ac ih Hi f oe Attia nal ie sia) ih Ve Hf We r my) i i\hi ? 0s Bue Typ ab) Wh itt my ray) ak vii) ; a itt ae * 1 ’ Dr Keg th ‘ sain bit he ihe 2K" wh v ity. mic} 3 a i ity Hi he ee is AN i ih Na ae Healy i Ne a sa nies} a il th Hy Hi sinh a a a tf i it iy i iy } 4% Ait weit th, In | an (i a iii a if ait aah Mt ae ‘ n ' Ant Mt { Hid ) hal ah uy at ann dina Ahh ana ult Rh ae LAN } hig’ ty bits etter { iy SRSA TY it Nasites Syst $3: of rau ea r heey ft tf hy | f i shh ue i 7 i y\ i ‘ p . a : nate {i i oH aii hhh th nat bb) . a He y a } if a {i ia 4th ik ae 1% fl He ith 2 = =: - Se er 8 ees Pie a re — ae = Se oe SE ee SP 2 arcs eee 3 1s = se et ee = ae eee apenas sc5 se Pm oie 5 ee WT = Dini = * %} Sac PEs age eee oes See ees - ih, ‘i ES ae ene Eee “ = z > SA ed aaa SS =e = Seay Ser ae ee sein = —~ 5 2 ae ee SS ee eee ry eS ines Sate ——S ; " nee Se = 3 oS = = 2 S a ae = - 3 Ea tr pee ez : at = = SS rg eS z eee = = ~ = = es = ; os Staal SS Saat es ns = os Pik ee _ aie SS ee = = ee =o" =a : ae Sere Sao SS = ~~ =e SS hi yaa i; Pits ABU PED oH jj Hh rt aa i) hy. tN f i ene EDA AEDT BH eee tS a -* pane SS Sete: : ee — Se : ~ iat 2S me oe : = aesease: Sse a Ease ae SS —< gene Sa ge atm 7 oe SS ees ms sas re Se Sees a a eee Sr oe pete ne See Fe SEs . ae we ae _ i Fi “ aS Ca ES 5 : =e : L. 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’), ..
. 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
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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.
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Diamond Green.
Fie. 5.
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VIN GMMEY OR
Phil. Mag. Ser. 6, Vol. 21, Pl. IV.
Fie. 6.
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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
»
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a
=
i ig
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Fs {
4
=)
<
oe by
ed Les es
: oe hia
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tite
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\ : : : , £
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th roe
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i 7 var an
he _ _ Bd ns cu 4,
wl a) SVE me eon a
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Pcs aes PS
Part
me S ¥ . ae
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7 or She
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a
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ra Tey Pane : '
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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