 22 one 120 aN | ie 6:5 2 ne 78% (0) 125 1 120 YN on 0 678 92) aN iN 125 Were ayy 1) ae 20 a0 ieeeat GaN Be 63 | 2 N, 48 | 3 .N (?) 125 12 cae | 100 ae oN 6:8 Aah 6 y eas 0) . 25 1 3 7 | 3) ae 125 Tae ; T0*| 0 nee | 125 eee Bal ee Dee 120 aN 68 |) oN 1 ON Question of Valency in Gaseous Ionization. 769 catches observed with this drop was a single. When large drops were used the chance that the same @ particle would ionize two adjacent molecules, and thus throw two separate ions simultaneously upon the drop, became larger, and yet, in general, we caught only singles even with drops which were so large that a § particle in going the length of a diameter of a drop would have to pass through at least $8 molecules. The drop shown in Table V. has a diameter which is 20 times the mean free path of an air molecule, and 20 OT latter be regarded as a point, yet out of 34 catches there is but one which could possibly be a double. This means of course that a 8 particle ionizes but a small fraction of the molecules through which it actually passes—a conclusion which can also be reached in other ways. With the use of drops whose diameters were ten times the mean free path of an electron we occasionally, as in our early experiments, observed double changes when the field was on, but since these were never observed with drops of one-tenth the size, the conclusion is inevitable that these apparent doubly charged ions were in fact two separate singly charged ions. The above observations represent only a small portion of those which we have made within the past three months, but they present the best evidence which we have been able to find for the formation of doubly charged ions. We would, however, point out this difference between our experiments and those of Townsend and Franck and Westphal. While their measurements have to do with the charges carried by ions at a considerable time after the formation of these ions our experiments have to do solely with the measurement of the charge which is freed from a neutral molecule in the act of ronization. If two single positive charges attached themselves after formation to a minute dust particle or other molecular aggregate, and thus formed a doubly charged ion, our experi- ments would not reveal the fact. Or, again, if X rays were capable of detaching more readily an elementary charge from an already singly charged ion than from a neutral molecule, and thus forming double positives (a very improbable hypo- thesis), our method would not be able to detect such doubles. We are extending investigations of the types herein described to other gases and to vapours. § 6. Conclusion. Our conclusion may be stated as follows :—Although we entered upon this investigation with the expectation of proving the existence of valency in gaseous lonization, we have instead =3°5 times the mean free path of an electron, if the 770 Mr. A. L. Fletcher on the Radioactivity of obtained direct, unmistakable evidence that the act of ionization of air molecules by both primary and secondary X rays of widely varying degrees of hardness, as well as by Sand y¥ rays, uniformly consists, under all conditions which we have been able to investigate, in the detachment from a neutral molecule of one single elementary electrical charge. Ryerson Physical Laboratory, University of Chicago, - February 14, 1911. XCI. The Radioactivity of some Igneous Rocks from Antarctic Regions. By Aryotp L. FietcuEr, B.A./., Research Assistant to the Professor of Gevlogy in the University of Dublin*. 4 vary question of the mean radioactivity of the Harth’s surface materials is of sufficient importance, and at the same time so far from being definitely agreed upon, that experimental data tending to increase our knowledge of the subject seem deserving of record. The accompanying analyses of rock-specimens from the Antarctic region of South Victorialand were determined upon a series of rocks collected on the recent Expedition under Lieutenant Shackleton, and kindly obtained and lent to me by Professor Joly. They embrace rock-types of varied chemical and petrographical characters. The determinations were made by the solution method of Professor Strutt in the manner described in a former communication. An examination of the constants of the two new electro- scopes used was made by the standard uraninite solution used in former experiments, and by a standard radium solu- tion sent by Prof. Rutherford. It was found that two expe- riments with Prof. Rutherford’s solution on the constant of one of the electroscopes read 0°63 and 0°62, and with Prof. Joly’s solution 0°61; and in the case of the other electroscope the consiant read 0°83 and 0:80 respectively with the two solutions. ‘The truth of the Uraninite Standard as used all along in the calibration of the electroscopes is therefore supported by these experiments. It is not without interest to note that on first comparing these standards, that coming from Prof. Rutherford gave a higher constant for the electroscopes referred to above, 2. e. 1-1 and 1°7 respectively. This was traced to insufficient acidifi- cation when diluting the standard solution sent. On adding an additional quantity of HCl and warming, the almost perfect agreement recorded above was obtained. The con- cealment of the emanation in the perfectly limpid solution * Communicated by the Author. on “I some Igneous Rocks from Antarctic Regions. (fa. before sufficient acidification, suggests possibilities when traces of radium emanation are sought for in bulky and chemically rich solutions— especially those which are examined in the alkaline state. The following results were obtained :— Radium Thorium | Radium Rock. Locality. gr. pergr. | gr. per gr. | Thorium a ee Ltion an Cane - | BES ee Mt. Erebus, McMurds Sound...... 23 o1Omac| bosoms lose lOm! L| Kenyt Waivay......... Mt. Erebus, Ross Sound ............ DIA 1-45 ,, NaS ch _| Alkali trachyte ...| Mt. Erebus (South cliffs) ......... DO, SOs Gi; Trachytic .¢ Ablceitic ag Cape Royds, Ross Sound _......... ieee. OH. 4, Onn IKeriyilava....<.s. Cape Royds, Ross Sound ......... 46.5, OM ins PO). Kenyt (coloured I red by geyser || McMurds Sound, Ross Sound...... AO2hae JERS ZZ action. : eed basic \ McMurds Sound, Ross Sound...... O03 O4G Lee. Bree’ \ McMurds Sound, Ross Sound...... Ores) ee) | Oaks) 5. Ose oe \ Cape Royds, Ross Sound ......... O00 EP) OOM. Seem Anes ei \ Cape Baray Ross) Soumd ie. a s.55. OAs ere? alll OO ies: Oa. ere : fie | Cape Royds, Ross Sound ......... O82 5s ON Oiaan 5 US : oe | McMurds Sound, Ross Sound...... O70 a, Oe TQ .| Erratic granite | block. “Lower ground” of 8. Victorialand| 0-20 By 0-14 ,, ose Another variety. | queried results Nica { excluding | V7 | * For description of rock-type see Quart. Journ. Geol. Soc. Lond. 1960, vol. lvi. p. 209. One of the most noticeable features of the above results is the remarkable constancy obtaining in the case of both radium and thorium among specimens from the same locality, and probably taking their origin from a common magma. This feature, einen IS as omeale marked in the ease of the thorium as in that of the radium, seems to be inde »pendent of the basicity or acidity of the anes Thus in the ease of Nos. 1, 2, and 3 in the above table, the figures are, within the limits of experimental error, identical, yet the rocks, although all from Mt. Erebus, are by no means similar eee fea eae i ee . Se — a as 772 Radioretivity of Igneous Rocks from Antarctic Regions. chemically or petrographically. Nos. 4 and 5 in the table, probably taking their origin in the same lava-flow, similarly show a striking equality in their radium and thorium contents. Some of the lowest results obtained were from erratic granites. These contained but a small quantity of biotite, and consisted almost entirely of orthoclase and quartz. With the single exception of No. 11, the granites showed a com- plete absence of precipitation, and are probably therefore approximately correct. Speaking generally, the results do not bear out the con- clusion of Farr and Florance in their research on the radium content of the Sub-Antarctic islands of New Zealand (Phil. Mag. Noy. 1909}, that the radioactivity depends roughly upon the acid or basic character of the rock. The mean value, 2°15, obtained for the basic rocks of Mt. Hrebus is in good agreement with the minimum value 2°38 found by them (doc. cit.) on a specimen of lava from the same locality. The radioactivity of No. 6 in the table is of especial interest. This rock is stained red probably under the action of infiltering geyser waters, and it seems probable that in this case the kenyt owes its relatively high radioactivity to this cause. In addition to the constancy shown in the quantities of radium and thorium present in the different groups of rocks in the table, these quantities seem to show a decidedly definite relationship between themselves, from the highest to the lowest reliable values obtained ; a rise in the radium present being almost invariably accompanied by a nearly corre- sponding rise in thorium present. This feature which, in the case of some weathered andesites examined, might find an explanation in the effects of the weathering action itself, cannot be similarly explained in the case of granite and the unweathered Antarctic rocks, where a similar relationship appears to hold. In the case, however, of the kenyt No. 6 in the table, the same cause which protably gave rise to its relatively high radium content appears at the same time to have propor- tionately increased the thorium content, for tlie ratio of radium to thorium present fairly approximates to the general mean ratio. Hence in this case the radioactivity, if due chiefly to the external effects of infiltrating waters rather than to any peculiarity inherent in its original magma, involves a similar relationship between the radium and thorium in those waters. In conclusion I desire to express my thanks to Prefessor Joly for his interest in the work. Geological Laboratory, Trinity College, Dublin, March 17th, 1911. ‘ He 4 3 On Water- Waves as Asymmetric Oscillations. 773 Notr.—A. Gockel (Jahrbuch der Radioaktivitat, vi. p. 003, 1910), examined various minerals found by Prof. Strutt to be relatively rich in radium, and obtained radium contents far below his values, accompanied at the same time by unasually high thorium richness. Gockel suggests the possi- bility of a source of error from the active deposit of thorium, consequent upon the introduction of thorium emanation into the electroscope. Some experiments were made to test the possibility of such an error affecting the estimation of radium in bodies rich in thorium. A solution containing as much as 0'1 gram thorianite, having been first boiled to expel radium emanation, was then treated in the same manner as that employed in the esti- mation of radium. Jour experiments were made using both slow and fast admission, in the latter case the gases being admitted into the electrosecope as fast as the safety of the gold leaf would permit. The whole of the gases were in this case transferred to the electroscope within two minutes subsequent to being cut off from the parent solution. In no case was any certain increase noticed in the rate of collapse of the leaves of the electroscope in three hours after admission. In view of the fact that no effect was observable in the ease of a solution so rich in thorium, it is safe to infer that in no case could the leak of an electroscope be noticeably affected as read three hours after the slow admission of such quantities of thorium emanation as are associated with amounts of thorinm of the order of magnitude dealt with in the estimation of rocks and minerals. v XCIT. On Water Waves as Asymmetric Oscillations and on the Stability of Free Wave-/rains. By ANDREW SrupHENson*. i, yee waves furnish a complex example of asym- metric oscillations, and it is natural to inquire whether they exhibit the marked energy absorption under direct force of double frequency, which is characteristic of the asymmetric system with one degree of freedom}. The problem is most simply considered as one of steady motion. Direct force may be applied to a deep stream flowing uniformly by a stationary periodic variation of pressure along its surface. Such variation will produce standing waves of equal length. Is this state of motion stable when the wave-length is half that of the free standing waves ? j For the purpose of testing the stability of a train of waves * Communicated by the Author. + “Ona Peculiar Property of the Asymmetric System,” Phil. Mae. Jun. 1911. 3 TTA Mr. A. Stephenson on for small disturbance of a given character, we seek the foree system necessary to maintain the distur bance. If the force when acting alone tends continually to damp it, we conclude that the steady motion is unstable for such variation. Thus in the present problem we seek the conditions under which a small standing train of double wave-length, and therefore of nearly free type, may be maintained by pressure variation of the corresponding period in such phase that its action tends to damp out the train without producing other change. As the discussion involves quantities of different orders of smallness, we shall discard the customary stream and velocity- potential functions in favour of more direct coor dinates, thus obtaining equations which are readily applicable to all cases in w hich? the magnitude of the amplitade is involved. 2. Let (2, y+ ‘”) be the coordinates of a particle the mean lev all of which is at distance y, positive upwards, from the undisturbed surface. Then the velocity in the stream-line of l mean level y is a/ i+ (2) pl y/a+®), where ¢ is a function of y, and its hor maa Nk eared components are (1+ 2) and ot M(1 ae =e a A displacement e ie is given by a displacement along the stream-line of Joni component doz, and a vertical displacement —— .6z2. For irrotational motion therefore dn a (2 dx = © dae PY does dx dy dy dy ye ny an é ip dy Ae a dy dy Lee dn - oa OMS dyn\dn d’n (1+ dy at (14+ { ih a) la? —2(1+ dudy —==( va e 1) = c rei af (1+ ( ¥ 0. ile The pressure is given by + a constant. 4, The forced motion due to a small simple pressure variation of wave-length 7/k is n= ae"! cos 2khv, Water- Waves as Asymmetric Oscillations. Ci and the equation of motion for any additional disturbance, ¢, is ae from (1) aE a tet con he + dake sin he 0E a 5 + take cos 2ha ae See Ake 4 sized ae : f 1 — 8ak*e"¥ cos yee + 8ak? sin orgy, SG) a esata dy de Putting C=acoske+ sin ka, where « and @ are functions of y, we have from (2) a=p(eY+ sake), B=q(e4¥—Zak e"). Now the pressure at the surface is proportional to = 4, E55 a Cos Deg ee + 2aksin 2he . ae dy dy da which is — p(ak—o) coskx+q(ak+¢) sin ka, where C= - (Lee ea): This pressure, being nearly of free period, tends to damp the disturbance if its phase i is —1/4: that is, “f p(ak—o)=¢7(ak+c). | o | must therefore be less than ak. Thus there is instability for any value of o numerically less than ak, the standing disturbance of wave tye ae +oa)} being magnified in one phase —tan7 ee aes *, and diminished in the nume- === rically equal phase of opposite sign. Interpreting the result in terms of progressive waves in still water, it is evident that if a periodic pressure variation moves uniformly over the surface, the forced train of equal wave-length constitutes an unstable state of motion if the ratio of the wave-length to that of the free wave of equal speed lies within a range 2ak about the value 1/2; for a small disturbance will result in a series of waves of double wave- length, which is continually magnified through the periodic pressure until the amplitude is large compared with the original motion. The process consists essentially in the continual enlargement of an asymmetric oscillation of approxi- mately free type by a direct force of fr equency lying within arange about the double frequency of free osci illation. For the purposes of experimental illustration it would be simplest to take the case of a stream flowing over a corrugated bed, 776 On Water- Waves as Asymmetric Oscillations. 3. Since the free wave of finite amplitude is appreciably asymmetric, 1t would seem possible from the foregoing that the free train might tend to magnify other relatively stationary trains of nearly the same wave-lengths and relatively smail amplitudes. This question is a partial test of the stability of a free train of waves. To determine the free wave of finite amplitude *, putting n=2coskx+P cos 2ke, where « and § are functions of y, we find ~ al" — Bat Th Ky) oe al! — 7 Baa!® — 38a! — 6h a’ B— Bey a Saaz Bb" Ar’ B=), t C= 2a —U), ¢ subject to the boundary conditions «= @=0 when y= — a, and T 4 — 2a! + 3a! 2B’ —30!? + 2haB—4 ara! = 0, CG ha 0 5 ¢ Boa 103 OP ean i? 3°=0, when y=0. Hence a = aed + 8 ad]2e%, [Se c Pan 5 {1+ Ha a’(1— 267") }. The equation for the disturbance, , is ie are ; ag pes io es ea a —~, + 2ake ¥ 4 08 kir( k oP) sin ke (a nan eA), leyazehen +) ibs ae ji Gs ae dg ey er +4 2 an 2 ke =? Re ay, {5 (Get ly? ae ie di? ee a) aC dt )} + sin 2 2h 3 7 Ally ole = * The method is evidently applicable when the depth is finite. In the case of long waves the process is necessarily different. We have then dy\ 1 ¥ uy (dn 1=M0TY ae TOT ! (a ) Ts (a), when small quantities of order higher than the second are neglected. Hence, putting y=—h, and substituting for the y devivatiyes from (1) and the surface condition we obtain a? w 3 (f= )% ee per Tp C mite eh °, the well-known equation for the contours. lee aa | _ On Water- Waves as Asymmetric Oscillations. 1a ‘he pressure at the surface is proportional to a 27,2 7 27,22 Jn Vealicons 26) de : dG —k(1+a7k?)€+(1+ 5 4 k?—3ak cos ka 5% k? cos Sher +aksin kv dai The solution C= (ety + : ake”) cos (k+s)e+ : ake cos (k—s)a + ake+ cos (2h+s)a+ake cos se—ake'*'¥ cos sic gives surface pressure (s—a?k*) cos (k+s)x—a’k’ cos (k —s\a: and t= ( etsy 4. 1 lee) sin (k+s)a— : ake sin (k—s) v + ak +99 sin (2h +s)a—akeC** sin su—ake Isl¥ sin sv the pressure (s— a?k*) sin (k+s)a+a7h? sin (k—s)a. Hence the disturbance +s A cos(k+s)e2+Osin (k+s)x} +e4-1B cos (E—s)a + D sin (k—s) x} is maintained by the pressure $(s—a?h3) A—a@h3 BY cos (k+s)a+4 (s—a2h3)C + a2? Di sin (k+s)x + 4(—s—a?h')B —ah® At cos (k—s)a+ {(s—ah3)D +a23C} sin (k—s)z. The pressure acting alone would tend simply to change the intensity of the disturbance if the two components are pro- portional to the amplitudes of the trains of corresponding wave-lengths, and if the phases differ by a quarter period from those of the trains ; that is, if (s—eP)A—eB __ (6— €R)C+AD C A (—s—@)B—@A _ — (—s— oh®)D +00 eR acTa (a OL ETRE GE Oe Ce alta aa B Hence G=—s83 gis therefore always complex, and the free train has no tendency to develop a periodicity of amplitude. Phil. Mag. 8.6. Vol. 21. No. 126. June 1911. 3 E a INDEX ro VOL XXI. << —_—_. Actinium, on the y-rays of, 30. Age-distribution, on a problem in, 435, Air, on the electrification of the, near the Zambesi Falls, 611. Airey (J. R.) on the oscillations of chains, 736. Aldis (A. C. W.) on a revolving table method of determining the curvature of spherical surfaces, 218. Alkali salt vapours, on the velocity of the ions of, in flames, 711. Allen (Prof. F.) on a method of measuring the luminosity of the spectrum, 604. Alpha particles, on the ionization of different gases by the, from polo- nium, 571; on the scattering of, by matter, 669. Anderson (Prof. A.) on the com- parison of two self-inductions, 608. Asymmetric oscillations, on water waves as, 773. system, on a property of the, 161. Atmosphere, on the ionization of the, due to radioactive matter, 26. Atomic weight, on the relation be- tween viscosity and, for the inert gases, 40. Atoms, on the law of chemical attraction between, 83; on the structure of, 669; on the number of electrons in, 718. Ayres (T.) on the distribution of secondary X rays and the electro- magnetic pulse theory, 270. Baeyer (Prof. O. von) on extremely long waves emitted by the quartz mercury lamp, 689. Bailey (E. B.) on recumbent folds in the Highland schists, 174. Barkla (Prof. C. G.) on the distri- bution of secondary X rays and the electromagnetic pulse theory, 270; on the energy of scattered X-radiation, 648. Barus (Prof. C.) on interferometry with the aid of a grating, 411. Bateman (f1.), some problems in the theory of probability, 746. Bessel’s functions as applied to the vibrations of a circular membrane, on, 53; on a physical interpreta- tion of Schlémilch’s theorem in, 567. Beta particles, on the scattering of, by matter, 669. Birds, on metallic colouring in, 554. Books, new :—Berkeley’s Mysticism in Modern Mathematics, 278; Curie’s Traité de Radioactivité, 390; Barbillion and Ferroux’s Les Compteurs électriques a Cour- ants continus et a Courants alter- natifs, 301; Turpain’s Notions fondamentales sur la Télégraphie, 391; Turpain’s Téléphonie, 391 ; Mann and Twiss’s Physics, 5838; Tutton’s Crystalline Structure and Chemical Constitution, 584. Bosworth (T. O.) on the Keuper marls around Charnweod Forest, 695. Boyle (Dr. R. W.) on the behaviour of radium emanation at low tempe- ratures, 722. Bromine vapour, on the destruction of the fluorescence of, by other oases, 509. Bury (H.) on the denudation of the western end of the Weald, 176. Campbell (N.) on a method of de- termining capacities in measure- ments of ionization, 42; on the common sense of relativity, 502; on relativity and the conservation of momentum, 626. - . INDEX. 779 Capacities, on a method of deter- nuning, in measurements of ioniza- tion, 42. Capillary tubes, on a method of cali- brating fine, 386. Chains, on the 736. Chapman (T. C.) on homogeneous Roéntgen radiation from vapours, AAG. Chemical attraction between atoms, on the law of, 83. Colouring, on metallic, in birds and insects, 554, Condensation nuclei produced by the action of light on iodine vapour, on, 465. Conduction of heat through rarefied gases, on the, 11. Cooke (Prof. H. L.) on the heat liberated during the absorption of electrons by different metals, 404, Cross (Dr. W.) on the natural classi- fication of igneous rocks, 174. Curvature of spherical surfaces, on a revolving table method of deter- mining the, 218. Cuthbertson (C.) on some constants of the inert gases, 69. Cylinder, on the uniform rotation of a circular, 342; on the motion of a, through viscous liquid, 706. Density, temperature, and pressure of substances, relations between the, 325. Dielectric sphere, on the damping of the vibrations of a, 488. Diffraction, on the photometric measurement of the obliquity factor of, 618. Discharge from an electrified point, on the, 585. Dubson (G. M. B.) on a revolving table method of determining the curvature of spherical surfaces, 218. Donaldson (H.) on the problem of uniform rotation treated on the principle of relativity, 319. Dust figures, on electric, 268. Earth inductor, on a new form of, 579. Elastic string, on an approximate theory of an, vibrating in a viscous medium, 742. Electric discharge in hydrogen, on the, 598. oscillations of, Electric dust figures, on, 268. waves, on the bending of, round a large sphere, 62, 281. wind, on the pressure of the, 591. Electricity, on the recent theories of, 196; on rays of positive, 225; on the discharge of positive from hot bodies, 6354. Electrification of the air near the Zambesi Falls, on the, 611. Electrified point, on the discharge from an, 585. sphere, on the initial accele- rated motion of an, 640. Electromagnetic pulse theory, on the distribution of secondary X rays and the, 270. Electrons, on the heat liberated during the absorption of, 404; on the number of, in the atom, 718; on the longitudinal and transverse mass of, 733. Energy, on Hamilton’s equations and the partition of, between matter and radiation, 15. Eve (Dr. A. 8.) on the ionization of the atmosphere due to radioactive matter, 26. Fit, on restricted lines and planes of closest, to systems of points, 367. Flames, on the velocity of the ions of alkali salt vapours in, 711. Fletcher (A. L.) on the radioactivity of the Leinster granite, 102; on the radioactivity of some igneous rocks from Antarctic regions, 770. Fletcher (H.) on the question of valency in gaseous ionization, 755. Fluid, on the uniform motion of a spnere through a viscous, 112. Fluid motion, on rotational, in a corner, 187. Fluorescence, on the destruction of the, of iodine and bromine vapour by other gases, 309; on the in- fluence upon the, of iodine and mereury of gases with different attinities for electrons, 314. Franck (J.) on the transformations of a resonance spectrum into a band-spectrum by presence of helium, 265; on the influence upon the fluorescence of iodine and mercury of .gases with different athinities for electrons, 314. 780 Friction, on the maintenance of periodic motion by solid, 161. Fry (J. D.) on the value of the pitot constant, 348. Gamma rays from RaC in the earth, on, 29; on the, of thorium and actinium, 180; on the effect of temperature on the absorption coethcient of iron for, 5382. Gaseous ionization, on the question of valency in, 753. Gases, on the conduction of heat through rarefied, 11 ; on the rela- tion between viscosity and atomic weight for the inert, 45; on some constants of the inert, 69; on the potentials required to produce dis- charge in, 479; on the ionization of different, by the alpha particles of polonium, 471. Geological Society, proceedings of the, 173, 279, 392, 698. Granite, on the radioactivity of the Leinster, 102. Grating, on interferometry with the aid of a, 411. Gray seu a G:) testing, 1. Griffith (O. W.) on the measurement of the refractive index of liquids, 301. Hamilton’s equations and the par- tition of energy between matter and radiation, on, 15. Hatch (Dr. F. H.) on dedolomiti- zation in the marble of Port Shep- stone, 173. Heat, on the conduction of, through rarefied gases, 11; on the, libe- rated during the absorption of electrons, 404; on the, of mixture of substances and the relative dis- tribution of molecules in the mixture, 535. Heat-waves, on the focal isolation of long, 249. Helium, on the transformation of a resonance spectrum into a band- spectrum by presence of, 265. Horwood (A. R.) on the origin of the British Trias, 892. Houstoun (Dr. R. A.) on magneto- striction, 78. Hughes (A. Ll.) on the ultra-violet light from the mercury arc, 393. Hydrodynamical notes, 177. on magnetic INDEX. Hydrogen, on the electric discharge in, 098. Inductor, on a new form of earth, 579. Insects, on metallic colouring in, 554. Interferometry with the aid of a erating, on, 411. Iodine, on the resonance spectra of, 261; on the influence of other gases upon the fluorescence of, 309, 314; on condensation nuclei produced by the action of light on vapour of, 465. Jonization, on the, of the atmo- sphere due to radioactive matter, 26; on a method of determining capacities in measurements of, 42 ; on the, of different gases by the alpha particles of polonium, 571 ; on the question of valency in gaseous, 753. Ions, on the energy required to pro- duce, 571; on the velocity of the, of alkali salt vapours in flames, MN. Iron, on the effect of temperature on the absorption coefficient of, for y rays, 582. Tron wires, on the change of resistance of, in strong magnetic fields, 122. Ives (Prof. J. E.) on a new torm of earth inductor, 579; on an approximate theory of an elastic string vibrating in a _ viscous medium, 742, Jordan (I. W.) on the direct mea- surement of the Peltier etlect, 454, Kleeman (Dr. R. D.) on the law of chemical attraction between atoms, 88; on relations between the den- sity, temperature, and pressure of substances, 825; on the heat of mixture of substances and the relative distribution of molecules in the mixture, 535. Lamb (Prof. H.) on the uniform motion of a sphere through a viscous fluid, 112. Lamp-black, on the reflective power OL G7. Light, on the ultra-violet, from the mercury arc, 393 ; on condensation produced by the action of, on iodine vapour, 465 ; on apparatus for the production of circularly polarized, 517. INDEX. 781 Lines and planes, of closest fit, on, 367. Liquids, on the measurement of the refractive index of, 301; on the motion of solid bodies through viscous, 697, Livens (G. H.) on the initial accele- rated motion of an electrified sphere, 649, Lotka (A. J.) on a problem of age- distribution, 435. McLaren (S. B.) on Hamilton’s equations and the partition of energy between matter and radi- ation, 15. Magnetic fields, on the change of resistance of iron and nickel wires in strong, 122. — testing, on, l. Maenetostriction, on, 78. Mauchly (S. J.) on a new form of earth inductor, 579. Membrane, on Bessel’s functions as applied to the vibrations of a circular, 53. Mercury, on the influence upon the fluorescence of, of gases with different affinities for electrons, 314, arc, on the ultra-violet light from the, 393. lamp, on extremely long waves emitted by the, 689. Merton (T. R.) on a method of cali- brating fine capillary tubes, 386. Meservey (A. B.) on the potentials required to produce discharge in gases, 479. Metallic colouring in birds and in- sects, on, 504. Metals, on the heat liberated during the absorption of electrons by, 404. Michelson (Prof. A. A.} on metallic colouring in birds and _ insects, ob4. Millikan (Prof. R. A.) on the ques- tion of valency in gaseous ioni- zation, 743. Minerals, on the ratio between ura- nium and radium in, 652. Mitchell (H.) on the ratios which the amounts of substances in radio- active equilibrium bear to one another, 40. Mixture, on the heat of, of sub- stances, 080. Molecules, on the heat of mixture of substances and the relative distii- bution of, 535. More (Prot. L. T.) on the recent theories of electricity, 126. Nicholson (Dr. J. W.) on the bend- ing of electric waves round a large sphere, 62, 281; on the damping of the vibrations of a dielectric sphere, 438. Nickel wires, on the change of re- sistance of, in strong magnetic fields, 122. Oscillations of chains, on the, 736. Owen (E. A.) on the change of re- sistance of iron and nickel wires in strong magnetic fields, 122. Owen (Dr. G.) on condensation nuclei produced by the action of light on iodine vapour, 465. Oxley (A. E.) on apparatus for the production of circularly polarized hicht, 517. Pealing (H.) on condensation nuclei produced by the action of light on 1odine vapour, 465. Peltier effect, on the direct measure- ment of the, 454. Periodic motion, on the maintenance of, by solid friction, 161. Photoelectric effect, on the normal _ and the selective, 155. ‘ Pirret (Miss R.) on the ratio between uranium and radium in minerals, 652. Pitchblende, on the rate of evolution of heat by, 58. Pitot constant, on the value of the, 348. Planes of closest fit, on, 367. Platinum-black, on the reflective power of, 167. Poll (Dr. R.) on the normal and the selective photoelectric effect, 155. Point, on the discharge from an electrified, 585. Polarized light, on apparatus for the produetion of circularly, 517. Polonium, on the ionization of dif- ferent gases by the alpha particles from, 571. Poole (H. H.) on the rate of evolu- tion of heat by pitchblende, 58. Positive electricity, on the rays of, 225; on the discharge of, from hot bodies, 634. 782 A IND) EX Potentials required to produce dis- charge in gases, on the, 479. Pressure, temperature, and density of substances, relations between the, 325. Pressure displacement of spectral lines, on the, 499. Pringsheim (Dr. P.) on the normal and the selective photoelectric effect, 155. Probability, some problems in the theory of, 745. Quartz mercury lamp, on extremely long waves emitted by the, 689. Radiation, on Hamilton’s equations and the partition of energy be- tween matter and, 15. Radioactive equilibrium, on the ratios which the amounts of sub- stances in, bear to one another, 40. matter, on the ionization of the atmosphere due to, 26. Radioactivity of the Leinster granite, on the, 102; of some igneous rocks from antarctic regions, on the, 770. tadium and uranium, on the ratio between, in minerals, 652. Radium emanation, on the relation between viscosity and atomic weight for, 49; on the behaviour of, at low temperatures, 722. Raman (C. V.) photegraphs of vibra- tion curves, 615; on the photo- metric measurement of the obli- quity factor of diffraction, 618. Rankine (Dr. A. O.) on the relation between viscosity and atomic weight for the inert gases, 45. Rastall (R. H.) on dedolomitization in the marble of Port Shepstone, 173. Rayleigh (Lord) on Bessel’s func- tions as applied to the vibrations of a circular membrane, 53; hydro- dynamical notes, 177; on a phy- sical interpretation of Schilo- milch’s theorem in Bessel’s func- tions, 567 ; on the motion of solid bodies through viscous liquid, 697. Refractive index of liquids, on the measurement of the, 301. Relativity, on the derivation from the principle of, of the fifth funda- mental equation of the Maxwell- Lorentz theory, 296; on the pro- blem of uniform rotation treated on the principle of, 319; on the problem of the uniform rotation of a circular cylinder in its con- nexion with the principle of, 342 ; on the common sense of, 502; on, and the conservation ofmomentum, 626. ; Resistance, on the change of, of iron and nickel wires in strong magnetic fields, 122. Richardson (L.) on the rheetic de- posits of Somerset, 279. Richardson (Prof. O. W.) on the heat liberated during the absorption of electrons by different metals, 404. Robinson (Dr. J.) on electric dust figures, 268. Rocks, on the radioactivity of some igneous, trom antarctic regions, 770. Roéntgen radiation, on homogeneous, from vapours, 446, rays, oh an apparent softening of, in transmission through matter, 699. Ross (A. D.) on magnetic testing, 1. Rossi (R.) on the pressure displace- ment of spectral lines, 499. Rotation, on the problem of uniform, treated on the principle of rela- tivity, 8319; on the problem of the uniform, of a circular cylinder in its connexion with the principle of relativity, 542. Royds (T.) on the reflective power of lamp- and platinum-black, 167. Rubens (Prof. H.) on the focal isola- tion of long heat-waves, 249; on extremely long waves emitted by the quartz mercury lamp, 689. Rudge (Prof. W. A. D.) on the elec- trification of the air near the Zambesi Falls, 611. Russell (A. 8S.) on the y-rays of thorium and actinium, 130. Rutherford (Prof. EF.) on the scatter- ing of a and @ particles by matter and the structure of the atom, 669. Sadler (Dr. C. A.) on an apparent softening of Réntgen rays in trans- mission through matter, 659. Schlémilch’s theorem in Bessel’s functions, on a physical interpre- tation of, 567. Searle (G. F. C.) on a revolving table method of determining the curvature of spherical surfaces, 218. TONED Ex 783 Self-inductions, on the comparison of two, 608. Sharpe (Dr. F. R.) ona problem in age-distribution, 435. Smoluchowski (Prof. M.S.) on the conduction of heat through rare- fied gases, 11. Snow (EH. C.) en restricted lines and planes of closest fit to systems or points, 367. Soddy (F.) on the y-rays of thorium and actinium, 130; on the ratio between uranium and radium in minerais, 652. Spectra, on the resonance, of iodine, 261. Spectrum, on the transformation of aresonance into a band, by pre- sence of helium, 265; ona method of measuring the luminosity of the, 604. Sphere, on the bending of electric waves round a large, 62, 281; on the uniform motion of a, through a viscous fluid, 112; on the damp- ing of the vibrations of a dielec- tric, 488; on the initial accele- rated motion of an electrified, 640. Spherical surfaces, on a revolving table method of determining the curvature of, 218. Stead (G.) on the problem of uni- form rotation treated on the prin- ciple of relativity, 319. Stephenson (A.) on the maintenance of periodic motion by solid fric- tion, 161; on a peculiar property of the asymmetric system, 166; on water waves as asymmetric oscillations and on the stability of free wave-trains, 773. Steven (A. I.) on an apparent soft- ening of Ronteen rays in trans- mission through matter, 659. Swann (Dr. W. F.G.) on the pro- blem of the uniform rotation of a circular cylinder in connexion with the principle of relativity, 342; on the longitudinal and transverse mass of an electron, 735. Taylor (T. 8.) on the ionization of different gases by the alpha parti- cles from polonium and the relative amounts of energy required to produce an ion, 571, Temperature, density, and pressure of substances, relations between the, 825; on the effect of, on the absorption coefticient of iron for y rays, 052. Thomson (Sir J. J.) on rays of posi- tive electricity, 225. Thorium, on the y-rays of, 130. Thornton (Prof. W. M.) on thunder- bolts, 630. Thunderbolts, on, 630. Tide races, on, 187. Tolman (Dr. R. C.) on the derivation from the principle of relativity of the fifth fundamental equations of the Maxwell-Lorentz theory, 296. Tubes, on a method of calibrating fine capillary, 586. Tyndall (A. M.) on the value of the pitot constant, 848; on the dis- charge from an electrified point, 585. Ultra-violet hght from the mercury arc, on the, 393. Uranium and radium, on the ratio between, in minerals, 652. Valency, on the question of, in gaseous lonization, 753. Vapours, on homogeneous Rontgen radiation from, 446. Vibration curves, photographs of, 615. Vibrations, of a circular membrane, on Bessel’s functions as applied to the, 53; on the damping of the, of a dielectric sphere, 458. Viscosity and atomic weight, on the relation between, for the inert gases, 40, Viscous fluid, on the uniform motion of a sphere through a, 112; on steady motion in a corner of a, 191. liquid, on the motion of solid bodies through, 697. —— medium, on an approximate theory of an elastic string vibra- ting in a, 742, Water waves as asymmetric oscil- lations, on, 773. Waves, on the bending of electric, round a large sphere, 62, 281; on the motion of, into shallower water, 178; on periodic, in deep water, 183; on water, as asymmetric oscillations, 773. 784 Waves, light-, on extremely long emitted by the quartz mercury lamp, 689, Wave-motion, on the potential and kinetic energies of, 177. -trains, on the stability Ou ixee, 773. Wilson (Prof. H. A.) on the velocity of the ions of alkali salt vapours in flames, 711; on the number of electrons in the atom, 718. Wilson (Dr. W.) on the discharge of positive electricity from hot bodies, 634. Wilson (W.) on the effect of tempe- rature on the absorption-coefficient of iron for y rays, 632. Wireless telegraphy, on the bending of electric waves in, 65. 4 “