PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY OF LONDON. SERIES A. CONTAINING PAPERS OF A MATHEMATICAL OR PHYSICAL CHARACTER. VOL. 211. LONDON : PRINTED BY HARRISON AND SONS, ST. MARTIN'S LANE, W.C., Jrhitfrs in ®rbinatg to fis MARCH, 1912 Q v. r iii ] CONTENTS. (A) VOL. 211. List of Illustrations page v Advertisement vii I. On the Absolute E.rpi vision of Mercury. By HUGH L. CALLEN!>AR, M.A., LL.D., F.ti.X., Professor of Physics >tt the Imperial College of Science and Tech- nology, S.W., and HERBERT Moss, RSc., A.R.C.S., Demonstrator of Physics at the Imperial College of Science and Technology, London, S. W. . page 1 II. Tin' Effect of Pressure upon Arc Spectra. No. 3. — Silver, X 4000 to X 4600. No. 4.— -Gold. By W. GEOFFREY DUFFIELD, D.Sc., Honorary Research Fellow in Physics in the University of Manchester. Communicated by ARTHUR SCHUSTER, F.RS. 33 III. Memoir on the Theory of the Partitions of Numbers. — Part V. Partitions in Two-dimensional Space. By Major P. A. MAcMAHOtf, R.A., D.Sc., F.RS. 75 IV. Sturm- Liouville Series of Normal Functions in the Theory of Integral Equations. By J. MERCER, M.A. (Cantab.), D.Sc. (Live)-pool), Fellow of Trinity College, Cambridge. Communicated by Prof. A. R. FORSYTH, Sc.D., LL.D., F.RS. Ill a 2 V The § //"* of Water and the Mechanical Equivalent of Jlre, jL 0- a to 80° C. By W. R BOUSFIELD, JO., A.C., ;MC BOUSED, A A Communicated b, Sir JOSEPH LARMOR VI The Vertical Temperature Distribution in the Atmosphere over England, and ' Reno*, on the Gene.-al and Local Circulation. By W. H. DINES, F.R.S. 253 VII. A Theory of Asymptotic Series. By G. N. WATSON, M.A., Fettow of Trinity College, Cambridge. Communicated by G. H. HARDY, F.R.S. ... 279 VIII. Tin- Constitution of tfte Alloys of Aluminium and Zinc. By WALTER RosENHAix, B.A., D.Sc., and SYDNEY L. ARCHBUTT, A.I. C., of the National Physical Laboratory. Communicated by R T. GLAZEBROOK, C.B., F.R.S. 315 IX. Memoir on the Theory of the Partitions of Numbers.—Part VI. Partitions in Two-dimensional Space, to which is added an Adumbration of the Theory of the Partitions in Three-dimensional Space. By Major P. A. MACMAHON, R.A., D.Sc., LL.D..F.RS. 345 X. Radiations in Explosions of Coal-gas and Air. My W. T. DAVID, B. A., B.Sc., Trinity College, Cambridge. Communicated by Prof B. HOPKINSON, F.R.S. 375 XI. On the Series of STURM and LIOUVILLB, as Derived from a Pair of Fundamental Integral Equations instead of a Differential Equation. By A. C. DIXON, F.R.S., Professor of Mathematics, Queen's University, Belfast 411 XII. The Kinetic Theory of a Gas Constituted of Spherically Symmetrical Molecules. By S. CHAPMAN, M.Sc.t Trinity College, Cambridge; Chief Assistant, Royal Oliservatory, Greenwich. Communicated by Sir JOSEPH LARMOR, Sec.R.S. 433 Index to Volume LIST OF ILLUSTRATIONS. Plates 1-4. — Dr. W. GEOFFREY DUFFIELD, on the Effect of Pressure upon Arc Spectra. No. 3.— Silver, X 4000 to X 4600. No. 4.— Gold. Plates 5-7. — Dr. WALTER ROSENHAIN and Mr. SYDNEY L. ARCHBUTT, A.I.C., on the Constitution of the Alloys of Aluminium and Zinc. [ vii ] ADVERTISEMENT THE Committee appointed by the Royal Society to direct the publication of the Philosophical Transactions take this opportunity to acquaint the public that it fully appeai-s, as well from the Council-books and Journals of the Society as from repeated declarations which have been made in several former Transactions, that the printing of them was always, from time to time, the single act of the respective Secretaries till the Forty-seventh Volume ; the Society, as a Body, never interesting themselves any further in their publication than by occasionally I'ecommending the revival of them to some of their Secretaries, when, from the particular circumstances of their affairs, the Transactions had happened for any length of time to be intermitted. And this seems principally to have been done with a view to satisfy the public that their usual meetings were then continued, for the improvement of knowledge and benefit of mankind : the great ends of their first institution by the Royal Charters, and which they have ever since steadily pursued. But the Society being of late years greatly enlarged, and their communications more numerous, it was thought advisable that a Committee of their members should be appointed to reconsider the papers read before them, and select out of them such as they should judge most proper for publication in the future Transaction; which was accordingly done upon the 26th of March, 1752. And the grounds of their choice are, and will continue to be, the importance and singularity of the subjects, or the advantageous manner of treating them ; without pretending to answer for the certainty of the facts, or propriety of the reasonings contained in the several papers so published, which must still rest on the credit or judgment of their respective authors. It is likewise necessary on this occasion to remark, that it is an established rule of the Society, to which they will always adhere, never to give their opinion, as a Body, upon nny subject, either of Nature or Art, that comes before them. And therefore the t hunks, which are frequently proposed from the Chair, to be given to the authors of such papers as are read at their accustomed meetings, or to the persons through whose hnndfl thev received them, are to be considered in no other light than as a matter of civility, in return for the respect shown to the Society by those communications. The liko also is to be said with regard to the several projects, inventions, and curiosities of various kinds, which are often exhibited to the Society ; the authors whereof, or those who exhibit them, frequently take the liberty to report, and even to certify in the public newspapers, that they have met with the highest applause and approbation, i therefore it is hoped that no regard will hereafter be paid to such reports and public notices; which in some instances have been too lightly credited, to the dishonour of the Society. PHILOSOPHICAL TRANSACTIONS. I. On the Absolute Expansion of Mercury. /it/ HUGH L. CALLENDAR, M.A., LL.D., F./'.S., I'rofessor of Physics at the Imperial College a 3 f n <: H I 0 \t J * c Fig. 2a. i«. 26. 8 mm. for 300° C., and necessitated a flexible connection between the hot and cold columns at the upper ends between the points e and b. The iron tube rectangles were firmly supported at the base, so that the lower cross tube cd was always very nearly horizontal. We are here concerned with the linear expansion only of the containing tubes, which will not affect the accuracy of the absolute values of the expansion of mercury, provided that adequate means are adopted for measuring the actual lengths of the hot and cold columns at each observation. The provision made by REGNAULT for this purpose was unsatisfactory, as he himself points out, especially in relation to his fourth series of observations, for which his apparatus was not originally designed. The essential point is that the tubes containing the mercury should be of small bore, and should be maintained accurately horizontal at the points where they emerge from the oil bath, and where the temperature changes from hot to cold. The method adopted for securing this result in the present investigation is shown in fig. 3. Steel tubes of 1 mm. bore were brazed with pure copper, using borax as a flux, into the upper and lower ends of the vertical steel tubes, 3'5 mm. bore, containing the mercury columns. The small bore tubes were bent round through a right angle, and were PROF. HUGH L. CALLENDAR AND MR. HERBERT MOSS .ver-Boere through holes accurately drilled in a brass plate A, which was clamped .gainst the vertical face of the T joint in the tubes containing the circulating «L Bnt.il Plai*. o) - Fig. 3. The mercury tubes up to the point A would thus be maintained at the temperature of the circulating oil. After traversing a distance of about 5 cm. in the air, the small bore tubes passed through a brass block B, being soldered into holes in the block drilled so as accurately to correspond with those in the plate A. The brass block B was carried by a rigid bracket CD, and was cooled by a water circulation as indicated in fig. 3. The plate A was adjusted so that the length of tube, AB, where the temperature changed from hot to cold, should be accurately horizontal. The same arrangement was adopted at each of the points, E, H, C, B, fig. 1, where the mercury tubes emerged from the circulating oil. The vertical heights of the hot and cold columns were measured by the steel tapes h, la, fig. 1, suspended from the upper brackets, and read by levelled telescopes. The effective heights of the columns were taken to 1* the vertical distances between the centres of the small bore steel tubes, which could be measured to about 01 mm., giving an order of accuracy of 1 in 20,000 in this fundamental measure- ment. The steel tapes were standardized by comparison with the standard invar scale, and were corrected for temperature at each observation. The gauge tubes, as shown in figs. 1 and 4, were mounted on a separate board in front of the apparatus so as to be protected from vibration and screened from the radiation of the hot columns. The temperature of the mercury in the gauge was estimated by means of four standardized mercury thermometers a, b, c, d, immersed in mercury contained in tubes of the same 9 gauge tubes, and placed at a distance apart equal to twice the distance ing the gauge tubes. The difference of temperature between the gauge tubes ON THE ABSOLUTE EXPANSION OF MEKCURY. 7 was taken as half the mean difference of temperature between the thermometers in a horizontal direction. A correction for this difference of temperature was applied to the columns of mercury in the gauge tubes below aa\t fig. 2b, i.e., below the level of the top of the cold column. This correction never amounted to more than 0'002 cm., and is included in the recorded values of h given in the tables. The mean temperature of the column representing the difference of level h was estimated from the vertical and horizontal temperature gradients indicated by the thermometers, and is denoted by t in the tables and equations. The accuracy required in the observation of the gauge temperature was about fifty times less than in the temperatures of the hot and cold columns. The platinum thermometers were annealed in place in the apparatus by heating the whole to 350° C. shortly after its erection before filling with mercury. This annealing reduced the resistance of each by 1 part in 3000. The apparatus was not heated afterwards beyond 300° C., and the thermometers showed no signs of further change. Owing to their great length the thermometers could not be tested satisfactorily, except at 0° and 100° C., and the value of the difference coefficient O'OOOISO was assumed to be the same as that found for other thermometers constructed of the same wire. The fundamental intervals of the thermometers were, for t\t 4'6456, and for ta, 4 '6412 ohms. Headings were taken to O'l mm. on the bridge wire, corresponding to 0°'002 C., giving an order of accuracy of 1 in 50,000 on the fundamental interval. The values of ti and ta recorded in the tables were deduced from the observed temperatures on the platinum scale pti} pta, by the formula, t-pt = 0'000150< (<-100), which may possibly be in error by 0°'03 C. (or 1 in 10,000) at 300° C. 3. Theory and Notation. % The following notation is adopted :— HI = 6/1 is the effective height of the cold column at a temperature t\. H2 = 6/2 is the effective height of the hot column at a temperature ta. c/H = Hj— HI is the effective height of the cross tubes at the air temperature t. h is the observed difference of level in the gauge tubes at the temperature t. h' = /i-0'OOOlS (h-dH.) (t-ti) is h corrected for dR and reduced to ti. The error of this reduction will not exceed 1 in 20,000 of h provided that the temperature of the gauge t is known within 0°'3 C., and that the temperature of the cross tubes does not differ by more than 4° C. from the gauge. The approximate value 0 '000 180 of the coefficient of expansion suffices within the same limits of accuracy provided that the difference of temperature t— ti does not exceed 50° C. The temperature of the gauge may be taken as known within 0°'l C. The cross 8 PROF. lin:il I.. r.U.I.KNDAR AND MR. HERBERT MO-SS till** M-l.lon, .lini-ml in temperature from the gauge by so much as 1° C., and the ,lir -f t.-mprrature t-d in the majority of the ol*ervations was less than 1° C. Tl»- minimi convetion to the value of h was therefore extremely small, and intr.Nlui-.-d v.-rv littl.- uiuvrtainty in the reduction. r,,iisul.-iiii^ the equilibrium of the columns, we have the hot column H2 at a t,-m|M-ratun- /,, together with the difference of level h in the gauge at a temperature f. lwlnno«-d liy tlit- cold column H, at a temperature tit together with the cross tubes '/If at a temperature t. The mean coefficient of expansion iaa between ti and ta in trims of the volume at tt, which is the coefficient most directly given by the observations, is easily obtained by reducing the columns to a common temperature t\. We thus obtain the equation -00018 («-*,)} = H, = H,-dH, . . (l) which, with a few simple approximations in the small terms involving h, reduces to A'), ......... (2) where // denotes the corrected and reduced value of h given above. The expatision between tt and t3 may further be expressed as a fraction of the volume at Oe C. by multiplying by the factor (l + 0'00018*1). The uncertainty of this reduction will not exceed 1 in 40,000 unless ti exceeds 25° C. Thus, Mo (f.-fi) = (l + 0'00018«,) A'/(H2-//), ...... (3) where (,a,)0 denotes the mean coefficient *, to ta in terms of the volume at 0° C. The expansion between 0° C. and ta in terms of which the results are generally tabulated, may readily be deduced by adding the expansion between 0° C. and fc to that between f, and t, in terms of the volume at 0° C. Thus, (4) but this involves a correction of quite a different order of magnitude, and requires the >f «ai to be accurately known, unless ti is very small. two last reductions may be included together in the formula A', ........ (5) but since the correction term ^lH, may be nearly as large as h', it is desirable to from the ON Till: AMSOLITK F.XI'ANSION OF MKKCURY. «) were taken with the cold column at a temperature in the neighbourhood of 20 < '. It makes a different ..... t' more than 1 in 500 in the fundamental coefficient a ....... 'ding as we assume HKHNA KIT'S value 0'0001795, or WULLNEK'S value ()'00()J814, for (In- mean coefficient l>etween d ( '. and 20° C. in reducing the observations. The uncertainty is greater at lower temperatures. REGNAULT states that he solved his formula by a method of successive approximation, but the approximation would evidently be unsatisfactory at low temperatures, and his calculations cannot be reproduced so as to make his results fit with his observations. REGNAULT himself was conscious of this difficulty, and endeavoured to avoid it by cooling the cold column with melting ice, but he appears to have abandoned this method on account of difficulties of manipulation. The apparatus employed in the present investigation was better suited for the purpose than REGNAULT'S, and a special series of observa- tions was successfully taken with the cold column in ice, and at — 10° C., for the accurate determination of the coefficient at low temperatures. But the majority of the observations were taken with the cold column at the atmospheric temperature, t>ecause this procedure, besides greatly facilitating the manipulation, made all the other corrections as small as possible, and in particular rendered the correction depending on c?H practically negligible, so that it was in most cases unnecessary to measure the length of the cold column at each observation. It is easily seen that a formula precisely analogous to (5) applies to the reduction of the observations to any convenient standard temperature t0, other than 0° C., namely, //), ...... (6) where Oa2, o«i, denote the mean coefficients between ?0 and ta, tr respectively expressed in terms of the volume at t0. Formulae (2) and (5) may be regarded as special cases of this more general formula in which /0 is replaced by t\ and by 0° C. respectively. The majority of the observations in the first series with the cold column at the atmospheric temperature in the neighbourhood of 20° C., were reduced to a standard temperature of 20° C. in the first instance, because the value of the coefficient at 20° C. in terms of the volume at 20° C. could be inferred with considerable accuracy from the observations themselves, and the difference (ti — 20) was comparatively small. The correction term (^ — 20) o«iH2 was of the order of 3 per cent, at most, and was itself known with certainty to 1 in 2,000. If the observations had been reduced directly to 0° C. by REGNAULT'S formula, this correction would in some cases have exceeded 30 per cent., and would have been most uncertain, since the mean coefficient from 0° C. to 20° C. could be obtained only by extrapolation. The corrections involved in deducing h' from h, were of the order of 2 or 3 parts in 10,000 only, and could not give rise to any similar uncertainty. With the apparatus above described, the expansion of mercury is obtained under a mean pressure of 2 '5 atmospheres, but the result will not differ from the expansion under a pressure of 1 atmosphere except in so far as the compressibility of mercury VOL. ccxi. — A. C 10 PEOF. HUGH L. CALLENDAR AND MR. HERBERT MOSS vark* with temperature. The compressibility of mercury, however, is so small that it would require a variation of 50 per cent, to affect the results appreciably even at 300* C. It is most improbable that the variation of compressibility with temperature is as great as this. It would have been necessary to apply a pressure of 2 or 3 atmospheres to the gauge to test this point satisfactorily, and it was not considered advisable to do this, since any accidental failure of any of the joints or taps under pressure at high temperatures might have involved dismounting and filling the whole apparatus afresh, and would have seriously interfered with the continuity of the observations. 4. Method of Filling the Apparatus. The adoption of the multiple manometer method, which was rendered necessary in order to avoid excessive length and pressure, entailed some difficulty in filling the apparatus. After adjusting the tubes in position, as shown in fig. 1, the small bore steel tubes at the top and bottom of the hot and cold columns were connected by horizontal glass tubes, as shown in fig. 5a, with taps attached at right angles. The glass tubes nearly fitted the steel tubes, and the joints were made tight by running in a mixture of beeswax and resin. The taps were suitably supported, and projected fanwise, upwards at the top, and downwards at the bottom, as shown in the side view in fig. 1. After the taps had been connected and the whole tested for leaks, the apparatus was heated to about 350° C., and evacuated, and dried by passing filtered air through it. When cool, the apparatus was evacuated as completely as possible, and mercury was admitted by connecting a reservoir to each of the lower taps in turn, until the level of the mercury rose nearly to the upper cross tubes. The apparatus was again evacuated by connecting the pump to each of the upper Fi«- to. Fig. 54. taps in turn in order to remove any air displaced by the mercury. The filling was iiifii completed and the gauge tubes conneotefl TJ V> c oJeth?lhuat> ^1 the leVd °Q, 7 "^ WCre d^W ^ ^Jm^btoToS ttjffifcST InTJchaT * "' aUd mei"CUry re"rV°irS e> f » the other side of the gauge. If the fevd ^Ld ^>ut ! ^y ON THE ABSOLUTE EXPANSION OF MERCURY. 11 inserting a glass plunger, without withdrawing or adding mercury, the levels returned to their previous values within O'Ol mm. in about a minute after the removal of the plunger, in spite of the great length of fine tube through which the mercury had to flow. If, on the other hand, the continuity of the mercury column was broken by a single air bubble in one of the fine tubes, the level could be altered by a centimetre or more on one side without any change taking place on the other. It was feared at first that the presence of air bubbles might be a serious source of error, but the effect produced was so immediately obvious that no uncertainty arose in this way. In spite of the care taken in evacuating the apparatus, some bubbles invariably appeared when the apparatus was first heated to high temperatures such as 200° C. to 300° C. after each fresh filling. These bubbles were removed as they appeared by altering the level of the mercury in the gauge, so as to reduce the pressure and drive the bubbles round into the open space where the tap was connected, whence they could be removed by applying the air pump. At the highest temperatures, from 200° C. to 300° C., it was found necessary on account of the expansion of the containing tubes, which amounted to nearly 8 mm. at 300° CL, to insert a flexible rubber connection, as indicated in fig. 5b, between the glass taps and the small-bore tubes on the cold side. The T-joint was placed close to the end of the small-bore steel tube on the hot side to facilitate the trapping of bubbles, which were most troublesome at the higher temperatures. Fortunately this trouble tended to disappear as the occluded or dissolved gas was removed by repeated heating of the mercury. 5. Method of Taking Obsei-vatiotis. After starting the water and oil circulations, the heating coils were connected to the electric-light mains. A current of 13 amperes at 100 volts sufficed to raise the temperature about 100° C. in an hour and a half. When the temperature approached the required point, the current was gradually reduced to the value which experience had shown to be sufficient to maintain the desired temperature. The current was then switched over to a large battery of accumulators, which made it possible to keep the temperature very nearly constant with slight occasional adjustments. But as the cold columns rose very slowly in temperature, at the rate of about a tenth of a degree in half an hour, the current was generally set to give a slightly greater rate of rise for the hot column during half an hour or so, followed by a slightly slower rate for another half hour, so that during the first period the difference of temperature and the difference of level might be slowly increasing, and during the second period slowly diminishing, at nearly the same rate. By taking observations in this way the effects of lag, if any, would be eliminated from the results. The actual readings of temperature and of difference of level were plotted on a large scale (10 cm. to 4 cm. on the bridge wire and 10 cm. to 2 mm. of the difference of level), c 2 II llt'GH L. CALLENDAR AND MR. HERBERT MOSS ..it a linn- lias,-, s.i that tin- omditions of'tlie experiment could be accurately followed, .in. I any d«-f»vt in tin- working of the apparatus, such as the appearance of an air Fig. 6. Fig. 7. h«- ' hi«h tnnperatures, immediately detected T. typio.,1 ,X;illlJ(lt.s, ;U1(, wi e the shown and ON THE ABSOLUTE EXPANSION OF MERCURY. 13 curves also served as a convenient means of graphic interpolation for deducing the simultaneous values of the observed quantities. Since all the readings could not be made simultaneously, attention was directed to obtaining them in quick and regular succession. The bridge was provided with mercury cup connections in place of the usual plugs, giving great improvement in quickness as well as in accuracy. The thermometers could be interchanged instantaneously by means of a mercury cup switch, without introducing any variable contact errors such as would have been unavoidable with screw connections. The telescopes for reading the mercury levels contained eyepiece micrometers divided to tenths of a millimetre, fitted with vertical screw adjustments, and were focussed and adjusted so that the millimetre divisions coincided with those of the standard invar scale situated between the mercury columns. By siiitably shading and illuminating the mercury columns, readings could easily be taken by inspection to O'Ol mm. A separate handle was provided for turning the vertical column carrying the telescopes through an angle of 2 degrees either way to verify the adjustment of the eyepiece micrometers on the invar scale. This method appeared greatly preferable in practice to the use of a filar micrometer, since it was never necessary to touch the telescopes when once they had been adjusted for a run. With a little practice, a single observer could take all the readings, and perform all the necessary adjust- ments, without any excessive haste or exertion. In reducing the observations, points were selected on the curves both with a rising and falling temperature difference, where the temperature conditions appeared to be most favourable. Points taken on the same day at nearly the same temperature always agreed so closely on reduction, that it was considered preferable to take short runs of about one hour each at two or three different temperatures on the same day rather than runs of long duration at one temperature. Runs taken at the same temperature on different days, separated often by many months, when all the conditions of observation were completely changed, afforded a much better test of the accuracy of the method, and were more likely to serve for the elimination of constant or accidental erroi-s than runs taken under constant conditions. Observations given in the tables under the same date at the same temperature were taken with a rising and falling temperature respectively, or otherwise differed materially in the conditions under which they were taken. G. Method of Reducing (lie Observations. The following example, showing the reduction of a single observation, will serve to illustrate the order of magnitude of the corrections involved. The corrections were worked to one figure beyond the limit of accuracy of reading, except that, in the case of the platinum thermometers reading to 0°'002 C., it was considered useless to express the temperatures beyond 0°'001 C., as this represented an order of accuracy U PROF HUGH L. CALLENDAK AND MR. HERBERT MOSS t»- times as great as could be obtained in reading the difference of level on the gauge: — Temperature difference November 2, 1908. 3 p.m. Current 7 '52 amperes. increasing. Box temperature observed, 2rO° C. Cold side. Hot side. Readings of platinum thermometers 129171 1656'82 Calibration and temperature corrections +0153 +0'228 Corrected bridge readings, R 1291'863 1657'048 Resistances at 0° C., R, 1194'052 1191-657 Differences, R-R, 97'811 465'391 Fundamental intervals, RIOO— RU 464'56 46412 Temperatures on platinum scale, pt 2T055 100'274 Corrections to gas scale (t—pt) -0'247 + 0'004 Temperatures tt and tt on gas scale 20'808 100'278 Mercury gauge readings :— Cold side. Hot side. Levels of mercury in gauge (cm. corrected) 511522 67'6196 Upper thermometers (at 52 cm. level) 21°'6 C. 21°'9 C. Lower thermometers (at 8 cm. level) 20°'95 C. 21°'0 C. Mean temperature of columns below 51 cm 21°'27 C. 21° '35 C. Correction for temperature difference, 0°'08 C — 0'0009 cm. Difference of level, h = 67'6196-511522 = .... 16 '4674 cm. Mean temperature of h, 22°'0 C. Correction of A to tt . -0'0035 Observed lengths of columns by steel tapes corrected . 192755 192'815 Effective height, H2 = 6^ = 1156'890 cm. dE = R2-R, = 0'360 cm. Temperature of rfH observed, 21°1 C. Reduction to f, negligible (0'00002 cm.). Corrected value of A, h' = A-0'00018 (h-dH) («-«,) = IG'4630. Expansion f, to tt, ^fa-tj = ///(H,-//) = (T0144358. Mean coefficient ft to t, in terms of volume at «,, ,«, = 0-Q00181645. To facilitate comparison between the different observations in which the cold *as at atmospheric temperature, the results were further reduced to a *mperature at 20' C. for the cold column, assuming the value of the \ n'r,8 1 ^ V°1Ume at 2°° °- to ^ °-0001805- The value lent could be deduced, with sufficient accuracy for the purpose, from M of observafcons, extending from 20° C. to 187° C This reductic »• »- effected by means of formula (6), p. 9, and involves the addUi n to /' the ON THE ABSOLUTE EXPANSION OF MERCURY. 15 Expansion from 20° C. to 100°'278 C. in terms of volume at 20° 0. = 0-0145838. Mean coefficient from 20° C. to 100-278° C. in terms of volume at 20° C. Q-Q145834 80-278 ,,,,. The further reduction to 0° C., involving a correction of 20 per cent., could not be effected satisfactorily until the conclusion of the third series of observations, and was not required for comparing the results of the first two series, which are therefore reduced to 20° C. in the tables given below. The corresponding observation taken on the same day, with the difference of temperature decreasing, was as follows : — • November 2, 1908, 3.30 p.m. Current, 7'50 amperes. Resistance box, 21° C. Bridge readings corrected for calibration and temperature ........ 1292*436 1657'310 Temperatures on gas scale deduced ... t\ = 20°'930 C. t-j = 100°"335 C. Lengths on hot and cold columns . . . . /i = 192755 12 = 192'815 Levels of mercury in gauge (corrected for scale) ........... /«, = 51-1552 h, = 67'G102 Gauge thermometers: upper 21°'35 C., 21°70 C. ; lower 20°'9 C., 21° C. Temperatures of cross tubes : upper 20° "7, lower 20° C. Corrected difference of level, h' = 16"4515 cm. Temperature of h, t = 21° 72 C. Reduction to 20° C. = 0'0001805 xO'930xHa = + 0'1942 cm. - 0-ou.Ms. Q-Q145958 80-335 = ° °00181685- The difference from the first observation would be explained by a lag of O'OOl cm. either way in the gauge reading, but is within the limits of accuracy of observation. * 7. Summary of Observations. The following tables contain a summary "of all the observations taken after the apparatus had been got into proper working order. Observations taken with the same upper limit of temperature t3 are grouped together to facilitate comparison, and the observations in each group are arranged in order of date. The first column gives „; IMIOF. IIHJII I- I'ALLKSIMR AND MI?. HERBERT MOSS tin- datr. Tli.- second column gives the observed value of the effective height of the |...t .-..liinm II. OOlWCted »'«>r scale error and temperature. The correction for the ,litt;.ivn( C li-n-rtli 'HI ..f the hot and cold columns was always negligible in the first two sriics. and tin- value of f lrvt-1 // in the gauge, corrected for errors of the standard invar scale, and for ilitf'.TfiK'f of temperature between the gauge columns. These corrections seldom ••eded O'OOl cm., the limit of accuracy of reading and the data for applying them could not have conveniently been included in the tables. The value of h is not corrected for the mean temperature of the column h itself, which is given under the heading / in the next column. The seventh column contains the value of the UpUMUn »«a('»— 20) from 20° C. to ta in terms of the volume at 20° C., calculated by formula (6), to the same order of accuracy as the values of//, namely, to one figure U-vond the limit of accuracy of reading. The values of the expansion are not directly comparable, because they include the small variations of t2. The last column is accordingly added, giving the corresponding value of the mean coefficient 2o«2 from 20° to tt in terms of the volume at 20° C. A variation of a tenth of a degree in t2 should produce a variation of about 2 in the last figure of this coefficient, so that the small variations of tt would seldom affect the last figure but one of the coefficient. The differences shown in this column exhibit the accumulated effect of all the possible errors of observation, including the effect of lag, to which many of the larger differences appear to be due. Since most of the observations were purposely taken in pairs, as explained above, in such a way as to exhibit this effect, with a view to detecting and eliminating it, it is probable that the accuracy of the final means is not seriously by this source of error. Series I. —Observations, 20° C. to 187° C. 1. •. H, _ n. t. 20*2(^-20). 20a2- ; K S (1) Observati ons at 68°' 5C. Oct. 24 . . „ 24 . . 1156-53 • 1156-53 18-335 18-690 68-766 69-070 10-4723 10-4598 18-89 18-92 •0088335 •0088879 •000181144 •000181127 Nor. 13 . . n !•' . . 1156-47 1156-47 17 '993 18-41* 68-693 68-755 10-5307 10-4568 18-87 18-94 •0088225 •0088356 •000181186 •000181224 Nor. 14 . „ H.. „ 14. . 1156-38 11 :,0-38 1156-38 18-200 1H-359 18-119 68-506 68-766 68-469 10-4503 10-4685 10-4608 18-74 18-64 18-77 •0087907 •0088361 •0087851 •000181229 •000181194 •000181249 ON THE ABSOLUTE EXPANSION OF MERCUKY. 17 Series I.— Observations, 20° C. to 187° C. (continued). I >;ite. H-. 18 PROF. HUGH L. CALLENDAR AND MR. HERBERT MOSS .Sm<« J.—Obtfrcatwnt 20° 0. to 187° C. (continued). !' 1908 Oct. 20 . . „ 20.. „ JO.. H* /i. «•! day. A leak in one of the oopper-braBdd joints was suspected, hut on dismounting the apparatus it was found that one of the solid drawn steel tuhes had apparently split in the process of manufacture, and had heen brazed up with ordinary spelter hy the makers so skilfully that the flaw had escaped detection when the apparatus was put together. In process of time the hot mercury had naturally found its way through the hrass. A completely new set of steel tuhes was accordingly fitted, which occasioned a good deal of delay. Owing to the pressure of other duties, the apparatus could not l>e got ready for work again till the end of June. Series II.— Obsercat,,,,,*, 187° C. to 300° C. Date. H.,. /.. /... h. /. »«.>('* -20). 20*2- 1 1909 June 28 . „ 28 . 1160-97 1160-97 (1 19-309 19-636 ) Observat 187-442 187-941 ions at 187 34-7558 34-7894 •c. 20-28 20-77 •0307267 -000183507 •0308173 -000183501 July 9 . „ 9 - 1160-97 1160-97 20-861 21-007 187-001 187-153 34-3608 34-3600 21-60 21-80 •0306553 •0306814 •000183563 •000183553 Means = | — 187-384 — — •0307202 •000183531 1909 June 28 . „ 28 . (2) Observat 1161-51 21-435 221-365 1161-51 21-203 221-122 ions at 221 41-2574 41-2592 °C. 22-91 22-53 •0370874 •0370463 •000184180 •000184198 July 9 . 1161-51 22-567 „ 9 . 1161-51 22-890 221-049 221-026 40-9783 40-9161 22-89 23-49 •0370484 -000184276 •0370496 -000184303 July 12 . 1161-54 22-039 221-162 41-1244 23-61 •0370753 -000184306 Menns = — 221-145 — •0370617 -000184253 1909 July 14 . .. H . 1162-17 1162-17 (3 23-951 24-235 ) Observat 260-041 260-041 ions at 260 48-6868 48-6286 °C. 25-80 26-30 •0444540 •0444512 •000185193 •000185182 July 20 . ,. 20 1162-17 1162-17 27-771 27-896 389-784 259-918 47-8494 47-8584 27-60 27-70 •0444048 •0444369 •000185225 •000185217 July 28 . 1162-17 22-659 259 • 7 s:, 48-9016 24-93 •0444085 •000185201 Means = — — 259-904 — — •0444308 •000186202 II •-' ,.,;MK. IH .:ll F, CALLKNPAR AND MR. HERBERT MOSS 5 , /[.-Oluervation*, 187° 0, to 300° C. (continued). b • it,. ii. /,. k. t. 2002 (<2- 20). 20a2- (4 i Observat oiis at 300° C. 1909 .Inly .Inlv29 . '.'9 . •J'J . lir-- 1162-86 1163-M 11 ()•.'• 86 24-035 300-608 23-897 300-020 24-022 299-993 24-101 300-089 57-0070 56-9050 56-8700 56-8690 26-38 25-12 25-22 25-22 •0522933 •0521818 •0521723 •0521875 •000186357 •000186350 •000186334 •000186325 «.lulv29 . . 29 . 1162-86 1162-86 25-010 25-123 299-841 300-003 56-6364 56-6404 25-74 25-75 •0521411 •0521678 •000186324 •000186311 Means = — — 300-092 — — •0521906 •000186334 * Apparatus allowed to cool between the two sets of observations. Confirmatory Series. As the determinations of the coefficient of expansion below 187° C. had all been made with the old set of steel tubes, whereas the determinations at temperatures above 187° C. had all been made after the apparatus had been taken down and re-erected with the new set of steel tubes, a short series of observations were taken to confirm the earlier results. Some observations were also taken at intermediate temperatures, but no use was made of these in evaluting an equation, as time did not permit of obtaining the steady state of temperature secured in runs of longer duration. The following is a summary of the results, reduced to 0° C. :•— Date. July 19, 1909 „ 19, 1909 „ 19, 1909 July 20, 190'.i „ 20, 1909 If 0«2 X t-,- o«2 observed. oa-2 Ccilculated. 136-270 •024898 •00018271 •00018272 150-255 •027499 -00018302 •00018299 168-009 •030804 -00018335 •00018336 81-140 •014748 •00018176 •00018173 240-933 •044577 •00018502 •00018502 IV n-snlts w,.,v not worked out beyond the limits of accuracy of the readings, but ; obtained with the equation calculated from the previous observations ON TllE Afi-SOLUTK EXPANSION OF MERCURY. 2l (see l>elo\v, p. 22) was considered sufficient to show that no systematic change had occurred in the working of the apparatus. In order to be able to reduce the olraervations with certainty to 0° C., and to obtain a direct value for the fundamental interval without extrapolation, it was necessary to take a series of observations with the cold column at a temperature as .near 0° C. as possible. This point has already been explained in a previous section. 'By surrounding one side of the iron rectangle containing the cold column with a jacket of melting ice, it was found possible to reduce the temperature to between 2° C. and 2°'5 C. By further cooling the cold column to —10° C. with a freezing mixture of ice and salt, while the hot column remained at the atmospheric tempe- rature of 16° C., it was possible to obtain a good approximation to the coefficient at 0° C. These observations entailed much greater difficulty in manipulation than the two previous series, but were valuable as giving direct evidence with regard to the expansion la-tween —10° C. and +20° C. Series III.— Observation* from -10° C. to 100° C. Date. H2> . 1,. t. O^i X f.,. 0«2. 1910 Jan. 5 . . . „ 5. . . „ 5. . . 1159-17 1159-17 1159-17 (1) 2-82 2-82 2-82 Observe 2-569 2-528 2-612 itions, 2°'' 37-890 37-933 37-886 '> C. to 3£ 7-3716 7-3939 7 • 3680 l°C. 14-42 14-41 14-81 •0068576 •0068696 •0068620 •000180986 •000181096 •000181120 Jan. C . . . 1159-17 2-82 2-346 37-259 7-2910 13-50 •0067481 •000181 113 Jan. 7 ... ., 7. . . 1159-17 1159-17 2-82 2-82 2-410 2-464 37-719 37-787 7-3733 7-3763 12-10 12-67 •0068325 •0068445 •000181144 •000181127 Menus = — — 37-746 — — •0068357 •000181097 1910 Jan. 5 . . . „ 5. . . 1159-47 1159-47 (2) 3-18 3-18 Observa 2-388 i-534 tions, 2° '5 68-798 68-968 G. to 68 13-8308 13-8385 '•5C. 14-87 14-50 •0124879 •0125222 •000181515 •000181565 Jan. 6 ... „ 6. . . 1159-47 1159-47 3-18 3-18 2-199 2 • 328 68-959 69-036 13-9040 13-8971 14-04 14-00 •0125188 •0125364 •000181540 •000181592 Jan. 7 ... „ 7. . . 1159-47 1159-47 3-18 3-18 -.'• is] 2-687 68-106 68-223 13-6671 13-6628 12-83 12-41 •0123640 •0123808 •000181540 •000181475 Mi'.-ms = , 68-682 __ _ •0124683 •000181538 PKOF. BOM U CA,.,.KN-.,AK ANI> M,, HBUMT MOSS | / 0*1 X i* Oa2- D h Hj. ,/H. d, \ 1 -I "1 — _ • i ' (3) Observations, 2°'5 C. to 100° C. 1910 J«n. 5. -. 1160-01 1160-04 3-75 3-75 2-494 2-576 100-778 100-703 20-4250 20-3932 14-87 14-91 •0183475 •0183342 •000182059 •000182062 1 • • Jan. 6 . . . e 1160-04 1160-04 3-75 3-75 2-338 2-407 100-547 100-626 20-4081 20-4090 14-65 14-56 •0183040 •0183180 •000182042 •000182040 Jan. 7 . . • „ 7. . . 1160-04 1160-04 3-75 3-75 2-609 2-703 100-442 100-505 20-3252 20-3241 14-16 14-11 •0182819 •0182987 •000182015 •000182068 — — 100-600 • — •0183140 •000182048 Means = — 1910 Jan. 10 . . 1158-66 „ 11 . . 1158-06 (4) Observations, -10° C. to + 1 2-37 -10-403 16-012 5-5220 2-37 -10-498 16-205 5-5870 ! 1 6°C. 14-33 13-70 i«.(d-«i). i*j. •0047765 •0048332 •00018082 •00018098 Means = _ -10-450 + 16-109 — •0048049 •00018090 J _ : — _ - The last observations give the mean coefficient from -10°'5 C. to + 1G°'1 C., which is practically the same as the actual coefficient at 2°'8 C. The values are expressed by formula (2) in terms of the volume at -10°'450 C. When expressed in terms of tin- volume at 0° C. by formula (3), the values become Means . . . [,as]« (/,-/,) = 0'0047958. [,«,]„ = 0 "000 180572. 8. Empirical Foivnulre for the Expansion of Mercury. An empirical formula representing the first two series was calculated in the first instanct- liy the method of least squares. This method was adopted by WtJLLNER and BKOCH in reducing KMSNAULT'S observations, and was fairly appropriate in that case, because the main source of error lay in measuring the small difference of level in the gauge. In the present series of experiments, the fact that the difference of ON THE ARSOLUTE EXPANSION OF MERCURY. 23 level in tin- gauge could not lie read nearer than O'OOl cm. was an important limitation of accuracy at low temperatures, when the difference of level was small. But at temperatures l>etween 200° C. and 300° C., where the difference of level was 40 to 60 cm., the possihle errors in the measurement of the length and temperature of the hot columns became more important, and the order of accuracy was limited in a different way, namely, as a fraction of the whole quantity measured. For low temperatures, the differences between the observed and calculated values of the expansion itself were the best criterion of accuracy ; but for high temperatures, the corresponding differences tatween the observed and calculated values of the coefficient of expansion appeared to l>e a better guide in the selection of an equation. The formula obtained by the method of least squares was accordingly nuxlified from this point of view, but the modifications required were so slight as to Ix; almost within the limits of expei imental error. The following formula was finally adopted to represent the value of the mean coefh'cient ua< between 0° C. and t" C. in terms of the volume at 0° 0. :— ««» = [1805553 + 12444 (*/100) + 253'J () 221-145 •0408120 •0408121 ' + 1 -000184549 -000184549 + 0 II (3) 259-904 -0482078 •0482134 + 56 •000185483 ! -000185505 + 22 II. (4) 300-092 •0559956 •0559900 -56 •000186595 : -000186576 -19 The ol»ervations are arranged in order of temperature. The first column gives the series and number corresponding to the previous tables of observations. The second gives the temperature t» of the hot column, except in the case of the first line, Observation III. (4), where the lower limit was -10°'450 C. in place of 0° C., and Ixith limits are given. The third and fourth columns give the observed and calculated values of the expansion (V,— V0)/V« between 0° C. and ta, except in the first line, wlwre the expansion (Va— V|)/V0 between t\ and t3 is given instead. The fifth column gives the differences between the calculated and observed values of the expansion multiplied by 10T. The last three columns give the observed and calculated values of tin- mean coefficient from 0° C. to t-3, and the difference (C— 0)xlO", except in the line, where the mean coefficients are from t\ to t2. y. Order of Accuracy of the Results. In comparing the results with the formula, it must be observed that the differences- in Tahlr I. are all calculated to one figure beyond the limit of accuracy of observation, namely. D'OOl cm., which corresponds to 10 in the difference between the calculated and ..Uened values of the expansion. Taking the observations at and below 100° C., tin- mean deviation of the observed expansion from the formula is only 11, which ron.-s|*,.i(ls with the limit of accuracy of reading. The corresponding differences in the \al.i.-s of the mean coefficients, given in the last column, are here without ''''• '•• '"•'"««*' it was obviously impossible to measure a short column of only ' •' '•'". to an onler of accuracy of 1 in 20,000 under the conditions of the experiment. ON THE ABSOLUTE EXPANSION OF MERCURY. 25 Taking the observations above 100° C., the mean deviation of the values of the mean ,-,„///,•/,,,/ ,,f .•\|.;Hivi,,n fVi.ni tli.- telow 100° C., and with an order of accuracy of 1 in 10,000 above 100° C. Since positive and negative differences occur almost alternately, and are little, if at all, greater than might naturally lie expected from the limits of accuracy of the various readings, it does not appear that any great advantage could l>e gained by the adoption of a more complicated formula, or by any more elaborate reduction «>i rr|>etition of the experiments. The mean deviation of the individual observations at each point is about twice as great as the deviation of the mean results from the formula. The individual observations are affected by accidental errors of refraction through the glass of the gauge tubes, and by errors of lag, which would disappear to some extent in the means. Correction for lag would have made the observations agree with each other much better in most cases, but the correction could not always be applied with certainty, and it was therefore preferably omitted from the tables. 10. Comparison with Previous Results. It may be of interest to compare the results of the present investigation, as expressed by formula (8), with some of the formula? which have previously been employed to represent the expansion of mercury. REGNAULT assumed a linear formula for the mean coefficient, namely, Oa« = {179007 + 2523(^/100)} x 10"9. He appears to have relied chiefly on the observations at the higher temperatures, and the formula does not represent his ol>servations satisfactorily at temperatures Ijelow '150°C. BKOCH, in reducing RKGXAULT'S results, assumed a panilx>lic formula of the same type as formula (8) for the mean coefficient. He also introduced a correction for the conduction of heat along the cross tubes, which were not quite horizontal in 1 1 KUNAULT'S fourth series of observations, in order to reconcile the results of the fourth series with those of the first three. The formula deduced by BROCH was as follows : — n<*i = {I817920+175(f/100) + 35ir6(?/100)s} x lO'10. VOL. COXI. — A. E I PROF. HUGH L. CALLENDAR AND MR. HERBERT MOSS CHAPPUIS gave a formula of a similar type, to represent the results of his observations by the weight thermometer method between 0° C. and 100 U. •' = {l816904-l-295r266(//100)+11456-2(?/100)3} x 10' Thia formula has been extrapolated by EUMORFOPOULOS ('Boy. Soc. Proc.,' A, vol. 81, 1 1. :»S9, 1908), but extrapolation in such a case would be somewhat unreliable. The'following table gives a short comparison of the above formulae with formula (8), •bowing tin- values of the mean coefficient multiplied by 109, together with the differences from formula (8) : — TABLE II. — Comparison of Formulae. Temperature. CAI.I.KNDAR and Moss (8). BROCH. Difference. CHAPPUIS. Difference. REGNAULT. Difference. *0. 40 181094 181855 + 761 181755 + 661 180025 -1069 100 182054 182161 + 107 182541 + 487 181530 - 524 140 182795 182506 - 289 (183323 + 528) 182536 - 259 200 184060 183232 - 828 (185683 + 1623) 184055 5 240 185004 183857 -1147 (187591 + 2587) 185063 + 59 300 186574 185005 -1569 (191116 + 4542) 186577 + 3 A comparison of these differences with those given in Table I. on p. 24 in terms of the same unit, illustrates the state of uncertainty which existed with regard to the expansion of mercury in the year 1907, and may be taken as sufficient excuse for the publication of the present work. A similar comparison is shown graphically iu a slightly different manner by the curves in fig. 8. Since it would be impossible to plot the expansion itself graphically on an adequate scale, even by the copper-plate method employed by REGNAULT, the quantity plotted in fig. 8 is the difference of the expansion from lineality, or the difference (a-0'000182054) of the mean coefficient a from the fundamental coefficient multiplied by t. The heavy line with the large circles and crosses © represents the results of the present series of observations. The deviations from the curve on this scale scarcely exceed the thickness of the line. The dots surrounded by small circles 3 ivpresent RBOXAULT'S actual observations. The broken lines represent the formulas of HK..NAI I.T, WuLLNER, and BROCH. It is evident that the curve representing our ' results also represents REUNAULT'S observations, as reduced by himself, much better than tli,.,y are represented by any of the other three formulae. The difference between the curves given by REGNAULT and WULLNER arises chiefly from the uncertainty alluded to on p. 9 in reducing REQNAULT'S observations to 0° C. The great Aion of BROCH'S curve from the others at high temperatures appears to arise Chiefly from the correction which he introduced in the endeavour to reconcile ON THE ABSOLUTE EXPANSION OF MERCURY. 27 REGNAULT'S fourth series with the first three. This correction produces a much larger deviation than tin- original discrepancy. It must he admitted, however, that a deviation of the type assumed by BROCH was quite possible so far as the evidence of •ooo^o II > > •ooota eqnoju. t Wall. Mon curve cwrve 300 Fig. 8. the observations went. It is satisfactory to find that the error was not so serious as lie supposed, and that the results of the present investigation are in such good general agreement with REGNAULT'S work. E 2 PROF. HUGH L. CALLENDAK AND MR HERBERT MOSS 11. (uri-i'i-tad Value of the Boiling-Point of Sulphur. The preliminary results published in the " Note on the Boiling-Point of Sulphur " in September, 1909, were affected by a small error in the fundamental interval, which at that time was uncertain, as the observations of Series III. had not then been taken. As pointed out in the note in question, a small error of this type is practically without effect on the result, owing to the manner in which the fundamental coefficient enters int-> (lit- formula for the correction of the gas thermometer. The final value of the lx>iling-j)oint of sulphur given in the note is not changed by more than 0°'01 C. by tlic error in the fundamental interval itself. Unfortunately the corresponding error in the coefficient b, though much smaller, produces a larger error in the result, namely, 0°'06 C., but this is still within the limits of error of the gas thermometer. The following are the corrected values : — The final corrected values of the ratios of the densities of mercury at 0° C., 100° C., and 184° C., given by formula (8), are as follows :— D0/D100 = 1-0182054, D0/D184 1-0338016. The observations taken with the weight thermometer in March, 1900, as reduced by EUMORFOPOULOS, assuming BROCK'S reduction of REGNAULT'S observations, gave the following values of the coefficients expressing the expansion of the bulb :— « = 2387xlO-8, 6 = 0-42 x lO'8. Our final corrected values of the expansion of mercury give the following : _ « = 2377x10-", b .-= T37 x 10~8. The correction to be added to the results of EUMORFOPOULOS for the boiling-point of sulphur, calculated by the formula given in the previous note, is dt= +1°-03 C. in place of dt == + 0°'97 C., as previously found by the preliminary reduction. tly speaking, thjs correction applies only to the gas-thermometer observations th the same bulb as that used for the weight-thermometer determinations. the bo,ling-point of sulphur found with this particular bulb in March, The addition of the above correction would raise this result The final mean obtained by EUMORKOFOULOS from observations with be, of winch the expansion was not directly determined, was t = 443°'58 C <>uld ra.se the corrected value of the boiling-point to t = 444-61 C. But since were not treated in exactly the same manner as the first bulb, it is I". Creator we.ght should be attached to the first result. The uncertainty •nnometar determinations at this point is of the order of 0°'l C and ON THE ABSOLUTi; KXI'ANSION OF MERCURY. 29 then- docs not si-fin to In- any sufficient reason for changing the final value of the boiling-point of sulphur on the scale of the constant-pressure air or nitrogen thermometer from that given in the previous note and assumed for so many years, nrmiely, t = 444°'53 C. 12. Explanation of the Tables of Expansion. The accompanying tables of the expansion of mercury from — 30° C. to 309" C., together with the table of differences on the opposite page, make it easy to calculate the expansion from 0° C. to any other temperature within the given limits. If one of the limits be not 0° C., the volume at each limit must be found, and the difference taken. The following examples will make the use of the tables clear :— (1) To find the expansion from 0° C. to 221°145 C.- Expansion from 0° C. to 221° C 0'0407846 Difference for 0°'l C. at 220° C 190 „ 0°'04 C. „ 220° C 76 „ 0°'005 C. „ 220° C 10 Expansion from 0° C. to 221°'145 C 0 '0408 122 The values found in this way from the tables will, in general, be correct to 1 in the last figure, or 0°'001 C., as given by formula (8). (2) To find the expansion from -10°'450 C. to -1- IG°'109 C.- Volumeat-10°C ........... 0'9981957 Difference for -0° '4 C. at-10°C ..... 722 „ -0°'050C. „ -10° C ..... - _ 90 Volume at -10° '450 C .......... 0*9981145 Volume at + 16° C ........... 1 "0028922 Difference for 0°'l C. at 16° C. „ 0°'009C. „ 16° C ...... + Volume at +16-1090 Volume at -10-4500 .......... 0'9981U5 Expansion between limits ...... 30 £ rc. 30 -20 10 0 10 20 30 40 50 60 70 80 90 PBOF. HUGH L. CALLENDAR AND MR. HERBERT MOSS TABLE HI.— Expansion of Mercury from -30° C. to 309° C. V + 1805553^) 10-s+12444 ( Degrees. •99 45939 47738 49537 51336 53135 63937 65738 67539 69340 71142 81957 83760 85563 87367 89171 1-0000000 01806 03612 05418 07224 18068 19876 21685 23494 25303 36163 37974 39785 41597 43409 54285 56099 57913 59728 61543 72437 74254 76072 77889 79707 90621 92441 94261 96082 97903 1-01 08836 10659 12483 14307 16132 27086 28913 30740 32567 34395 45371 47201 49032 50863 52695 63693 65527 67362 69197 71032 54935 56735 58535 60335 62136 72944 74746 76548 78351 80154 90975 92780 94584 96389 98195 09031 10838 12645 14452 16260 27112 28922 30732 32542 34352 45221 47033 48846 50659 52472 63358 65173 66989 68805 70621 81525 83344 85162 86981 88801 99724 01546 03368 05190 07013 17957 19782 21607 23433 25259 36224 38052 39881 41711 43541 54527 56359 58192 60025 61859 72868 74705 76541 78378 80216 cS I <°C. 30 20 10 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 » 82054 83892 85730 87570 89409 1-0200455 02297 04140 05983 07826 18897 20744 22591 24438 26286 37383 39234 41085 42937 44789 55913 57769 59625 61481 63338 91249 93089 94930 96771 98613 09670 11515 13360 15205 17051 28134 29983 31832 33682 35532 46642 48495 50349 52203 54058 65196 67054 68912 70771 72630 74490 76350 78211 80072 81934 93114 94979 96845 98711 00578 1-03 11788 13658 15529 17400 19272 30512 32387 34263 36140 38016 49289 51169 53050 54932 56815 83796 85659 87522 89835 91249 02445 04312 06180 08049 09918 21144 23016 24889 26763 28637 39894 41772 43650 45529 47409 58697 60580 62464 64349 66234 68119 70005 71892 73779 75667 87005 88897 90789 92682 94576 1-04 05948 07846 09744 11642 13541 24949 26853 28757 30661 32566 44010 45920 47830 49741 51652 77555 79444 81334 83224 85114 *96470 98364 00259 02155 04051 15441 17342 19243 21144 23046 34472 36379 38286 40193 42101 53564 55476 57390 59301 61218 63133 65049 66965 68882 70799 82318 84240 86163 88087 90011 1-0501569 03497 05426 07356 09287 20885 22820 24756 26692 mso 40268 42210 44153 46097 48041 59721 61670 63620 65570 67522 72718 74637 76556 78476 80397 91935 93861 95787 97713 99641 11218 13150 15083 17016 18950 30568 32506 34446 36386 38327 49986 51931 53878 55825 57772 69474 71426 73380 75334 77289 ON THE ARSOLUTE EXPANSION OF MERCURY. TABLE.' of Differences for Fractions of 1° C. 31 fj Differences for tenths of 1° C. x 107. j s m 2 g pl & § Tenths of a°C. I H H rc. •1 '2 •3 •4 •5 •6 •7 •8 •9 rc. -30 180 360 540 720 900 1080 1260 1440 1620 -30 -20 180 360 541 721 901 1081 1261 1442 1622 -20 -10 180 361 541 722 902 1083 1263 1443 1624 -10 0 1S1 361 542 723 903 1084 1265 1445 1626 0 10 181 362 543 724 905 1086 1267 1448 1629 10 20 181 362 544 725 906 1087 1269 1450 1631 20 30 182 363 545 726 908 1089 1271 1452 1634 30 40 182 364 546 727 909 1091 1273 1455 1636 40 50 182 364 546 729 911 1093 1275 1457 1639 50 60 182 365 547 730 912 1095 1277 1460 1642 60 70 183 366 549 731 914 1097 1280 1463 1646 70 80 183 366 550 733 916 1099 1283 1466 1649 80 90 184 367 551 734 918 1102 1285 1469 1652 90 100 184 368 552 736 920 1104 1288 1472 1656 100 110 184 369 553 738 922 1107 1291 1475 1660 110 120 185 370 555 739 924 1109 1294 1479 1664 120 130 185 371 556 741 926 1112 1297 1482 1668 130 140 186 372 557 743 929 1115 1300 1486 1672 140 150 186 372 559 745 931 1117 1304 1490 1676 150 160 187 373 560 747 934 1120 1307 1494 1681 160 170 187 374 562 749 936 1123 1311 1498 16B5 170 180 188 376 563 751 939 1127 1314 1502 1690 180 190 188 377 565 753 941 1130 1318 1506 1695 190 200 189 378 567 755 944 1133 1322 1511 1700 200 210 189 379 568 758 947 1137 1326 1515 1705 210 220 190 380 570 760 950 1140 1330 1520 1710 220 230 191 381 572 762 953 1144 1334 1525 1715 230 240 191 382 574 7"65 956 1147 1339 1530 1721 240 250 192 384 576 767 959 1151 1343 1535 1727 250 260 192 385 577 770 962 1155 1347 1540 1732 260 270 193 386 579 773 966 1159 1352 1545 1738 270 280 194 388 581 775 969 1163 1357 1551 1744 280 290 195 389 584 778 973 1167 1362 1556 1751 290 300 195 390 586 781 976 1171 1367 1562 1757 300 Mean coefficient of expansion between 0°C. and/'C. •000180205 0317 0433 0555 0682 0814 0951 1094 1241 1393 1551 1713 1881 2054 2231 2414 2602 2795 2993 3196 3405 3618 S83fl 4060 4288 4522 4761 5004 5359 5507 5766 6030 6299 6574 32 PROF. CALLENDAR AND MR. MOSS ON EXPANSION OF MERCURY. The actual coefficient at any temperature is seldom required with a high order of accuracy. It may be obtained from the tables with sufficient accuracy by taking the difference of the volumes for a range of 5° C. on either side of the point where the coefficient is required, and dividing by 10. E.y., to find the coefficient at 300° C., we have Volume at 305° C. = 1 '0569474, „ 295° C. = 1-0549986, Difference/10 = Coefficient of expansion at 300° C. = 0'00019488. [Note added Fel>i->i«rii 13, 1911.. — It should be observed that the expansion of iiiiTcury is here expressed in terms of the scale of temperature, based on the platinum resistance thermometer, proposed by CALLENDAR ('Phil. Mag.,' December, 1899, p. 519) at the meeting of the British Association at Dover. This scale assumes the formula given on p. 7 alxrve for reducing the readings of a platinum thermometer to the gas-scale, and is equivalent to assuming the value 444°'53 C. for the boiling-point of sulphur. It was admitted that this value might require a correction between + 0°'3 C. and 0°'5 C. to reduce it to the absolute scale, but, as this correction depended on the extrapolation of experiments between 0° C. and 100° C., it was considered inadvisable to alter the existing standard scale of platinum thermometry until further experiments had been made with helium and argon at high tempera- tures. Many writers now adopt values ranging from 444 '8 to 445 '0 for the boiling- point of sulphur. This may lead to some confusion unless a definite convention is established. Until the correction to the absolute scale has been determined with greater precision it would be preferable to retain the old scale.] II. The Kffect oj Pressure upon Arc Spectra. No. 3.— Silver, X 4000 to X 4600. No. t.—Gold. By W. GEOFFREY DUFFIELD, D.Sc., Honorary Research Felloiv in Physics in the University of Manchester. Communicated by ARTHUR SCHUSTER, F.R.S. No. 3.— Silver, X 4000 to X4600. [PLATE l.] (Received January 9, —Read February 4, 1909.) CONTENTS. 1. Preliminary 2. The apparatus 3. The behaviour of the silver arc under high pressures 4. The photographs (1) Method of exposure (2) Description of the plates •• 5. The broadening of the lines 6. The structural character of the wings of some broadened lines . . 7. The displacement of the lines (1) Method of measurement (2) Table of the displacements (3) The relation between the pressure and the displacement . 8. Changes in relative intensity under pressure 9. Series of lines in the silver spectrum 10. The banded spectrum produced under pressure (1) The structure of the banded spectrum . (2) The broadening of the bands (3) Table of wave-lengths and intensities of bands . (4) Relative intensities under pressure 11. The continuous spectrum 12. Influence of the electrical conditions upon the nature of the spectrum Origin of the banded and continuous spectra. . . Summary of results. (See ' Roy. Soc. Proc.,' A, vol. 84, p. 118, 1910.) VOL. CCXI. — A 472. 20.3.11 34 DK. W. GEOFFREY DUFFIELD ON THE , IWmtmwy-THK effect of pressure upon arc spectra was first investigated by HUMPHRY and MOHLER,* and later by HUMPHREY^ They found that, m general the lines broadened, were diaplaced towards the red end of the spectrum, and showed a greater tendency towards reversal. HUMPHREYS' work has dealt more fully with metals other than silver, of which the following table comprises all previous measure- inents .under pressure : - A. 8 atmospheres. 9-75 atmospheres. 12-5 atmospheres. 13 atmospheres. 3280-80 3383-00 0-029 0-028 0-034 0-032 0-027 0-032 The displacements are in Angstrom units. The following investigation was undertaken for the purpose of extending the work to higher pressures. The spectrum of the silver arc has been photographed under pressures ranging from 1 to 201 atmospheres in the region X = 4000 to X = 4600 A.U. 2. The Apparatus. — The arc was formed between silver poles, diameter f inch, within the Pressure Cylinder (designed by Dr. PETAVEL, F.R.S.), which had previously been used for the investigation of the effect of pressure upon the Iron Arc,} and the Copper Arc. § The light passed through the window in the side of the steel chamber, and was reflected by the mirror system (previously described), which enabled the image of the arc, which was very unsteady at high pressures, to be continually focussed upon the slit of the 21^-foot Rowland Grating Spectroscope in the Physical Laboratory of the Manchester University. The Second Order Spectrum was employed, the dispersion being 1'3 A.U. per 1 mm. An increase in pressure was obtained by the admission of air into the cylinder from a gasholder, suitable valves and gauges being interposed. The arc was fed by current from the Corporation mains, which supplied 100 volts continuous, and this was reduced to about 50 at the terminals. 3. The Behaviour of the Silver Arc under High Pressures. — The continuous current arc between silver poles in air at atmospheric pressure was maintained without difficulty until the poles got thoroughly hot, when the convection currents became so violent that they frequently blew it out. The striking of this arc was more easily accomplished than in the case of iron or copper, where the formation of a non-conducting oxide necessitated breaking through this layer before the current could pass. * HUMPHREYS, ' Astrophys. Journ.,' VI., p. 169, 1897. t HUMPHREYS, ' Astrophys. Journ.,' XXVI., p. 18, 1907. t W. G. DUFFIELD, 'Phil. Trans.,' A, vol. 208, p. Ill, 1908. § W. G. DITJ-FISLD, 'Phil. Trans.,' A, vol. 209, p. 205, 1908. EFFECT OF PRESSURE UPON ARC SPECTRA.— SILVER. 35 The ease with which the arc burned at any pressure seemed dependent upon the amount of air present ; when the cylinder was freshly filled the arc burned steadily for some time, though not often for longer than one minute, without requiring the poles to be brought together, but later on the arc required more attention and frequent re-striking ; if, however, the gases were allowed to escape from the cylinder and fresh air introduced, steadiness was again obtained. A plentiful supply of air thus appears to be necessary for steady running, but the cooling of the poles is also a factor that makes for success. Not only did the steadiness of the arc decrease at any particular pressure as time advanced, but also the brightness, the brilliance of the image on the jaws of th'e slit gradually waning as the arc burned, until a fresh supply of air replaced the old. And that the readiness with which air has access to the poles is connected with the brightness is further borne out by the increased brightness as the pressure is increased, provided that arcs newly supplied with air are compared ; for instance, the arc under a pressure of 200 atmospheres, when first struck, is very much brighter than the arc under a pressure of 50 atmospheres when first struck. This indicates also that the temperature of the arc under high pressure is very much greater than it is at normal atmospheric pressure. Photometric measurements of the intensity of the light emitted under different pressures have been attempted, but the intermittent nature of the arc has not permitted accurate determinations. Distinct changes were observed in the colour of the arc as the pressure was increased ; at low pressures it maintained its characteristic greenish appearance, but at the highest pressure reached, 200 atmospheres, it was, when the air was fresh, as white as a cart>on arc ; at lower pressures, or when the arc had been burning for some time, the greenish colour returned, and my assistant pointed out to me two distinct tints besides the pure white, namely, yellowish-green and green ; sometimes as the arc flickered about the poles in an irregular manner these would appear in rapid succession, possibly dependent on the varying length of the arc gap as it moved from point to point on the electrodes. The silver contained a trace of lead, which is characterised by the line at 4058 "04, and on one or two of the photographs at atmospheric pressure the cyanogen band at 3883 is visible and is due to the silver having been melted in a carbon reducing atmosphere. As is usually the case when metallic arcs are burned in air under pressure, some nitric acid was formed within the cylinder, and the air issuing from it gave the characteristic smell of nitrogen peroxide. The appearance of some finely divided particles of iron in the window-tube after the arc had been run for some time at high pressures needs explanation, but it is considered to have either been introduced with the air, which was stored in an iron cylinder, or to have scaled from the steel rods which carried the silver electrodes, though this must have been greatly obviated by the discs of asbestos which were strung on the silver poles and were a good fit in the cylinder. F 2 I)R W. OKOPFREY DlTFFIELD ON THE The arc was more easily maintained between 50 and 200 atmospheres pressure than between 1 and 50 atmospheres. Of all the photographs taken, that at 200 atmospheres gave least trouble, the arc burning steadily for several seconds with dazzling brightness. Extension to higher pressures would be quite possible if the difficulties attendant upon the fracture of the glass and quartz windows could be overcome. As has already been described* in connection with experiments made with the copper arc, the straining of the windows resulted in splinters of glass breaking off and ruining the surfaces, and sometimes in the complete breakdown of the window. The present pressure cylinder cannot be safely used at higher pressures, but a similar but stronger one should present no great difficulties. The running of the arc itself is a simple matter. 4. The Photographs: (1) Method of Exposure. — As in the previous work with the iron and copper arcs, the comparison spectrum under atmospheric pressure was photographed in the central strip of a plate (20 inches long by 2j inches broad) with the spectrum under pressure above and below it. To ensure that no accidental displacements were produced the comparison spectrum was photographed before and after the one under pressure. The arc was operated by the writer and the mirrors by an assistant. The following photographs have been obtained : — Atmospheres. No. of photographs. Atmospheres. No. of photographs. 5 1 60 1 10 1 75 1 20 2 80 1 25 1 100 1 40 1 120 1 60 1 200 1 Plates : Imperial Flashlight. Developer Imperial Pyro-Metol Standard. Exposure 0 minutes at 5 atmospheres to 5 minutes at 200 atmospheres; but this ;tle indlCation of the relative intensities because the width of the slit was different in nearly every exposure. (2) Description of the P/o^.-Pkte 1 illustrates the behaviour of the silver arc Ferent pressures. The plate includes the region X = 4020 to X = 4320 .tographs which are full-size positive reproductions of the originals •n order of increasing pressure from the top at 1 atmosphere to the ) atmospheres, the central strip of. each being at normal atmospheric pressure. 1 , A, EFFECT OF PRESSURE UPON ARC SPECTRA.— SILVER. 37 To facilitate reference to the lines arbitrary letters have been assigned to them, beginning alphabetically at the more refrangible end. The prominent features are :— (1) The broadening of the lines ; (2) Their displacement towards the red end of the spectrum ; (3) The structure which Incomes apparent in the wings of the strong lines under pressure ; (4) The gradual disappearance of the line spectrum ; (5) Its replacement by a banded spectrum ; (6) The development of the banded spectrum into an almost continuous spectrum. The lines seen on the photographs are :— a 4055 '44 1st sub-series. [6 . . .' . . 4058-04 Lead.] c 4212'! 1st sub-series. d 4311-28 e 4476-29 2nd sub-series. (a) 4055-44. Silver. 1st subordinate series. At normal atmospheric pressure it is broad and covers about 13 A.U. It is unsymmetrically reversed, its reversed portion being slightly on the violet side of the centre of the bright line. At 5 atmospheres* the line spreads over 17 A.U. or more, and is so broad that it might almost be called a band. The reversed part is badly defined, and its centre cannot be determined. Its wings now present a marked structure, and are resolvable into a number of well-defined lines, in which there is no obvious regularity. The shadings on the two wings are different from one another. At 20 atmospheres the line has spread out over a greater range and can be distin- guished on the red side as far as 24 A.U. from its original position. The structural appearance, which extends as far as the wings are visible, is well marked, though the individual lines are broader. The broadening is much greater on the red side. At 25 atmospheres the line spectrum of silver has almost disappeared and given place to a banded spectrum. Between 25 and 200 atmospheres, this line is indistinguishable against this background of banded or continuous spectrum. (b) 4058-04. Lead? At normal atmospheric pressure this line is fine and sharp and apparently superposed upon the less refrangible wing of line a (4055'44). From 5 to 25 * Throughout this paper, unless expressly stated to the contrary, the pressure is the excess above 1 atmosphere. gg DR. W. GEOFFREY DUFFIELD ON THE ataMVto*. it remains a definite line, but has broadened (about 0*5 A.U at 25 Atmospheres) and is displaced. At 80, 100, 120, and 200 atmospheres the photographs show no trace of its presence. (<•) 4212*1. Silver. 1st subordinate series. At normal atmospheric pressure it is broad and covers 16 to 20 A.U. (according to the amount of exposure). It is asymmetrically reversed, its reversed portion being slightly on the red side of the centre of the bright line, thus differing from a (4055-44), the other member of the 1st subordinate series. At 5 and 20 atmospheres its behaviour strongly resembles that of line a ; it has broadened and is distinguishable 60 A.U. from its original position ; the reversal which is just visible is displaced towards the red end of the spectrum. The wings are shaded into a number of lines, which are, however, differently spaced from those into which line a, is resolved. Bettveen 25 and 200 atmospheres, this line cannot be distinguished from the banded or continuous spectrum which in this region takes the place of the line spectrum of silver. (d) 431T28. Non-series line.* At normal atmospheric pressure the line is fairly sharp, being only slightly broadened towards the red. At 5 and 20 atmospheres it is seen to be slightly broadened towards the red, though it is still comparatively sharp ; its displacement towards the same end of the spectrum is also obvious. At 25 atmospheres the line appears to be the violet head of a band stretching some distance towards the red. At 60 and 80 atmospheres the line at ordinary atmospheric pressure seems to mark the violet edge of the band, which becomes less sharp and less obvious as the pressure is increased. It is now doubtful if the band to the right of the line has any causal connection with it. Between 100 and 200 atmospheres, this band has become submerged in the continuous spectrum that now dominates the photograph. . (e) 4476-29. Member of the 2nd subordinate series. At normal atmospheric pressure it is broadened towards the red. At 5 and 20 atmospheres its width has increased and it has suffered displacement towards the red end of the spectrum. At 25 atmospheres it can still be distinguished, though its intensity has been greatly reduced. Its displacement, though large, is difficult to At 60, 80, and 100 atmospheres no sign of this line appears, though, if any radiation of a period close to its original period existed, it should be visible, because the here were three slit* in the comparison shutter where this line occurs, so that three small pieces of c pressure appeared upon the photograph. Two of them may be seen on the 0 atmospheres. They permit the comparison line to be easily placed parallel to the iMe crow-wires in the measuring machine. EFFECT OF I'KKSSURE UPON ARC SPECTRA.- SILVER. 39 nebulous flutings do not occupy this particular part of the plate. At 200 atmosphere* tlic spectrum is, as in other parts of the plate, nearly a continuous one. 5. The broadening of the Lines. — From the photographs we learn the following facts within the region X 4000 to X 4600 : — 1 . All silver lines broaden under pressure. 2. The broadening increases with the pressure, but different amounts of exposure make it difficult to determine if the increase is continuous and linear with the pressure. 3. The broadening of the two 1st sub-series lines is, at normal atmospheric pressure, unsymmetrical relatively to the superposed absorption line ; increase of pressure increases the width of these lines, and their wings are then seen to consist of numbers of fine lines which merge into one another at higher pressure. 4. The broadening of the 2nd sub-series line (4476 '29) at normal atmospheric pressure is unsymmetrical, being greater on the red side. Under pressure this is more pronounced, but the line remains definite though its intensity diminishes. 5. The broadening of the non-series line is distinctive because the line appears to become the violet edge of a band which stretches further towards the red as the pressure is increased, but at the highest pressure the violet edge has lost its character as the head of a band. 6. The distinction that was observed in the same region of the copper spectrum is here also apparent : — the 1st sub-series lines become hazy under pressure, resembling hazy bands, and finally leaving only a cloudy banded appearance on the plate ; the 2nd sub-series lines remain definite lines without abnormal broadening, though they ultimately disappear through a gradual weakening process. 7. No relation has been found between the original intensity of a line and its width under pressure. 8. The magnitude of the broadening of the 1st sub-series lines is at 20 atmospheres as great as 120 A.U. 6. The Structural Character of the Wings of the Members of the 1st Sub- Series under Pressure. — The photographs of lines a and c at 5 and 20 atmospheres show that the wings of those lines are of complex structure, being composed of a number of closely packed lines which are comparatively fine at 5 atmospheres, and, though broader at 20 atmospheres and somewhat merged into one another, are still recognisable as the same lines. There is no obvious regularity in the shadings on the wings, which are dissimilar for the two lines and also for the two wings of the same line. As the pressure increases, the wings of the lines extend outwards, and at 20 atmospheres some of the patches of light are separated from the original lines, forming what is ultimately a band spectrum. On one photograph taken at each of the pressures 10 and 20 atmospheres the 40 DR. W. GEOFFREY DUFFIELD ON THE structure is not apparent, either because it is not invariably present in the arc or because the slit was too wide. Upon this photograph the reversals were rather better defined than upon the reproductions. 7. The DiitjiliKviiH'iit of the Lines: (l) Method of Measurement of the Photo- ffraphs.—llie Kayser Measuring Machine was used, the setting being always made between parallel threads as accurately as possible upon the most intense portions of the lines under pressure, and advantage was taken of the astigmatic property of the grating of narrowing a line at its extremities. Thirty-six settings were made upon each line on each plate, 18 with the plate in one position and 18 with it in the reversed position. When there was not good agreement between the readings this number was exceeded. (2) Table of the Displacements. — The first column contains the arbitrary letters assigned to the lines to facilitate reference to them. The second column gives the wave-lengths according to KAYSER and RUNGE'S tables. The subsequent columns give the displacements at the corresponding pressures, the use of italic figures, e.g., 0'220, for displacements, indicates that the line is reversed. An asterisk (*) denotes that the line is reversed but that measurements could not be satisfactorily made at that pressure. In all cases the displacements are towards the side of greater wave-length. The pressures are the excess above 1 atmosphere. Figures in brackets indicate that the measurements were made with difficulty. TABLE I. 5. 10. 20. 25. 30. 40. 80. 100. 200. a P e d t 4055-44 4058-04 4212-1 4311-28 4476-29 * 0-030 0-087 0-099 0-141 0-220 0-052 0-209 0-128 0-220 (0-444) 0-080 0-233 0-188 0-332 (ob (0-120) (ob (0-410) (0-558) [iterated (0-145) lite rated (0-433) (0-650) on all on all (0-838) photogr photogr (oblit (oblit aphs aphs erated crated ]Pb. 1 [This table has been revised.— December, 1910.} (3) The Relation between the Pressure and the Displacement.— From the infor- s supplied by these measurements it is gathered that an approximately .ear relat.onsh.p holds between the pressure and the displacement, as it does in the and copper but the rates of displacement with the pressure are not .tmgu.shab , The higher rate belongs to the series lines a, c, and e, which apkced more than the non-series line d. 'I.;,,,; ,„ /Ma(,W Intemity under Pre>mre.~lt has already been pointed for hrc^dened .,»«, the - intensity • is an indefinite term. If the intensity EFFECT OF PRESSURE UPON ARC SPECTRA.- SILVER. 41 per unit of area of the plate be considered, it seems true that the members of the 1st sub-series lines a and c, Plate 1, have diminished in intensity relatively to the other two silver lines. Though if the whole intensity of the radiation in the region of the plate occupied by the broadened lines be considered, the relative strengths of the lines are very difficult to determine. Under pressure, all the lines in this part of the spectrum become obliterated. It may be here remarked that the complete obliteration recorded by the photographs with the large Rowland grating are not entirely confirmed by photographs with a 1 m. grating. These latter sometimes show faint indications of lines under high pressures where none are apparent in those taken by the 2l£ feet instrument. This is due to the fact that the lines do appear in the spectrum under pressure at the moment of extinction of the arc, and possibly at the moment of striking it (c/. p. 44). With large dispersion the duration of these lines is insignificant compared with the total length of exposure necessary to affect the plate, but when the dispersion is smaller the exposure is less, and the interval during which the lines shine out is comparable with the total exposure necessary. The lines to vanish first are those of the 1st subordinate series (a, X = 4055'44 and c, X = 4212 '!), which are of great intensity at normal pressure. In this respect, and also because they become dissipated before they are obliterated, their behaviour is very similar to the 1st subordinate series lines in the corresponding region of the copper spectrum, 4022 and 4063. The line of the 2nd subordinate series (e, \ = 4476'29) weakens as the pressure is increased, and is only just visible at 40 atmospheres. It has disappeared at 60 atmospheres. The manner of its disappearance allies it to the two lines in the copper spectrum, also members of the 2nd subordinate series (4480'6 and 4531 "0), which also become weaker as the pressure is increased, but without abnormal widening. The non-series line (d, X = 4311 "28) is clearly seen at 25 atmospheres, and at 40 atmospheres it seems to be the violet head of a band, but it is very doubtful line' and the band are in any way causally related-photographs at higher pressure suggest that the band is independent of this line, which has at 60 atmospheres cea to°exist. This non-series line differs from the non-series copper lines examined; the latter have not been found to undergo complete obliteration, t the " sharp" series are weakened as the pressure is increased. 9. Series of Lines in the Silver Spectrum.-The classification of the lines of the silver spectrum has been attempted by the same methods that have for the copper spectrum. KAYSEB and RUNGE'S* classification is based upon the relationships between frequencies of the lines, and the series have become known as the * KAYSEU and RUNGK, ' Uber die Spectren der Elemente,' vol. V., 1892. VOL. CCXI. — A. ° 42 DR W. GEOFFREY DUFFIELD ON THE and Second Subordinate Series, according to the particular formula which includes all its members. The Zeeman effect, which has been examined by RUNGE and PASCHEN,* shows distinctive behaviour for certain lines, and should afford additional data for their classification, but unfortunately only one of the lines dealt with in this paper has been investigated by their method. Pressure is also effective in bringing to light characteristic differences between groups of lines, and studies of their behaviour as regards displacement, broadening, changes in relative intensity, and reversal have been made with this end in view. For silver this method does not yield such definite results as were obtained for the copper spectrum, but they are nevertheless of considerable assistance in this classifi- cation. The following table summarizes the methods of forming series in the silver spectrum : — Magnetic field. Pressure effect. Displace- Frequency relationship, K. and R. RUNGK and PASCHEN'S separation, a multiple of— Broadening under pressure. Reversal under pressure. t Change in intensity under pressure. 1 atmosphere (omitting readings at 5 atmo- spheres). a 4055-44 1st sub-series _ Very broad. Reversed f Obliterated ~) 0-022? Structure first, apparent in < though V e 4212-1 1st sub-series — wings. Resembles Reversed originally [ strongest J 0-021? band. d e 4311-28 4476-29 2nd sub-series 0-92 Moderate Moderate Non-reversed Non-reversed {Less T easily ^ obliterated J 0-009 0-017 Fhough the above table shows great resemblances under pressure between the ehaviour of a and c, they differ at 1 atmosphere in the manner in which the sorption line is superposed upon the bright line, and the photographs show that rings of the two differ greatly in structure ; this difference was to have been from the differences shown by them in a magnetic field. No such les under pressure were, however, discovered for the corresponding copper lines 4022, 4063. 10. The Band Spectrum under Prmwre.-The prominent feature of these • RUNGK and PASCHEN, « Astrophys. Journ,' vol. XVI., p. 123 1902 disappear at almost the same pressure, the 2nd sub-series line has been weakened the non-senes line because its original intensity was greater. EFFECT OF PRESSURE UPON ARC SPECTRA.— SILVER. 43 experiments is the replacement of the line spectrum of silver by a banded spectrum when the pressure is large. A necessary condition for the production of this new spectrum is that the arc burns steadily, because, as has just been pointed out, at the moments of extinction some of the lines have been observed to flash out instan- taneously, even at 100 atmospheres. On the negatives the bands are not always easily distinguishable, but greater contrast has been obtained by using a slow Velox paper, which enables the detail to be more carefully studied. (1) The Structure of the Banded Spectrum. — The photographs show a number 01 bright isolated patches or bands, separated by dark intervals. The bright bands are symmetrical about their centres, and are generally very broad, fuzzy and ill-defined, but a few are as narrow as 2 or 3 A. U. There is no easily recognisable regularity in the spacings of these patches of light, or bands, and their intensities differ considerably and show no dependence upon their widths. The resemblance of this spectrum to an absorption spectrum is discussed under a separate heading, p. 48. If the bands be observed visually they present a novel appearance. In place of the steady spectrum which is observed with other metals, there is a constant flickering of the spectrum and waves seem to pass across it, and the bands appear more like tongues of coloured flames which are violently disturbed by a breeze. Horizontal striae crossing the spectrum were also frequently noticed. The whole phenomenon is thought to be due to violent convection currents in the vapour surrounding the arc, which, being of varying density, cause continually changing amounts of refraction of the rays which pass through it on their way to the window. The constant play of shadows over the inner surface of the window, which is easily visible because of the fine deposit upon it, indicates that some such action as this is in progress, and the striae may be due to shadows falling upon the slit. The Behaviour of the Banded Spectrum under Pressure. — The bands first appear on the photographs at 20 atmospheres, where many are resolvable into a number of very fine lines, but, as it was thought possible that longer exposure at a lower pressure might show a trace of them, the normal silver arc was photographed with an exposure ten times longer than usual, and with a very wide slit (£ mm.) ; but there was no sign of bands in this experiment though the slit included both poles and the centre of the arc. (2) Broadening. — As the pressure increases the bands strengthen and become a more definite phenomenon; many grow broader (cf. 45697, which increases from 8 A. U. at 25 atmospheres to 13 A.U. at 80 atmospheres and 26 A.U. at 100 atmo- spheres), and several, which are separately distinguishable at 25 atmospheres, become merged into one another at higher pressures. At 100 atmospheres the bands have become so broad that they almost constitute a continuous spectrum, though the characteristics of the banded spectrum are not entirely lost. o 2 44 DR W. GEOFFREY DUFFIELD ON THE A photograph of this part of the spectrum has been taken at 200 atmospheres, and here the spectrum looks still more continuous. The breadths of the bands at different pressures are given in Tab] II., p. 4 measurements are in Angstrom units. (3) Table of Wave-Lengths and Intensities of Bands. -This is given on pp. 45-47 (4) The Relative Intensities of the Bands under Pram*.— Though the intensities of the bright bands are increased by pressure, it does not produce any marked change in the relative intensities of adjacent bands. What is, however, noticeable is the gradual increase in the intensities of the bands that are distant from the strong lines a and c relatively to those in their immediate neighbourhood, e.g., band at 4162 has increased in intensity relatively to 4199 and 4215, which are near the line c at 4212. For other examples see Table II. There is an outward extension of the luminosity from these lines as the pressure increases. A line not shown on the photographs in the red (ride p. 50) seems also to be concerned in the production of the spectrum. 11. The Continuous Spectrum. — The highest pressure available for these experi- ments was 200 atmospheres, and one photograph was obtained at this pressure, but no bands are discernible against the background of continuous spectrum. Too much risk would be run by pushing these experiments further with the existing apparatus, but it would be of interest to observe if this continuous spectrum persists at still higher pressures and behaves like the continuous spectrum derived from black body radiation, i.e., if the maximum of intensity is displaced towards the region of short wave-lengths as the temperature increases, or if the energy remains localised in the centres of a hidden banded spectrum from which the continuous spectrum has been developed. 12. Influence of the Electrical Conditions upon the Banded and Continuous Spectra. — It has already been remarked that at the moment of extinction the bright lines have been seen to flash out and replace the bands. The electrical conditions accompanying the sudden breaking of the circuit occasion this change. Visual observations suggest that at such times the quantity of material taking part in the discharge is diminished ; for example, when the arc is burning steadily under pressure, one of the yellowish-green lines is broad, almost resembling a band, and is strongly reversed, yet at the moment when the arc is extinguished this line narrows down to the width of the reversal and flashes out as a bright line. By varying the voltage of the discharge, interesting information regarding the electrical conditions necessary for the production of the banded spectrum should be forthcoming. The Origin of the Banded and Continuous Spectra.— The resemblance of the banded to an absorption spectrum requires discussion ; hitherto it has been treated as a emission spectrum. It is necessary to inquire into the possibility of the bands mg the result of a continuous radiation from the hot poles of the silver and suffering absorption by the surrounding vapours. 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Q>. f^ d OO OQ oo toos^'^to^os Sb"555 1 8 |^ to os •"$* o ^ - -~ - --. •<£ ^4 ^4 •<(• •j ^ 9os^.QOoO>0-«' i? s 2 S »5 S 8 ^2" •** ^ "^ **(• ^*" r -«• (M CM OO ^ . -r it ' ' oo O eo O • 8 •§" •£ 8 8 8 " I J ** 5 5 * •* •* ^ •a o « *5 "-^ — S gf gc 5 £ B tmospheres r-*- IO CO IO OO IO tO IO •) to -4* os o t^» t^- **• J ^ J ^ J J | 2 2 g « 2 Z o o <-> o o o <3< os oo d* s< {/(«)} possesses an asymptotic expansion, valid for the same range of values of arg x as the expansion for f(x), with characteristics k, I, p», , (a0+ f ), when | £| S c be since this series is absolutely convergent when | £ | = c, we have I |A.|c", {/(«)} into the series provided |R,jSc. Since |R0|s E\x\ ~\ the expansion will be valid when |*|iy and E\x\~lSc. Substituting for R,, R,8 ..... R,- ^ in Theorem II., we get ...... (9) where rf. = /J0, U0 = S A.R,- and wlien n > 0 *=„?,*— *« U.= SA..A+ S A.R.-. m = » m = n+1 The quantities .6., .8. are the same as those which occur in equation (6). ", Po = P + &, where S is an arbitrary positive quantity as small as we please. Mil. G. N. WATSON: A THEORY OF ASYMPTOTIC SERII •> 293 Consequently, if n > 0, |4,|< 2 IA.IA-X— ^(foi+ijp- : < i \-IH(AX/t-)-r(A-n+l)/}" ....... (9A) m = 1 If AX ^ c, we have From this result, combined with (9x), we see that, if AX ^ c, we take pa, the inner radius of*, {f(x)}, to be the smaller of the quantities p, pA\/c. If AX = c, we have and if S > 0, we have up* < K, (p + 8)*, where K, is a finite quantity depending on p and 8, but not on n. That is to say, if AX = c, p + 8 is a possible radius of , and k is a possible grade. Further, HI = 1 10 = »-H _ _ _ ,11+1 « = «+i i|ar| I! Hfi-^l"1. Using the inequalities B, \J (•*")} • 6. Theorem IV. — Suppose that for a certain range of values of arg (x+a), f(x+a) possesses an asymptotic expansion in negative powers of x+a valid when |x+a|>y with coiixttiitt.* iiinl characteristics A, B, k, I, p, +^+ ... where a; 6. = a.-.-A . a . a.-i+.-iC, . a2 . a,_3- . . . MK. G. N. WATSON: A TIIKORY OF ASYMPTOTIC SKUIKS. 295 and S. = R. + (-)"a,Y1 + (-)'"1«3Y3+ ... -»„¥„ so that |&.|±=KI +.-iC,|o| |o..,| +.-,C,|a| | a,., | + ... ^ Now T{k(n-r)+l} s where V1 = '8856 ... if k < 1, T?, = 1 if *£ 1. Consequently IM^A^r^+iKp+H}"- ....... (IOA) Also ^Br(fa+l)a» VA n!|a|-' L Ja i|(n-r)!L And for the values of r under consideration (since k £ Z), Tlierefore ff Ll «r," ........... ." . (10B) From (IOA) and (10B) we see that k, I, ply o-, are characteristics of the expansion of /(a;+a) in descending powers of x. We have now proved all the theorems which seem to be of importance concerning asymptotic series in general. We proceed to discuss properties of analytic functions of which asymptotic expansions are given. PART II. — ANALYTIC FUNCTIONS DEFINED BY ASYMPTOTIC SERIES. 7. We first propose to consider the question of the uniqueness of an analytic function which is defined by means of an asymptotic expansion possessing given characteristics. The discussion will be based on the result of the following important lemma, of which we shall give a proof before proceeding further : — Lemma. — Let f(x) be a function of x which is analytic* in the sector defined by * Soe (iii.) on p. 280. ,,,,; MR. O. N. WATSON: A THEORY OF ASYMPTOTIC SERIES. x\*br+* where X> 0; and let the region in which f(x) is •• --«— x ...i.-~~ distance from the yc w »<* P*» <«"•• «• boundary of the sector does not exceed 2A, where A > 0. Le, x . fc Hri >/"", I*"*** «Ae «ctor and *« to the sector, then- exists an inequality of the form | denotes the nth differential coefficient of /. But on the contour |/(0|oint in an assigned region of non-zero area (not including the origin) and describes a closed circuit round the origin, the initial and final values of F( 0. Put y = 0, and we get, since F (y) is a constant, 0 = * If the imaginary part of y is positive or zero, we may take 0 S arg y ^v, and the upper sign is to he taken in the ambiguity ; if the imaginary part of y is negative or zero, we may take 0 i arg y i -f, and we take the lower sign ; we have already discussed what happens when y = 0. t HKOMWICH, 'Theory of Infinite Series,' p. 437. J Ibid., p. 434. 2 Q 2 300 MR. G. N. WATSON: A THEORY OF ASYMPTOTIC SERIES. In like manner, 8ince f V»* is convergent, we may show that, when n is any finite integer, &- F (?/) = I ffM («.»/) e~}' dt when y == 0, cfy" Jo and /*•» denotes the »**" differential coefficient of/ Putting y = 0, we get 0 = /<"> (0) rt-» d«, ^e., /(B) (0) = 0. Therefore /(y) is analytic when |?y| < 2 A and all the differential coefficients of f(y) vanish when y = 0 ; that is to say, /(y) is a pure constant, which we wil Furthermore, by the definition of f(y) L< Aexp(-|y|) when |argy|Ssfir+X, for all values of |y| , no matter how large ; since exp (- \y\ )* 0 as jy|*- 0°. we infer that L = 0. We have thus proved the lemma, that 8. We are now in a position to discuss the uniqueness of an analytic function possessing an asymptotic expansion with given characteristics for a certain range of values of the argument of the variable. The theorem, stated precisely, is as follows : Theorem V.—Let there be two functions /, (?;), /2 (x), which are analytic in the region dejined by the inequalities and let them be such that in this region they possess the asymptotic expansions x where, for all values ofn, Then, ?//8-«>ir/, Let the i-egiou in which x is permitted to lie be called the region C. Since MR. G. N. WATSON: A THEORY OF ASYMPTOTIC SERIES. 301 we have for all values of n, provided x lie in the region C. Now choose n to depend on x in such a way that so that we may put »-!->{ !•*->! }"_*, where 0 :£ 0 < 1. Now let y he the greater of the two quantities (1 +/)' y', a S arg j- :S /8 l>e called the region C7. When x lies in the region C', we have ln= {\x(l+l)-W> 1. * But when /« > 1, hy the asymptotic expansion of the gamma function, logr(Jn+l) = where J does not exceed a finite quantity independent of n. Consequently, when x lies in the region C', I/I CO-/. (•<•)! exp [- | *r-' 1 «-*]. Putting /i (•*')— ./»(•«) = ./•((>£), we want to show that if yi (x) is analytic in the region C' and subject to the inequality | /, (x) |< B! | «r- 1 -»«« exp [- | o j-) = 0. Let us put* x = ory1 and/3 (.r) =J\ (y). Then/4(y) is analytic in the region, C", in which \'J\> *-"(/)"> «sUrgy dyi (y) is subject to the inequality (when y lies in C") If 1 2: J, we see at once that in C" |/,(y)| l-« if I<* T^t v be the greater of the quantities ( y» - * * arg ^ A ^e region C we see from 0 ; and let /, (y) =/»: Then /(z) is certainly analytic in the sector jarg^-^+X, and at all points at a distance not greater than 2A from the boundary of the sector ; for when z hes m the region just specified, y certainly lies in the region C2. Also, in the region specified for z, |/(2)|27r, we could infer that f,(x) is identically zero on account of the theorem that " a non-convergent series cannot represent asymptotically the same one-valued analytic function for all arguments of /."t Tliis theorem w not applicable, because /, (4 />(.«) may not be analytic inside the circle |*j-yj a multiform function may have an asymptotic expansion valid for a range of values of arg x greater than 2ir. Thus, the generalised hypergeometric function formula J (a) V(l - P) ,F, (a ; /•; /) + I1 («+ 1 - P) r (p ' l)*a+'"P iF, (a - p+ 1 ; 2 - p ; ») is valid when |argz| vL t HROMWICH, ' Theory of Infinite Series,' p. 335. J See BARNES, ' Cambridge Philosophical Transactions,' vol. 20, p. 260. § The range of validity being taken less than 2ir. MR. 0. N. WATSON A THEORY OF ASYMPTOTIC SERIES. 303 Let us suppose that the expansion is valid when I jf|i y and a ^ arg ,r -^ ft, where /J-a = 7T+2X and 0 < X < £TT. Putting : = .rexp { — £ (a + £)t} and /(.r) = F(z), we see that F(z) possesses an asymptotic expansion of the form* where |a.|sA.n! when |argz|s ^TT + X and |z]iy. We notice that where J, = B . n! er"y~'. Let tin- region in which tin- asymptotic expansion of F(z)is valid he called the region D. Let L be a contour formed hy the following lines :— (i) The portion of the ray arg z = — (far+0) for which |z|iy|fl, 0 being an arbitrary quantity, as small as we please, such that 0 < 6 < X. (ii) The major arc of the circle |z| = y|<| terminated by the points (iii) The portion of the ray arg z = £TT + 6 for which | ; | S y j £ | . us consider the function tj>(t) defined by the equation when; |arg<| S X— 0. We observe that when z is at any point on tin- contour L, z/t lies within or on the Ixmndary of the region D. Now let us define functions «/>„ \jt3, ..., fa ..... by the system of equations The path of integration is supposed to be taken along the ray from v to infinity which, when produced backwards, passes through the point w = 0; we deduce by continued integration that the asymptotic expansion of \(>n(r) is ,/, M = ' .ti xtm n\v (n-U)!r» (n f MI): ,~ where provided that v lies within the region D or on its boundary. * This function F will not be confused with the F of Section 7. 304 MR. G. N. WATSON: A THEORY OF ASYMPTOTIC SERIES. Also, we notice that, by CAUCHY'S theorem, 1 f i — . i ""Vcte = the residue of «~"~V at the origin = — . ZTTIJL n\ By making use of the equations (15) in conjunction with this last result we get in succession on integrating by parts : . fc (z/t)]L + -L. • (16) each of the terms in square brackets vanishing at both ends of the contour. We shall now estimate the value of At all points on L we have a (z/t) ,dz for any Vftlue P, + nlz/t n\z/t *or brevity we put On the arc of the circle we have and l«P«|Sexp|y<|, where „ varies from -fr-Q to MU. O. N. WATSON: A THEORY OF ASYMPTOTIC SERIES. Consequently, for the integral round the arc, 305 On the ray arg z = \n + 0, we put where r varies from 1 to oo; so that for the integral along this ray, « cosec 0exp {— |y£|sin 0}; and we get the same inequality for the integral along the other ray of L. Combining our results we get (16A). |s ^ — 0 and |/ 1 < o-"1, tve may expand the integral Remembering that p^cr, we see from this formula that, if \t \ < cr~l, then uniformly as n * oo. That is to say, when |arg )/* the f'm-m where 11, * 0 as n*-oo. In other words, when | arg t\^\—0 and t \ < a"1, the integral represents r."iti:i.s associated function defined by tlie series converging when |«| 0) (n = 0). n > 0 n = 0 ,... (17) where K, K' are finite quantities independent of t and n. from an obvious modification of the thcorem 8tate(() converges when |/|(t)\ < K" when- K" is finite and independent of t. Also when 1 1 \ S (i+ 3) o-"' we have | (t) \ < K"' where K"' is finite and independent of t. Accordingly, when !/|^^or~', we use the formula the contour of integration heing a circle of radius Hence we get, when |<|s {tr~\ de Combining these results with (17), we see that for all values of t such that either |arg t\ < X — 6 or \t\^ ± y, and •/<. is any assigned integer, = 0, if y, > y ; and the function (t) is analytic in the region in which either of the inequalities (\.)\t\ y, where y, > y. If also It {2 exp ( — //*)} > yi, we can see that f •(-!•) (18) where /i is any ciu;mtity such that 0 < p. < X — 0, p. < J-jr. Ecjuation (18) gives the analytic continuation of F, (z) over the whole of the area fur which R {2 exp ( — t/n)} > y,. Let ff be a small quantity such that 0 < ff < /n. 2 K 2 BOG MR <;. X. WATSON: A THEORY OP ASYMPTOTIC SERIES. Then F, (z) is certainly analytic in the region (see figure) in which both inequalities I2| > y\ cosec (fji—tr), are satisfied. In like manner, when yi,R{zexp(i/i)}>yi, L- and we can deduce that F, (2) is analytic in the region in which both the inequalities |z| > yi cosec (/i-^), -^n-ff < argz < ^ + 6'-^ are satisfied. That is to say, F, (2) is analytic in the region in which ^-^), |argz| the inequalities |2| are satisfied. Now consider the function F, (z) in the region in which l*l>(ri+l) cosec (/n-fl'), jargj We may define F, (2) in this region by the equation F>(2)=f J ( where the upper sign may be taken if arg z - 0, and the lower sign if arg 2 ^ 0 By repeated integration by parts we get ,. From have where (1?A) a" thu hlte^ted Parts vanish at the upper limit; and we -if e JJfTM) Ml; i;. N. WATSON: A THEORY OF ASYMPTOTIC SKKIKS. Now y, 1 1 1 - H (zt) :£ - 1 1 1 ; and hence from (17x) 309 • : ; so that, applying the formula just obtained to and Ka is independent of n. But we have | S, | = SB+1, we get But we can find a finite quantity K independent of 7i such that ^ — /Ji\\» < ^ when n is a positive integer ; and a fortiori < K ; therefore, since p s (y, + 1) cosec (/A— ff), larg ej < %ir + ff, we have asymptotic expansions of the form (1'J) Taking 1=1, and writing 2cr for (yi + 1 ) cosec (p. — ff), |urgz| < ^n + ff, we have proved that F(z)=f z(t)c-",/t. .'(T».) But z (t) e~" dt is an analytic function of 2 when R {2 exp (—»/*)} > y,, where J(-»«) y, is any quantity greater than y ; hence, by the theory of analytic continuation, we have .•iiid more generally, if — (X— 0) < v < X — 6, we have* P| )=| z(()e-:l y,. ia analytic along the ray arg t = - v. a,0 MR. 0. N. WATSON: A THEORY OF ASYMPTOTIC SERIES. X,,, draw the circle |*|-»! and draw the tangents to this c,rcle at - y exp { + ,- (X-2*)} in the dictions of the rays arg z = ± {fcr+X-2*} respective], . "if - 1* any point to th, ri«ht of the curve formed by these tangents and the arc of th,. Circle job^ th,ir extremities (see figure), we can find a real quantity „ such that ]»/! < X-0, and such that R{zexp(-tV)} > yi. For such a value of z we have " summed" F(z) by the equation which we have just proved, viz. : F (z)= f .!<—> /n o /(*) = F(2)- we see that f(x) is summable by an integral of the same nature as BOREL'S integral ; the formal result is hardly worth writing down, since it usually happens that a = — ft, so that z = x. 10. We may also show, by the methods of Section 9, that if we are given a function (t) defined by the series where |a.| < A .n! />", and if (t) have no singularities in the region | arg t \ S X, and it when |argt|sX, |^(<)| < K exp {y|«| }, where K is a constant, then the function F(i) defined by the equation F(z)= f" has an asymptotic expansion in powers of 1/2 valid when |argz| < ^ir + \— 6 (where 8 > 0), provided that 1 2 1 be sufficiently large ; and that unity is a grade and outer of the expansion, and p is a radius. For, when |argt|SX— 6, we have the contour being a circle ot radius p,"1 sin 0, where p^> p; assuming that when \t | a pt~l, !<£(*) | < K, we find without difficulty that < n! K[exp {y\t\ +yPl~l sin 0}] {Pl cosec B}". * From the results proved concerning <£""(/) it follows that F(s) is " absolutely sunnnable." MR. O. N. WATSON: A THEORY OF ASYMPTOTIC SKItlKS. 311 Now suppose that we can find a n-.-tl quantity v such that |v y+ 1 ; then we may write F(')=f( , " \ — 9. and, on integrating by parts, we get i.e., Y(z) = al,+ ^ + ... + ^ Z - where |R,.z"| < (n+1)! K |<-zp (ypr1 sin ff)} {pt cosec 0}" \ exp {- \t\ }d\t\, •M-r) i.e., | R.Z" | < (n+ 1 )! K' (p, cosec 0)", when? K' is finite. Now so that. l . (n+l)! + (u-f 2)! K'(p, cosec 0)'}\z\—1. If p3 > pi, we have p- (n + 1 ) < K V. p," (n + 1 ) (n + 2) < K' >/, when; K", K'" are finite and independent of n. Therefore < B (pt cosec 0)' . n! | z | -•-'. That is to siiy, for values of z such that R{zexp(-tV)} where \v\ is less than or equal to X— 0, F(z) has an asymptotic expansion with grades equal to unity, a radius p, and an outer radius p, cosec 0, where p, is any quantity greater than p. This is, effectively, the result stated at the beginning of the section. 1 1 . We shall conclude by investigating the characteristics of the asymptotic expansion of a function connected with the "logarithmic integral," or "It" function, defined by the equation 312 ML O. N. WATSON: A THEORY OF ASYMPTOTIC SEKIES. When x is real and positive it is known that* "11 * * real and positive, e-li(e-) has an asymptotic expansion of which characteristics are A; = 1, p - i> 1=1, 0, on the path of integration | x + v \ > | x \ . If R(a-)<0, then |z+»|»|I(x)|. Thus, if R (*)>(), I RB I < | x \ "n"1 f »"e"' a!v £ | a; | "B"1 . n! . Jo Whereas, if i7r | x \ cos a, nn that |R»|Sseca.|x|~' "'. n!. Thus, if iargxjs£ir+«, the function exli(e~I) possesses an asymptotic expansion of which characteristics and constants are k= 1, p = 1, A= 1, 1=1, y W. KOSKMIAIN at the Royal Institution in June, 1910. VIM.. . . xi. A 178. -2 s 2 7.9.11 816 DR. WALTER ROSENHAIN AND MR. SYDNEY I, ARCHBUTT ON THE variety of zinc, kindly presented by Sir JOHN BRUNNER. The analySe8 of these materials are as follows :- Aluminium- Silicon .... Iron Copper . . • • Aluminium (diff.) . 0'17 per cent. 0-20 Trace. 9 9 '6 3 per cent. Zinc — The zinc used contained traces only of iron and silicon, and was of at least 99 '98 per cent, purity. The alloys were prepared by melting together weighed quantities of the two metals in salamander crucibles heated in a small furnace. For the preparation of the alloys rich in aluminium, that metal was first melted and the requisite quantity of pure ainc added to it, the zinc dissolving very rapidly TABLE I.— Calculated and Analytically Determined Composition of the Alloys. Calculated. Determined. Alloy. Zinc. Aluminium. Zinc. Aluminium. per cent. per cent. per cent. per cent. 99 99 1 — 1 '22 98 98 2 — 2-11 95 95 5 95-76 — 92 92 8 91-54 — MB 90 10 89-85 — 90 90 10 88-75 — H6 86 14 85-52 — B8 85 15 84-23 — 84 84 16 83-50 — 80 80 20 79-54 — 78 78 22 78-48 — 76 76 24 75-29 — 75 75 25 74-50 — 70 70 30 69-88 — U 65 35 66-71 — 60 60 40 60-27 — 55 55 45 55-57 — 50 50 50 50-52 — 45 45 55 46-40 — 40 40 60 39-13 — 35 35 65 33-21 — 30 30 70 29-37 — 20 20 80 20-54 — 15 15 85 15-50 — 10 10 90 10-64 — 5 5 95 5-46 - CONSTITUTION OF THE ALLOYS OF ALUMINIUM AND ZINC. 317 in the molten aluminium ; in the case of alloys near the zinc end of the series a previously prepared alloy of equal parts of aluminium and zinc was usually melted first and further zinc subsequently added. All the ingots upon which the results stated in the paper are based have been chemically analysed, and the results obtained are shown in Table I., p. 316. In the majority of cases it will be seen that the analytically determined composition agrees very closely with that aimed at } in a few cases where discrepancies of over 1 per cent, are found this is due to repeated melting of the ingots in question. During their preparation and subsequent treatment the alloys were carefully protected from contamination with foreign materials. The extent to which this was attained is illustrated in Table II., where the analyses of two alloys are given together with their composition (as regards impurities) as calculated from the materials used. TABLE II. Alloy. Impurities. Calculated. Determined. Iron. Silicon. Iron. Silicon. 90u 50 per cent. 0-020 0-100 per i-i-iii . 0-017 0-085 per cent. 0-03 0-09 per cent. 0-02 o-io In most of the experiments the melting-furnace was heated by gas, but in other experiments a small electric resistance-furnace was employed. Oxidation of the metals was minimised by keeping the temperatures of the melting- furnaces as low as possible consistent with the complete fusion of the alloys ; it was not found practicable to protect the surface of the alloys by a flux, or other covering, without risking contamination from substances liable to be reduced by the molten aluminium. The thermal study of the alloys was carried out by means of a series of cooling- curves taken by the " inverse-rate " method with the delicate potentiometer installed for that purpose at the Laboratory. The ingots used for obtaining these cooling- curves weighed 300 gr. and were cooled in a furnace which was placed in a closed water-jacketed steel vessel, the thermo-couple being introduced into the alloys through suitable apertures in the top of the water-jacket. The cooling alloys were thus entirely protected from draughts or other extraneous disturbances. In a first series of cooling-curves the rate of cooling was such that when the alloys 318 DR WALTKR ROSENHAIN AND MR. SYDNEY L. ARCHBUTT OX THE were hot the temperature fell at a rate of 1° C. per second, the rate of cooling decreasing as the alloys cooled, until it became 1° C. in 12 seconds. It was found that this rate of cooling was too rapid and for the data used in the present paper a series of curves were obtained at much slower rates of cooling, ranging from a rate of fall «>f temperature of 1° C. in 4 seconds when the alloys were hot (near 650° C.) to a rate of 1° C. in 20 seconds when the alloys were cool (near 200° C.). Tli is slow rate of cooling was adopted because it was found that certain of the reactions or inversions which occur in these alloys are to a considerable extent suppressed by the more rapid rates of cooling ; even the rate adopted does not allow tln-M- reactions to be completed, but they occur to a sufficient extent to allow unmistakable signs of their existence to appear on the cooling-curves. For the purpose of studying the alloys under conditions of complete equilibrium, it was, therefore, necessary to adopt the method of microscopic examination of samples which had been subjected to prolonged heating. As already indicated, the temperature measurements were made by means of a thermo-couple and potentiometer. One thermo-couple, composed of wires of platinum and platinum with 10 per cent, of iridium, has been used throughout the research ; this junction has never been broken or repaired, and it has been carefully protected from injury by such causes as unnecessarily high temperatures or contact with furnace gases or other injurious substances. At the beginning of the research, and frequently during its progress, this thermo-couple has been calibrated by using it to determine the freezing-points of a series of standard pure metals. In these calibra- tions the thermo-couple was used in the same small furnace, and with the same fire- clay protectors, as were employed in taking the cooling-curves, and similar rates of cooling were used. The temperature-E.M.F. curves obtained from these calibrations could therefore l>e applied direct to the readings of the cooling-curves without the introduction of any corrections. At the conclusion of the research the thermo-couple gave readings which agreed with the original calibration within 1° C. The use of very slow cooling, and the fact that the alloys were allowed to solidity in a perfectly undisturbed manner, would seem to involve the risk that the observations might be vitiated by errors due to surfusiou. Surfusion phenomena can, however, be detected readily with the apparatus used in this work, by the occurrence of a rise of temperature following the first arrest of cooling ; although small rises of temperature of this nature were frequently observed, the large mass of alloy used, mid tli« slow rate of cooling, always resulted in a prolonged arrest at a definite maximum temperature, and the manner in which the observed points fall upon smooth curves or upon straight lines shows that errors from surfusiou have been avoided. In the study of the cooling-curves, the cooling of the same ingot has in many cases IH-.-M repeated three aud even four times in order to eliminate the possibility of mistaking small irregularities due to experimental error for minor heat evolutions' CONSTITUTION OF TMK Al.M>YS OF AI.I'MINUM AM> /INC. 31!) Eleven typical examples <>t' tlirs. >ling-curves are reproduced in figs. 1 and 2. The numbers attached to each curve refer to the alloys as numbered in Table I. ; these stfc «J ; \ \ 1 ^ 7 VU>JL /* U >f Fig. 1. M •*,»•»•*•»* Fig. 2. numbers approximately represent the percentage of zinc contained in the alloys. The actual observations on which these curves are based are given in an appendix. 320 DR. WALTER HosKMIAIN AND MR. SYDNEY L. ARCHBUTT ON THE Curve No. 100 i» the cooling-curve of pure zinc. This merely shows the arrest due to the freezing of the pure metal ; no evolutions of heat can be observed at lower temperatures down to 200° C. No. 99 is the cooling-curve of an alloy containing 98 '8 per cent, of zinc; here a small arrest is visible at 376° C., but nothing further. In No. 98 (97 '9 per cent, zinc) three arrest points are to be observed : the highest is obviously the temperature of initial freezing, the second is related to the lower of the two points in the previous curve and occurs at the same temperature, while the third, at about 240° C., is a faint indication of a heat evolution which becomes more marked in the following curves. In the next, No. 95 (9576 per cent, zinc), we have two large points very nearly merged together, one occurring at 383° C. and the other at 380° C. As the time •observations taken for these curves were made at intervals of 5° C., it would not have been possible to separate these two points had the observer's attention been confined to taking the data for the curve ; in practice, however, the exact temperature at which the actual arrest of cooling occurs is carefully noted, and as an interval of 3° C. is represented by a distance of 24 mm. on the scale of the instrument, the two arrests in the present alloy were readily distinguished. Curve No. 95 also shows a decidedly more marked peak at 257° C. ; it will appear later that this is connected with the points at 240° C. on No. 98, the lower temperature in the former case being an example of the manner in which the temperature of a reaction is apt to be lowered when that reaction is rendered faint by the extreme dilution of the constituent in which it occurs. Curve No. 90 (8875 per cent, zinc) again shows three distinct peaks: the temperature of the initial freezing has now risen to 426° C., the second point still occurs at the constant temperature which lies between 377° C. and 381° C., while the third point, now very large, is again seen at 257° C. Passing on to No. 84 (83'5 per cent, zinc) we find a curve with four points ; the temperature of initial freezing has risen further to 450° C., and we again have arrests at 381° C. and 257° C. In addition to these, however, there is now a very small point occurring at a temperature of about 425° C. ; this again is the first sign of a heat evolution which will be met with in a more vigorous form in succeeding alloys. In No. 76 (7 5 '29 per cent, zinc) we again have four points. The faint reaction noticed in No. 84, at 425° C., has now developed into a marked heat evolution at 443" C., the temperature of the faint reaction in No. 84 being depressed by dilution. We still have a small point in No. 76, at 380° C., and the peak at 257° C. is still well marked. In No. 55 (557 per cent, zinc) the character of the cooling-curves has changed considerably ; the heat evolutions at 443° C. and 257° C. can still be traced, but the peaks found in previous curves at 380° C. are unrepresented here ; on the other hand, a very small, but quite definite, peak is found at a temperature between that of initial freezing and that of the reaction at 443° C., and appears to be the indication CONSTITUTION OF TIIK ALLOYS OF ALUMINIUM AND XI NT. 321 of a third group of reactions. It will be seen that a series of small heat evolutions can be traced throughout the succeeding alloys up to and including pure aluminium, forming a generically connected series with the point shown in this curve, but presenting tin- curious feature that they occur at successively higher temperatures as the concentration approaches pure aluminium. ( 'urve No. 45 ( I*', I per cent, zinc) shows the same four points which have just been described in Curve 55, but the peaks at 443° C. and 257° C. are now much smaller. In ( 'urve No. 35 (33'2 per cent, zinc) only the two upper points persist, all the lower points having disappeared ; the same remarks apply to Curve No. 05. In further reference to the points occurring in Curves 55 to 05 it should be pointed out that the apparently feeble character of this heat evolution arises in part from the fact that it occurs so soon after the commencement of freezing of the alloys ; that portion of the cooling-curve upon which this peak is shown slopes backward very steeply, and a very considerable evolution of heat is therefore required to arrest and even to reverse this steep slope. This circumstance somewhat militates against accuracy in the determination of these peaks, the actual peaks shown on the curves being in some cases not very much larger than experimental perturbations which now and then occur. In order, however, to show definite evidence of the existence and constant occurrence of these heat evolutions, three cooling-curves of the same ingot are reproduced in fig. 3 ; a comparison of these three curves will clearly demonstrate St SSI '.f 'C Fig. 3. tin- constant occurrence of this heat evolution in spite of the apparent smallness of the peak on eacli individual curve. The group of cooling-curves, as shown in figs. 1 and 2, cannot readily be translated into an equilibrium diagram, even with the aid of the additional intermediate curves which were obtained by the authors, and a larger amount of experimental evidence VOL. 00X1. —A. 2 T 322 DR. WALTER ROSENHAIN AND MR SYDNEY L. ARCHBUTT ON THE was required in order to render intelligible the nature of the various heat evolutions and their mutual relations; it will, however, be more convenient to describe this further experimental work by first giving the equilibrium diagram finally arrived at. It will then be possible to discuss each group of alloys and the reactions which they undergo, and to give in detail the evidence upon which the position and interpretation of each line of the diagram is based. The equilibrium diagram of the aluminium-zinc alloys is shown in fig. 4, where the concentration of aluminium is plotted horizontally and temperature is plotted ft* aw- f>. 4. 34 t r*1 603- .fil'C MO "6 if. H fJ • Fig. 4. vertically in the usual manner. The observed temperatures of arrests or retardations on the cooling-curves are shown as crosses, while the results of quenching experiments are indicated by dots surrounded by small circles or squares. The points plotted include all those observed. The liquidus, or curve of initial freezing, is represented by the line ABCD ; this * only very slightly from the corresponding curve in the diagram of SHEPHERD,* e diagram of SHEPHERD did not show the small break in the liquidus at the The evidence for the existence of this break in the liquidus lies in the first the accuracy with which the observed points on the liquidus have been SHEPHERD, « Aluminium-Zinc Alloys," 'Journal of the Physical Society,' 1905, 9, p. 504. CONSTITUTION OF THE ALLOYS OF ALUMINIUM AND ZINC. 323 determined ; this is such that a continuous curve from B to D would lie further from the observed points than the limits of experimental error would allow, particularly as the points on the liquidus near the point 0 have been repeatedly determined with special care. The existence of a break at C is further confirmed on theoretical grounds by the evidence to be described presently, proving that the phases which commence to crystallize along the branches BC and CD are two distinct bodies. The solidus curve of the diagram consists of the lines AE, EB, BF, FG, GH, HD. The evidence for this statement is to a large extent microscopical ; the method of prolonged amif.-iling followed by quenching has been adopted throughout and details of the evidence obtained in this way, particularly with regard to the lines GH and HD, will be given below. The line EBF, lx>th as regards temperature and as regards the position of the point B, agrees closely with the eutectic line shown in SHEPHERD'S diagram, but ;m important difference is found with regard to the end of this line on the aluminium side. On referring to the cooling-curves in figs. 1 and 2, it will be found that the arrests along the line EBF are still marked in Nos. 78 and 76, yet the eutectic line in the diagram is drawn only to the point F, corresponding to a concentration of 78 '3 per cent, of aluminium. The reason for this discrepancy lies in the fact that although alloys cooled with moderate rapidity exhibit a eutectic heat evolution to a considerable distance to the right of the point F, yet both microscopically and pyrometrically this eutectic can be made to disappear entirely by exposing the alloys to a temperature of about 430° C. for a considerable period of time. In order to establish this point, ingots of a series of alloys lying on either side of the point F were maintained for about 100 hours at a temperature near 430° C., and cooling-curves of these ingots were subsequently taken at a rate similar to that described for the series of curves illustrated in figs. 1 and 2, the observations being taken from a temperature of 430° C. to about 200° C. The curves of alloys, Nos. 80, 78, and 75, are shown in fig. 5, the curves marked a, b, and c l>eing those taken by ordinary slow cooling from fusion, while those marked d, e, and f are those taken after the heat treatment just described. In No. 80 the eutectic heat evolution has not been entirely removed by the treatment, but in Nos. 78 and 75 the well-marked peaks of the ordinary curves have entirely disappeared after the treatment. This indicates that the existence of stable eutectic is limited by a concentration lying between 78 and 80 per cent, of aluminium, and probably lying quite close to 78 per cent. In order further to ascertain the concentration at which the eutectic disappears the method of measuring the heat evolutions in successive alloys was adopted. Although every precaution was taken to render these measurements as accurate as possible, it must be recognised that the method itself is not capable of any high degree of accuracy, the reason being that although the cooling-curves generally give a definite 2x2 3'24 DR. WALTER ROSENHAIN AND MR. SYDNEY L. ARCHBUTT ON THE indication of the commencement of an evolution of heat, no sharp indication of the , ,„/ of such an evolution can be obtained.* In the present experiments „ mgots used each weighed 150 gr, and all of them had been annealed together at . temperature of 430" C. for about 100 hours, experiment having shown that was sufficient to secure complete equilibrium. Fig. 5. These ingots were cooled in an electrically heated furnace placed inside a water- jacket, and the initial temperature as well as the rate of cooling were kept as nearly uniform as possible throughout the series. The resulting curves were plotted on a large scale and the peaks of the inverse rate curves were measured with a planimeter, or, in some cases, the areas were determined by plotting on thin card, cutting out the peaks and weighing the pieces of card on a sensitive balance. The observed areas of the peaks depend upon the choice of a point on the cooling-curve to represent the end of the heat evolution, and it has already been pointed out that this choice cannot be made with any degree of certainty. In the present series of curves these points were inserted by estimation and the endeavour was made to place them on corre- Bponding parts of the curves for all the alloys. The areas of the peaks thus measured represent approximately the quantities of heat evolved by each of the alloys and, within the limits of accuracy of the method, these may be taken to be proportional in the present case to the quantity of eutectic undergoing solidification in each alloy. These quantities as ordinates are plotted on abscissae representing the concentration of the alloys in the upper curve of fig. 6. It will be seen that the points thus obtained lie with considerable accuracy upon the straight line which cuts the zero line at a concentration of 78 '8 per cent, of zinc, this concentration representing the limit of the eutectiferous alloys as determined by this method of extrapolation. It will be shown later that the termination of the * See "Observations on Recalescence Curves," by W. ROSENHAIN, 'Proc. Phys. Soc.,' 1908. -TITUT1ON OF THE ALLOYS OF ALUMINIUM AM> /.INC. 325 eutectic line at this concentration is really due to the formation of a definite aluminium-due compound whose existence is confirmed by other evidence. at ilo'C ' ~ .*'. t .•« Fig. 6. We have now to consider the nature of the heat evolutions along the lines CH and IL of fig. 4. The fact which has already been stated that in slowly cooled alloys eutectic is found to the right of the point F, while this eutectic disappears when full equilibrium has been obtained, points to the conclusion that there is in these alloys a gradual reaction at some temperature above that of the eutectic line, and that this reaction, when allowed to complete itself, results in the total absorption of the liquid phase which would otherwise have solidified along the dotted line FP. The existence of a scries of heat evolutions along the horizontal line CH at once suggests that such a reaction is represented by these heat evolutions, and this view is confirmed when it is rememlxM-ed that in curves of more rapid cooling these heat evolutions are much less marked, an observation which indicates the occurrence of a gradual reaction. We have, further, the discontinuity in the liquidus curve at the point C, which also suggests that we are here dealing with a compound which is produced by a reaction of the solid which had begun to crystallize along the branch of the liquidus CD, with the liquid phase which is present when the temperature of the line CH is reached. It is well known that such a reaction taking place between a solid and its mother liquor results in the formation of sheaths of the new compound, and that these 326 DR WALTER ROSENHAIN AND MR. SYDNEY L. ARCHBUTT ON THE sheaths, by separating the two reacting bodies, considerably retard the further progress of the reaction, which can then only take place by diffusion through the sheath. An indication of the composition of this compound has already been obtained by the determination of the end of the eutectic line, which has been found to lie at a concentration of about 78 per cent, of zinc, and this corresponds approximately with the composition of the compound Al»Zn3 which contains 7 8 '3 5 per cent, of zinc. It was not found possible to apply the method of approximate quantitative estimation of the heat evolutions along the line CH for the purpose of determining the concen- tration at which this heat evolution attains its maximum value. In the first place, the peaks representing these heat evolutions on the cooling-curves lie so near the large peaks representing the initial freezing of the alloys that the precise shapes of these smaller peaks cannot be ascertained even with that degree of approximation which was reached in the other case ; further, this reaction is not complete when the alloys are cooled even at the slowest rate practicable for cooling-curve observations, and no reliance can be placed on the quantities of heat evolved by a series of heat evolutions which might proceed to a greater or lesser degree of completion in the various alloys of the series. In the absence of the kind of evidence just referred to, it is fortunate that it is possible to show an intimate connection between the heat evolutions which take place along the line CH and those which occur at a temperature of 256° C. along the line IL. This connection was established in the first place by means of cooling-curves of a number of the alloys taken first by allowing them to cool at the standard slow rate from fusion, and again after prolonged heating at 430° C. It was found that the size of the peak at 250° C. increased very much after the alloy had been exposed to prolonged heat at a temperature just below that of the line CH. This is illustrated in fig. 7, where cooling-curves of alloys Nos. 70 and 75 are given. The peaks on the cooling-curves of these alloys when cooled slowly from fusion are shown on the curves marked (a) and (<•), while the corresponding peaks on the same alloys after prolonged annealing at 430" C. are shown on the curves marked (b) and (d). The curves them- selves show that the rate of cooling was practically identical in all four cases. This observation indicates that the magnitude of the lower heat evolution is directly dependent upon the extent to which the reaction along the line CH has been allowed to take place, and the inference is justified that the heat evolutions along the line IL are due to a reaction occurring in the compound which is formed along the line CH. rhis inference will be more completely established by the microscopic evidence to be described below. The connection between the heat evolutions along the line IL and those along the line CH having been established, an attempt was made to utilize the measurement of the quantities of heat evolved by the reactions at 256° C. for the purpose of locating the maximum of that reaction, a maximum which would coincide with the maximum of the heat evolutions along the line CH. An attempt to do this was first made by CONSTITUTION OF THE ALLOYS OF ALUMINIUM AND ZINC. 327 taking a series of cooling-curves of alloys cooled slowly from fusion under careful standard conditions, the heat evolutions being measured in the manner described in connection with the eutectic line EF. These determinations give the points repre- sented in the second curve of fig. 6, and it will be seen that these points indicate the » t * Timt /tltrmti Tii u» Fig. 7. existence of a well-defined maximum lying exactly under the point C. This un- expected result can, however, be explained in a simple manner when it is remembered that the compound whose reaction is being measured crystallizes freely from the liquid along the line BC of the liquidus, while to the right of the point C this compound is the result of a slow reaction between a solid and a liquid. In these circumstances it is not surprising to find that in a series of alloys cooled at a standard rate the quantity of this compound would increase steadily as we pass from the concentration of the point B to that of point C, but that immediately on passing beyond C the quantity of the compound would decrease. The maximum obtained from such a series is therefore a misleading one, and does not represent the composition of any phase. The authors believe that this is an observation of some importance in the application of the methods of investigation styled " Thermal Analysis " by 328 DK. WAI/I'KK I;<>>KNI1AIN AND MR. SYDNEY L. ARCHBUTT ON THE TAMMAN, since it indicates that the results of such heat measurements, even \vhrn allowance is made for their unavoidable inaccuracies, must be considered with great caution before they can be accepted. In the present case the difficulty was overcome by measuring the heat evolutions on the series of cooling-curves already described in connection with the measurements of the peaks along the eutectic line EF; these curves were taken from ingots so treated that the reaction along the line CH had been allowed to reach completion, ami on these curves the maximum should truly represent the quantity of the compound present in the alloy. The data from these cooling-curves relating to the reaction along the line IL, plotted in the usual manner, are shown in the third curve in fig. 6; this curve shows a maximum near a concentration of 77 '2 per cent. zinc. It will be seen that tins agrees nearly, but not quite accurately, with the end of the eutectic line, as indicated on the upper curve of this figure, which falls at 78 '8 per cent. zinc. The slope of one of the lines in the lowest curve of fig. G, however, is somewhat flat, so that a small alteration of one or two of the observed points would displace the maximum to an extent quite comparable with this discrepancy. The authors, therefore, feel justified in saying that both the end of the eutectic line and the maximum of the heat evolutions along the line IL are consistent with the existence of a definite compound Al2Zn3. The determination of the position of the point H and of the curved portion of the solidus from H to D, having been based entirely on microscopical evidence, will be described in connection with that portion of the work, and we pass on to consider the line of points shown on the diagram in fig. 4, mostly lying within the area CDHG. It will be seen that these points lie on a perfectly smooth curve, starting from the temperature 638° C., at which a minute heat evolution is observed in most varieties of aluminium, including the comparatively pure variety used in preparing the present series of alloys.* Samples of aluminium have, however, been obtained which do not show this heat evolution, and this fact would suggest that the thermal change in question does not occur in the purest varieties of aluminium, but that its occurrence is determined, and ita intensity accentuated, by the presence of certain other elements. One of the authors, in a recent research on the alloys of aluminium and manganese, found a series of small heat evolutions starting from this temperature in nearly pure aluminium, and continuing at the same temperature, but with increased intensity, with successive tions of manganese until the concentration of the point Mn3Al is reached.! In the present series of alloys the addition of zinc also appears to intensity the heat •This point has frequently been observed in aluminium, and special attention was drawn to it by 1 - \v e Failure of the Light Engineering Alloys," ' Faraday Society,' 1910.) IIAIS and F. C. A. H. LANTSBERRY, "Ninth Report to the Alloys Research Committee of 5chanical Engineen-On the Properties of some Alloys of Copper, Aluminium, and CONSTITI'TION OF Till-: AI.I.oYS OK Al.rMIMl'M AM- /INC. 329 evolution slightly, but its principal effect is the gradual depnttioa of the temperature at which tin- change takes place. It was thought at first that a curve joining these points near the aluminium end might represent tin- solidus of the alloys, while the points near the other end of the series were suspected of tunning another horizontal line running into the liquidus and indicating the presence of another alumininin-/.inc compound. These heat evolutions, however, although very minute are very definite, and the temperatures at which they occur are so well defined that tin- experimental evidence, is sufficient to negative the supposed existence of a hori/.ontal line in this case. The certainty with which these minute peaks recur on repeated cooling-curves of the same alloy has already been illustrated in fig. 3, where three successive cooling-curves of the alloy No. 35 are plotted side by side. The supposition that the upper part of the curve might represent the solidus was also negatived by the indications of quenching experiments which, by demonstrating the absence of an upward step in the solidus between H and D, further negative the supposed existence of a compound whose formation might lie represented by these am The explanation which at first sight suggested itself for this Hue of arrest-points is that they are due to an allotropic change in aluminium, or in the phase rich in aluminium, and that this change occurs at lower temperatures with increasing concentration of zinc. If the line of the points in question lay entirely below the solidus, this explanation might be regarded as feasible, but in the alloys from about 3 per cent, to 66 per cent, of zinc, these points occur very definitely a)>ove the solidus and a transformation or inversion of any kind in a solid phase which is surrounded by its mother liquor can only occur at a constant temperature since the composition of such a solid phase at a given temperature is the same for the whole series of alloys in which it occurs, and the same inversion must occur in all such Ixxlies at the same temperature. Meta-stable conditions cannot be invoked to explain away this difficulty, since the meta-stable formation of "cores" in such alloys implies the presence of a solid containing less zinc than the average composition of the alloy, and since the line of points in question slopes upwards towards the aluminium end of the diagram these meta-stable cores would undergo transformation or inversion at a hnjlii-r temperature than that which would occur under equilibrium conditions in the same alloys. These considerations, therefore, afford no explanation for the depression of these points with increasing concentration of zinc. The oidy explanation which the authors feel justified in suggesting is that the occurrence of these points, or rather their continual depression with increasing zinc content, may be due to the action of a third component substance which enters into the alloys as an impurity. Such an impurity is found in the form of 0'20 per cent, of iron present in the aluminium employed, while a certain amount of silicon is also presrnt in the alloys. The author's suspicion has fallen principally upon iron since a few isolated crystals of an iron aluminium compound have been observed in most of \OI.. i 'I'M.- -A. - I WO DR. WALTER ROSENHAIN AND MR. SYDNEY L. ARCHBUTT ON Till. the alloys. Some experiments have been made by taking cooling-curves of a few ternary alloys of aluminium, zinc, and iron with a view to establishing the connection between the small heat evolutions under discussion and the presence of the compound referred to. It would seem, however, that the iron-aluminium diagram itself requires to be worked out afresh and that the relationships between these three metals are of a complex kind, so that a great deal of further experimental work will be necessary before the truth of the suggested explanation can be tested. In view of these circumstances the authors have not thought it desirable to delay the publication of the present research until this further and larger investigation could be completed. They therefore content themselves with indicating this series of heat evolutions on the diagram, but without drawing a line through them or offering more than the tentative explanation outlined above. We now pass on to the consideration of the micro-structures of the alloys. As has already been indicated, much of the evidence for the lines of the diagram, and particularly for their interpretation, is based upon microscopic evidence, and the micro-structures will be described from this point of view. For the purpose of microscopic examination small specimens were, as a rule, cut from the ingots used for cooling-curve determinations. In some cases complete vertical sections of the ingots were examined in order to avoid risks of error from segregation. The polishing of the specimens near both ends of the series proved very difficult. Most of the specimens were etched with a 10-per-cent. solution of caustic potash — in one or two cases weak nitric acid was used. Commencing at the zinc end of the diagram we have first a very narrow area marked in fig. 4 as representing alloys consisting entirely of the meteral a. This designation includes pure zinc and the solid solution of aluminium in zinc. The exact limits of the solubility of aluminium in solid zinc have not been finally determined, although an alloy containing 1 per cent, of aluminium has been found to show signs of the presence of eutectic after 24 hours of annealing at a temperature just below the freezing-point of the eutectic. This observation indicates that unless the rate of solution is extremely slow the limit of the a region lies within 1 per cent, of pure The line indicating the limit of this region is therefore shown as a dotted line only, and has, for convenience, been drawn further to the right than the evidence justifies. The diagram next indicates a group of alloys consisting of the meteral a embedded in eutectic, the constituents of the eutectic being termed a and ft above the line IJ, and a. and y below that line. The general structure and appearance of one of these s shown in Plate 5, fig. 8, under a magnification of 150 diameters. This the alloy containing 98 per cent, of zinc slowly cooled, and shows a with ies of eutectic. If such a specimen is quenched from a temperature 256° C.) the appearance of the micro-structure under moderate magni- tion, such as that of fig. 8, is as shown in that figure, but under higher CONSTITUTION OF THE ALLOYS OF ALUMINIUM AND ZINC. 331 magnifications there is a distinct difference l>etweeii such a quenched alloy and one which has been slowly cooled. It is found that while the ft, or dark-etching, constituent of the eutectic is homogeneous in the quenched alloy, in the slowly cooled specimens the ft meteral has undergone decomposition and exhibits a duplex structure which may take the fonn either of parallel lamellae or of minute granules. Fig. 9 shows the structure of this decomposed eutectic under a magnification of 1000 diameters ; the large dark areas represent the structure of the eutectic while the minute granulation in these areas indicates the decomposition of the ft constituent. The photomicrograph in fig. 9 is actually taken from an alloy containing 95 per cent, of zinc, but the same features are found in all the alloys in which the eutectic is present. The typical structure of the eutectic alloy, seen under moderate magnification (200 and 150 diameters respectively), is shown in figs. 10 and 11. It will be seen that the eutectic exhibits in a very beautiful way the characteristic features of well- defined eutectic alloys, and it will further be noted that the relative areas of the dark and light constituents are not very widely different. When it is realized that this structure is found in an alloy containing only 5 per cent, of aluminium, this obser- vation alone strongly suggests that the components of the eutectic are not in reality zinc and aluminium, but zinc and a compound of zinc and aluminium containing a considerable proportion of zinc. Eutectics lying near one end of a binary series are not unknown in other groups of alloys, but they do not exhibit a well-balanced laminated structure unless a compound comes iuto play. The micro-structures found in alloys containing less than 95 per cent, of zinc, that is to say, lying just to the right of the eutectic concentration, further support the view that the ft meteral, which forms the dark-etching constituent of these alloys, is a definite compound. These micro-structures are illustrated under a magnification of 200 diameters in figs. 13 and 14. These two figures show the presence of strikingly characteristic dendritic crystals, which possess a strong tendency to assume six-rayed forms in which angles closely approximating to 60 degrees are of frequent and typical appearance. This is a striking characteristic, particularly in view of the fact that in the great majority of metals the dendritic branches tend to fonn rectangular systems. The microscopic evidence thus strongly supports the pyrometric evidence for the existence of a definite compound, and grounds have been given above for the view that this compound is represented by the formula A L /n The close relationship between the six-rayed crystals just described and the eutectic surrounding them is also illustrated in an interesting manner by fig. 14. It is there clearly evident how the crystallization of the eutectic has radiated from the various branches of tin- denilrite, resulting in the formation of an interesting pattern. The photomicrographs, figs. 13 and 14, having been taken from slowly cooled specimens of the alloys, are not free from signs of the reaction represented in the 2 U 2 \ 33-J DR. WALTER ROSENHAIN AND MR. SYDNEY L. ARCHBUTT ON THK diagram by the line UK. The material of the six-rayed dendrite, even under this moderate magnification, is seen to be far from homogeneous, and under higher magnifications it is found to have undergone the same kind of change as that which has already been described and illustrated for the ft constituent of the eutectic. Fig. 16 (x!50) shows the micro-structure of alloy No. 90, containing 89'85 per cent, of zinc, as it appears when the alloy is cooled from fusion in the furnace in about 45 minutes. The photograph shows a larger proportion of the ft meteral than that seen in fig. 14 embedded in eutectic, but the edges of the ft masses show rims of a changed material arising from the reaction indicated by the line UK of the diagram. The existence of these rims proves that this reaction is rather slow, and the facility with which it can be inhibited by quenching further confirms this view. In fig. 17 (x 150) we see the micro-structure of alloy No. 86 (85'52 per cent, zinc) cooled rather more slowly than the specimen represented in fig. 16. This specimen (fig. 17) has, in fact, been cooled at the rate adopted for the observation of cooling- curves. The proportion of ft is larger than in fig. 16, but the change which resulted in the formation of "rims" in fig. 16 has here taken place to a much larger extent, so that the original ft body has almost entirely disappeared. Under the magnification of fig. 17 (150 diameters) the resulting structure of the "changed" ft is not very evident, but a portion of the same specimen is shown in fig. 18 (Plate 5) under a magnification of 500 diameters, and there the uniformly dark ft is seen to have become broken up into a laminated stnicture closely resembling the pearlite seen in steel. On account of this resemblance, the authors propose to describe this structure as " pearlitic " ; it will, however, be seen that in this case the duplex stnicture arises from the decomposition of a compound and not, as in the case of pearlite, from the decomposition of a solid solution. The nature of the reaction represented by the line UK can now be understood — it is the decomposition of the definite compound ft into two phases which phase-rule considerations prove to be identical with those called a and y on the diagram, the former being solid zinc saturated with aluminium and probably containing less than 1 per cent, of that metal, the latter being aluminium saturated with free zinc, the exact proportion not Ixjing quite definitely determined, but probably lying in the cinity of 40 per cent. The phase-rule considerations just referred to are that, in any field or area of a binary system, two phases only can exist in equilibrium. In the field BJKF we have the phases a and ft present. Now, it ft is decomposed, as e .Microscope proves that it is, into two phases, say X and Y, then in the field below have a + X+Y present, and it follows that either X or Y must be ith a. Similar considerations applied to the fields to the right of the line . show that the other phase must be identical with y. The alloys whose micro-structure is shown in figs. 13 to 18 all belong to the group the line BC. We now have to consider the alloys to the right of In these the phase first separating from the liquid is y and not ft. On CONSTITUTION OF THE Al.I.oys OF ALUMINIUM AND ZINC. 333 reaching the temperature of the line CGH the y crystals react with the liquid according to tin- equation y + liquid = £(Al,Zn,). Up to the line GFK, which represents the composition of the compound this iv.-i.'tion, even if completed, leaves a residue of liquid which solidifies as eutectic. The amount of this eutectic, however, decreases until it vanishes at the composition of the compound. The microscopic evidence supporting this statement is of considerable importance because it constitutes a strong confirmation of the existence of the definite compound Al»Zna ; it is, therefore, given in some detail. We have first in h'g. 19 (x!5()) an alloy containing 80 per cent, of zinc and, therefore, lying just to the left of the line (rFK; this has IM-CII maintained for Hve hours at a temperature just below the line CH and has then been slowly cooled. The photomicrograph shows the presence of small quantities of eutectic. In this respect this alloy is in contrast with one containing 78 per cent, of zinc, in which the eutectic disappears entirely under the same treatment, the resulting structure being perfectly homogeneous, like that shown in h'g. 26. The structure shown in fig. 19 represents an aggregate of /3+ eutectic, except that in consequence of slow cooling the reaction along JK has taken place and the dark- etched ft Ixxly is in reality duplex, consisting of a. and y. The reaction y + liquid = ft which takes place along the line CH, being a reaction between a solid and its mother liquor, results in the formation of sheaths of ft surrounding the y crystals. In the alloy of fig. 19 the prolonged heating at 430° C. has obliterated the sheaths, but if a similar alloy is cooled comparatively quickly the existence of these sheaths is very clearly seen. In fig. 20 ( x 150) we have an alloy of the same group which has been quickly cooled down to a temperature just above 256° C., and has then been quenched in order to prevent the decomposition of the ft sheaths. These are very clearly seen in the photograph as dark edges surrounding the relatively light bodies of primary y. The same alloy slowly cooled from fusion to the ordinary temperature is shown in fig. 21 ( x 150) ; the cooling having been slower m this case the original sheaths of ft are hardly visible, almost the whole of the dark dendrites having l>een transformed into ft when the alloy passed the line CH. At 256° C., however, the rate of cooling was not slow enough to allow the decomposition of ft to be completed, and the structure therefore shows dendrites of ft merely decomposed at the edges. In fig. 22 ( x 300) we have the same alloy, this time after prolonged heating just below 430° C., and then cooled more slowly than in the previous example. Here both the formation and decomposition of ft have been completed, and a higher magnification would reveal the completely pearlitic structure of the dark areas of this photograph ; in the photograph the duplex character of the dark constituent is evident in many places. In contrast to this we have fig. 23 334 DR. WALTER ROSENHAIN AND MR. SYDNEY L. ARCHBUTT ON THE ( x 300) which represents an alloy in which the formation of ft has been. completed by p o old heating just helow 430° C, but the decompose of ft has been prevented £ q3ching thf specimen just above 256" C. The grey areas seen to this pho o- graph are perfectly nlogeneous and show no signs of pearhtic struc ure, even under hfehJT magnification. These five photomicrographs, showing the variation m £ Zto of an alloy lying just to the left of the line GFK, confirm the digram of fig. 4 and its explanation as put forward by the authors. We now pass to the alloys lying just to the right of the line GFK. Fig represents alloy No. 78 containing 78'48 per cent, of zinc magnified 150 diameters. The specimen in this case has been slowly cooled from fusion and is therefore in a meta-stable condition ; dark cores of primary y are surrounded by sheaths of decon osed B leaving a residue of eutectic. The decomposition of the ft body is not obvious in this photograph, but is very clearly seen in fig. 25 representing a port, of the same specimen as fig. 24 but magnified by 500 diameters. Here the laminated structure of the rims of decomposed ft is very evident. In fig. 26 ( x 150) we have the micro-structure of alloy No. 77, which in this respect is typical of all alloys containing less than 78 per cent, of zinc, after annealing for a considerable time 430° C. Mowed by quenching from a temperature just above 256° C. The specime. exhibits a perfectly homogeneous structure merely diversified by a few holes and cracks. If this structure were met with only in the alloy of the composition of compound Al,Zn,, the structure might be described as that of the pure ft meteral, but as a matter of fact this same homogeneous structure is exhibited by all the alloys containing less than 78 per cent, of zinc which have been treated in the same way. It follows from this observation that the ft body forms a solid solution with the y body, since to the right of the line GFK the reaction y + liquid = ft must leave a residue of y. For purposes of distinction the authors have termed this solid solution of ft and y the $ meteral. The ft which is dissolved in this S body, at any rate so long as it is present in any considerable quantity, still undergoes decomposition at the temperature of the line JKL, and the phases present below this line are again y + «, but of course the amount of a decreases rapidly with decreasing concentration of zinc. It has not, however, been found possible to determine microscopically the concentration at which a ceases to appear in the alloys. Fig. 27 shows, under a magnification of 600 diameters, the appearance of fully decomposed ft or S rich in ft. The photograph represents alloy No. 77 (7 7 '3 per cent. of zinc) after prolonged heating, first at 430° C. and then just below 256° C. The beautifully laminated structure and its striking resemblance to the pearlitic structure of steel will be recognized at once. It is interesting to notice that this laminated structure is only produced when the alloy is cooled slowly through the decomposition temperature and is then held for some time just below that temperature. If the specimen be quenched from above and be then heated to a temperature just below 256° C. and held there for a considerable time, the resulting micro-structure, although CONSTITUTION OF THE ALLOYS OF ALTMINII'M \NI> ZINC. 335 definitely duplex, is granular and patchy, recalling the structure of sorbitic and troostitic steels rather than the pearlitic structure shown in fig. 27. Although the rates of cooling and heating involved are decidedly slower in the present case, the behaviour of the /3 meteral appears to present some interesting analogies to that of carbon steels. To illustrate the behaviour of the group of alloys lying somewhat further to the right of the line GFK, the photomicrographs of figs. 28 to 31 represent the micro- structure of alloy No. 75 (74'5 per cent, of zinc). Fig. 28 (x!50) represents the structure of a specimen of this alloy which has been cooled from fusion to 470° C., held at that temperature 30 minutes and then quenched. This temperature lies just above the line CH, and the micro-structure accordingly consists of crystals of y embedded in a finely-granulated matrix which represents that part of the alloy which was liquid at the moment of quenching. In fig. 29 ( x 150) we have the structure of a specimen of the same alloy which has been cooled from fusion to 430° C., held at that temperature for 30 minutes and then quenched. In this structure we see an approach to homogeneity ; the black areas represent holes or cavities in the specimen, but there is no sign of the presence of liquid at the moment of quenching, nor is any eutectic present. This structure represents a stage of the reaction y + liquid = (5, where the reaction, although still incomplete, has progressed to a considerable extent. Fig. 30 ( x 150) represents the structure of a specimen of the same alloy which has been kept at 430° C. for about 2 hours, this treatment being followed by quenching from that temperature ; here the reaction named above has been allowed to complete itself, while the decomposition which would otherwise have set in at 256° C. has been prevented by quenching ; the resulting structure is therefore homogeneous and is typical of the alloys consisting of S. The same alloy, when slowly cooled after prolonged heating at 430° C., again shows the typical pearlitic structure resulting from the decomposition of S or ft. This is illustrated in fig. 31 under a magnification of 300 diameters. The persistence with which meta-stable eutectic is found in the alloys of this group when slowly cooled is illustrated by fig. 32 (x 150) which represents alloy No. 70 in t his condition, which corresponds closely to that illustrated for alloy No. 78 in fig. 24 ; we have cores of primary y with rims of decomposed S enclosing areas of eutectic. The same alloy, after heating for a considerable time at 430° C. followed by slow cooling, is illustrated in fig. 33 (x!70), where the pearlitic structure can again be seen while there is total absence of eutectic. The micro-structures of alloys lying to the right of the point H present few points of interest so far as the constitution of the alloys is concerned ; they exhibit the characteristic features of alloys consisting of a single solid solution possessing a long range of solidification. The typical formation of cores in such alloys is illustrated in fig. 34 (x 150), which represents the structure of a slowly-cooled specimen of alloy No. 1st ; it' heated for any length of time to a temperature below that of the s.. I id us. »3B DR. WALTER ROSENHAIN AND MR. SYDNEY I, ARCHBUTT ON THE this alloy and all those lying to the right of the point II Income perfectly homogeneous. It will be seen in the diagram that the lines GH and KL have been continued as dotted lines to the aluminium end of the diagram. This has been done because it may be supposed that, if final equilibrium is attained in the freezing process, the solid solution will react at the temperature of the line GH with the zinc which it contains to form the compound Al3Zn3, and this in turn will undergo decomposition when the temperature of the line KL is reached. This supposition, however, involves the assumption that the y solid solution contains the zinc as " free" zinc, while in the S solid solution the zinc is present in the "combined" state, i.e., in the form of molecules of the compound Al2Zn3. Just to the right of the line GFK of the diagram there can be no doubt that we have in the $ meteral a solid solution in which the compound /8 is the solvent and free aluminium is the solute ; this solid solution, however, is merely one end of a continuous series whose opposite extreme is represented by a solid solution in which aluminium is undoubtedly the solvent and the compound the dissolved body. The question arises whether the phenomena of dissociation or ionisation which are known to occur in dilute aqueous solutions also occur in these solid metallic solutions. Where the solution is a strong one, i.e., in the present case for alloys containing more than 35 per cent, of zinc, there is ample evidence that the compound ft preserves its identity in the solid solution, but for lower concentrations there are two reasons why experimental evidence is not available. In the first place, for all alloys lying to the right of the point H in the diagram, the reaction between the two constituents of the y solid solution must take place in entirely solid metal. Such reactions are not unknown, but in most cases they are very slow, and in the present instance the reaction in question is known to be slow even when one of the reacting phases is a liquid. It is, therefore, not surprising to find that even at the slowest rates of cooling which are possible for the observation of < •»"ling-curves the reaction in these alloys takes place too slowly to be indicated by a heat evolution. The absence of arrest-points along the line GH beyond the point H is, therefore, no definite evidence that the reaction does not take place gradually if tun.- !*• jrivim at a favourable temperature. Further, microscopic evidence is difficult to obtain since microscopically the y and S solid solutions are both homogeneous bodies. The only test would be that of tracing the effects of the decomposition at the temperature of the line KL by looking for the appearance of the a phase in the annealed and slowly-cooled alloys. An endeavour was made to do this, and the think it probable that the a phase can be traced up to a concentration of aluminium us high as 80 per cent., but it must be admitted that this method is not •ry reliable iii this instance. The reason lies in the fact that even the homogeneous this series never present a perfectly clean polygonal structure— the surfaces of the crystals always show a certain amount of marking or pattern, while minute CONSTITUTION OF Till! ALLOYS OF ALUMINIUM AND ZINC. 337 crystals of impurities are often scattered about the specimens. This " dirty " appearance of the sections may be due partly to the great difficulty of polishing them perfectly, but it certainly makes it difficult if not quite impossible to state definitely at which concentration the appearance of a given phase ceases — it becomes difficult to decide whether certain markings are due to the causes mentioned above, or whether they indicate the separation of the a phase along the line KL. For these reasons the lines in question have been drawn as dotted lines only, and it is prot>able that for the higher concentrations of aluminium they merely represent theoretical possibilities. As has already been indicated, the position of the solidus curve of these alloys has been determined by a series of quenching experiments. These were preferred to the • •in ploy merit of heating-curves because of the greater certainty of results from quenching, although a number of heating-curves were also observed. The quenching i-\pri imcuts in question consist, as is well known, in quenching a series of specimens from diUI-iviit t rmperatures, and subsequently examining the micro-structure of the quenched specimens in order to ascertain whether the presence of liquid at the moment of quenching is indicated or not. In the present series of alloys it was, however, essential that the specimens of alloy should be brought to a condition of equilibrium before being quenched, and for that reason a prolonged heating at a definite temperature was often required before the actual quenching operation itself could be carried out. The quenching apparatus devised by one of the authors* was employed for this purpose throughout the research. The special advantages of this apparatus made themselves felt in a very marked degree in carrying out this work. The specimens are heated in vacua, and this made it possible to employ very small specimens without fear of oxidation ; these small specimens, generally measuring about 6 mm. cube, remained perfectly bright and were cooled with very great rapidity. The temperatures of the specimens, up to the moment of quenching, was measured by means of a thermo-couple placed in a small hole drilled to the centre of the specimen of metal, and since in this apparatus the specimen is quenched by a powerful stream of cold water which strikes the specimen in the furnace itself, and carries it out of the furnace with the water, the quenching-temperatures were known as accurately as those of the arrest-points on the cooling-curves. The first series of quenching experiments were undertaken in order to prove that the line GH really constitutes part of the solidus curve ; this fact is important not only for fixing the position of the solidus curve itself, but also as supporting the evidence for the existence of a definite compound at the concentration where the step-up in the solidus is found. The result of one of these quenching experiments has already been (It-scribed and illustrated in connection with alloy No. 75 (74'5 per * W. K .si MI UN, " The Metallurgical Laboratories at the National Physical Laboratory," ' Journ. Iron and Steel Inst.,' 1908, I. VOL. OCX I. — A. 2 X 338 DR. WALTER ROSENHAIN AND MR. SYDNEY L. ARCHBUTT ON THE cent, of zinc) in figs. 28 and 29. The former figure shows the structure of this alloy quenched from a temperature above the line GH, and the presence of liquid at the liniment of quenching is quite evident from the presence of a finely-granulated matrix in the section of the quenched specimen; on the other hand, in fig. 29, the structure is that of the same alloy quenched from a temperature just below the line <;il. ami here, although perfect homogeneity has not been attained, the absence of liquid is quite evident, although some cracks and cavities are found, but these are present in all quenched specimens of this somewhat brittle alloy. The results of further quenching experiments are indicated on the diagram by dots surrounded by small circles in those cases where liquid was found and by squares in those cases where the alloy proved to have been completely solid. A much larger number of quenching experiments were made than those shown on the diagram, but only those have been plotted which lie close to the limiting temperature, since the others are of no direct importance in determining the solidus, and merely served as guides for the choice of fresh quenching temperatures. For the determination of the solidus curve from H to D over twenty quenching experiments were made, but only fourteen are plotted. The solidus curve has been drawn in such a manner as to pass between the nearest circles and squares (indicating liquid and solid respectively), some account being taken of the amount of liquid which was indicated by the observations on the various specimens. In order to illustrate the appearance of " liquid " in the quenched specimens, photo- micrographs representing the results of two quenching experiments are shown in figs. 35 and 36. Fig. 35 (x!50) refers to alloy No. 50 (49'8 per cent, of zinc) quenched from a temperature of 467° C. The presence of liquid is indicated iu this case by the broadening of the crystal boundaries and the presence of minute fusion spots in the crystals themselves. Fig. 36 ( x 150) shows a similar alloy quenched at a temperature just above the solidus. Here the indications of liquid are confined to small traces in the crystal boundaries, and there are no fusion spots. General Conclusions. The general conclusions arrived at as a result of the pyrometric and microscopic investigations of the aluminium-zinc alloys, which have been described above, may now be summarized. They are represented graphically in the equilibrium diagram of fig. 4. The addition of aluminium to zinc first depresses the freezing-point until a eutectic "»mn is reached at a concentration of 95 per cent, of zinc and a temperature of Further addition of aluminium raises the freezing-point until a concentration 5 per cent, of zinc is reached where there is a small but distinct break in the sing-point curve. From this point the freezing-point curve rises smoothly until CONSTITUTION OF THE ALLOYS OF ALUMINIUM AND ZINC. 339 freezing-point of pure aluminium !H reached. The phases which commence to i-r\ stallize along the four branches of the liquidus curve just described are — (1) The a phase, which is zinc, or zinc carrying in solid solution less than 1 per cent, of aluminium. (2) The ft phase, which has been shown to be a definite compound of aluminium and zinc, probably of the formula AljZnj. This compound is characterized by a well- marked tendency to crystallize in six-rayed dendrites ; it has only a limited range of stable existence, being decomposed on heating above 443° C. into y and liquid and on cooling below 256° C., breaking up into tin- two phases a and y. The product of the decomposition of this Compound, or of its solid solutions with the y phase, are found to assume a typical laminated structure closely resembling that of the " pearlite " of carbon steels. The ft compound is not capable of forming solid solutions \\itli ;m excess of zinc, but with aluminium it forms a continuous series of solid solutions cxtfinling up to pure aluminium itself. (8) The y phase, which is a solid solution of aluminium containing up to approxi- mately 40 per cent, of zinc. Between concentrations of from 40 to 85 per cent, of y.mc this solid solution reacts with zinc which may be present in either the liquid or solid form, to produce the compound AlaZns; from 78'3 to 85 per cent, of zinc the product of this reaction is the free ft compound, with a residue of liquid, but between 40 and 78'3 per cent, of zinc the resulting compound dissolves in the residual solid y phase, forming a series of solid solutions which have been termed the 3 phase. Between the concentrations named, this solid solution undergoes decomposition on cooling to the temperature of 256° C., separating into the a and y phases. It is probable, although it has not been- definitely proved, that both these reactions are continued in the alloys lying between the concentrations of 40 per cent, of zinc and pure aluminium. The researches described in the present paper have been carried out in the Department of Metallurgy and Metallurgical Chemistry at the National Physical Laboratory. The authors are indebted to members of the chemical staff", particularly Messrs. G. BARR and L. L. BIRCUMSHAW for carrying out the numerous analyses required in the research. As has already been indicated, they constitute a portion of a larger scheme of investigation of the alloys of aluminium which is being carried out at the Laboratory under the auspices of the Alloys Research Committee of the Institution of Mechanical Engineers. The authors desire to express their appreciation of the interest taken in this research by Dr. R. T. GLAZEBROOK, C.B., F.R.S., Director of the Laboratory. x 'J 340 DR. WALTER ROSENHAIN AND MR. SYDNEY L. ARCHBUTT ON Till! 8 . X 04 i « u; O o o Q g I •? OH o PU x h : to I- i § s s sa i I H I I I I I M •* co in « t— neooooinin'tsoi— oint-t— i~o>oo5C>>— — s-ioic^-^ SI 1O •* 00 •* -* -H i— "I i-H r-l r-l t-. p-H tf\ r-l i-c "M 71 (M Cl 1M S-l «ot~toi-aoooooo3Oo>coi^toc^ I I I I I I I I I I I I I I I J I I 1 I Illlcocococococococcc-; t-- oo i— i— ao oo t- o» rJO O O O OO 00 00 s § s CD ••* IM t~ «— 1-- CONSTITUTION OF THE ALLOYS OF ALUMINIUM AND ZINC. 341 O w H •- -'• " » * S'H 10 lO lO lO IO oo tt ip — on • ~ * ~ '/ — ti :" n ^« *5 CO O O> O Ol — i~ I- t- o — i O 0 — IS O — I I I I I I I I I , « o — ^ ft n I I I I I O5QOOJOO — icc-iojaooc-ieoujt-o — wt~<3i'M-«'oo-*i«i~iot--«wowc»oaO"«oioinototc«s>t'-i~t~«aoo»o»OOO — — (Mi-t-t^'C'^'^iO I gQOt-t- ^,^.^. ) O ~ OO w — I I I— « «O I I I I I t-t- 1- II I I I ocoo>— ' — ««— '^>ooo^"O»'«««a>o*t— oo Tl PR. WALTER ROREXHAIN AXD MR. SYDXEY I, ARCHBUTT ON THE DESCRIPTION OF PHOTOMICROGRAPHS (PLATES 5-7). PLATE 5. (Fl(!S. 12 AND 15 OMITTED.) Fig. 8. The a phase with small quantity of eutectic in alloy 98. x 150. 9. Decomposed /? phase in eutectic in alloy 95. x 1000. „ 10. Typical structure of eutectic. Alloy 95. x 200. „ 11. Typical structure of eutectic. Alloy 95. x 150. ., 13. Hexagonal dendrite of /3 phase (Al2Zns). Alloy 95. x 200. „ 14. Dendrites of /3 crossing at angles near 60 degrees. Alloy 94. x 200. „ 16. Alloy 90 cooled from fusion in 45 minutes. /? embedded in eutectic and partially decomposed at the edges, x 150. „ 17. Alloy 86 cooled at the standard rate used for cooling-curves, ft almost completely decomposed. x!50. „ 18. Alloy 86, under higher magnification, showing the laminated structure of decomposed /?. x500. PLATE 6. Fig. 19. Alloy 80, slowly cooled after annealing at 430° C., showing the persistence of eutectic. x 150. „ 20. Alloy 84, quickly cooled to a temperature just above 250° C. and then quenched, showing sheaths of /? surrounding primary y. x 150. „ 21. Alloy 84, slowly cooled from fusion, showing the more complete formation of ft and its subsequent partial decomposition, x 150. „ 22. Alloy 84, annealed for a considerable time at 430° C. and then slowly cooled, showing the complete formation and decomposition of /?. x 300. „ 23. Alloy 84, annealed at 430° C. and then quenched from a temperature just above 250° C., shoving the complete formation of /2 and the prevention of decomposition by quenching. x300. „ 24. Alloy 78, slowly cooled, showing dark cores of primary 7 surrounded by sheaths of decomposed /3 and a residue of eutectic. x 150. Alloy 78, similar to fig. 24, under higher magnification, showing laminated structure of decomposed /?. x 500. „ 26. Alloy 77, annealed at 430' C. and quenched just above 250° C., showing the homogeneous structure of ft or 8. x 600. „ 27. Alloy 77, annealed first at 430° C. and then just below 250° C. Laminated or pearlitic structure of decomposed /3 or 8. x 600. CONSTITUTION OF THE ALLOYS OF ALUMINIUM AND ZINC. 343 PLATE 7. Fig. 28. Alloy 75, quenched from 470° C., showing y crystals surrounded by " liquid." Etched with nitric acid. x 150. „ 29. Alloy 75, annealed for 30 minutes and quenched at 430° C., showing absence of " liquid " and approach to homogeneity, x 150. „ 30. Alloy 75, annealed for 2 hours and then quenched at 430' C., showing homogeneous S. x 150. .. 31. Alloy 75, annealed at 430° C., showing " pearlitic " structure of decomposed S. x 300. „ 32. Alloy 70, slowly cooled, showing cores of y, sheaths of partly decomposed ft, and residues of eutectic. x 150. „ 33. Alloy 70, slowly cooled after annealing at 430° C., showing " pearlitic " structure, x 170. .. 34. Alloy 40, slowly cooled, showing cores, x 150. „ 35. Alloy 50, quenched at 467° C., showing presence of " liquid." x 150. „ 36. Similar alloy to fig. 35, quenched just above the solidus, showing slight traces of liquid. x!50. [ 345 ] IX. Memoir on the Tkeori/ of the Partitions of Numbers. — Part VI. Partitions in Two-dimensional Space, to which is added an Adumbration of the Theory of tiie Partitions in Three-dimensional Space. By Major P. A. MACMAHON, R.A., D.Sc., LL.D., F.R.S. Received June 13,— Read June 29, 1911. Introduction. I RESUME the subject of Part V.* of this Memoir by inquiring further into the generating function of the partitions of a number when the parts are placed at the nodes of an incomplete lattice, viz., of a lattice which is regular but made up of unequal rows. Such a lattice is the graph of the line partition of a number. In Part V. I arrived at the expression of the generating function in respect of a two- row lattice when the past magnitude is unrestricted. This was given in Art. 16 in the form GF ( oo • « 6) = (!)+*»*' (»-b) I remind the reader that the determination of the generating function, when the part magnitude is unrestricted, depends upon the determination of the associated lattice function (see Art. 5, loc. cit.). This function is assumed to be the product of an expression of known form and of another function which I termed the inner lattice function (see Art. 10, loc. cit.), and it is on the form of this function that the interest of the investigation in large measure depends. All that is known about it a priori is its numerical value when x is put equal to unity (Art. 10, loc. cit.). The lattice function was also exhibited as a sum of sub-lattice functions, and it was shown that the generating function, when the part magnitude is restricted, may be expressed as a linear function of them. These sub-lattice functions are intrinsically interesting, but it will be shown in what follows that they are not of vital importance to the investigation. In fact, the difficulty of constructing them has been turned by the * 'Phil. Trans.,' A, vol. 211, 1911. VOL. GXIXI.— A 479. 2 Y 9.9.11 346 MAJOR P. A. MxcMAHON: MEMOIR ON THE formation and solution of certain functional equations which lead in the first place to the required generating functions, and in the second place to an exhibition of the forms of the sub-lattice functions. To previous definitions I here add the definition of the inner lattice function when there is a restriction upon the part magnitude, and it will be shown that the generating, lattice, and inner lattice functions satisfy certain functional equations both when there is not and when there is a restriction upon the part magnitude. There are two methods of investigation available. We may commence with a study of the Greek-letter successions (Art. 6, et seq., loc. cit.) from which the lattice functions are derived, and having obtained the functional equations which they satisfy, proceed thence to those satisfied by the generating and inner lattice functions ; or we may reverse the process, and, by a prior determination of the equations apper taining to the generating functions, arrive at those satisfied by the lattice and inner lattice functions. Both methods have been of service. The results, herein achieved, are complete so far as the lattice of unequal rows and the particular question under consideration are concerned. They are elegant and algebraically interesting. In proof of this, it will suffice to say that the generating function is unaltered when the lattice is changed into its conjugate. The subject thus swarms with algebraical relations which are established intuitively. Other results are obtained of a more general and extensive character which mark out the path of further investigation. Art. 1. I recall that for the lattice of two unequal rows, containing a, b nodes respectively, the established results are Inner lattice function = IL ( ; a, a-1) ; (3a)IL(oo;a,a,a) = (a)IL(oo; a, a, a-1); (a+2b) IL( oo ; a, b, 6) = (a+2) IL (oo ; a-1. b, 6)+(b) IL( » ; a, 6, 6-1) -(a+2) (b) IL ( oo ; a- 1,6,6-1); (a+b+c)IL( oo; c, 6, c) = (a+2) IL( oo; a-l, b, c) + (b+l) IL( ), as may be readily seen by putting the expression in the form We are therefore justified in putting IL( oo ; a, b, c) equal to the expression with an added term which contains the factor (c). Write, therefore, By working out several particular cases I was led to the conjecture F ( « : «> M = jj) {*>-(a-b)-.r-'(b-c)} ; and I then found that the expression does, as a fact, satisfy the functional equation. Art. 9. Having thus, beyond doubt, established the forms of IL ( oo ; a, b) and IL ( oo ; a, b, c), I proceed to a study of the functional equations. In the equation = (a+l)IL(oo;a-i,6)4-(b)IL(cx5;«,6-l)-(a-fl)(b)IL(oo;a-l>6-l), put X-{IL(cx>;a,6)-(a-fl)IL(oo;a-l,6)}=V1(oo;a,t); 352 then MAJOR P. A. MxcMAHON: MEMOIR ON THE which is of the same form as the original equation. Hence, if IL ( oo ; a, 6) be a solution of the equs ar-{IL(a,;a,&Ha+l)IL(oo;a-l,&)} is also a solution. exhibiting the new solution as the result of the performance of a certain operation upon the original one. Again put, in the original equation, a-{IL(cD ; a,fc)-(b)IL( a» ; a, 6-1)} = V2(°o ; a, 1) = O.JL ( » ; a, 6), and we find so that another solution is V,( oo ;«,&) = 0»IL(»; a,&). I write further O- 1L ( oo ; a, 6) = a;--*{IL ( » ; a, />)-(a+l) IL ( oo ; a-1, &) ; a, &-l) + so that, from the functional equation itself, and it is easy to verify that O0O(, = Oai. Art, 10. Since we know one solution of the equation H.^*>i^3,OT more conveniently (l) + a^+1(a— b)> or better still (a+l)-* (b), we may at once apply the operators O0, O6. Operating 8—1 times successively with Oa, I find O.-1 { and TIIKORY OF TIIK I'AKTITIONS OF NTMIll 353 We may therefore take (a+1) and (b) as the two fundamental solutions, and cle;n ly we may always multiply a solution by any function of x which does not involve " ->i //. 'I'll.' lin.il r\|irfssion of IL(oo ; it, b) which I adopt is (»+l), (b) 1, 1 I I. ( oo ; a, 6) = -J- s , 1 x, 1 and we will find that, expressed thus as the quotient of two determinants, it is generalizable. I might now, knowing d posteriori the expression for IL ( oo ; a6<-), proceed in a simpler manner than what follows ; but I think it better to put l>efore the reader the actual course that the investigation took. Art. 11. In the functional equation satisfied by IL(oo ; a, b, c), which may be written IL( oo ; a, b, <-)-(b+l) IL( oo ; a, 6-1, r)-(c) IL( oo ; «, 6, c-1) -(c)IL(oo; «-l, 6,c-l) + (b+l)(c)IL( oo ; a-1, 6-1,, •-!)}, I write V, ( oo ; a, b, c) = x-"-' {IL ( oo ; a, b, e)-(b + l) IL( oo ; a, 6-1, c) -(c) IL( oo ; a, b, c-l) + (b+l) (c) 1L( oo ; o, 6-1, c-1} ; and thence derive the relation (»+¥+•) V, (»; a, M = (a+2) V, ( oo ; «-l, 6, c) + (b+l) V, (oo ; a, b- 1, <-) + (c) V, ( oo ; a, 6, c-1) -(a+2) (b + 1) V, ( oo ; a-1, b- 1, r)-(a+2) (o) V, ( GO ; o- 1, b, ,-l) -(b+1) (c) V, ( oo ; «, 6-1, t-l) + (a+2) (b+l)(c) V, ( » ; a-1, 6-1, c-1). Comparini; this with the functional equation it is clear that V, ( oo ; a, 6, c), as defined, is a solution. I'r.KVfiliiii: similarly \\«- tindsix solutions \\liidi I i-xliihit as .>j,.Tat ions |ifrfo|-[ii.-,l upon IL ( oo ; a, 6, c) as follows :— x- {IL( oo ; a, b, o)-(a+2)IL(oo ; a-1, 6, c)} = O.IL( oo ; «, 6, c) ; *'• {IL( oo ; a, 6, <•) -(b+1) IL( oo ; a, 6-1, c)} = OJL ( oo ; a, 6, c)«; a:-{IL(oo;rt,ft,c)- (c) lL(oo;a,6,c-l)} = OJL ( oo ; a, 6, c) ; VOU CXJXI. — A. 2 Z MAJOR P. A. BUoMAHON: MEMOIR ON THE . • ' 1 + (b+l)(c)IL( oo ; a, 6-1, r-1)} = OJL( oo ; a, b, c) ; »;«)^--l)-(a+2)IL(oc;a-l>M + (c)(a+2)IL( oo ; a-1, 6, c-1)} = OJL(eo ; a, 6, c) ; IL(«;«-l,M)-(b+l)nJ(oc;a)fc-l,) + (a+8)(b+l)IL( oo ; o-l, 6-1, c)} = 0*IL(ao ; a, b, <•). I further write , N O.JL ( oo ; a, 6, c) = x— »-{IL( oo ; «, b, r)-(a+8)IL( oo ; a-1, b, c)-(b+l)IL( co ; a, ^-1, c) -(c)IL( oo ; a, 6, c-1) +(a+8)(b+l)IL( oo ; a-1, 6-1, c) + (a4-2)(c)IL(oo;a-l,6,(-l)+(b+l)(c)IL(oc;a>6-l,o-l) _(a+2)(b+l)(c)IL(oo ; o-l, 6-1, c-1)} ; and it is easy to establish the operator relations oaotoc = i> o.o^eu 0.0. = o«, o4oe = ofc, OaO,, = 0,0^ = 0,0^ = 0^=1. Art, 12. I now operate with these operators upon the known solution of the functional equation. To clear it of fractions I multiply throughout by (l)'(2). Operating m times in succession with Oc I obtain the result ^+> (b) + jJ«+»+4 (a-b) b-l)-^+3(2a-2b)} (c) {(1) +aj»«(a-b)} (c-1) (c). Whence I conclude that P, = (l)-xa+1(2)(b+l)-xa+!((a+l)+x6+!1(b)+xa+26+4(a-b)' are solutions of the functional equation. I find that but new solutions are obtained by operating upon PI, Pa, and P3 with Oa and O6. THEORY OF THE PARTITIONS OF NUMBERS. 355 iiii,' with 04, m times successively, upon P, I obtain and I draw the inference that (c-l)(c)(b+l) and (c-l)(c)(a+2) are solutions. Further, operating with Ot, m times successively, upon P,, I obtain and it thence appears that (a+l)(a+2)(c) and (b)(b+l)(c) are solutions. Again, operating with Ot, m times successively, upon P,, I obtain and the conclusion is that (a+l)(a+2)(b+l) and (b)(b+l)(a+2) are solutions. No other fundamental solutions are obtainable by operating with Oa, O6> and Oc upon P,, Pa, and P3, and clearly we have no need to consider the other operators because of the relations between them. We have thus six fundamental solutions Art. 13. The known solution of the functional equation from which these solutions have been derived can now be expressed in terms of these. Since it has been found that O." (l)a (2) IL ( oo ; a, 6, r) = Pl-*-+'P,+*-'-+3PSl we have (l)a (2) IL ( oo ; a, 6, c) = P.-*?, and, putting m = 0 in results obtained above, it appears that P, = (a+l)(a+2)(b+l)-*(b)(b+l)(a+2), P, = (a+1) (a+2) (c) -*> (b) (b+1) (c), 2 z 2 356 whence or MAJOR P. A. MxcMAHON: MEMOIR ON THE IL ( oo ; «, 6, c) = +*3(b) (b+1) (c)+z3 (c-1) (c) (a+2)-*4 (c-1) (c) (b+1), (c) -=- a-, 1, 1 x 1 *3, j_ O C. 4- 1 d a + o and herein putting a-6+1, a-o + 2, a-d+3, 6-e+l, 6-rf + 2, c-d+1 separately equal to zero, we in each case find two columns becoming identical and the deter- minant vanishing. Hence the sum of the coefficients has the proper numerical value. Art. 18. As a second test I will show that the quotient of determinants becomes unity on putting a = 6 = c = d. The numerator determinant becomes (a (• x(a+2)(a+3) x>+3) of x (a+2) (a) (a+1) (a+1) x (a) 1 Transform this by taking For New First Row- 1" Row+xa(l+x+x8)x2nd Row+x2"+1(1+x + x2)x3rd Row + x^x 4th Row, For New Second Row— 2nd Row + x" (1 + x) x 3rrt Row+ x2""1"1 x 4th Row, 3rd Row+xax4th Row, For New Third Row- and it becomes i 1 r r3 i, i, .1, & x, 1, 1, x X3, X, 1, 1 X6, X3, X, 1 and thus the quotient of determinants is unity. This verifies numerous particular cases. Art. 19. A third test is to show that the quotient of determinants has the value when 6 = c = a, 1 ' ' '"' i-6 , ar1 , x , 1 , •••» x * • * • ja, ,<->, JM. JW. .... i • being figurate numbers of order 3, 1 7%e Restriction on the Part Magnitude. Art. 21. I pass on to consider the case in which the part magnitude is restricted by the integer I. I take as my point of departure the functional equation (\-PI) (1 -p2) ... (1 -p.) GF (I ; a,, o» . . .', a.) = xs° GF (I- 1 ; als o,, . . ., a.), and, by means of the relation . » Ll (t ', Qi\, Cl2, . . . , dnj convert it into a functional equation for the lattice function. For the orders 2 and 3 we have = (a+b) {L (I • o-l, 6)+L (/ ; a, fc-l)}- a+b) L (Z ; o-l, 6-1) ; = (a+b+c) {L(Z ; o-l, 6, c) + L(Z ; a, fc-1, c) + L (« ; a, 6, c-1)} (»+b+c-2)(a+b+c-l) (a+b+ c) L(l ; o-l, 6-1, c-1) ; THEORY OF THE PARTITIONS OF NTMI'.KRS. and in general (l-plX)(l-paX)...(l-p.X)L(l; tt,,afc ...,a.) = x*°L(l-l ; a,, a,, ...,«.); wherein /> is the symbol of Art. 3, and symbolically X" = Art. 22. Also, from the relation L(/; a,, a,, .... am) n_1)(n_1) ...(aa+n_2) ..... .(l).. for the orders 2 and 3 we have L(Z ; a-1, 6-1) = (I+a+2) (l+b+l)(l+c) IL(1 ; a, 6, c) (I) (1+1) (1+2) IL (I- 1 ; a, b, c). While in general, if r, be a symbol such that ^ IL (/ ; a,, «„...,«.-!, ...,«.), S i n — 8/ rI)...(l-»-.)lL(/;aI, «„..., a.) Art. 23. I propose to obtain solutions of these functional equations. In order to ascertain the form of the required solutions it was necessary to examine several particular cases appertaining to the order 2 ; the result was the conjecture that - i M----i (2)(3)...(a+l).(l)(2)...(b) VOL. CCXI. — A. 3 A 362 MAJOR P. A. MAcMAHON: MEMOIR ON THE This expression was found to satisfy the functional equation, so that certainly IL (I ;«,&) = 1 +oj*+1 • , y1, ' £y; ; and then, observing that we may write l (a+1), (b) IL ' "(1) (l+a+1) x(fi } (l + 1) and remembering the nature of solution when I = oo , it became clear that we should seek solutions for the order 2 of the forms (a+1) v (b) F . • ' * r" (l+a+1) where F, is a function of I to be determined in each case. I therefore substitute fo"*"1) . F, for IL (I ; a, I) in the functional equation and (l+a+1) arrive at the relation (l)F, = (l+l)F,_i; from which I deduce F, = 0+1), yielding for me the fundamental solution (l+a+1) Similarly I find that another fundamental solution is 0>) (I) (l+a+1)' and, in terms of these two solutions, I find :o'6) = (i)t Art. 24. This simple exposition for the second order clearly points out the path of investigation for the third order. For, guided by the six fundamental solutions when / = oo, it is natural to seek for solutions of the functional equation of the six types (l+a+1) (l+a+2) (l+b+1)' . (l+a+l) (l+a+2) (l+b+1)' (c-l)(c)(a+2)F, (l+a+1) (l+a+2) (l+b+1) ' (l+a+1) (l+a+2) (l+b+1) ' (l+a+1) (l+a+2) (l+b+1) ' (l+a+1) (l+a+2) (l+b + 1) ' where Ff is a function of / to be determined in each case. THEORY OF THE PARTITIONS OF NTMMKIIS. :;•;:; Substituting the first of these for IL (I ; a, 1>, c) in the functional equation of order 3 I find F _ but the solution of the equation F, = <£jF,_! is clearly F, = <£,<£;_ !<£,_j ... so that, in the present case, F -(1+1) (1+2) (0(1+1) (l- -- a • (i_2)a so that I obtain a fundamental solution Art. 25. Similarly I arrive at five other fundamental solutions, (a+1) (a+2) (c) (l)(l+l) (1+2) (b)(b+l)(a+2) (l)(l+l) (1+2) (c-1) (c) (b+1) (l- and I next seek, guided by previous work, to construct the function IL (I ; a, b, c) by a linear function of these six solutions. It is natural to write (a+1) (a+2) (b+1) (l+l)2 (1+2) -x (b) (b+1) (a+2) (1) (1+1) (1+2) -x (a+1) (a+2) (c) (1) (l+l) (1+2) +xs (c- 1) (c) (a+2) (l)a (l+l) +x* (b) (b+1) (c) (l-l) (1) (1+2) -x* (c-1) (c) (b+1) (1-1) (I) (l+l), with a denominator (l)a (2) (l+a+1) (l+a+2) (l+b+1), 3 A 2 ;!C,4 MAJOR P. A. MAcMAHON: MEMOIR ON Till: which, in determinant form, is ar(a+2)(l) , l-l)(l) , a: (I) (l+l) , divided by (I)2 (2) (l+a+1) (l+a+2) (l+b+1) ; and it will be shown that it is, in fact, the expression of IL (I ; a, b, c). Art. 26. In a general manner we may take the following view- Recalling a previous notation for the order 3 , A, = , C8 = (c- Cx = (c), A0 = B0 = C0 = 1 ; and taking a product where p, q, r denotes some permutation of the numbers 2, 1,0, suppose _ _ (l+a+1) (l+a + 2) (l+b+1) substituted for IL (I ; a, b, c) is the functional equation. The result on reduction is F - and now writing (0 (l+l) (1) (1-1) (1+2) (l+l) (I) adinf-6 '*»! Fj = apbgcr ', and the inner lattice function IL (I ; a, b, c) is, to a divisor (I)3 (2) (l+a+1) (l+a+2) (l+b+1) TIIKORY OF THE I'AI.TITK >NS OF NUMBERS. prbs, equal to the determinant 365 and this determinant is clearly A3 B3 a;C3 C,(l+2) Art. 27. This is evidently a perfectly general process and suffices to establish that a solution of the functional equation of order n is a determinant of order u of which the constituent in the slh row and tili column is r\ } (at+s-t+l) ... K-f n-t) . (l-s+t+1) ... (l+t-1) ; and when this determinant is divided by we have, as will be proved, the expression of Art. 28. To establish this we may apply a series of tests. Thus, take the expression of IL (I ; a, b, c, d) x(l)(a+2)(a+3), (l+l)(b+l)(b+2), (l+2)(o) (c+1) , x (1+3) (d-1) (d) divided by It clearly reduces to IL ( oo ; a, b, c, d) when I is put equal to <» . Moreover, it can be shown that when d =• c = b = a, the determinant involves I and a always in the combination l + a; for consider the determinant in question when I is put equal to —a. 366 MAJOR P. A. M.vcMAHOX: MKMDIR OX TIIH •b S a £ ft 2 rH 1 a G B ^^ e a B ^3 1 7 1 T ^ + ^ 'e? ^ 03 T 1 c > T* 1 i CO oS ^T ci A rH 1 7 cS rH M cS ,5 .2 II £ ° 0 0 -j- ^ ^ rH jL - jj « 1 e ^Pj + cS I rH rH rH -}- rH ^ _ ^ w ^ ^O rH '-3 54-1 ^**i «4H ^* -^ -H>- S ^U * ,5, - — cS -)- "caT JJ. -^j- o •^ «" 1 cS ~~^ - — ^ ^-' ^g. o V3 c5~ ^J J, i _1_ *~~* | rH C^ O4 ,"*"! '^ a • rH^ CT i cS rH - — i 1 c^ 1 § O ^Pj i CM V J^ ej I T >s 1 cS « ^^i i^ ^ 03 .2 rH .^2 I 1 o • ->j d P< -^ * ^*£ ** " C3 ^ o "'^— ^ tj '*«-^^' CQ .2 w • rH i^t CO * * •» J x— s 1 -— -^ -^> '-p CO cS ^*^ ^_^ , ^ rH CL oa TH gj ^H ^_^ + jL + cd ^_ ^> ^ «§ .2 cS rH **~ ^^ I C^ + ^ eS ^ rH rH -^- to ™ n 1 M O) ^ x -S i eS rH rH ^-^ "• CJ x-*** >^— ^ C^-< Li rH i ^-*v '~J _ r-^ rH ^ X 41 0 "^~" ** "T" "^5 CO -TH" -i- "J 1 ^ ? i 1 ^ a> a> -o ° d ^ -S ^ «C + | X fl vS -5- e£ 3 T" "pi 1 °* •£• rH >— ' TH ^ --^ c3 ^3 "2 1 — ' ^ 2~^ ^-" ^ ™^~ cS 1 ."~ u c3 3 _j 4l- 08 | 08 ^ ^_^ - _J a H " — ' 1 1 ^ CO ^ ^ ^ ^~v 'S 's - • *— - H ra" ' _$ -)- CO CO OJ g E x- v. H ^ + + + 'a ^2 -2 co" - - • cS "* G ^ X X X *S -*J S S -5 7 ^.^ co^ 1 "*"* 1 OS rH rH ^ "5 ^ of X PI co" CO + i ^ •^ vS ^i. c8 ^ jg a> x-. ^ 7 c^ c3 rt "oT If rH + I I ^^"^ fr^ **"^^N O "^ 8 3 CD GO • i—* rH s^^ hrl HH ^5 (c i _j_ ^ i i 1 H . « - ^ ^7 : 'JT ^ ^ TH X + TH + •g ^ ^ TH . : + X~s ^ 1 2. x— - ^_^ £ ~ — • ^ ± + oS I-1 - — - TH a H ^ + H •^ ^— *s * — •»" °S • ^ ~— " ~~~? 9 ££• -5, y" «" v~' f^ X"**- *- _ " - o y • K . ^ , ^p S TH '. ^ TH . . • ^- •»« ** ~n • . „ • •2 ^ •^ ST "^ TH" ^ TH i ^ •£• X e" ^ ^^ —• ' ^^ ^-~> 03 3 • + ^ 1 [ + TH T" A ! e^ H •iT -«-> '3 •-g ^ ~T | i ^ • ^-< ^ cr1 CE] 00 • ^ ^ ^ — 0) TH + TH TH CO~ s £ e-T « 0 1 -C s"*^^ ""^-^l -i— 1 1 j .2 : ; ^ II £, ^ vS ^ rH~ a _o eg g •^ »-< . — . . c5^ ^_^ ^^ -*j - — c« • | '• + J «" ,—, > !> •,7 ± r ^" -2 . g «2 .2 -P ^j r* * 3 •M H 3 alt the second > «2) ..., the nth > «„. Denoting the generating function of these by , a2, ...,«„; n), X T1IKORY OF THE PARTITIONS OF NUMBERS. 369 X 0 (1)(2)(3) GF(«Ac,e*;4) x X»(c) X» x , XM(b+l), X31(c)(c+l) 0 b (!).. .(5) (a+1), (b)(b+l), x , X«(b+l), 0 ,' x 0 , 0 , 0 0 xX41(d-l)(d) x , XM(c+l) , X4S(d) 0 , x , X43 GF(a,b,c,d,e;!>) 5+1), .r3(d-2)(d-l)(d)(d+l), x>-3)(e-2)(e-l)(e) X31(c)(c+l) , xX41(d-l)(d)(d+l) , .r'X51(e-2)(e-l)(e) Xsa(c+l), , X«(d)(d+l) , xX( x X43(d+l) X5a(e) 0 x XM and tlie law is evident. Art. 32. In general, supposing the lattice to l)e in the plane of xy, that of the paper and the axis of z perpendicular to the plane of the paper, if we project the partition on to the plane of yz, we obtaiu a partition at the nodes of a lattice of I rows in which the part magnitude in the slh columns is limited by the number ar The general formula for GF(Z; a,, aa, ..., a,) is remarkable from the fact that GV(l ; a,, «„ .... a.) = GF(Z ; 6,, 6,, .... 6.), where (at, aa, ..., a,), (blt l>3, ..., bm) are any two conjugate line partitions. ,'• Adumbration of the Three-dimensional Theory. Art. 33. 1 conclude this Part by pointing out a path of future investigation into the Theory of Partitions in space of three dimensions. I consider a complete or incomplete lattice in three dimensions, the lines of the lattice being in the direction of three rectangular axes of x, y, z -4 respectively. Just as an incomplete lattice in two dimensions is defined by a one- VOL. ccxi. — A. 3 B 370 MAJOR P. A. MACMAHON: MKMolR ON Till. dimensional partition whose successive parts specify the successive rows of the lattice, so an incomplete lattice in three dimensions is defined by a two-dimensional partition whose successive rows specify the successive layers of the lattice. I shall suppose these layers to be in or parallel to the plane of xy which is the plane of the paper, and the axis of z to be perpendicular to the plane of the paper. Descending order of magnitude of parts placed at the points of the lattice is to be in evidence in the three directions Ox, Oy, Oz. Art. 34. Consider the simplest case of a complete lattice, the points forming the summits of a cube. The two-dimensional lattices a^/Si/?,, a2a2;82/J2, in and parallel to the plane of the paper are superposed to form the three-dimensional lattice. Suppose that the first 8 integers are placed at the x points of the lattice so that descending order of magnitude is in evidence in the directions Ox, Oy, Oz, e.y., one of 48 such arrangements is as shown. I associate with the first and second rows of the first layer the letters a,, /8i respectively, and with the first and second rows of the second layer the letters «a, /82 respectively, and then from the illustrated arrange- ment of the first 8 numbers I derive a Greek-letter succession in the following manner : — I take the numbers in descending order of magnitude and write down the Greek letter with which the position of each number is associated : thus the arrangement above gives 87654321 Art. 35. In this Greek-letter succession we have to note (i.) A /3 which is succeeded by an a, (ii.) An a which is succeeded by an a with a smaller suffix, (iii.) A /8 which is succeeded by a /3 with a smaller suffix. If a letter which is thus noted is the sth letter in the permutation I associate with the permutation the power x*1, and taking the sum of these powers in respect of the whole of the permutations associated with and derived from the lattice I obtain the lattice function 2.x2'; and, following the reasoning of Part V., Art. 6, I derive the generating function for partitions at the points of the lattice, the part magnitude being unrestricted, viz., THEORY OK THK PARTITIONS OK NUMBERS. 371 Siniil.-.rly, from the ( Jreek -letter successions \\hidi involve / n»ted letters, I derive the sub-lattice function of order t, and thence, by previous reasoning, arrive at the ^em-rut ing function when the part magnitude is restricted by the integer I, I. 22 . 22) = SL<( °° ; 22 ; 22)(l-t-|-l)(l-t-l-2) ... (l-t+8) Art. 36. The method is generally applicable to any incomplete lattice in three dimensions. I work out in detail the case in which the points of the lattice form the 8 summits of a cube, in order to show that the result obtained, in Part II., Section 7, in quite a different manner, is verified. That result, with modified notation, was Generating Function where P(*)» R(a;) = 2zIO+2zll+3xI>+2.c18+2a;14. I shall now show that are, in fact, the sub-lattice functions of orders 0 to 4 which appertain to the lattice formed by the summits of a cube. I write down the 48 permutations of the Greek letters and over each the arrange- ment of the first 8 integers from which it is derived, the lower layer of numbers being placed to the left :— 3 B 2 MAJOR P. A. MACMAHON: MKMOIR ON THE 87 65 43 21 86 75 43 21 86 54 73 21 87 54 63 21 85 74 63 21 al«2& I «laS 87 64 53 21 87 43 65 21 84 73 62 51 87 63 54 21 87 63 52 41 *i«h*iPi&»\*&Ps, 87 65 42 31 87 62 54 31 85 64 73 21 86 74 53 21 86 43 75 21 84 63 72 51 86 73 54 21 86 73 52 41 86 53 74 21 86 53 72 41 87 53 64 21 «2/32/32, 87 53 62 41 86 75 42 31 86 72 54 31 85 73 64 21 85 73 62 41 86 54 72 31 86 52 74 31 87 54 62 31 87 52 64 31 85 74 G2 31 85 72 64 31 87 64 52 31 87 42 65 31 84 72 63 51 87 62 53 41 85 63 74 21 85 63 72 41 85 64 72 31 «2 85 62 74 31 86 74 52 31 86 42 75 31 84 62 73 51 86 72 53 41 a,aa 86 52 73 41 87 52 63 41 85 72 63 41 85 62 73 41 THEORY OF THK I'AKTITIONS OF NUMBERS. 373 )+ a;11, A dividing line has been placed after each letter that has to be noted. Thence, by the rules given, Lo(oo;22;22)= 1, L, ( oo ; 22 ; 22) = 2x'+2x3+3a;4+2:r5+2z9, L,(t» ; 22; 22) = La( oo ; 22 ; 22) = L1(oo;22;22) = xw, supplying a complete verification of the work in Part II. We have, therefore, GF(/; 22; 22) - r (L+ll v JL+ 8) L (1). ..(1+7) L (l-l)...(l+8) (1)...(8) (1)...(8) (1)...(8) Q-2). .. We have evidently, potentially, the complete solution of the problem of three- dimensional partition, and it remains to work it out and bring it to the same completeness as has been secured in this Part for the problem in two dimensions. This will form the subject of Part VII. of this Memoir. [ 375 ] X. Radiation in Explosions of Coal-Gas and Air. By W. T. DAVID, E.A., B.Sc., Trinity College, Cambridge. Communicated by Prof. B. HOPKINSON, F.R.S. Received May 24,— Read June 29, 1911. IN the first part of this paper results of experiments are given on the radiation emitted during the explosion and subsequent cooling of mixtures of various strengths and densities of Cambridge coal-gas and air. Bolometric measurements were made of that part of the radiation from the hot gaseous mixtures which was transmitted through clear plates of fluorite, quartz, plate glass, and water (contained between two plates of glass). The fluorite (6 mm. thick) transmits very approximately 95 per cent, of the total radiation emitted by the gas ; the quartz (also 6 mm. thick) transmits about 70 per cent, of the radiation from water vapour and cuts off a very large proportion of that from COa ; the water cell transmits practically only luminous radiation. It has been therefore possible to estimate fairly accurately the total radiation emitted by the gas, and, roughly, the proportions emitted by water vapour and by COj, and also the amount of energy in the luminous radiation. The radiation emitted in the explosion of a 25-per-cent. mixture of hydrogen and air has also been measured. The second part of the paper consists of an investigation into the diathermancy and emissive power of the hot gaseous mixture after explosion. The conclusions drawn from these experiments offer an explanation of many of the peculiar results given in the first part. Prof. HOPKINSON has already shown that the heat lost by radiation in explosions of coal-gas and air is a considerable fraction of the total heat of combustion of the gas. Recently he has made a very complete investigation of this radiation loss in 15-per-cent. mixtures.* He exploded mixtures of the same strength in a vessel of •bout f cub. ft. capacity whose walls were silver-plated, first, when the walls were highly polished, and then when these same walls were coated with a thin layer of dull black paint. In the first case the maximum pressure developed was about 3 per cent, greater and the subsequent rate of cooling much slower than in the second * 'Roy. Soc. Proc.,' A, voL 84, p. 155. VOL. ccxi. — A 480. 20.10.11 376 MR. \V. T. DAVID ON THE RADIATION IN case. By measuring the radiation emitted by the hot 'gaseous mixture (by means of a bolometer protected by a plate of fluorite) he shows that these results are in a large measure, if not entirely, due to this radiation being reflected back by the polished walls and reabsorbed by the gas, this reabsorption of the radiation being ultimately realised, at any rate in part, as pressure or translational energy. No other work has been done on radiation in gaseous explosions. Some interesting measurements, however, have been made on the radiation emitted by flames and by heated gas. Those made by Prof. CALLENDAR on flames and by Prof. PASCHEN* on heated CO» are particularly interesting and will be referred to later on in this paper. Description of Apparatus. The explosion vessel used in these experiments consists of a cast-iron cylinder, 30 cm. in diameter and 30 cm. long, on to which are bolted two end plates. It is shown in section in fig. la together with the bolometer holder H. Fig. 1. The bolometer was cut into the form of a grid from a circular disc of platinum about £ mm. thick, weighing 0'25 gr. per sq. cm., and had, therefore, a thermal capacity equivalent to O'OOS gr. of water per sq. cm. (the specific heat assumed to be 0*032). Its resistance measured about O'llS ohms at 15° C. The temperature coefficient was measured and found to be 0'0036. The bolometer was mounted on a hollow cylinder of wood (W in fig. l) and was pushed into the gunmetal holder H, which carried at its inner end the plate of diathermanous substance. The holder H was screwed into a boss on the end cover of the vessel and was tightened up with the lock nut L. In the experiments described in the first part of this paper the bolometer was placed close up to the diathermanous plate, so that all the radiation from the hot gaseous mixture, which was transmitted through the diathermanous substance, fell on its blackened surface. In order to measure the amounts of radiation absorbed by the bolometer it was necessary only to measure its rise of temperature. This was KXPLUSIONS OF COAL-GAS AND All!. 377 liy recording (lie rise of its electrical resistance, which is proportional to its rise of teni|>erature, as it warmed up. The arrangements used to record its rise of resist, mice \\i-ro in principle exactly the same as those used by HOPKINSON for determining (lit- rise <>f resistance of the copper strip in his Recording Calorimeter.* The method consists in passing through the bolometer a constant continuous current sullicient to produce a convenient difference of potential at its terminals, balancing this dillerence by means of a source of constant E.M.F., and recording by means of a mirror galvanometer the rise of potential which occurs when the bolometer gets wanned up. The deflection of the galvanometer is proportional to this rise of potential difference, which, since the current through the bolometer is constant, is pro- portional to the increase of its resistance and, therefore, to its rise of temperature. The connections are shown diagrammati- cally in fig. 2. The current C, passing through the bolometer B is taken from a battery of storage cells giving about 100 volts. In this circuit are included a bank of lamps, L, and ammeter, AI. With this arrangement the small variation in the resistance of the bolo- meter as it gets warm has no appreciable effect on Ci since the resistance of the lamps L is very large compared with that of the bolometer. The current C|, which is measured by the ammeter A!, is adjusted to a convenient value before the experiment. The terminals of the bolometer are connected also to the galvanometer G through a resist- ance, R, in which a constant current, C2, is maintained by means of two storage cells, (In- direction and magnitude of C2 being such that the potential difference at the terminals of the bolometer before the experiment is balanced, or approximately balanced. The relation between the rise of potential difference at the terminals of. the bolometer and the galvanometer deflection was found by passing small currents through the bolometer, when it was at a known temperature, and noting the galvanometer deflections, C2 having been reduced to zero during the calibration. The recording galvanometer used in these experiments had a resistance of about 3'5 ohms. It was of the suspended coil type, with a fairly stiff phosphor-bronze Fig. 2. NOTE. — (', was kept constant during each experiment, but was, in general, different in different experiments. VOL. CCXI. — A. * ' Roy. Soc. Proc.,' A, vol. 79, p. 140. 3 C 378 MR. W. T. DAVID ON THE RADIATION IN suspension, having a period of alx>ut ~£$ second. The field was produced by means of an electromagnet, magnetised nearly to saturation value. For recording the pressures a Hopkinson Optical Indicator was used. The mirrors on both the indicator and the galvanometer were connive, and by means of them two spots of light were reflected on to a photographic film, revolving at a known speed, care being taken that the two spots of light were on a line at right angles to the direction of rotation of the film. By this means both the rise of temperature of the bolometer and the pressure of the gas were recorded at the same time. The inflammable mixture was introduced into the Vessel in the following way : — The vessel was first exhausted by means of an air pump, and the quantity of coal-gas required to give the mixture a certain strength was admitted ; air was then allowed to rush in bringing the pressure up to atmospheric. The gas when let in to the explosion vessel at low pressure quickly diffused throughout the whole space, and the air afterwards rushing in at high velocity thoroughly stirred up the mixture. To make certain that the mixture was homogeneous, it was allowed to stand for about half-an-hour before firing. This is a method of mixing recommended by Mr. DUUALD CLERK.* In all the following experiments the mixture was fired by means of an electric spark at the centre of the vessel. Fig. 3 is a print from an actual record taken during the explosions of a 'J'8-per-ctMit. mixture of coal-gas and air when the bolometer was protected by the plate of fluorite. Curve P gives the rise of pressure (measured downwards from the atmosphere line P0) 0-6 0-56 0-5 0-45 0 '4 0 "35 0 '3 0 '25 0 "2 0 "15 0 '1 0 '05 0-00-9 0 '8 -^ second 1 '0 Fig. 3. and curve G the galvanometer deflection (measured upwards from the zero line G0). On the pressure curve P, 1 mm. deflection corresponds to a rise of pressure of 4 ll>s. per sq. in., equivalent to a rise of temperature of 80° C.t On the curve G, showing the galvanometer deflection, 1 mm. corresponds to a rise of temperature of the bolometer of 1'36° C., or, since the thermal capacity of the bolometer was equivalent * 'The Gas, Petrol, and Oil Engine,' vol. I., p. 156. t Temperature before firing 14° C., barometer 760 mm. of mercury, contraction of volume on combustion 2 • 5 per cent. EXPLOSIONS OF COAL-GAS AND AIR. 370 to 0*008 gr. of water per sq. cm., to an absorption of heat of 0'0109 calories per sq. cm. In order to apply a correction for the loss of heat by the bolometer as it warms up the record was continued for some time after the radiation from the gas was inappreciable.* From the rate at which the galvanometer deflection decreased the rate of loss of heat by the bolometer was determined. In this way it was found that in this particular record the loss amounted to 6'1 per cent, of the heat in the Ixilometer at 0'5 second after ignition, and 13'5 per cent, at 1 second after ignition. RESULTS OF EXPERIMENTS. — PART I. The following radiation measurements were made in explosions of mixtures of Cambridge coal-gas and air of various strengths and densities. Measurements of the radiation received by the walls per sq. cm. of surface were made at three different places on one of the end covers, A, B, and C, as shown in fig. \b. The amounts measured in the three places were distinctly different and showed peculiarities which will be discussed later. In all the experiments about to be described in this section the interior surface of the explosion vessel was painted over with a thin layer of dull black paint, so that practically all the radiation emitted by the gas was absorbed by the walls, t Two or three records were generally taken under the same conditions ; those taken on the same day gave precisely the same results. Mixtures of Various Strengths fit Atmospheric Density. The results shown in figs. 4-11 refer to experiments made with the fluorite window. An allowance of 5 per cent, has been made for the absorption of the fluorite,| but no allowance has been made for reflection from tin- blackened surface of the bolometer. The curves in these figures, therefore, show the radiation surl><'etween 2/i and Bji. I A clear plate of fluorite, from 5 mm. to 10 mm. thick, transmits very approximately 95 per cent, of incident radiation up to 8/i. This figure was checked in the following way : — The radiation was measured when the window consisted of a quartz plate only, and also when the plate of fluorite was placed in front of the quartz in explosions of identical mixtures. In the latter case the radiation measured was almost exactly 95 per cent, of that in the former case. There is very little energy in the emitted radiation of wave-length greater than 8/1. 302 380 MR. W. T. DAVID ON THE RADIATION IN of surface, in the three positions A, B, and C, as ordinates with times from ignition as abscissae in explosions of approximately 9'8-per-cent. mixtures. These curves were taken from those on the films traced by the recording galvanometer after allowing for a small loss of heat from the bolometer. This loss was determined for each film in the manner shown previously. The curves marked B, b were taken from the record, a print of which is shown in fig. 3. T/me offer Fig. 4. Gas-temperature curves are also shown in the same figure, corresponding to the radiation curves. These curves were deduced from the pressure curves on the films by means of the equation pv = R0, after allowing for a 2'4-per-cent. contraction of volume (which occurs in the combination of a 9'8-per-cent. mixture of Cambridge coal-gas and air). They give the mean absolute temperatures of the gaseous mixture, assuming it to be a perfect gas, or, at any rate, having the difference of its specific heats at constant pressure and constant volume independent of the density and temperature. Fig. 5 shows the rates at which the walls are receiving heat by radiation in calories per sq. cm. per second at the three places A, B, and C, plotted against the mean absolute temperatures of the gas. These curves have been obtained from the radiation curves in fig. 4 by differentiation. It will be noted that the top parts of the end cover receive more heat by radiation than the bottom parts, or, in other words, the hot gas at the top of the vessel radiates more strongly than the colder gas at the bottom. EXPLOSIONS OF COAL-GAS AND AIR. :J81 The dotted curves A' and B' in fig. 4 are the differentials of the radiation curves A and H in the same figure. These curves show very plainly that the gas at the top <>(' the cylinder radiates much more strongly than that at the bottom during explosion (except for a moment just before the attainment of maximum pressure) as well as SSOO WOO S3O0 tSOO /fOO tOOO Gas Temflem/are — °C absotuff Fig. 5. during cooling. The most interesting point, however, shown by these curves is that the rate at which the gas emits radiation is less at the moment at which the mean temperature of the gas is a maximum (which is the same as the moment of maximum preasure) than it is some little time before. An examination of the curves shows that the rate of emission of radiation is a maximum about 4*0 of a second before the attainment of maximum temperature, the "time of explosion" being 0'18 second. At this point the temperature of the gas is about 1600° 0. (abs.), and pressure about 65 Ibs. per sq. in.', the maximum temperature and pressure subsequently attained being 1700° C. (abs.), and 70 Ibs. per'sq. in. respectively. Prof. HOPKINSON,* experimenting with a very much larger explosion vessel (of G'2 cub. ft. capacity), found that in a 10-per-cent. mixture of coal-gas and air the flame completely fills the vessel about 3*0 second before maximum pressure is attained, the "time of explosion" being 0'2f> second. The preasure at this point was about 70 Ibs. per sq. in. and the maximum pressure reached -£$ of a second afterwards 82 ll>s. Thus it appeal's that the maximum rate at which the gas emits radiation occurs very approximately at the moment when the flame completely fills the vessel.t Experiments were next made with 15-per-cent. mixtures. The results are shown in fig. 6, the radiation curves again showing the amount of radiation received in calories per sq. cm. of surface at various times after ignition in the same three positions, A, B, and C. The corresponding gas-temperature curves are also shown in the same figure ; these were deduced from the preasure records after allowing for a * ' Roy. Soc. Proc.,' A, vol. 77, p. 389. t I have several records which show that the gas radiates most strongly some time before the attain mi'iit of maximum pressure in explosions of from 10-per-cent. to 12-per-cent. mixtures. The period ot the galvanometer is not sufficiently low to determine definitely whether the same thing happens in stronger mixtures. MR. W. T. DAVID ON THE RADIATION IN 800 400 -2 0 0 •J -4 -S -6 -7 T/me after Ignit/on - seconds. Fig. G. •9 1-0 3'6-per-cent. contraction of volume which occurs in the combination of a 15-per-cent. mixture. Fig. 7 shows these radiation curves differentiated, the rates of radiation in calories per sq. cm. per second being plotted against the mean absolute temperatures of the gas as was done for the 9'8-per-cent. mixtures in fig. 5. At the moment of maximum pressure the rate of receiving heat by radiation at the 5-0 4-0 W/x/ures. ( firs/- s, *• — *j ^4oo 2200 2000 isoo teoo tioo Gas Jf/nfitrafure — °C aiso/uk. Fig. 7. 1SOO 1000 KX PLOSIONS OF COAI^GAS AND AIR. 383 centre of tl ml rover (position C) is alioiit .VI <-alories per sq. cm. per second, at the t-.|i (position A) it is 4'85 calories, and at the bottom (position B) it is 475 calorics, at a slightly lower maximum temperature. As cooling proceeds the centre still continues to receive more radiation than either the top or bottom parts of the end rover, and the top more than the bottom, until the gas temperature falls to about 1 800° C. (abs.), when every part of the end cover seems to be receiving radiation at approximately the same rate and continues to do so until a temperature of something like 1300° C. (al».) is reached. After this the top receives more radiation than either the centre or bottom and the centre more than the bottom, owing to the hot gas at the top of the vessel emitting more powerfully than the colder gas at the bottom of the vessel. In fig. 8, Curves I. and III. show the results of later experiments made with 15-per- cent, mixtures and 13-per-cent. mixtures respectively, and Curves II. and IV. show the results of the previous experiments made with 15-per-cent. mixtures and !)'8-per- I! •« " I* ^ j< zcw t-o /600 -8 (SCO 800 4 400 offtr Tyni//on - Fig. 8. cent, mixtures (from figs. 4 and G). The Curves H refer to experiments made with a i'.r)'4-per-cent. mixture of hydrogen and air (see p. 392). The radiation curves in this figure are the means of those taken with the bolometer in positions A, B, and C. The corresponding gas-temperature curves are also shown. The experiments with the 9'8-per-cent. mixtures and 15-per-cent. mixtures (first series) were made within a week of each other, during which time the calorific value of the coal-gas was probably Curves!^ — 15% mixfures (second ser/ti ) Cf/nf scr/'et ) ~ t<5"/o Wx&rct Curves HE. — 9-8% frmi /arcs ffrom ^ CvrresH. -25-4%mMurts 1280 0-91 24-2 0-7o 1060 0-960 25-7 1-0 930 0-98 26-1 TABLE IV. — 13-per-cent. Mixtures. Initial Pressure, Atmospheric. Heat of Combustion of Coal-gas present in Vessel = 14,230 Calories. Time from ignition. Mean absolute temperature of gas. Mean radiation received by walls per sq. cm. Total loss of heat by radiation per cent, heat of combustion. 0-07 o-i max. temp. 2170 2080 0-12 0-23 3-7 7-1 0-15 1920 0-38 11-7 0-2 1780 0-48 14-8 0-25 1660 0-56 17-2 0-5 1280 0-74 22-8 0-75 1070 0-8 24-6 1-0 940 0-81 25-0 After 1 second from ignition the radiation emitted is very little. At this time each mixture has radiated about the same proportion, viz., 25 per cent., of its heat of combustion. The amount of heat lost by radiation up to the moment of maximum temperature is roughly proportional to the product of the third power of the maximum tem- perature (absolute) attained into the " time of explosion." This will be seen from the following table, where Iim is the radiation absorbed by the blackened wall per square centimetre up to the moment of maximum temperature, 6m the maximum temperature (absolute) attained in the explosions, and t the "time of explosion." VOL. ccxr. — A. 3 D MR. W. T. DAVID ON THE RADIATION IN TABLE V. Mixture strength. '• 0,,, RM- Rm 0,,? x t. per cent. 9-7 0-18 1700 0-17 1-92x10-'° 15-0 0-05 2360 0-11 1 -67 x 10-i° (1st series) 15-0 0-05 2410 0-128 1-82x10-'° (2nd series) 13-0 0-07 2170 0-12 1-68x10-'° The curves in fig. 9 are the differentials of the radiation curves shown in fig. 8 plotted to a gas-temperature base. They show the mean rates at which radiation is received by the black walls of the vessel (in calories per square centimetre per second) from the maximum temperatures attained by the various mixtures after explosion down to 1000° C. (abs.). These curves show that the weaker mixtures radiate much 2600 24-00 /60O 1400 TemAera/ure. - "C aiso/att 2200 2000 tSOO Gas fSOO 100O Fig. 9. more powerfully in the initial stages of cooling than the stronger mixtures do when they have cooled to the same temperatures, although there is very much more radiating gas present in the latter mixtures. Later on in the cooling the radiation from all the mixtures is very much the same at the same gas temperatures. The chain-dotted curve is a 0* curve, where & is the mean absolute temperature of the gas, made to coincide with the radiation curves at the low temperatures. EXPLOSIONS OF COAL-GAS AND AIR. i nt Mi.rtures of the. SHHW tif Kxperiments of the same kind wen- then made with 1 5-per-cent. mixtures at \ ;n i<>us densities. The results of these experiments have been collected into fig. 10. Tlit- radiation curves in this figure are the means of those taken with the bolometer in positions A, B, and C on the end cover. Curves I. refer to 1 5-per-cent. mixtures at one-and-a-half atmospheres density and Curves III. to 15-per-cent. mixtures at half an atmosphere density. Curves II. in the same figure, referring to 15-per-ceut. mixtures at atmospheric density, are the same as Curves I. in fig. 8, and are included in this figure for purposes of comparison. 1600 SOO 100 0-2 0 0 •3 -4 -5 -6 Time offer fynifcon - seconds. Fig. 10. Tables VI. aud VII. have been prepared from Curves I. and III. in this figure ; for Curves II., see Table III., p. 385. It will be noted that the denser mixtures emit a rather smaller proportion of their heat of combustion up to the moment of maximum pressure than the thinner mixtures do; this is so because the denser mixtures have a slightly greater opacity than the thinner mixtures. In comparing the loss of heat by radiation during cooling it is to be remembered that the rate of cooling of the thinner mixtures is greater than that of the denser mixtures; had the rate of cooling of the mixtures been the same the thinner mixtures would have radiated off a far larger proportion of their heat of combustion than the denser mixtures. 3 D 2 388 MR. W. T. DAVID ON Till; KA1HATION IN TXRLE VI. — 15-per-ceut. Mixture. Half Atmosphere Initial IVssmv. Heat of Combustion of Coal-gas present in Vessel = 8000 Calories. Time from ignition. Mean absolute temperature of gas. Mean radiation received by walls per sq. cm. Total loss of heat by radiation per cent, heat of combustion. 0-05 2270 0-061 3-3 o-i 2020 0-2 11-0 0-15 1790 0-29 15-9 0-2 1600 0-35 19-2 0-25 1440 0-39 21-4 0-5 1030 0-47 25-7 0-75 810 0-49 26-8 1-0 700 0-492 26-9 TABLE VII. — 15-per-cent. Mixture. One-and-a-half Atmospheres Initial Pressure. Heat of Combustion of Coal-gas present in Vessel = 24,190 Calories. Time from ignition. Mean absolute temperature of gas. Mean radiation received by walls per sq. cm. Total loss of heat by radiation per cent, heat of combustion. 0-05 2400 0-14 2-5 0-1 2210 0-425 7-7 0-15 2040 0-615 11-3 0-2 1890 0-75 13-6 0-25 1765 0-843 15-3 0-5 1350 1-065 19-3 0-75 1140 1-143 20-7 1-0 1010 1-158 21-0 The curves in fig. 11 are the differentials of the radiation curves in fig. 10. They give the average rate at which the 15-per-cent. mixtures of the various densities emit radiation during cooling. It will be noticed that the denser mixtures emit much more radiation than the thinner mixtures, especially at the moment of maximum pressure and in the initial stages of cooling. The emission, however, is not in proportion to the density, but varies more nearly as the square root of the density. • Rough Analysis of the Radiation Emitted by the Gaseous Mixture. The various radiation curves in fig. 12 show the radiation which is transmitted through clear plates of fluorite, quartz, and plate glass, and through j in. of water (contained between two plates of glass) in explosions of 15-per-cent. mixtures at atmospheric initial pressure. -, i IK COAL i:.\.s AM> All;. SB* F» 6-0 SO r id-o f i&0 1 i 0 ro n \ f5% mixtures I ~ j Z atmosfi/ierei mi 'fiat 'pressure JL — / a/mosp/ierc * » TIL ~ fe atmosphere \ \ \ \ \ tl U v-jzr V Vv \N$ ^ ^ ^ 240O 2200 2000 S80O /6OO Gas Tefn/xraftire - °C aAso/uff Fig. 11. •fS Ttme -S -2$ -3 fynifion ~ Sccor/(fe. Fig. 12. •J5 45 •6 MR. W. T. DAVID ON THK RADIATION IN An allowance of 5 per cent, has been made for the absorption of the fluorite, but no allowance has been made for the absorption of the other diathermanous substances. The quartz plate transmits about 50 per cent, of the total radiation. The Bunsen flame has two strong emission bands whose maxima are at 2'8/x and 4'4/«. The 4'4yu band is due to CO2 alone ; the 2'8/x band is due to water vapour and also to CO2. The quartz plate (6 mm. thick) would transmit about 70 per cent, of the 2'8/x band and would almost entirely cut oft* the 4'4/x band.* It is highly probable, however, that with the high pressures in explosions the bands broaden out, for SCHAEFER has shown that there is a widening of the absorption bands of CO3 when the pressure is increased. If the emission and absorption bands are similar the quartz will transmit from 10 per cent, to 20 per cent, of the 4'4/u bands, t The 2'8yot band also broadens out and the quartz transmits about 65 per cent, of it.J It is not difficult from these observations to estimate roughly the proportion emitted by the water vapour and by the CO* lloughly, one may take it, the hydrogen emits from 50 per cent, to 60 per cent, of the total radiation, the remainder being, of course, emitted by CO2. There is nearly two and a half times as much water vapour present in the mixture as there is COa, so that, speaking somewhat loosely, the CO2 emits about twice as strongly as the water vapour does volume for volume.§ Bolometric measurements with the window of quartz were also made in the different positions A, B, and C, and also with different strengths of mixtures, and in each case the radiation transmitted through the quartz was always the same proportion of that measured with the fluorite window. The plate glass (|- in. thick) transmits about one-third of the total radiation emitted by the gaseous mixture. The glass probably cuts oft' most of the radiation emitted by CO3 and transmits about 50 per cent, of that emitted by the water vapour. The water-cell almost entirely cuts off all the radiation emitted by the gas (see * See ' Transmission Spectrum of Quartz, Coblentz, Infra-Red Spectra,' Part VI., p. 45. t See absorption spectrum curves of CO2 in SCHAEFEK'S paper (' Ann. der Phys.,' 16, 1., p. 93), and also the transmission spectrum of quartz. t This result was obtained from a comparison of the amounts of radiation from a hydrogen and air mixture received by the bolometer when it was protected first by the plate of fluorite and then by the plate of quartz. This gaseous mixture contained after explosion only steam and nitrogen, so that the radiation emitted was almost entirely of wave-length in the neighbourhood of 2 -8/1. § The radiation emitted by the gaseous mixture is almost entirely due to the H2O and CO* which it contains. The mixture contains about 8 '5 per cent, of C02 and 20 per cent, of H2O, the remainder being almost entirely N. It is interesting to compare this result with those of R. VON HKLMHOI/T/ on the radiation from hydrogen, carbon monoxide, marsh gas, ethylene, and coal-gas flames. He found that the CO2 produced in the CO flame emitted about 2 "4 times as strongly as an equal volume of water vapour produced in a hydrogen flame, and shows that this ratio is preserved in flames whose products of combustion contain CO, and steam. The flames in these experiments were just rendered non-luminous by adjusting the air supply, and the temperatures of all of them were probably pretty much the same. i:\ri.osioNs OF COAI.-UAS AM> AII;. 391 fig. 12). Water 1 cm. thick entirely cuts off all radiation of wave-length greater than 1'2/x or 1'3/u, and is most transparent in the visible part of the spectrum (0'4/u to 07^).* The water-cell therefore cuts off all the radiation peculiar to heated CO2 itinl water vapour (which is of wave-length between 2/» and 5ft) and transmits practically only the luminous radiation. The water-cell continues to transmit radiation for about one-tenth of a second after the attainment of maximum pressure, and it seems probable that the mixture is luminous during this period.! The total loss of heat in tin- explosion of this mixture due to the emission of luminous radiation is ;il>out 0'25 per cent, of its heat of combustion. Table VIII. has been prepared from tin- Curves H in fig. 8. These curves refer to experiments made with a 25'4-per-cent. mixture of hydrogen and air. The radiation curve is the mean of those taken with the bolometer in positions A, B, and C on the end cover. The hydrogen used in these experiments was supplied by the British Oxygen Company, guaranteed 98 per cent. pure. TABLE VIII. — 25'4-per-cent. Mixture of Hydrogen and Air. Initial Pressure, Atmospheric. Heat of Combustion of Hydrogen in Vessel = 1C, 320 Calories. Time from ignition. Mean absolute temperature of gas. Mean radiation received by walls per sq. cm. Total loss of heat by radiation |>er cent, heat of combustion. 0-017 2400 0-018 0-5 0-05 2200 0-15 4-0 0-1 1920 0-285 7-7 0-15 1700 0-37 10-0 0-2 1530 0-425 11-4 0-25 1400 0-47 12-6 0-5 1100 0-57 15-5 0-75 910 0-59 15-8 1-0 810 0-60 16-1 The total amount of radiation emitted up to the moment of maximum pressure amounts to 0'5 per cent, of its heat of combustion, the maximum temperature l)eing 2400° C. (abs.) and the time of explosion 0'017 second. A 1 5-per-cent. mixture of coal-gas and air whose maximum temperature also reached 2400° C. (aim.) emitted up to the moment of maximum pressure rather more than 3 per cent, of its heat of combustion, the time of explosion in this case being 0'05 second. * For the transmission spectrum of water, see E. F. NICHOL'S paper, ' Phys. Rev.,' I., p. 1, 1896. t On looking at the explosion of a 15-per-cent. mixture in the same vessel through the window, the bright flash appeared to the eye to last for about \ second (the " time of explosion " of the mixture in the vessel being only Js second). Of course if water transmits even a small projx>rtion of radiation of longer wave-length the above statement is not justifiable. 392 MR. W. T. DAVID ON THE RADIATION IN The total amount of radiation emitted by the hydrogen mixture amounts to about 16 per cent, of its heat of combustion. A 15-per-cent. mixture of coal-gas and air having the same maximum temperature emitted 26 per cent, of its heat of com- bustion ; in the latter case, however, the rate of cooling of the gaseous mixture after explosion is much slower than that of the hydrogen mixture. The Curve H in fig. 9 is the differential of the radiation Curve H in fig. 8. It gives the average rate at which the blackened walls of the explosion vessel receives radiation in calories per square centimetre of surface per second from the hydrogen-air mixture during cooling. Table IX. compares the emission of the 25'4-per-cent. hydrogen mixture and that of a 15-per-cent. mixtiire of coal-gas and air in the same vessel and at the same mean gas temperatures. The hydrogen mixture after explosion contains 30 per cent, of water vapour and the coal-gas mixture contains 8 '5 per cent, of CO2 and 20 per cent, of water vapour. The densities of the two mixtures are very nearly the same. TABLE IX. Emission calories per sq. cm. per second. Mean absolute gas temperature. Hydrogen mixture. Coal-gas mixture. 2400 5-0 5-4 2200 3-75 3-85 2000 2-55 2-7 1800 1-75 1-8 1600 1-15 1-12 1400 0-70 0-G2 The hydrogen mixture emits just as strongly at high temperatures as the coal-gas mixture does ; at lower temperatures the hydrogen mixture emits rather more powerfully. This, at first sight, seems rather extraordinary, in view of the results given on p. 390, for in the hydrogen mixture there is no C02 and the quantity of water vapour is only 50 per cent, greater than that in the coal-gas mixture. We shall see presently (p. 397) that the water vapour is more transparent to the radiation which it emits than is the mixture CO2 . H2O to its radiation. This may to some extent account for the equality of the emission in the two mixtures. Probably also there are larger temperature differences in the hydrogen mixture. The much quicker rate of cooling suggests that the temperature gradient in this mixture is greater than that in the coal-gas mixture, so that at the same mean gas temperature the hottest portions of the hydrogen mixture may be at higher temperatures than the same portions of the coal-gas mixture. EXPLOSIONS OF COAL-GAS AND AIR, .'593 PART H.--Tm: DIATMI -;i;\i V.VCY AND EMI->I\I. POWER OF THE HOT GASEOUS MIXTURE AFTER EXPLOSION. After the experiments just described had been made some of them were repeated with an explosion vessel of the same shape and size whose interior surface was silver- plated and therefore reflecting. It at once appeared that the gaseous mixture when exploded in this vessel emitted radiation much more strongly than a mixture of the same strength exploded in the vessel with black walls. Curve AP, fig. 13, shows the radiation absorted by the bolometer per square centimetre when it was protected by the plate of quart/* during the explosion and subsequent cooling of a 15-per-cent. mixture of coal-gas and air at atmospheric density in the vessel with reflecting walls. Curve AB in the same figure shows the same thing when the walls were black. The corresponding gas temperature curves (TP and TB) are also shown. The maximum gas temperature reached after explosion is about 3 per cent, greater and the subsequent rate of cooling much slower when the mixture is enclosed in the vessel with reflecting walls than it is when the mixture is enclosed in the vessel with black walls. This is in agreement with Prof. HOPKINSON'S recent experiments, t In the following table the second and third columns give the rate at which that part TABLE X. — 15-per-cent. Mixtures of Coal-gas and Air. Quartz Window. Bolometer close up to Quartz Window. Mean absolute Rate of absorption of radiation by bolometer in calories per sq. cm. per second Ratio Column II. to temperature of gas. Column III. Walls reflecting. Walls black. "C. 2300 4-95 2-6 1-9 2200 4-1 2-2 1-85 2000 2-55 1-45 1-75 1800 1-55 0-9 1-7 1600 0-85 0-5 1-7 of the radiation from the gaseous mixture which is transmitted through the plate of quartz is absorbed by the bolometer per square centimetre when the walls of the vessel are reflecting and blackened respectively. These figures are taken from the * No measurements were made of the radiation transmitted through the fluorite window when the walls of the vessel were reflecting. The platinum bolometer, having only a thermal capacity equivalent to 0 • 008 gr. of water per square centimetre, would have been heated up to a temperature of over 300* C. with a 15-per-cent. mixture, and the correction to be applied for the loss of heat by the bolometer would have baen so great as to make the results unreliable. t 'Roy. Soc. Proc.,' A, vol. 84, p. 155. VOL. coxi. — A. 3 E; 394 MK. W. T. DAVID ON TIIK KADIATION IN dotted Curves AP' and AB' (fig. 13) which are the differentials of the radiation Curves AP and AB. The fourth column gives the ratio of the figures in the second column to those in the third column. 1200 800 400 •05 •/$ v? Time offer -ZS •35 •2 •45 Fig. 13. From this tahle it will be seen that the gaseous mixture when enclosed in the vessel with reflecting walls radiates from 70 per cent, to 90 per cent, more strongly than the same mixture does when enclosed in the black- walled vessel. With the object of analysing this effect measurements were made of the radiation emitted by a small sectioned cylinder (or more correctly a cone of small solid angle) of the gaseous mixture of different (effective) lengths, first, when the walls were made reflecting, and, secondly, when the walls were black. These experiments were carried out with the silver-plated explosion, vessel. The same bolometer holder was used, but the bolometer was placed some distance behind the plate of diathermanous substance, and the interior of the tube, into which the bolometer is pushed, blackened over so as to prevent radiation from any point in the gas outside the cone reaching the bolometer by reflection from its surface. The emission was measured from two lengths of the gas, viz., 30 cm. and an effective length of 59 cm., these lengths being chosen because they could be conveniently obtained in the vessel. The vessel was 30 cm. in length, and when the walls were black the first length was directly obtained. The second length, which is KXI'LOSIONS OF COAL-GAS AND All:. nearly double the first length, was in effect got by polishing a circular patch of silver of about (! in. diameter opposite the bolometer on the other end cover, and so reflecting the 30 cm. tack upon itself. Were the silver perfectly reflecting a virtual length of 60 cm. would have been obtained by this means, but having regard to its imperfect reflecting power the 30 cm. was increased by about 97 per cent, or 98 per cent.,* this bring! th<^ equivalent length down to 59 cm. When the walls of the vessel were reflecting, the 30 cm. length of gas was obtained by blackening a circular patch of about 6 in. diameter on the end cover opposite the bolometer. This made it impossible for radiation from any point outside the cone of gas to reach the bolometer. When this black patch was rubbed off and the silver polished, the 30 cm. was virtually increased to 59 cm. as explained above. In this case, however, it was possible for radiation reflected from the silvered walls surrounding the bolometer to reach the bolometer after again being reflected from the opposite end cover. In order to prevent this a ring of black paint was placed on the walls round the bolometer. In all the following experiments the bolometer was in position A on the end cover. It was placed at a distance of 12 '5 cm. behind the fluorite, so that the solid angle subtended was 0'062. Records of the pressure of the gas and rise of temperature of the bolometer during explosion and subsequent cooling of mixtures of coal-gas and air were taken in the usual way. From these records curves of mean gas temperature and of radiation emitted by the cone of gas were obtained with times after ignition as abscissae. In what follows these radiation curves have been differentiated (with respect to time), and the rates at which the cone of gas emits radiation plotted against the mean absolute temperature of the gas. Fig. 14 gives the results of these experiments for 15-per-cent. mixtures of coal-gas and air with the fluorite window, an allowance of 5 per cent, having been made for the absorption of the fluorite and 5 per cent, for reflection from the blackened surface of the bolometer. The results have been divided by 0'062 so as to give the emission from unit solid angle. Prof. CALLENDAU calls this the " intrinsic radiance." Curve A shows the intrinsic radiance from 59 cm. of the hot gaseous mixture from the maximum temperature reached in the explosion down to 1300° C. (;iks.), when the walls of the vessel were reflecting. Curve B. — 30 cm., walls reflecting. Curve C. — 59 cm., walls black. Curve D. — 30 cm., walls black. t Curve S. — 15 cm., walls black. * Following the figures given by HAGEN and RUBENS for the reflecting power of silver for radiation of wave-length between 2/* and 5/x ('Z. fur Instr. Kunde,' 22 (1902), p. 52). t The intrinsic radiance from 15 cm. reflected back upon itself, by means of a small polished silvered (il.ite placed opposite the l>olometer, was also measured ; the results were precisely the same as those shown in Curve D. When this plate was painted black the record giving Curve S was taken. 3 £ 2 396 MR. W T. DAVID ON THE RADIATION IN On comparing Curves A and B we find that when the walls are reflecting, the radiation emitted by 59 cm. is over 30 per cent, greater than that emitted by 30 cm. From Curves C and D we find that when the walls are black, 59 cms. radiate well over 20 per cent, more heat than 30 cm. do until the temperature falls to about 2200° C. (abs.) (i.e., for about -£Q second after the attainment of maximum pressure). f/ in N s\ A5 %r>ih Init Curr M \fvres ial f eA- B- c - D - S ofCoa/Oas ana *rcssvre — afnt^ tStycais— ***/!* A,r. bfier/c . ref/ethng. ft b/acK 1U V. • jj 6 f ^ ^^ & *a £ tJ \ \ \ QCc sorption is closely ol>eyed.* He finds practically the same coefficient of absorption, viz., 0'054, for two distinct states of flame (at different temperatures) produced in Meker burners by varying the air supply. Reducing my results to atmospheric pressure, on the assumption that the radiation and absorption of a layer of gas whose thickness is inversely proportional to the density is constant, I find that at -^ second after the attainment of maximum pressure the coefficient of absorption in the 13-per-cent. mixture is O'OOS (temperature of gas 1960° C. abs.), and in the 15-per-cent. mixture 0'0072 (temperature of gas 2210° C. abs.), whilst at the moment of maximum temperature it is in both mixtures only about yfr of the value found by CALLENDAR for flame. This extremely high transparency of the gaseous mixtures at the moment of maximum pressure, and in the initial stages of cooling, cannot be wholly due to the higher temperatures reached in the explosions, for the transparency of the 13-per-cent. mixture in the initial stages of cooling is much greater than that of the 15-per-cent. mixture at temperatures which the 13-per-cent. mixture has in this epoch, as will be seen from the following table. This TABLE XIV.— Walls of Vessel Black. Mean absolute temperature of gas. Per cent, transmission by (}f x 30) cm. of 13-per-cent. mixture. Per cent, transmission by 30 cm. of 15-per-cent. mixture. °C. 2100 2000 1800 22 17 10 17 14 10 table gives the proportion of incident radiation transmitted by (ff x 30) cm. of the 13-per-cent. mixture and 30 cm. of the 15-per-cent. mixture; a layer of the 13-per-cent. mixture (if* 30) cm. thick contains the same amount of absorbing gas (COj. H2O) as 30 cm. of the 15-per-cent. mixture. The following table gives the observed values of the intrinsic radiance from different thicknesses of the 15-per-cent. mixture after explosion (from Curves C, D, and S), and directly under them are given the values for the same thicknesses calculated by * Prof. CALENDAR'S paper on "Radiation from Flames" is given in the 'Third Reprt of the B.A, Committee on Gaseous Explosions,' Appendix A, p. 1$. F.XN.I>MI>\S OF COAL-HAS ANI> AII; 401 means of the formula where RX is the intrinsic radiance from x cm. of gas, R, is tin- intrinsic radiance from an infinite thickness, and K is the coefficient of absorption per cm. The sixth column in this table gives the value of K at various temperatures, and the last column gives the value K would have were the gas expanded down to atmospheric pressure, assuming, of course, that the transparency of a thickness of gas inversely proportional to the gas pressure is independent of the pressure.* TABLE XV. — 15-per-cent. Mixtures of Coal-gas and Air. Walls black. Fluorite Window. Mean absolute temperature of gas. Intrinsic radiance, calories per second. Coefficient of absorption, K. K, reduced to atmospheric pressure. From 15 cm. From 30 cm. From 59 cm. From infinite thickness. °C. 2300 | observed 4 • 2 calculated 4*18 6-1 6-12 7-45 7-4 7-75 0-052 0-0069 2200 | observed 3 '75 calculated 3 • 7 5-35 5-35 6-4 6-4 6-66 0-054 0-0075 2000 | observed 2 • 65 calculated 2-66 3-7 3-72- 4-25 4-28 4-4 0-062 0-0095 1800 | observed 1 • 75 calculated 1-88 2-55 2-53 2-85 2-82 2-86 0-072 0-0122 The close agreement between the observed and calculated values shows that the exponential law of absorption is closely obeyed in the gaseous mixture during cooling. Table XVI. gives the values which the intrinsic radiance from 1 cm. would have were the gaseous mixture perfectly transparent. The intrinsic radiance corrected for absorption from 1 cm. of the gaseous mixture is the limit of RX/J? when x = 0 in the formula R, = R»(l— *. from -. from Sfcmt 'rom SO Cms. . abs. -from =\p®(J.,-, <1> = j ( satisfying (3) and such that, when x = a, = 0, and 4> = 1 will be denoted by 0 (x, a), (x, a) ; if, when JS = a, 0 = 1, and $ = 0 they will be denoted by \[s(x, a), ^ (x, a). If $ = A, $ = B, when x = a, then the equations (3) become = A+ 1 p dx, Jo .0 and have, according to the known theory of integral equations, a unique solution which must be = At//- (-f , «) (.r, l>) = $ (a, b) ^ (x, a) + * (a, h) (x, a), ^ (.r, 6) = ^ (a, 6) >/r (», a) + * (a, &) tf> (x, «), 0> (;r, b) = $ (a, b) * (x, a) + 0> (a, &) * (x, a), * (.»•, />) = V' («, &) *• (.*', a) + ^ (a, &) * (.r, a). Other important relations are ^. (x, b) (a, c)- '/> (.r, t-) v> («, &) = ^ (», a) ^ (?>, c), V' (•»', &) V' («, <')-f (•'•, <0 V' («, '>) = (x, a) * (<', ft), and so on. 4. If a second pair of equations of the type (3) is taken where /, a-', A', ', <£' take the places of p, cr, X, $, , we have *'}<** ....... (4) by the formula for integration by parts, and similarly '}<&, ....... (4') PROF. A. C. DIXON ON STURM-LIOUVILLE HARMONIC EXPANSIONS. 413 These integrals are indefinite, each carrying with it an additive constant tc» \»- detexmioed by trial of some particular \alueof.r. The fonnnla (5) includes a great variety of particular results of which many arc well known in the theory of the equation (l). Km- instance, take , »/>' to It- $(x,a), (.'•,/') so that p = p', X = X', a- = (/>,«) = -tf> (",/>), and similarly ,/, (/,, «) = * (a, b), * (/,, «) ^ -* (,r, />), and (r, «) (.1-, tt) — ^ (a-, a) * (.£, a) = 1 . Again, taking 0, ^' to Ije 0 (.r, n), ^'(a:, />), we have , a) '(.r, h) + (, a) is taken as equal to ^ (/*,«); in it the term depending on ) (hi* interval can be so divided info a finite number of sub-intervals, and a function (x) constant in each sub- ('' iufcrrnf can be no chosen that (f''— fa')3 •••, and take an integer p such that Z ,V«> Ar-t(r=0, l,a, .... n). 414 PROF. A. C. DIXON ON STURM-LIOUVILLK HARMONIC EXPANSIONS. Let er,m be the part of Sr>at which is also in \\. Put = «i in $i,m(m = 1, 2, ..., />). except where the value Oo has been already assigned = ... = «r in 5r, „,(?» = 1, 2, ...,p), except at such points as belong to <$0,m, fV»» ••-, ot-j»»(*B = 1, •••>/') where the value has been already assigned, and, lastly, = L in the rest of the domain. Then \f(x) — tf> (:*•)) < (U— L)//i in all parts of Sr,m that are not in l\(r = 0, 1, ..., n; m = 1, 2, ..., jt>), that is in intervals E whose sum is Now » /< r = 0 HI = I II II 2 2 r = ii i» = /» ii it 2 X 2 A,-(»t+l)e> (6-a)- and, therefore, 2 2 e <(»+l)e, /• = o w = i E > (b-a)-2(n+l)e. Hence the points where \f(x} — (x)\ > (U— L)//t form a set whose measure < 2(n+l)e, and -a) (U-L)2/»2-f (U-L)2 2 (n+\) *, In this expression we may suppose e = yt"2 and make ?i as great as we wish so that the whole is arbitrarily small. The most advantageous value for 0 (x) in each interval is the average of f(x) over the interval, for when c is to be a constant in the interval (x0, JL\) the least value of is given by putting f^i /(.<•) dx -r- (xt- - r0 G. The proof that has been given indicates a particular method of subdivision, say the method A, but any other method, say B, may also be used. To prove this, let PROF. A. C. DIXON OX STUBM-LIOUYILLE HARMONIC EXPANSIONS. 415 «i, aa, ... be the points of a subdivision AI, according to A, for which the value of the integral is < e. Take a subdivision B,, according to B, in which the sum of the intervals containing ai, a2, ..., an is < e : then in tin- other intervals of B, the value of made < * , while for the intervals containing a, a, the value of the integral is not more than (U— L)'*. Hence for the subdivision B! the integral can IKS made lews than 1 1 lilt is, arbitrarily small. Hence the subdivisions of (a, 6) may IK- taken all equal, or according to any other method, provided that the greatest of them tends to zero. Moreover, the square of \'fx-tf>x\dx .'a is less than f* (I i— a) (Jx—dacf doc, .'a and, therefore, f'' \f.c-x\dx Ja is also made arbitrarily small. If L is positive, then i: _L .J_ ./> •• '(.'• and is also arbitrarily small. We may, in fact, say that is arbitrarily small. This is the property that will l>e immediately useful. It may l>e extended to an unlimited function fx when r* f f I \f.i-\dx and .* a .' f dx exist. For take a limited function ftx which is equal to f.r when N i (f-<'Y — 1/N N being a certain positive number, and is 1 for other values of x. N may be so chosen that f* l/r~ f \x\dx J a and t 1 d.r lx)th Then the function tj>x, constant in each of a system of sub-intervals, can l>e so determined that 4 M; n;or. A. c. DIXON* ON STIMIM uoi'vn.u-: HARMONIC KXI-ANSIONS. Then, by help of the inequality, it follows that In the same way, if \\fx)'dx exists, <{> may be so determined that f* (fx—fatfdK < e, Ja by help of the inequality 7. Thus, for the purpose of approximation, we are to divide the domain of a- into sub-intervals, all tending to zero, and in each sub-interval put for p a suitable mean among its values over the sub-interval, which may be conveniently called a local average of p and denoted by r. Suppose, then, that fx pz \7/ ?-U dx, U (x, a) = U = 1 - — dx, .a Jo 1' v (.r, «) = v = 1 + f*>-V dx, V (.<-, a) = V = - f - <1s. .'a • " I' In each sub-intervral u, v, U, V are solutions of the equation and are, therefore, of the form A exp.r v/— X + Bexp (—x v—\). A, B are constants through each sub-interval, but are changed at the passage from one to another. It is of great importance to ascertain whether they can increase or decrease indefinitely, and we shall now prove that they cannot if the total fluctuation of log r is uniformly limited,* that is, if the total fluctuation is always less than u fixed finite quantity at all stages of subdivision of the domain. 8. At every internal point of a sub-interval v, V have differential coefficients, and at the points of division their derivatives, upper and lower, on the same side are equal. * This does not imply that the total fluctuation of log p must be limited. For instance, p may lie 1 at all rational points and 2 at irrational points, then r = 2 everywhere. PROP. A. C. DIXON ON 8TURM-LIOUVILLE HAKMOMC EXPANSIONS. 417 Since ««• ,r J — = rV = -/• dor, '/./ . .'» /• the integral is a coiitinuottt function of x, it follows that the discontinuity in l"g y >s equal to that in log /•. Also /• has no discontinuity, being equal t» ' / ./ 1+ ["rVcfcs. . n Let the imaginary quant itirs conjugate to X, ;•, ... In- denoted by X, f, ..., and let if = rr, \/ — \ = a + i/9, a being positive. Also write D for d/d.r. Then in each sub- inter\;il Also, Hence, when JT = d, the value of (I)*+4f?) ir is U(a2+/3:'), or 2JX', and that of its derivative is 0. At any discontinuity when ;• is changed to / (both are real) the derivative is multiplied by /•'//• and (D'-i l/-f') ir itself consists of two positive terms of which the first is unchanged while the second is multiplied by (f'/r)1. Thus the effect on (I):l+4/3:') ir is to multiply it by a quantity In-tween 1 and (r'/r)3. No\\. if in any interval a function // satisfies the diH'erential equation Day = 4a;//, and if at the beginning of that interval // lies between A cosh 2a (jr— a) and B cosh 2a (./• — ti), while I)// lies l)etween 2Aa siuh 2a(.r— a) and 2B« sinh 2a (x— a) where A, B, j-— a are real and positive, then these same statements must hold good throughout the interval. For suppose B > A, then Bcosh 2a(.r— a)— y and y— A cosh 'la. (.r—a) are both functions satisfying the and at the beginning of the interval their values and derivatives are all positive; it follows from the differential equation that the values and derivatives will increase, and therefore be positive throughout the interval. In the first sub-interval (D'+Af?) u< must be 2|\|cosh 2a(x— a) and its derivative 4a|X ! sinh 2a (x— a). At entrance to any later sub-interval of the domain ("./') (« < 6) this function and its derivative are multiplied by quantities of which we know that each lies between 1 and T* where T is the ratio of increase in r. Thus throughout all the sub-intervals (D*+ 4£*) w = 2P | X | cosh 2 I and IIa that of all the factors < I among tin- quantities T*. If log r is of uniformly limited total fluctuation then HIT" and II:,r"' arc limited, :ind so, therefore, are P, Q. y. Now = PJX I cosh 2oc («-«), and heu.ce |Xr2| and [Di'l* are severally less than this. Also, the real part of y--\vDv is oa = iQ|x|sinh2a(a,--«) ....... (7) Hence \v\ and jDr -r v\\ are both <{Pcosh2a(a— a)}1''2, but sinh 2a (x-a) {P cosh 2a (./•-«)}""' 2- Now P, Q cannot increase or decrease indefinitely, and therefore, if we do not allow a to tend to zero, we have that \v\ and |Dr -f- \/\ \ bear to expa(.r— a) ratios which are limited in both directions ; and this is true, both when the sub-intervals are increased in number and also when | X | is increased indefinitely. In the same way it may be seen that the ratios of |«\/X and jDu| to exp «(.'•— ft) are limited in both directions. When a = 0 the argument shows that j»\/X|, Du\, jt'j, and Dr -r- v/XJ are limited above, but not below, and, in fact, we know that each of the four is capable of vanishing. 10. The condition that log r should be of uniformly limited total fluctuation is necessary, for if it is not fulfilled P may be indefinitely great or small, and it is conceivable that u\/\ and D?<, for instance, shoiild become very small together, in the same way that a pendulum would be practically stopped if its velocity wen- suddenly reduced in a constant ratio at every passage through the lowest position and increased in the same ratio at every time of reaching one of the extreme positions, when, of course, the increase would be of no effect. 11. Hence if in (5) we take the limits to be a, b and put />' = r, a local average of p, o-' = 0, X' = X, = 0 (./•, a), <{>' = u (.r, b), we have = (b,a)—u(b,a) (.,-, ,() ,/(.,-, !,)-<,$ (.<•, a) »(.,-, b) I'ROF. A. C. DIXON ON STITRM-LIOUVII.LK II \1;M<)NIC KXI'ANsloN^ < 'han<,nn<,' 0 into \(f, or u into /-. .,r U>th, we him- similarly *(6,a) ! o(a,b) = - (/>,«)+ U (/-.") 1 1 '.i '•• " > v < •'•- % a) U (r, - r) * (,-, «) V (,', These are ezpntliona for the errors committed when n, r, U, V are taken as the values of 0, \ff, 4>, ^. Ijet /u denote the upper Itmindury of the ratios of ] %/X i^ (•*•!. -r»)- " (-''I. -ro)} | , | (^i, A)-U (o-i, a-o) ! , | ^ (.*•„ .,-„)- r (x,, a-,) | , I X'w {^ (a-,, a-.)- V (.r,, r0)} | tr> c'xp « (.r, — .r,,) for values of ./•„, .'•, such that a 2 r,, < .r, 2 I. Let the symlx)! // denote equality in order of magnitude so that P//Q means that the ratios !'/(,), Q/P are both limited. Then in the expressions for the four en-ore, since h^.r^a, \}J»(.r,h), *•(.,-,/,), U (./-,/>), X-iaV(.r,/>) are all at most //exp «(/>-.r), while X1!l^ (.'-,<-/), \js(.r,rt), *(./', a), X'1-''*!' (.«•, a) are all at most // (l +/x) exp a (.1-— rt). Hence such a product as * (.r, a) U (x, /*) or \^ (^, a) '< (r, 1>) is // ( I +M) exp a (/>—'<) at most, ,-ind f j(P- r) * (a?, «) U (r, />) + X ( 1 - I) v, (.,-, a) « (.r, /v) [ ./ .'« I V r/ is at most // w (I +/x) «XP « (/'—«) where r* Also |*|<2z i« sui)jK)sed to lx- finite, and therefore Jo r* 1 4- rr(.c,) — a) 420 PROF. A. C. DIXON ON STURM LIOUVILLK HARMONIC EXPANSIONS. Thus the error in X1 'J (b, a) is at most // (l+M){«r|x|1/a+|X I -12}exp «(£>-«), and the same may be proved for the errors in + (?>,«), *(&,a), X1 »*(/>, a); also we may put Xi, XD for b, a. Hence, at most and or may lie as small as we please. If we make ra/f\~l we have M//M-", so that in each of the four cases the error is of this order relatively to the true value. 12. Since u, i\ U, V do not depend on a-, the error produced by neglecting or altering a- is also of the order of X"1'3 in comparison with the true value. From (5) by putting p = //, a- = tr', and making X' approach X, we can deduce such results as Jx~ * (b' a} = J! + (x> a) * (x> b} dx> 1 thus proving that , i//-, $, ^, have everywhere finite differential coefficients with respect to the complex variable X. They are therefore holomorphic functions of X all over the plane. 13. It is sometimes useful to know that and i/r cannot tend to destroy each other in such an expression as 3> + k\{s, where k is positive. To see this, begin at the other end of the interval (a, b). The argument of §§ 8, 9 proves that the real part of \/— X?' («, b) Dv (x,b) is — i | X | Qt sinh 2a (b— x) where Q, is positive and limited both ways. In this put x = a, and subtract from a multiple of the result of putting b for x in (7). Since v (a, b) = U (b, a) and V (a, b) = —V (/>, a) the real part ot x/^X {U (b, a) + kv (b, a)} V (6, a) is thus found to be + sinh 2a (*,-«), where r0, 1\ are the values of r at a, b respectively. Since k, Q, Q,, r0, n are positive, and Q, Qu r0, t\ are limited in both directions, the ratio of U (b,a) + kv(b,a) to expa(6 — a) is also limited in both directions. Combining this with the results of §11 we have the following theorem: — The values of \(b,a), \{s(b,a), (b,ci), ^(b,fi)/\/\ and aho &(b,a)+k\(r(b',a) irli<>r<- I: /* — '<), be »f loii-fi- .!/•»/.•/• ///,/c.vx « r»-//«/.v /-. :«•/-,!. 14. Hie values of u (6, a), !"(/>.'<)... can be written down M follows : — • L'-l tin' succi->M\e intervals into which /< — ^ is divided Ix- denoted by #,, rt,, ..., ft,, and tin- valurs of /• in those interval-, respectively liy /'i, /';,, ..., /'. : also let ,\ = — /*. Then tt(6,o).^±to±^Hsfid2fflL^i^^ f2»3 . e3)'3 ... f.-i?',., where *,, e2 ..... e,, are all ±1 and 2 refers to the 2" ways of taking them. The product in the numerator of the fractional factor contains it— I hinoiiiial factors. exp " exp |/i ... f,, It may, in fact, lx- veritieil that •!>< dU C2 ~ so that », U satisfy the diffen-ntial e(|uations assigned ; also, when 0n = 0 the expressions reduce to the corresponding ones for u— 1 intervals, and therefore u, U are continuous throughout as functions of 1, ; lastly, at the beginning of the second interval u = j sinh /#,, U = cosh /#,. This completes the verification for «, U, and V, /• may l>e treated similarly. The approximations generally used in the treatment- of the equation (l) might lie derived from these by neglecting all the terms except those in which e, = t, = ... = f., that is, the terms which contain one or more of the differences r, — r3, >»— rn, ... as factors, and in the two terms which are left putting Sx/rl/vTi lor <•», + >•»+!. Thus « (l>, a), for instance, would reduce to \/>\rn sinh I (h — a). This simplified form is clearly not admissible unless all the differences /•, — /•,. /•,.-/•.-,, ... are small. 422 PROP. A. C. DIXON ON STIMfM LIOUVILLE HARMONIC EXPANSIONS. 15. Now suppose p to l>e limited both ways, and let F(t, x, R) denot.- If '( ii) where pt means the value of the function p of the argument t, and the path of integration in the X-plane indicated by (R) is a circle of radius R with centre at the origin, R l)eing such that this path does not pass through a point where •*• (l, 0) = 0. /)(F (t, x, R) is a symmetric function of t, x for ,(t,l)-^(t,V) + (x,l)}(1\ = \ (x, t) d\ = 0, .(R) since (x, t) is a holomorphic function of X everywhere (§ 12). It will now be proved that F (t, x, R) satisfies the conditions of HOBSON'S convergence theorem (' Proc. L.M.S.,' ser. 2, vol. 6, pp. 350-1), that is— (1) Its absolute value does not exceed a certain quantity F for all values of t, x such that t ; ~ x 2t /u. and for all values of R. f* (2) F (t, x, R) dt exists for all values of a, h such that 0 :£ a < 1> ^ 1 and for each Ja value of x in the interval (0, l) which does not lie between a— /j. and I> + n; this integral, moreover, is less than a positive number A, independent of a, b, x. (3) A -> 0 when R -> o°. 1G. A change in the value of R does not affect ¥ (t, x, R) unless it changes the number of zeros of ^ (l, 0) enclosed by the circle : hence we may suppose the circle to cross the real axis on the positive side at a point T, where v(l, 0) is zero. Thus, at the point T, V(l, 0), and therefore also *(l, 0), are //v/X, since y|2+ Dt'|2/|A cannot tend to zero. Again (§ 12) l ,l)dx, ...... (8) which is limited when a is limited,* whatever .the value of /3. A distance // v/X can therefore be assigned such that within that distance of T >^(l, 0) -r %/X| does not approach zero, but exceeds a certain fixed quantity independent of R. Beyond that distance from T on the path it has been proved already (§11) that *•(!,()) //X1 * exp a, so that this is now proved for the whole path. The numerator \[, (x, 0) ^ (t, 1 ) (f > ./•) is // exp ax . exp a ( 1 — /), so that the subject of integration in F(/, x, R) is // X~l a exp a. (.t — t\ that is, X~l a exp ( — «/*) at most, * We still take ,/- X = a + i/i; thus a is limited for points within a distance // v/X (or f (lie path tin- factor exp( — O./UL) is so small that the contribution to the integral is negligible. ** i I lence F (/, if, It) is at most // I exp (-a/u) da, that is, -. • " ft. This is the first of Hoi WON 's conditions. Ag.-tin In the first term of this expression the first term of the subject of integration contains the factor which is of the order of expa(x— a), that is. at most exp ( — a/*) when we take ./• <«-/*. Also X~1/2 d\ // da as before and X""*//^'1*. Hence the contribution of this term is // — y= at most, and the same is true of the second part of the first term. In the other term the integral of jo-j is finite, and the integration with respect to t is over a finite range, so that these two elements do not affect the order of magnitude : the factor ^ (x, 0) ^ (t, 1 ) -=- x* ( 1 , 0) is of the order of X'3* expa(z-<), that is, IT3'* at most, even when // = 0 : the length of path in the X-plane is 2xlt. Hence the contribution of this term is //It"12 independently of /u if a-Sa. On account of the symmetry tatween .r and t, like results can Ixj deduced if .<•> b + n. Thus the second and third of HOBSOX'S conditions are fulfilled. 17. Again, so long as u-^a the value of \fs(x, 0)*(a, l) -i- *(l, 0) is limited, and so is that of \^(x, 0)*(fc, l) -r- *(l, 0). Hence the first term on the right in (9) is limited, the integral of d\ being 2ir. The second term has been found to tend to zero when It is increased, and therefore F(«,*,R)c& .'a is limited if x^a < If, or similarly if ,rZzl>> a. When >i < jc < // we may write *-J>f, so that j F(f, .r, R) dt is the sum of two terms, each limited, and is itself limited for • a all values of a, b, x, R. This covers one of HOBSON'S further conditions (§ 4 of his paper, p. 361). 424 PROF. A. C. DIXON ON STURM LlorVILLK HARMONIC EXPANSIONS. It is not clear that the two integrals £**>(*, *,B)o°, but their difference does so, and, in fact, if a QO except . 0) + * Or. !)*(*. 0) > J(R) A* (1 which d\ I x = ~2nr- Jon A 18. It is now possible to prove that tf fix) ** continuous at x and of limited total fluctuation in a neighbourhood of x then ¥(t,.r>li)dt ....... (11) Jo For ( I ) this holds when f(x) has a constant value (§ 17) ; (2) The contribution to the integral from values of t not lying between x±/u. may be ignored, the values ic±/u lying within the neighbourhood where f(x] is of limited total fluctuation (HoBSON s convergence theorem) ; (3) By the second mean-value theorem, if f\(x) is monotone, where 0 ^/*, < /u.. In the last expression the second factor is finite (§ 17), while the first can be made as small as we please by taking p small enough, if fi is supposed continuous. The same holds for the integral from x— /x to x. Thus if f(x) is of limited total fluctuation between X±/JL and is, in fact, the sum of two functions /i, /2 which are monotone between those limits, and continuous at .r, 2,7r/(x) =f(x) x Lim rF(«,*, R->-oo JO ,B>*+jr% r + p+ r i IJO Jl-n .'j- J*+!iJ f f^-,1 ,v ,-X+M rl 1 + + + {/^-/^}F(« L Jo • x-n Jx .V+». J PROF. A. ('. l»IXi>\ ON STI'IIM l.lorvil.l.K HARMONIC i:\lv\\slo\s. 425 where tin- last eight integrals all tend to /.fro when K is increased without limit, and, therefore, (II) K -> • • " \\ hidi was to IK- proved. From this result one of the expansions of SrritM and Liorviu.E can be deduced by considering the singularities of the subject of integration in F (t, x, R), that is, the values of X for which >fr(l,0) = 0. . When *(l,0) = 0 we have \!s(x,0) and \Js(x,l) the same but for a constant factor, since *(l,0)*(aj,0)-*(l,0)*(*,0) -*(*,!) ....... (12) Also , it A .'i> p 1 1. Mice, the residue of V(t,X, It) is - 1 y, (r, 0) y, (t, 0) - f 1 {+ (x, 0)}'dx., pi •'» p and we have I OH'rfar . . . (13) //"' xiumnntion referring to the infinite series of ralnex of \ for which ^(l,0) = 0, t tiken in ascending order of magnitude; thus f(jr) ix expanded in a series of f n,ift ions

ranittltes at each of the extreme rallies 0, 1. In order to investigate the validity of the expansion when x = 1 we need to discuss which fF(«,l,B)«ft Ja Here the only term of importance is the second part of the rirst integral, which has the value — 2nr. From this it follows, in the same way, that the expansion holds good at the upper limit, and a like result can be proved when .r = 0 ; in each case it is supposed that f(x) is continuous and of limited total fluctuation in the neighbourhood. The course of the proof, moreover, shows that the series is uniformly convergent so long as x lies within an interval which is contained within another interval in which /'(.'•> is continuous and of limited total fluctuation. It ha« l>een supposed that p is limited lx>th ways. When this is not so, but the integrals of p and - exist, the argument still applies if we detach the factor — from f Pt vol.. rex i. — A. 3 i 426 PROF. A. C. DIXON OX STI'UM l.K >r\'IU,K 1 1. \ UNION 1C KXI'AXSIOXS. ',x, R) and group it with . The same method can lie applied when F (/, x, R) has the more general value where E, G, H, K, L are real constants, with the one proviso that when K is 0, GH is positive or zero, so that the terms in Q involving G, H cannot tend to destroy each other (§ 13). Q cannot vanish except for real values of X if GH — EK — L2 is zero or positive,* a condition which includes the proviso made. Thus it still follows that when 0 < x < 1 f(x) = — Lim — . J V ' H^x2«rJo .« (14) the summation referring to all the values of X for ichich S2 = 0, . . ......... (15) these values being taken in ascending order of magnitude. Sufficient conditions for the validity of this expansion are those already stated, namely, that f(x) should be continuous at x and of limited total fluctuation ivithin some neighbourhood containing x as an internal point. The restrictions placed on the functions a-, p in the fundamental equations (3) are that — (i) p shall be positive ; (ii) The integrals of \ a- \ , p, — shall exist ; P (iii) A local average of p shall have a logarithm of uniformly limited total fluctuation for some method of division into sub-intervals of the fundamental domain * For this and other results see 'Proc. L.M.S.,' ser. 2, vol. 3, pp. 86-90, and vol. o, p. 420; the former passage contains a discussion of the case of equal roots. . A. G. DIXON ON STURM -UOUVILLK HAI.'Mi >MC F.XP \NSIONS. 427 (o, I). Here the term "local average" means a function r constant in each sub- iritrrval, and such that fl rl \p-r\dx and . ji J ii 1 __1 P r dx tend to zero when the greatest of the sub-intervals does so. If - is limited, r may be the actual average of p in cadi sub interval. Tl xpansions discussed by LIOUVII.LK and STI I;M are those for which in the present notation I. = 0, GH = EK. Thus K»(t,x)~ and since = 0 when iJ = 0, the typical term may be written as a multiple of • K^ {.r, 0) + Jfy (j; 0) or K^ (.r, 1 ) - G0 (.'", 1 ) iiiditU-iriitly, that is, it satisfies the equations (:{) and is such that when .r = 0, K«I> = H^>, when .r = 1 , K = — (!<», unless x, t are each equal to one of the limiting values 0, 1. Now the difference of the two expansions for f(x) is that is. Lim - H-> Liin ~- \f(t) { F (t, .*-, R)-F, (t, x, R)) !ill and F— FI is finite. The whole is therefore arhitrarily small and, in fact, zero; thus if the one expansion is valid for f(x), so is the other,* provided that f(x} is continuous at .r. 21. Again, the error produced in one of the functions ,/,. \>,, <|>, ^ by neglecting or altering a is relatively of the order of X"12, and thus it follows that if F, F, are functions of the present type (§ 19) the same except for a change in or then Y(t,x, R) — FI (t, u", R) is limited. Similarly, then, the expansibility of f(x) is not affected by a change in the value of o- in the fundamental integral equations (3) so long as ||o-|(£x exists and f(x) is continuous at x. 22. In order that the more general expansion may hold at the limits 0, 1, some further conditions are necessary. It does not seem worth while to discuss these in detail, but they are satisfied— (1) When /(I) =/(0) = 0 ; and (2) When /(l) =/(o) and G = H == L = 1, K = E = 0. this being the case of a periodic function expanded in a series of periodic terms, since (17) is the condition that functions , <£ may satisfy the equations (3) and have the same values at both ends of the interval (0, l). On account of the periodicity there is no occasion to distinguish between the two end-points or between these and the other points of the domain. 23. It is known that the Fourier constants «„, bn of a function f(x) are such as to give the least possible value to {f(x}— £«„ cos nx— 2,bn sin nx}" dx J — * and there is a similar theorem for the present expansions. The condition that there may be functions £(x), £(x) satisfying (3) and fulfilling the boundary conditions • . - (18) is '(1,0), h This result and that of §21 were first given for the usual LKM vi.i.i.i. scries \>\- .1. MK.KCKR (sec • Roy. Soc. Proc.,' A, vol. 84, pp. 573-5, and 'Phil. Trans.,' A, vol. U11, p. 147). PROF. A. C. 1HXOX MX VITRM UOUVILLE HARMONIC EXPANSION^ 429 which reduces to if ft is a riMit of the equal inn (15) (19) When GH— EK— L1 is ptwtitive, fti, fa, the two roots of ( I!)), ;ire conjugate complex quantities, and they may still be considered so when GH — EK = L* and 0,, 0, are rrnl and equal. Let £ ,, l>e the two corresponding solutions of the fundamental equations (3) ; these are ulso conjugate. Let coefficients anil l>e so determined that '<„, l>n are conjugate for all values of n, and that . (20) is the least possible. Here the different values of X satisfying the equation (15) and the corresponding values of £ ., are distinguished by suffixes. Thus the two factors in the subject of integration are conjugate imaginaries and the integral is essentially jxjsitive. From (5), § 4, we have '•(») 3.(«) »/> H,(o) \rli — EK1 /. I- from (18) Similarly, = 0, since 0,0, = GH-EK. rl I for any unequal suffixes »», n. Hence it readily follows that (20) is a minimum when «. = f - /(•»•) •>• Or) dx -5- | - & (-r) I, (x) dx, fi i .-' I 1> — \ - f(r\t latltLr -=-1 - f ( r\ « iv\ilv "» — J VJ / »• V*1'/ u-* • »» V*1/ 'J* V1* / •' i Jo n Jo p whilf tin- gfiirral term in the expansion of./'(x) (§ 19) is 430 PROF. A. C. DIXON ON STURM LIOUVILLK HARMONIC FA'PANSIONS. • with these values of an, bn, since it may be verified that when A = A,, € (.r) „ (t) { Ky, ( 1 , 0) + H0 ( 1 , 0) \ -.= € ( I ) n ( " ) ('•' (>, •'•) + ( L - &) '/• (.'', 0 } , and theivfoiv. by addition. <» (t,x) is the same as but for a constant factor, since 61 + 62 = 2L. (Compare ' Proc. L.M.S.,' ser. 2, vol. 5, p. 473.) 24. The integral (20) may now be written » = 1 .Op a form which shows that it decreases continually as m increases, and therefore tends to a definite limit when m increases without limit. Also fri ^ * \\ - £n ('i') 'In (»') (•» P •i -j ri ^ = - /'(•'•) f» (.'<;) dx x -/(*) ^,, (x) dx .0/0 Jon Jo Jo pj.pt which shows that the integral (20) is equal to I ' I (fxy dx+ J_ I1 f Lf(x)f(t) F (t, x, II) dx dt JO p Z(7T Jfl Jit pz if Jl is so chosen that the path (II) encloses \\, \2, ..., X,,. Since then Lim | F(t,x,li)dt = - o the limit to which (20) tends when R is indefinitely increased is the limit of -^-\l\l-(Jx-ftYV(t,x,}i)dxdt ..... v - (21) 4(7T JO JO px 25. From this form it is possible to prove that the limit is zero. First, let the domain (0, l) be divided into intervals in each of which /(.<•) is constant. Then when x, t are in the same interval the contribution to the double integral (21) is zero. When x, t are in different intervals (ic0> ^i), (fn, t\),fx— ft has a A. C. I>IXON ON S-ITKM UOI'VII.I.K ll.\i;M"\l< I.Xl'ANsIuNS 431 constant \alue s<> long as those intervals do not change, and the corresponding part of tin- double integral is a constant multiple of In tliis expression take tin- Irniis of',,.(/,.r) separately. Taking tli«' coefficient of K we lia\f „, 0)- \ * (*„ 0)+ 1 A A.r,, CC; -*(ap.°)*('>i) and of P the different sub-intervals. It has now also l)een proved that the second term can IM- diminished indefinitely by a proper choice of at, fij, ... and by taking in great enough. f1 1 Hence \Jx—-yX\'dxt or (20), is also arbitrarily small. Now (20) has its minimum Jo p value for any given value of m when «i, ..., «„ have the values assigned in §2:}, and this minimum value must tend to zero when m is. increased without limit. the expansion of f(.r), found in § 19, may Ite, written in the form c £n, >ln have the meanings assigned in §2:3 find f1 i r1 i - /(•*) n» (•*) 'I* + I - . 0 ) J9 p . = 1 1 ' ~ /(•'•) ft (*) *r + f ~ ft W * (*) '/-'"- Jo p Jo p 432 PROF. A. ('. IUXOX ON STl'KM l.loi: VIl.hK HARMONIC KXl'ANSIONS. rnfitcii of il,, liu itrc i-nnjiKjah- vmagin&riet, «nf this intiyral. cc if x t, and thus it follows that To this expression the methods of Dr. J. MERCER ('Roy. Soc. Proc.,' A, vol. 84, p. 573, and 'Phil. Trans.,' A, vol. 211, pp. 134.^) may be applied, hut his idea of the bilateral limit cannot be used without some modification, since we have no reason to believe that even at a point of discontinuity where f(x ±0) exist their mean is represented by the present expansions. [ 433 ] XII. 'ike Kinetic Theory of a Gas Constituted of Spherically Symmetrical Molecules. By S. CHAPMAN, M.Sc., Trinity College, Cambridge; Chief Assistant, Royal Ohservatory, Communicated by Sir JOSEPH L ARMOR, Sec. Jt.S. Received May 29, — Bead June 29, — Received after Revision and Alteration as indicated in the text, with the addition of Part III., pp. 460-483, November 9, 1911. CONTENTS. Page Introduction 433 Part I.— General theory 434 Part II. — On certain special forma of molecular interaction 453 Part III. — Theoretical results, and comparison with experimental data 460 Introduction. THE principal kinetic theories of a gas proceed either on the hypothesis that the molecules are rigid elastic spheres, or that they are point centres of forces which v;iry inversely as the fifth power of the distance. MAXWELL has worked out the consequences of the latter hypothesis in his well-known theory,* which is unrivalled in its high degree of accuracy and (after some improvements by BoLTZMANNt) in its perfection of mathematical form. All the quantities not taken account of in the theory (such as the time occupied by molecular encounters, and the effect of collisions in which more than two molecules take part) are properly negligible under ordinary conditions. The theory has the disadvantage, however, that the underlying hypothesis is highly artificial (being chosen chiefly on account of mathematical simplifications connected with it, rather than from any physical reasons), and does not represent the real facts at all adequately. * 'Phil. Trans.,' 1867 ; 'Scientific Papers,' vol. iL, p. 23. For convenience we shall refer to a gas of the type there contemplated as a Maxwellian gas. Of course, its molecules possess no internal energy, t ' Vorlesungeu iiber Gastheorie,' vol. i. VOL. CCXI. — A 482. 3 K 9-3.12 434 Ml;. S. CHAPMAN OX THE KINETIC THEORY OF A GAS The other hypothesis referred to seems to be much more in agn>nnrnt with fact, but its consequences have been worked out less accurately. The method which has almost always been used is the one originally devised by CLAVSIUS and MAXWELL ; MAXWELL abandoned it later, however, as it had " led him at times into grave error." In spite of its apparent simplicity, numerical errors of large amount may undoubtedly creep in in a very subtle way. Hence the theory of a gas whose molecules are elastic spheres remains in a rather unsatisfactory state. As a " descriptive " theory (to use MEYER'S apt term) it has, however, served a useful purpose ; the general laws of gaseous phenomena have been developed by its aid in an elementary way, which has conduced to a wider diffusion of knowledge of the kinetic theory than would have been possible if the sole line of development had been by the more mathematical and accurate methods used by MAXWELL and BOLTZMANN. In this paper I have applied the latter methods, with an extension of the analysis, to the elastic-sphere theory among others. In fact, I have obtained expressions for the viscosity, diffusivity, and conductivity of a gas without assuming any properties of the molecules save that they are spherically symmetrical. Many known laws are thus proved more generally than in any former theories, but the formulae so obtained cannot in all cases be put into a really useful form without a knowledge of the nature of the molecules. The supplementary calculations required to complete the general formulas of Part I. of this paper are carried out in Part II. for three special cases, viz., rigid elastic spherical molecules, molecules which are centres of repulsive or attractive force varying inversely as the nUl power of the distance, and rigid elastic spherical molecules surrounded by fields of attractive force. In Part III. the general formulae are completed in these cases and discussed in their relations to the results of former theories and of experiment. PART I. — GENERAL THEORY. 1. Statement of the Problem. We shall deal with a gas composed of two kinds of molecules, which are all supposed to be spherically symmetrical ; m, m', v, v will denote their masses and the number of each kind per unit volume respectively. Similarly the velocity components of typical molecules of the two kinds will be denoted by («, v, w], («', v', w'}. Let Q be any function of the velocity components of a single molecule (e.g., momentum, energy). At any point (x, y, z) let Q be the mean value of Q, so that Q = I 1 1 Qf(u, v, w) du dv dw, where f(u, v, w) is the function which expresses the law of distribution of the CONSTITUTED OF SPHERICALLY SYMMETRICAL MOLECULES. velocities among the molecules m.* Thus the aggregate value of 2)Q for the <• il.,- ill/ ,1: molecules m in a volume element dxdydz at the point considered is given by I-'-! (",,, '•„, "•„). ( " ,,', '•.', //•„') l>e the mean values of (u, >\ w), («', r', «/). Except in considering diffusion, we shall suppose that the mean velocities of each system of molecules is the same, so that u0 = >/„', »'„ = v9', w0 = w0'. We shall write U = u— u^, U' = tt'-ti,,', and so on, so that U = V = W = 0, U' = V"' = W7 = 0. * The general equation of transfer of Q, independent of the action of external forces ist here =r- denotes the "mobile operator" (-r + 'Wo— +V0-z- +w0;r-) of the hydro- Dt \dt 3a5 9y tizl dynamical equations. The only term of the above equation which needs explanation is AQ, which denotes the rate of change of Q due to the molecular encounters ; thus the increase in 2Q which is produced in the element dxdydz by collisions \ in time dt is dx dy dz dt AQ. The calculation of AQ is the immediate object of our investi- gation. The motion of the mass centre G of two colliding molecules remains unaltered by the encounter, and this point may be taken as the origin of a system of uniformly moving axes, relative to which each molecule will describe an orbit in a plane through G ; the two orbits will be similar to each other and symmetrical with respect to the line of apses. If the molecules move with sufficient velocity to carry them beyond the range of each other's action, the orbits will each have a pair of asymptotes ; the asymptotes of the paths while entering on collision are parallel and separated by a distance p (say), and the effect of the collision is to turn the direction * Thus the number of molecules m in a volume element dx dy dz which possess velocities whose three components lie between u and u + du, v and v + dv,w and w + dw, is v /(«, v, w) du dvdw dxdydz; this property rl>its (where ^TT— x is the angle between an asymptote mid the line of apses), the molecules travelling away from one another along the second pair of asymptotes with the relative velocity unchanged. The angle x is a function of p and Va which depends upon the nature of the molecular forces. The velocity components (>/,.,, t>12, w12) of a molecule m after collision with a molecule m' can therefore be written down,* from merely geometrical considerations, in terms of the original components (u, r, iv), (u1, v', it/), m, m', x> and e, the latter being the angle between the plane of the orbits (which contains V0 and p) and a plane containing V0 and parallel with the axis of x. Thus m' (2) »r, = u+ - — [2 (u'-u) sin2x- :» i we must, of course, not put ua = v0 = w0 = 0 till the differentiation has been performed. Thus, when Q = wa+vs+w* we have Q = u03+v0i+w03+\J3 + V3+ W, so that r-3^- = TT3^ = ^- = 0, and the equation of transfer becomes* (9) where we have written q for ^ U'+V^+W2; An (u3 + v*+tt?) vanishes because the energy of the molecules, being wholly translational, is unaltered by the encounters of the molecules m among themselves. In writing out these equations of transfer, it is customary and sufficient to neglect the mean values of odd functions of U, V, W in comparison with the mean values of even functions, and also to neglect the differences between the mean values of f<>rr<'sj)), and taking ?«„ = t»0 = w0 = 0, we find that — = 5q, r-^ = =-*• = 0, so that the equation of transfer takes the form or (since, as we shah1 see later, on p. 447, the left-hand side of this equation is of a lower order than the terms on the right) (10) 5»tf |Z = ^.u(u'+vi+wi\. We next put Q = u2 and Q = uv in order to consider the phenomenon of viscosity ; in this case, of course, we must take into account the mass velocity of the gas. The substitution of these values of Q in the general equation is quite straightforward,! so that we set down the results at once : — (11) _| \ dx dy fit->\ /3vn . 8mA (12) vq ( 3-* + --^) = 1 \9aj oyj We now turn to the calculation of Aw2 and &u (u* + v2 + w2\ ; the value of A?«> will not be calculated directly, but obtained from Aw2 by transformation of co-ordinates, and the value of A18(tt2+va+ti^)0 will ultimately be eliminated from our equations, and so need not be calculated. First of all we shall calculate the values of ^12Q from equations (2)-(5) for substitution in (8). Since, however, several terms of §12Q, disappear on integration with respect to e, it will be most convenient if we immediately calculate <512Q de. Jo 3. Values of ^Q de. Jo In calculating ^12Q we shall find it convenient, partly for immediate brevity, but much more for the sake of a subsequent transformation of the variables of integration in (8), to write (13) (m+m')(U'-U) = X, (w+m')(V'-V) = Y, (m+m')(W-W) = Z, (14) mU+m'U' = X,, mV+m'V^Yi, mW+m'W = Z,. * In which /(«,»,») = * \* t See JEANS' treatise, §§ 336, 338. CONSTITUTED OF SPHERICALLY SYMMETRICAL MOLECULES. 439 • Thus X*+Y'+Z* = (m+m'YVf, and (15) (m + m') (m2lP + m'2U'*) = V,* + mm'V0*, writing V,' = X.'+Y.'+Z,* Also, we have (16) (m + m')U = X1-A'X, (m+m') V = Y.-JPY, (m+m') W = Z,-LJZ, (17) (m+m')U' = X, + /tX, (m+m')V' = Yl + /,-¥, (m+m') W = Z^fcZ, m + m" Since the mass velocities of the two systems of molecules are the same, we have (m+m')(u'—u) = (m + m')(U'— U) = X, and two similar equations. Considering Q = «*, it is easily seen that r$i2uade = (uia3—u*)de Jo 772- r/ / \ / / t\ f\ ' Q / r TT a *\ f i \tt • o ^ n -jra [(u —u) (mu + mu') Sir sin8 x+m { V0— 3 (w — M)*} TT sin* 2XJ, where we have expressed the result in terms of sin* x and sin* 2X ; on making the above transformations we obtain (18) rXu'd«= m> /.,I uTxr- -^ y (m + m )~ We next consider <$,.,« («*+va+wa)0; we write «18 = u + a— a' cos (e— w,), where a = 2&'(w'— M) sin*x, a' = kf\/V^—(u'—uf sin 2X, and similarly for ?»„ and wia. Since we have 1 f5** «12 (ttu1 + via3 + w,/) (if = (M + a) 2 (u + a)a+% (u + a) 2«'2 + a'2 (« + a) «' cos w,. ZTT Jo It is easy to show that a'2 (u+a) a' cos w, = /^/w {« (Y8+Z»)-i'XY-wXZ| sin* 2X, and from the last three equations it readily follows that (19) rsiau(ua 2m/S-7T5 (3X2XX.-X.2X*) x sin* 2X. (m + m')* 440 MR. S. CHAPMAN ON THE KINETIC THEORY OF A V0, and p, and consequently also the values of the above integrals, depends on the law of molecular interaction, which will difl'er according as the collision in question takes place between two molecules m, two molecules m', or one of each kind, we distinguish between the three cases by adding the suffixes 11, 22, 12 respectively to Q'(V0) and Q"(V0). Thus (20) A,X = + V ( Y2 + Z2- 2X2) Q"12 ( V0)] f(u, v, w) f (uf, v', w') du dv dw du' dv' dit/. (21) A12M(«a+V2+^)0 = - In the case of a Maxwellian gast the expressions Q ( V0) are independent of V0, so that the integrals just written can be evaluated in terms of mean values of functions of U, V, W without any knowledge of the functions f(u, v, w), f (ur, v', w'). In general, however, this is not possible ; we require some knowledge of these functions, which express the law of distribution of velocity, in order to make further progress. 4. The Law of Distribution of Velocity. It is well known that in a gas which has had time to attain a uniform state the functions/,/' have the respective forms w k The factors 4n-V0 and irV0 are added merely for convenience. t Also in the case of a gas whose molecules are point centres of force which attract one another according to the fifth-power law. This case, among others, is considered in Part II., and probably is nearer to actual fact than MAXWELL'S case. — Note added October, 1911. CONSTITUTED OF SPHERICALLY SYMMETRICAL MOLECULES. 441 which we shall denote by fv, f'u. When there are slight inequalities of temperature and mass motion in the gas, we shall supjxmo that /=/u{l+F(U,V,W)}, /=.A{l + F'(U',V',W')}, where F and F' are expansihle in the fonn of power series (without the constant terms) in the variables (U, V, W) and (U', V, W) respectively. The coefficients will be small quantities which are functions of the velocity and temperature, and their derivatives, at the point considered. Thus F and F' represent the small disturbance from the normal law of distribution, caused by the slight lack of uniformity in the gas. MAXWELL and BOLTZMAITN* considered that the terms of the first three degrees are sufficient for the adequate n-|nvM-]itatii>n of the disturbed state, and I shall follow them in this assumption. Thus we write (22) F (U, V, W) = (2/mt)1" (a1U+o>V + o,W) + 2Am (i2anUa+22a12UV) where the factors (2hm)113, (2hm), (2/mi)s/a are added merely for convenience in the integration. We shall have a similar equation for F', in which m, alt a,, ... are replaced by m', a',, a'2, .... . Since, by definition, III f(u> v' w) du dv dw = l> jff f (u'> v'> w/) du> dv> dd = 1, we have l+«n + «a + «3s = 1» l+a'n + a'aa + a'a = 1, or (23) an + Wja+ass = 0, a' '„ + a' 'ffl+ a' '33 = 0. * By MAXWELL in his memoir "On Stresses in Rarefied Gases," 'Phil. Trans.,' 1879, or 'Scientific Papers,' vol. ii., p. 681 ; by BOLT/MANN in his ' Vorlesungen iibcr Gastheorie,' vol. i., p. 185. In each case the assumption was made in connection with a Maxwellian gas, but there are good grounds for believing that it is equally valid in general. As neither MAXWKI.I, nor DOI.T/MAXX considered it necessary to give any justification for their procedure, I deliberately followed the same course, more especially as the attempt to m.-ike the step perfectly rigorous would have necessitated the introduction of much mathe- matical analysis which would be out of place here. I should also mention that EXSKOG ('Phys. Zeitschrift,' xii., 58, January, 1911) has made an attempt to determine directly the form of the function F(w, r, «•), applying methods of integration, similar in many ways to those used in this paper to evaluate AQ, to an equation arrived at by BOI.T/.MAXN (' Vorlesungen,' vol. i., p. 114). From the expression for F (», *•, i/-) thus obtained EXSKOG deduces values of the coefficients of viscosity ;'ii. The various component^ of partial pressure due to the first gas, pix, pzy, &c., are given by , pxy = since p = vm ; the mean hydrostatic pressure p is given by P = %(pIZ+Pyy+P:*} = (3 + «n similarly for the second set of molecules. In substituting^' in the equations (20), (21), we shall write (by equation 15) since FF', being the product of two first order small quantities, is negligible. * It should be noted that the above expression for F is of the lowest degree consistent with the satisfaction of the requirements. The function F must provide for small changes in the mean values of even functions of U, V, W, and also of odd functions, both these changes being of the same order. The terms of the second degree do this for the even functions, an, «22, ^33 being of the first order ; it might at first sight be thought that the terms of the first degree would provide for the odd functions of U, V, W, but this is not so on account of the conditions U = V = W = 0. Hence the terms of the third degree must be present, and their coefficients must be of the same order as oj, au, &c. — Note added October, 1911. See the note on p. 483. CONSTITUTED OF SPHERICALLY SYMMETRICAL MOLECI'l.ls 443 f>. lii'il urtin, i of tin- The evaluation of the expressions (20) and (21) is facilitated by changing the variables (U, V, W), (U',V, W) to (X, Y, Z), (X,, Y,,Z,) in accordance with equations (16) and (17). The Jacohian of transformation is easily found to be (m + m')~*, and the limits remain as before, viz., — oo and + «. The two integrals are now of the form A jjj jjj ,' Q (Vo) ^ (X, Y, Z, X,, Y,, Z,) ( 1 + F+ F') dX dY dZ dX, eZY, dZ, where ^ is an integral polynomial in the given variables ; F and F' are also polynomials in these variables. Evidently only those terms in the product of >/<• and (l + F+F') which are of even degree in each of the six variables separately will give any result upon integration. It is easy, though tedious, to pick out these terms ; evidently if -0- is of odd degree in the six variables combined, only those terms of F and F' whose a-coefficients have an odd number of suffixes will need to be considered ; similarly, if >/r is of even degree, we need only consider terms in F which have au, aia, ... for coefficients. Having picked out these terms we are left with a Dumber of integrals of the form :'*o(v0): 1 dX dV dZ dX, dY, dZ,. The integration with respect to X,, Y,, Zt can be carried out immediately — most conveniently by changing to polar co-ordinates. We do the same also in the case of the variables X, Y, Z changing to the variables V0, 0, . The integration with respect to the latter two variables is simple, and there remains an integral of the form J1J' where a = hmm! ,/ > , or %hm' according as Q(V0) or A has the suffix 12, 11, or m + m 22 respectively. We shall consider this more particularly later ; at present it is sufficient to denote it by a symbol. As we shall only need to consider the three cases n = 2, n = 3, n = 4, it i». perhaps, most convenient to denote it by P, R, S respectively, instead of adopting a more general notation. To distinguish between the different cases arising f'n un the various functions Q ( V0) occurring in the integrand, we add the same dashes and suffixes to P, R, S. Thus we have integrals P'ia, P'n, P",,, and so on, corresponding to the cases when in the above integral n = 2 and 1} ( V0) has the special forms Q'12, Q'n, Q"n, and so on : similarly for R and S. Though the execution of the processes indicated is rather lengthy, it is quite simple and straightforward, so that, without entering into the details of the calculation, we 3 L 2 444 MR. S. CHAPMAN ON THE KINETIC THEORY OF A GAS shall quote the results forthwith. After simplification by means of the equations (23) and (24), we obtain the formulae '2 /0(-\ A 2 (25) A,,n - /_-x 3/2 1/3 (26) /. «' ^P' , 3 Aram'2 /On CT'U\ p,, 1 - a „) P „+ -.- ^^ (- + ^r) B «J • (m'^-m""^)} Fu] where for convenience we have written k^ and ka for the quantities given by 7, 2 jTOm i2 z, _ 2 vmmf\2 S'18 "1 — 5" ~ / TV ' 2 — \ 5 ~~ rl TV~ ' / T^/ ) /l/2 — \ 5 — I ] m + m Jr 12 \ m+m ] By putting m = m', v = v, a = a' in the above equations we get also the following : — (28) ' Antt2=_4^(^)3/2^a11Il"1, / f) f\\ * / 2 i 2 i 2\ 4 ' By transformation of co-ordinates we obtain the following equations from (25) and (28) :— (30) A18wt» _w, / mm'2 /Amm'\3/3ro/ /\/ , \-nt ... o hmm'2 = — 47T ' VV -, T— -*TVS- — -u— u- — -- - (m+m')3\m + mV L m + m'\m m' M'2 / mm (3D* ' AllW= * All the formulae (25)-(31) were in the present form before the paper was revised, with the exception of (26). Their calculation was given almost in full, and was performed by essentially the same methods as those explained above. The work was made rather more lengthy than necessary by the use of an unBymmetrical transformation of the integrals in place of that given in equations (13)-(17), to which Pwas led while endeavouring to simplify the calculation of A12w(w2 + ip + w2)0. The calculation of Aw(w2 + vz + w2) was also complicated by the fact that the gas was not assumed to be at rest ; but on revision, despite these simplifications, I decided to omit the routine calculations altogether. Equation (26) was not given in the paper as originally written because it is connected with the conductivity of mixed gases, and I was then unaware of any experiments on the subject which would make so tedious a calculation worth the while. I have since found such experimental data, and have therefore worked out the formula (26), the results of which (as will be seen in Part III.) show very satisfactory agreement with the observations. — Note added October, 1911. CONSTITUTED OF SPHERICALLY SYMMETRICAL MOLECULES. 445 6. Calculation of Aw«. Finally we turn to consider A,.,«, the calculation of which is rather different from that of the foregoing quantities because ?/„ and ?/„, ?'„ and t''0) «.'„ and «/„ are now different. We require Aiatt in order to determine the coefficient of diffusion. It is evident from equation (2) that rS13ii de = -, (•'*'—»«)> ^o IHTH so that / fft m + m' Auw = W/-J2— , fff fff Q'ta(Vu)(n'-u)S(u,v,w)j'(u',tf,u/)dudvdwdu'dv'fl»'. m-\-m JJJJJJ In the present case we may neglect the small deviation from MAXWELL'S law of distribution of velocities, and thus avoid making any assumption as to the form of F. This is so because in this case (unlike all the preceding cases considered) the unit term of (l + F+F') in the expression for ff leads to a term in the final result, so that the terms which would result from the inclusion of F and F', being multiplied by the small coefficients a and «', are negligible to our order of accuracy. The term arising from the unit in (l + F + F') is itself a multiple of «'„— «„, which we assume to be so small that its squares and higher powers are negligible ; u'0— ua being thus a small quantity of the first order, the terms arising from F and F' are of the second and higher orders of small quantities. Thus we now have to / h. mm 's <„•-„•„>») i2»v>»y/\:t/-' e" where we have written (x, y, z) for (u— u', v—v', w—tt/), so that V0a = Changing the variables in the above integral to (x, y, 2) in place of n', t/, «/, and recalling that f< J -go \i/a -nP+M^ = (- I "K>l4a- w on integration with respect to «, i', ?//• we obtain / m' f hmm' Aun = - i hmm' l?/a fflV^ nr \ -*^,Voeo. *+,.«) , , , { —x\ i/ia ( V0) xe »*-' dx dy d [ir(m+m)i JJJ m + j/i where we have written (V,,2— 2pV0cos \+p3) for 2(x+tt'0— «0)2; thus -tt'0)a, pVl)coB\ = x(u'0-ul))+y(v'0—v0)+z('u/()-ivl)). 44(5 MR. S. CHAPMAN ON TIIH KINETIC THEORY OF A GAS Since, as has already been mentioned, p2 is a small quantity of the second order the term e 1"+m' may be neglected, since it is equal to unity to our order of approxi- mation. Transforming the variables x, y, z in the last integral to polar co-ordinates. we have , \ hmm' ITffvw /v \ Aiaw = -„„'- — i —*\ V03Q'12( V0) e \tr(m + m')} JJJ , m' \ hmm' -'- — i 012 0 ')} JJJ y cog x rfy g.n 0 cQg ^ ^ ^ o n\ \ by putting x = V0 cos 6, &c. It must be remembered that X is a function of 9 and , since cos X = ^C±J Cos 0+ ^^-° sin 0 cos + Wfl~W/0 sin 0 sin . All the terms of the exponential series occurring in the integrand of the last integral are negligible (on account of the factors /o2, />3, and so on) except the unit term and the term of the first degree. The unit term leads to a null result on integration with respect to 9 and tf>. The first degree term alone contributes to the final result, which (in our previous notation) is easily seen to be /o«\ -1/2 / tn mm / , \-nl (32) ; ^.f^^—- (W'0-«0)F,, If we put m = m', u0 = u'0, v = v we find that Anw = 0, as is otherwise evident, since the momentum of a system of molecules is unaltered by their mutual encounters. Having now calculated all the values of AQ which we require, we proceed to substitute them in the various special forms of the equation of transfer, in order to obtain expressions for the coefficients of viscosity, diffusion, and conduction in simple and mixed gases. 7. The Coefficient of Viscosity of a, Simple Gas. First considering a gas composed of molecules of one kind only, we substitute for A?<2 (which in this case equals An«2) and A?/v from equations (27) and (30) in the special equations of transfer (ll) and (12) respectively. We get R// = . + _a + _ 2 «., 20 \Sx tiy dz/ 80: r/ 20 Remembering that «„ = --(pzz-p), «J2 = —p^ (see p. 442), and comparing these CONSTITI'TKI) OF Si'HKHU'ALLY SY.M MHTKICAI. MOLECULES. 447 equations with those of a viscous fluid whose coefficient of viscosity is /z, viz. , we obtain the following expression for n : — (33) M = 4sU!=^ l 5 8. 7%e Coefficient of Thermal Conductivity of a Simple Gas. We next substitute for A«(tta +v*+wil)0 in equation (10), and so obtain '* 5*7 = 4- 4«" 2ir/ )8 R"n on substituting for a, from the equation on p. 442. This equation is simplified if we recall the value of n from equation (32). Thus* 15/u in the equation (l)), we get If we now substitute for If we compare this with the equation of conduction of heat in a gas at rest, whose thermal conductivity is & — viz., with 1)6 i (where Cv is the specific heat at constant volume) — and remember that q is directly proportional to the absolute temperature 6, we find that (34) » = tpC.. D * We can now see why the term v =- U (Ua + V2 + W2) on the left-hand side of the equation preceding equation (10) is negligible. The ratio of its coefficient v to the coefficient of U (U2 + V2 + W*) on the right hand (viz., in J^u (u* + p2 + 1^)0) is now seen to be p/p, which is an exceedingly small quantity in all gases under normal conditions. 448 MR. S. CHAPMAN ON THE KINETIC THEORY OF A GAS 9. The Coefficient of Diffusion. The general equation of transfer (l) is true for any system of molecules in a gas, whether in the presence of other systems of molecules or not, provided that no diffusion is taking place (this restriction arises in the elimination of the external forces,* where it is assumed that AM = 0, which is true, of course, only when the restriction just mentioned is satisfied). The form of the equation of transfer which is applicable to the more general case now under consideration, where diffusion is taking place, ist x.y,z cx m where X is the component of external force. If we put Q = u, and suppose that there are no external forces, and that the motion has assumed a steady state, the equation becomes where we have neglected products of u0, v0, iv0 as being small quantities of the second order; similarly we have neglected such terms as UV, and have given j/mUQ its proper value p. The equation just obtained applies to the first system of molecules, p being the partial pressure due to this system ; there is a similar equation for the second system. Since the temperature is supposed to remain constant, we have Now, from equation (31), it follows by symmetry that so that |i = - ¥ = « w> jB=^(*=rfr ,-i.jufc (?/u_?0 Fji. *"11* s-nr* /*v\ ..I. rt-^T ' \ /v\-i I /w-i' / \ V v/ j« « ^ ' .'• cx Ihe total flow of molecules of the first kind per unit area per unit time is clearly vu0, and also (by the definition of the coefficient of diffusion DJ2) is equal to -D12— . dx * See JEANH' treatise, p. 279. t Ibid., p. 278 ci)NsTiTrTKi» OF SPIIKKH AI.I.Y SYMMETRICAL MOLECULI> Hence n ^" ',' it d» T» vv ' -D»£' '"' = ~D"a^==D»^' by which tin- preceding equation may be reduced to dv , 8 mm' /Amm'V _,/,„, / ,\ TJ n 9" 5— = + S - ;( - -.) IT "J2A (v + v ) ruUu r— . 3x m+m'Vm+mv 3z Tliis gives us* as the expression for Du /oc\ ™ •» itt/W» + m'V/SI 1 (35) D'-' = 1 n *" 1 -- 7 / . /MV • \ Amm / (v + v) Pu By putting m = m', v = /, we get the following expression for the coefficient of diffusion D,, of a gas into itself: — /Ji \-7ft Dn = pT-- 10. The Coefficient of Viscosity of a Compound Gas. As in dealing with the case of diffusion, we must now use an equation of transfer for each system of molecules. Writing where (since the various systems of molecules are not supposed to be diffusing through one another, so that u0 = u'u, &c.) E and F are the same for both systems of molecules, by equations (ll) and (12) we have An* = A«r = i^F, Att'r' = v'q'Y. In the expressions already obtained for A,,ws, Au«v (viz., (25) and (29)) there occur terms containing the coefficients «„ a',. As we have already seen in discussing the conduction of heat, these coefficients depend upon the existence of variations of temperatures in the gas. We shall here suppose that the gas is at a uniform temperature throughout, so that the said terms will disappear. * Since this paper was written I have found that the expression (35) had already been obtained by L ANGEVIN (' Ann. de Chimie et de Physique,' (8), v., 245, 1905), who applied it to the motion of electrons in an electric field. The present proof is shorter than LANOEVIX'S. ENSK arid substitute for P'12 in terms of the coefficient of diffusion D12, the equation for A12M2 is simplified to ""' 2 1 "" jX ±s — *- — * - — — - — -- -— - v + v h2mm' (m + m') D12(\ m Again, by means of the expression obtained for /*, the equation (27) for Auw2 reduces to We get similar equations for Auw'2 and A12w'2, writing p.' for the coefficient oi viscosity of a gas composed only of the second kind of molecules. Hence, remembering that AQ = AUQ + A12Q, we have E = Aau + Ba'n , E = C«u + Da'n , where 1 (i\k m'\ - " ' ' 1 __ \_l\_V\ '-( D = - - __ __ _l+k™_ _*_ - ' ' 2V' We next substitute for an and a'u their values — piz, ^p'xx respectively in the V V above equations, and solve so as to obtain pzz+p'zz (the total normal pressure parallel to Ox) in terms of E. After a little reduction we find that _/ A'+B'+C'+D' A'D'-B'C' where ^ CONSTITUTED OF SPHERICALLY SYMMETRICAL MOLECULES. 451 In these last expressions w and w' are the specific gravities of the two component gases referred to a standard gas whose density at the pressure and temperature of the compound gas here considered is po ; and G is given by (38) G Since the partial pressures of the component gases are directly proportional to v, v v ,_ v'\ OT P== 2h'P :=2V'W Ptl+P'*! = ~ j Zh+k ^W+ Ii»±£E KJ+ 1 1 ^'+ Ui+i !2^p. At\ UT/ L IVW (jr/mfjL } (i, \ W 1 Similarly we may show that Pzy+p'zy is the same multiple of F. Hence, recalling the denotation of E and F, and comparing the said equations with the equations of pressure in a medium whose coefficient of viscosity is M«> as in § 7, we obtain the following equation : — /oq\ (89) 11. The Coefficient of Theivnal Conductivity of a Compound Gas* From equations (26) and (29) we see that &u(ua+v*+wi)a and Aw' (tt"+v'a +«/*)„ can be expressed as in the following equations : — Aw where A, B, C, D are written for convenience in place of some rather long expressions which can easily be written down from the equations cited. Remembering the values of a^ and a\ as found in § 4, and substituting from equation (10) for A?t(w2+tx'+wa)0, the last equations may be written 2= ,(A U(U2+V3+Wa)0 +BU/(U/a+V"+W")0), ox 5v>q> = j (c u(ua+v»+w2)0 +D uxu^ * Added October, 1911.— See the footnote to p. 444. 3 M 2 1.V2 MR. S. CHAPMAN ON THE KINETIC THEORY OF A (IAS Since q = l/2//m, so that mq = m'q', on solving for U(U2 + V2 + W2),, we obtain (AD-BC) U(U2 + V2+W2)0 = 51 \»yi ffftj (AD-BC) U'(U'2+V'2+W'2)0 = 5A-_c. \m m / da; If we multiply equation (9) by m, and add the corresponding equation for the second system of molecules, we get 3 („+„') ^ = _ 2 {*m U(U2+V2+ W2)u + /m' U' z,y, i since the sum of the remaining terms m A12 (?ta + v2 + W2)0 + m' A12 (?t'2 vanishes. This follows from the principle of conservation of energy, for the last expression represents twice the rate of change of the combined energy of the two systems of molecules due to their mutual collisions, which is evidently zero. We substitute the values already obtained for U (U2 + V2 + W2),, and U'(tJ'"+V"+W'*)0 in the last equation, and compare the resulting equation with the equation of conduction of heat in a medium at rest. As in the case of a simple gas (§ 8) we obtain the following expression for the thermal conductivity 3-12 :— m . AD-BC In this equation we must substitute the proper values for A, B, C, D. I shall not enter into the details of the calculation, which is rather long and complicated, but will simply quote the result in the simplest form. It must first be mentioned that (Cv)i2, the specific heat of the compound gas at constant volume, is connected with the same constants for the component gases, viz., C,, and C'r, by the following relation : — (39 A) (c.)u=-pt<€-+pyp' pw+pw using the notation of § 10. The formula finally obtained is (40) = M CON'STITUTEI) OF SPHERICALLY SYMMETRICAL MOLECULES. 453 \\linv E, F, F,, G are given by the following equations, in which the quantities / , /,,. /., have the values assigned to them in equations (27) and (37) : — E M*-&(l-*.)-8(*,-*b)-' ' (41) W G = %k-S(l-kl)-$(kl-k3) + ™ w w This completes the first part of the paper. We proceed now to the evaluation of the quantities P'12) R',.,, k, kt, and kt in some special cases. PART II. — ON CERTAIN SPECIAL FORMS OF MOLECULAR INTERACTION. 12. The general expressions which we have obtained in §§ 7-11 contain four integrals, P',2, R'12) R"12, S',2, given by the equations (42) ' Fu = f V()- £ R',, = From these we obtain the equations (45) 71 ^, (46) k = |, ^ = f , k2 = ft, so that in this case the constants k, klt k2 (defined by equations (27) and (37)) are numerical constants ; in general, they are functions of the temperature (i.e., of h). 14. Molecules which are Point Centres of Force. We next consider the hypothesis that the molecules are geometrical points endowed with inertia and repelling or attracting one another with forces which are functions only of the distance between their centres. Let 012 (r) be the mutual potential energy of two molecules m, m' at distance r, and let us write 'l2 (r) in place of - —r- 012 (r). The first two integrals of the equations mm of motion of the second molecule in terms of co-ordinates r, 6 with the first molecule as origin are as usual where A and B are constants. Eliminating the time from these equations we get CONSTITUTED OF SPHERICALLY SYMMETRICAL MOLECULES. 455 Taking the direction of one asymptote for initial line this has the integral Since Vu is the relative velocity before collision, and p is the perpendicular from the origin on to the asymptote, we have so that f = L -i/a dr- The apse is given by the vanishing of the expression in square brackets. If r0 is the positive root, we have since x is the angle between the asymptote and the apsidal radius vector. For some purposes it is more convenient to transform the variable of the last integral to »;, where q =p/r- We thus get >;„ being the least positive root of the expression in square brackets. No further progress can be made without knowledge of the form of ^»'13 or ^ia. The simplest and most natural form to consider is that corresponding to the case in which the molecules attract or repel one another with a force varying inversely as the ttlath power of the distance. In this case K K > ~' (n..- 1)V' ' where K is a constant depending on the nature of the molecules. Hence where* 2K,a(m+m;) .-, -^-(n_1j|v0^" J " Thus x is now a function of a only. * When the forces are attractive the sign of a""1 is reversed. In the text the case considered ia that of repulsion. 456 MR. S. CHAPMAN ON THE KINETIC THEORY OF A GAS We now transform the variable in the integrals Q', S2" from p to a so tlmt 4 p dp = K',2V0 "»-' a da. where (47) K'12 = Thus we have \3 ( V0) = K'uV,,1"- 4,r sin2 x . a da, Jo "12 ( V0) = K'^V,,1'^ P TT sin2 2X . a da. The definite integrals in the last two expressions are pure numbers depending on nia alone.* We shall denote them by X' (n12) and X"(n12) respectively. We thus havet so that, as in the case of molecules which are elastic spheres, Q" is merely a constant multiple of 0' If we substitute the last expressions in the integrals for P, R, S (see equations (42)) and remember that Cx hmm' „ ., .lv"'"'~= we find that 2 (48) . 12 i - 7- -- - \hmm'J \ HU— 1 - J-K' \'^« W ^ ^"""^T'/Q 2 \ p// xrr/ ,// v/ 2 V""^ - i^K) 8-' lii = ^KiiX{Wii) where * And on whether the molecules repel or attract, there being one value for each case, corresponding to each value of n. t These formulae are true whether the forces are repulsive or attractive. The only difference occurs in the value of the numerical constants X' and X". CONSTITUTED OF SPHERICALLY SYMMETRICAL MOLECULES. 457 Kn being the constant of attraction or repulsion between two molecules m in the same way as K13 is the constant for two unlike molecules. Thus k, /,-,, /•., an- in this case, as in the preceding case of § 13, numerical constants. The calculation of \' and X" for various values of n from the definite integrals already indicated would afford an interesting theoretical investigation, but as actual molecules do not conform very closely to our hypothesis, the practical importance of the matter would hardly justify the labour. MAXWELL calculated their values, however, by quadrature, in the special case n = 5 (the forces being repulsive), and found that* (49) X' (5) = 2-6595, X"(5) = 1'3682. 15. Rigid Elastic. Spheres which Attract One Another. It is an undoubted fact that molecules attract one another at small distances (as is manifest from the force of cohesion, to take but a single example). By considering the effect of such forces in bringing about collisions between molecules (regarded as rigid elastic spheres) which would otherwise pass by one another, SUTHERLAND! had great success in explaining the variation of viscosity with temperature. His treatment of the problem, while very suggestive, and forming an important contribution to the kinetic theory, laid no claim to rigour. In applying the present methods to the study of the same problem, we shall use the notation ia, ^>'12 of §§ 13 and 14 ; u(r) will now represent the mutual potential due to the forces of cohesion. As before, we consider the path of the centre of the second molecule relative to that of the first. When the apsidal distance exceeds o-4-o-', no collision takes place, and the deflection 2x is given by the same equation as in § 14, viz., by (50) X = f [l-^+ V-i Jo L *o When, however, the apsidal distance is less than d!p +,4,rV0 f sin2 x . Jo ,Jo _pv0 Jpv •0 where in the latter integral x is given by equation (50). We denote the latter integral by f'12 (Vo)> the form of the function depending on the law of attraction between the molecules. Thus v» ( vfl) = -v0 Or+Trff+v.') + 4xVo/'12 ( VB). vo Similarly we have Q"»(v.) = ^YO (*+°V(»+ yo2) +7ry0/^13 (v0). where /"»(V0)= f sm*2x.pdp. Jrvo Substituting these values in the integrals P, E, S, we find that F12 = 7r(^+(r7JrV05e"'^;Vo^V0+62 f°° V03f which I was aware, for which i> is known was mercury vapour, referred to by MEYER.* While a number of polyatomic gases obeyed his law fairly closely, this monatomic gas alone tunned a striking exception. Koc.nf lias determine' 1 fl tor mercury vapour at 273° C. and 380° C., and also at the much lower temperature 203° C. ; SCHLEIER- MAOHERf has determined ^ at 203° C. ; these data, together with the theoretically calculated value of Ct,, lead to :5-15 as the value of./! MKYKK raises some weighty objections against the accuracy of the data, viz., (i) that KOCH'S three values of /z show an unlikely amount of variation with the temperature ; (ii) that the conden- sation of saturated mercury in the capillary tub;- probably affected the determination of n ; and (iii) that it is uncertain whether the vapour is completely monatomic at. 203° C. While the resulting value of f is certainly unreliable, it is hard to conceive of the experimental errors being so great as to explain the difference between T6027 and 3' 15. So far as it goes, it tends rather to the support of the present theory. Quite recently, however, I have found that SCHWARZE had disproved MEYEK'S theory nine years ago, by showing that / = 2' 5 for the true monatomic gases argon and helium.§ As these determinations are important in this connection, a few details of them will be given. SCHLEIERMACHER'S method, now generally accepted as the best, was used to determine ^, heat l>eing conducted by the gas from a heated platinum wire. The gases were very carefully prepared and purified, || and in the reduction of the observations (which were made at two temperatures) due corrections were applied for the heat lost by conduction along the wire and for the temperature drop at the walls. The following values of Su were obtained :— Argon 0-00003894, Helium 0'0003386, Air 0'00005690, the last named being determined as a check on the apparatus, and agreeing well with * ' Kinetic Theory,' p. 205. t ' WIED. Ann.,' 1889, xxxvi., p. 346. J Ibid., 1883, xix., p. 857. § \V. SCHWARZE, 'Inaugural Dissertation,' Halle, 1902; 'Ann. d. Physik,' 11, p. 303, 1903; ' Pbys. Xi'itschrift,' 4, p. 229, 1903. MKHI.ISS (' Halle Diss.,1 1902) made an earlier determination of ^ for argon by the STEFAS-WiNKEl.MAXX method, and found / = 2-44; this result was in such striking disagreement with MEYEK'S theory that SCHWARZE investigated the mutter very thoroughly, both for argon and for helium, by the more accurate method due to ScHLElKUM. \CHF.K. The result was quite unexpected, and neither observer seemed inclined to regard it ;ia a confirmation of MAXWELL'S hypothesis, but rather as being due to a numerical defect in MI.YI.KS theory — which view is nearer the truth, according to the present theory. || The helium contained £ per cent, impurity (probably of neon); the same matetial was used by Seiiri.i/r. C Ann. d. Phys.,' 6, p. 303, 1901) to determine the value of p used above. 464 MR. S. CHAPMAN ON THE KINETIC THEORY OF A GAS WINKELMANN'S* value (T0000568 and MuLLER'st value 0'00005G. MEHLISSJ obtained 0-000038 for argon. SCHULTZE§ determined the following values of MO for argon and helium, using the same materials as SCHWARZE : — Argon 0'0002104, Helium 0'0001891. Using the following values of Cc : — Argonl 0'0740, Helium|| 0'7142, SCHWARZE thus finds these values of/: — Argon 2-501, Helium 2'507. I proceed to discuss briefly the case of polyatomic gases. It is immediately obvious that the present theory does not apply here, for Gv in actual fact differs widely from its value for the monatomic gas contemplated in the foregoing calculations (this is not to say that /is not equal to -ij- for any polyatomic gas, but merely that our theory leaves the question perfectly open). The same remark applies to any theory which supposes the molecules to be devoid of internal energy — in particular to MEYER'S theory. But the latter has derived all its support from polyatomic gases. MEYER'S views on this point underwent some changes. In 1877** he seems reluctantly to have accepted the theory (strongly upheld by STEFAN and BOLTZMANN) that the internal and translational molecular energies travelled at different rates (the latter most rapidly), so that the conductivity would be less for a gas whose molecules possess much atomic energy than, for a similar gas with little atomic energy. In 1899,tt however, he held that the conductivity is the same for both kinds of energy, and supported this view by an unsound argument based on the law ot equipartition. This enabled him to assert that / is equal to T6027 for all gases, and so obtain all possible support from the data for polyatomic gases. More modern data would give much less support to the theory, as we shall see. But such disproof is unnecessary, for SCHWARZE'S experiments conclusively show (i) that MEYER'S value * ' WIED. Ann.,' 48, p. 180, 1893. t Ibid., 60, p. 82, 1897. J ' Halle Diss.,' 1902. § ' Halle Diss.,' 1901 ; ' Ann. d. Phys.,' 5, p. 140, 1901, and 6, p. 302, 1901. || Calculated from the formula fR/Jni ; m = 0-1439 for the helium used, and J was taken as 427. This leads to the above value of /, given in ' Ann. d. Phys.,' 11, p. 303, 1903. In ' Phys. Zeitschrift,' 8, p. 229, 1903, J was taken as 424, which made / = 2-490. f Cc for argon is calculated from Cp (determined as 0-1233 by DITTEXBERGER, 'Halle Diss.,' 1897) and 7 (determined as 1-667 by NIEMEYER, cf. 'Smithsonian Physical Tables,' 1910, p. 232). Thus Cfjy = 0-0002104. ** 'Kinetic Theory of Gases,' 1st edition, 1877. ft Ibid, 2nd edition, English translation, 1899, pp. 291-296. CONSTITUTED OF SPHERICALLY SYMMETRICAL MOLECULES. 465 of f is theoretically unsound, and (ii) that the internal energy does not travel at the same rate as the translational energy. Hence the agreement of MEYER'S theory with experiment must be accidental. I'.oi.T/M \\v, who opposed Mi.\ i:i;'> \ie\v, developed a theory of conduction* on the basis of M \.\\\ KI.I.'S hypothesis, taking into account the internal energy. Heobtained the foll<>\\ ing relation between / and y, the ratio of the specific heats, This reduces to MAXWELL'S law when y = $, but the formula, as we shall see, is not borne out by the experimental data. I have not attempted to work oxit a theory of conduction in polyatomic gases, and shall t)e content with pointing out how (in a general way, and with some marked exceptions, which may, however, be due to faulty data) / tends to be larger or smaller (while always less than 2) according as a gas has less or more internal energy. The table below gives the values of &0 for all the gases for which determinations are available, together with y, Cr, //, and the values of / calculated from them, and also f as calculated from BOLTZMANN'S formula. The gases are arranged in increasing order of y, i.e., in diminishing order of /3, the ratio of internal energy to total energy. Gas. y.t C*t /*o x 107. a« x 107. (observed). / (BOLTZMANN). Ethane 1-2191 0-3024 843 4957 1-94 0-82 Ethylene 1-248 0-274 944 §98 1-53 0-93 Carbon dioxide . . . Nitrous oxide .... Methane 1-300 1-304 1-3162 0-1477 0-1483 0-4515 1388 1381 104 3078 3508 7467 1-50 1-71 1-59 1-12 1-14 1 -19 Ammonia Nitric oxide .... Oxvcren 1-3178 1-394 1-402 0-397" 0-1665 0*1563 960 1680 1900 458" 451" 57811 1-20 1-61 1 -95 1-19 1-48 1 -51 -'•JV&C11 Air 1-405 0-1695 1721 56910 1-95 1-52 Nitrogen Hvdrosreii . 1-405 1-402 0-1738 2-427 1670 854 569" 3871" 1-96 1-87 1-52 1 -51 Carbon monoxide . . 1-409 0-1730 1628 499» 1-77 1-53 * TOGO. Ann.,' 157, 1876, pp. 457-469. The theory is partly empirical, being an adaptation of MAXWELL'S formula in which / = £. BOLTZMANX states (p. 468) that the numerical coefficients would have to be altered if any other molecular hypothesis were adopted. Our theorem shows that this is not true, at any rate when y -• • • t These values of y are taken from JEANS' "Dynamical Theory of Gases " (pp. 220, 221), except where the contrary is indicated. 1 LANDOLT and BORNSTEIN'S tables ; observed by MULLER. 2 'Smithsonian Physical Tables,' 1910; observed by MULLER. 8 Ibid. ; by WULLNER. J These values of C0 are taken from JEANS' treatise (p. 218), except where the contrary is indicated. They are due to WIEDEMANN and WULLNER. VOL. CCXI. — A. 3 O 4(56 Mil. S. CHAPMAN ON THE KINETIC THEORY OF A GAS 4 Calculated from y. 6, ' Calculated from Cp and y, as given in the Smithsonian tables. ~ ZIEGLER, ' Halle Diss.,' 1904. STEFAN ('Wiener Sitz.,' 72, II., p. 69, 1875) had found &„ x 1 k% = f 5> MAXWELL'S repelling molecules (48 and 49) — • k = 0771, k, = 1, k, = I, Attracting rigid spheres (52) — 7. _«i + ic»/e * A8 ' where (as we shall see in § 21) the constant C13 must be determined by experiments CONSTITUTED OF SPHERICALLY SYMMETRICAL MOLECULES. 467 on the variation of the coefficient of diffusion (of the gases concerned) with temperature. The formula for the coefficient of conduction in a mixed gas (equations 40 and 41) seems very complicated, but the calculations from it in any particular case are quite simple. Fortunately we have experimental data for testing the law. WACHSMUTH,* at Halle, has recently determined the conductivity of mixtures of argon and helium in various proportions ; the gases being monatomic, our formulae are properly applicable. WACHSMUTH himself undertook the research, at Prof. DORN'S suggestion, in order to determine f in the formula ^ = fnQ,, taking the value of n from TANZLER'S experiments (which will l>e discussed in the next section) on the viscosity of mixtxires of argon and helium. He found that f so determined for the mixture was greater than §, approaching a maximum (about 4) when there was 60 per cent, of helium in the gas, and falling to $, as SCHWABZE showed, when either gas was eliminated. This fact is interesting, but, in the absence of any explanatory theory, does not lead to anything further. WACHSMUTH also found that the observations could be represented to within 2 per cent, or 3 per cent, by the formula s jy ^'3 = l+Ap'/p + l + Bp/p' (modelled on a similar formula for the viscosity of gases which we shall consider presently). As there were only four observations and two empirical constants the fact is not very remarkable ; when p = 0 of course S13 = W, and when p' = 0 we have £>,.) = S-. A much better agreement was obtained by allowing the empirical constants A and B (obtained by the method of least squares) to be imaginary. In this case the expression reduces to the quotient of a cubic by a quartic homogeneous expression in the variables p, p' ; as WACHSMUTH remarks, the relation is good as an interpolation formula, but that is all. We proceed to determine the value of §ia from our formula. Taking oxygen as the standard gas (for which pa = 0'001429 at 0° C. and normal pressure) we have w (for argon) equal to 1'224, and w' = 0'125. The values of MO (*'•«•, M at 0° C.) determined by SCHULTZE (loc. cit. on p. 463), viz., 0'0002104 and 0'0001891 respec- tively, will be used ; also the previously given values of Cv. The coefficient of diffusion D^ for argon and helium has been determined by SCHMIDT and LONIUS (references given in § 21) ; reduced to 0° C. and normal pressure, in C.G.S. units it is 0'650.t We now have all the necessary data for the calculation of &12 in the case of rigid spherical molecules and of Maxwellian molecules. For the case of attracting spherical molecules we need also the coefficient C12 ; this has not been determined, to * 'Halle Disa.,' 1907; ' Phys. Zeitschrift,' 7, p. 235, 1908. The method was that due to SCIILKIER- MACHKR, and the appiratus was that previously used by SCHWARZE and GUNTIJER, alre ady quoted, t As explained in § 21, this value is uncertain to within 2 per cent, or 3 per cent. 3 o 2 468 ME. S. CHAPMAN ON THE KINETIC THEORY OF A GAS my knowledge, but the value 100 (intermediate between the values of Cu for argon and helium) is near enough for our purpose, as the constant does not have much weight in the formula. From the equations (41) it is an easy matter to calculate the constants E, F, G, Ej, Fj, Gj ; we find the following values* for the three special kinds of molecules considered : — Rigid spheres — E = 6'66, E! = 3-93 xlO4, Maxwellian molecules — E = 4-10, Ej = 2'42 xlO4, Attracting spheres— E = 5'38, Ej = 3'17 xlO4, We have written Ej and G, thus becomes F = 2073, Fj = 376 xlO4, F = 17-56, F! = 2'65 x 104, G = 10'GO, G! = 0704 xlO4. G = 10-65, 0706 xlO4. F = 19'0, F! = 313 xlO4, G = 10-56, G! = 0700 xlO4. in place of EW//U and Gw'/fj.' respectively. Our formula Ep2+~Fpp' + Gp'2 where (GV)12 is given by equation (3 9 A). The following table gives the values of p and p' and the observed values of S12,t from WACHSMUTH'S paper, together with &13 calculated according to the above three hypotheses :— THERMAL Conductivity of Mixtures of Argon and Helium. S-o x 107 (calculated). P f?' ^A V 1 (V (argon). (helium). \ v^y /12" (observed). Rigid Attracting Maxwellian spheres. spheres. molecules. 1-000 o-oooo 0-0740 389 389 389 389 0-730 0-270 0-0984 741 675 723 797 0-546 0-454 0-1257 1077 957 1054 1230 0-153 0-847 0-3094 2320 2208 2370 2550 0-0539 0-946 0-4952 2939 2750 2850 2900 o-oooo 1-000 0-7142 3386 3386 3386 3386 h These and nearly all the other calculations in Part III. have been performed with a slide rule, which is accurate enough for the purpose. t S>,2 was determined at two temperatures; we are, of course, using the value reduced to 0° C., as given by WACHSMUTH. CONSTITUTED OF SPHERICALLY SYMMETRICAL MOLECULES. 469 tin- n;it mv r,t' th" .Ml'-iii-iti..!]. til-- •gneoteoi i- extramdy good in ••'!! tin-re cases (of course the values of ^ for the pure gases are taken from the formula 34). The hypothesis that the molecules are rigid elastic spheres surrounded hy fields of attractive force gives notably the best results, however, the agreement being within 3 per cent, in every case — which is little, if any, more than is introduced by uncertainty of the data. The above are the only available data for monatomic gases, so far as I know. WASSILEWJA* has determined the conductivity of mixtures of hydrogen and oxygen, and, of course, we have the conductivity of air ; but as these are mixtures of polyatomic gases our formula does not apply to them, so that they will not be discussed here. In conclusion, it is hardly necessary to remark that our formula for the conductivity of a mixed gas reduces to that for a pure gas on putting p or p' equal to zero. Smce (as we shall see) D13, the coefficient of diffusion, varies inversely as the total pressure, puD12 is independent of the pressure, as also are all the other quantities entering into E, F, G, FI ; hence the coefficient of conduction of a mixture, like that of a pure gas, is independent of the total pressure. This has been experimentally verified in the case of air. t 19. The Coefficient of Viscosity of Mixed (rases. The viscosity of mixed gases has been much studied, both theoretically and experimentally. It is especially interesting on account of the curious fact, first noticed by GRAHAM ('Phil. Trans.,' 1846), that the addition of a moderate amount of light gas (hydrogen, in the case mentioned) to a much more viscous and heavy gas (carbon dioxide) may actually increase the viscosity of the latter. The same phenomenon was commented on by MAXWELL in a Bakerian Lecture (1866), the gases receiving particular mention being air and hydrogen. The principal formulae which have been deduced for the viscosity of a mixture are due to MAXWELL,]: PULUJ,§ SUTHERLAND, || and THIESEN.T Of these only the first is based on an adequate proof, but as it is a particular case of my own formula (got by putting k = A.J/A, = 0771) it does not need separate discussion. PULUJ'S formula is based on an ingenious adaptation of the earlier theories of viscosity of a pure gas, and although, like those theories, it is. only approximate, and in some ways not * 'Phys. Zeitschrift,' 5, 1904, p. 737. t MEYER, ' Kinetic Theory,' Chapter IX.; TODD, ' Roy. Soc. Proc.,' A, 83, p. 19, 1909. I ' Scientific Papers,' vol. ii., p. 72. § 'CARb's Repertorium,' xv., p. 590; ' Wien. Sitzungsber.,' 1879, Ixxix., Abth. 2, pp. 97, 745; see also MEYER, ' Kinetic Theory,' p. 200. || 'Phil. Mag.,' 1895, xl, p. 421. U 'Verb. d. Deutsch. Phys. Ges.,' 4, p. 238, 1902. 470 MR. S. CHAPMAN ON THE KINETIC THEORY OF A GAS satisfactory,* it nevertheless forms a useful interpolation formula. The expression is (mp + m'pT (PULUJ) -s''-*l2'3»i» SUTHERLAND, by an argument which though not rigorous is very interesting and suggestive, arrives at the formula r H ^ . , (SUTHERLAND, THIESEN), 1 + Ap ip 1 + which THIESEN also obtained by a different method of proof. While, however, THIESEN left the formula as an empirical one, SUTHERLAND strove to find expressions for A and B in terms of molecular constants. He found such expressions, but did so half empirically by a study of GRAHAM'S data. In the case of the gases which he considered, a very fair agreement was obtained. My own formula (39) can be written (53) M where E, F, G, Fl are given by equation (39) in terms of /x, //, D12 (the coefficient of diffusion), and a constant k which depends on the particular law of action between the molecules ; k is unknown, but may be expected to lie near or between the values already found for it in special cases (O'GO for elastic spheres, 0771 for Maxwellian molecules). Of the above formulae, that by PULUJ is the only one which is perfectly explicit. THIESEN'S expression is completely empirical, and is useful only as an interpolation formula. PULUJ'S relation shares the virtues and defects of the theory on which it is founded, and therefore prescribes a law of variation with temperature inconsistent with the facts for most gases. THIESEN'S law, on the other hand, does not give any information concerning variation with temperature, and the constants A and B must be empirically determined for each case. The formula (34) or (53), as we shall see later, completely expresses the relation between /x12 and the temperature, but, being quite general, it specifies a different law for each molecular hypothesis ; and while, conversely, the determination of k and the law of temperature variation from experimental data may lead to further knowledge as to the best molecular hypothesis, this very generality gives the formula a semi-empirical character. On theoretical grounds it is desirable that the success of equation (34) as an interpolation formula should be tested. Excellent experimental material exists for the purpose. ScHMiTrt has lately given a resumd of an extensive series of experi- h Thus it tacitly implies that the viscosity varies as the square root of the absolute temperature (since the theories on which it is based lead to this law). t 'Ann. d. Phys.,' 30, p. 393, 1909. Full references to the original sources of the data are there given. SCHMITT was apparently unaware of SUTHERLAND'S formula, but, of course, it is the same as THIESEN'S, if treated empirically. CONSTITUTED OF SPHERICALLY SYMMETRICAL MOLECULES. 471 meats on viscosity carried on at Halle, and has compared the results with the formulae of PULUJ and THIESEN. The former agrees remarkably well with the olwervations, the residuals seldom exceeding 3 per cent. THIESEN'S formula agrees even better still, the residuals rarely exceeding 1 per cent. This is not unnatural, as the constants A and B were found by the method of least squares to three or four significant figures. I have taken two of the longest tables given by SCHMITT, and have roughly found the value of k (by trial and error) which makes the formula (34) or (53) agree most closely with the observations. The results are given in the following tables* := — VISCOSITY of Mixtures of Argon and Helium at I'B" C. p (argon). P' (helium). /M2.107 (observed). Mu-107 (calculated) *- 0-692. 1-0000 0-0000 2220 2220 0-9507 0-0493 2231 2230 0-9093 0-0907 2243 2240 0-8571 0-1429 2253 2249 0-8074 0-1926 1 2266 2259 0-7705 0-2295 2264 2266 0-6846 0-3154 2266 2283 0-6119 0-3881 2303 2294 0-5337 0-4663 2299 2297 0-2915 0-7085 2280 2292 0-1921 0-8079 2226 2246 o-oooo 1-0000 1966 1966 VISCOSITY of Mixtures of Oxygen and Hydrogen at 15° C. p (oxygen). P (hydrogen). Ml,.107 (observed). (calculated) o-oooo 1-0000 878 878 0-0521 0-9479 1092 1076 0-0878 0-9122 1191 1191 0-1561 0-8439 1359 1370 0-3333 0-6667 1674 1672 0-5678 0-4322 1877 1876 0-8126 0-1874 1992 1975 0-9555 0-0445 2014 2007 1-0000 o-oooo 2014 2014 * Iii the first table the experimental constants used were as follows: /*x 10~7 = 2220, /*' x 10T = 1966, w = 1-224, w' ••= 0-125 (relative to oxygen), po (at 15° C.) = 0-001353, DU (at 15° C. and in C.G.S. units) =0-705 (from SCHMIDT'S experiments, rf. p. 478). The values of E, F, G, F! were 7-B97, 6-897, 1 -069, and 2 -776 x 104 respectively. In the second tabje the values of /*x 107 were 2014 and 878, w = 1, */ = 0-0629, p<> = 0-001353, D,8 (from JACKMANN'S experiments, cf. p. 478) = 0-760, at 15° C. and in C.G.S. unite. The values of E, F, G, FI were 11-34, 9-676, 1-049, and 5'205x 104 respectively. 472 MR. S. CHAPMAN ON THE KINETIC THEOKY OF A GAS The sum of the residuals l>etween columns 3 and 4 is 77 in the first case, 54 in the second* ; more accurate determinations of k would probably reduce these figures, but the agreement is already within the limits of experimental error. It will be seen that the values of k which have been found, viz., 0'69 for argon and helium and 0'65 for oxygen and hydrogen, are intermediate between the values corresponding to the elastic-sphere theory and MAXWELL'S hypothesis, a result which confirms our theory. It is impossible to tell what special molecular structure these numbers indicate until the integrals of Part II. of this paper are worked out for other cases ; it may be that a very slight modification of one of the hypotheses there considered would explain the figures. The above tables, and others which might be given, show that our formula agrees well with observation, with a value of k accordant with the theory ; its success is therefore more significant than that of THIESEN'S formula, with its two empirical constants. It may be noticed that THIESEN'S expression can also be put into the form (53), with the speciality, however, that a relation exists between the three independent constants of (53) reducing them to two. Nevertheless, this agreement in functional form is sufficient to explain the success of THIESEN'S expression as an interpolation formula over the limited range (0, l) of p and p'. It may be concluded that, as far as the latter purpose goes, PULUJ'S formula, though not theoretically well founded, is the best (when the values of ^ and ft! at the temperature considered are assigned), because it is very simple in form, quite definite, and sufficiently accurate for most purposes ; while THIESEN'S formula requires the knowledge of two empirical constants, and my own formula is not so well suited for numerical calculation, f The study of the variation of the viscosity of mixed gases with tne temperature is best deferred till the analogous question has been discussed for simple gases; the subject will therefore be briefly discussed in § 22. 20. The Coefficient of Viscosity of a Simple Gas. The formulae which have already been discussed do not contain any reference to the internal structure of the molecule, so far as this could be avoided, but are h The corresponding figures for THIESEX'S formula are 55 and 51, and for PULUJ'S formula 126 and 187. Cf. K. SCHMITT, 'Ann. d. Phys.,' 30, p. 393, 1909; pp. 408, 406. t The form of the expression (53) is well suited to explain the phenomenon mentioned at the beginning of this section. It shows that /*i2 is really a weighted mean between p,, p.' and F/Flt the weight depending on the ratios E : F : G and varying with p and p'. For mixtures of C02 and H we have F/Fj > p > // ; also as E : G, or 1 + kw[w : 1 + kw'jw is large, on account of the largeness of w/w, /*' is given very little weight in the mean except when p is very small. Therefore, for moderate values of the ratio p/p', the value of pi-, lies between F/Fi and /*, and may exceed the latter considerably. The same thing may be noted to a small extent in the above argon-helium table. From the figures already given, we have F/Fj = 2482 x 10"7, which Is greater than 2220 x W~", the viscosity of argon at the temperature considered. CONSTITUTED OF SPHERICALLY SYMMETRICAL MOLECULES. 473 expressed as relations between directly measurable quantities. This enabled us to verify that our conclusions were borne out by the results of experiment. The equations might have been written otherwise, however, so that the formulae should express S, S,,, /zla in terms of unknown molecular data, which might then be found from the experimental values of ^ and /«„ ; there would still be some check on the theory by the comparison of the values of the molecular constants found from different formulae. The reference to these constants was avoided by substitution of /u and D12 into the formulae, which were thus rendered much simpler ; no such simplification can be made in the expressions for M and D12, and as these involve the molecular data in the least complicated form, they are much the most suitable for the purpose of determining the diameter of molecules. Our expression (33) for the viscosity of a simple gas is ~ ih'm'VunJ R"n is a function of the temperature only, so also is h (since 2h = 1/R0) ; and m, the mass of a molecule, is a constant for any gas. Hence we have obtained a perfectly general proof (for the case of a monatomic gas) that the coefficient of viscosity is independent of the pressure and density of a gas, and depends only on the temperature. This remarkable law was first discovered by MAXWELL. If we substitute the values of R"n, which we have already found in special cases (see equations 45, 52, and 48), we get the following special laws:— (54) M = -L- ,A x . i = 59mx e (elastic spheres), (55) M = 5,m (5 of -J— (attracting spheres), ''jr\m / - L>n 6 (56) /u = Atf"*"-" (centres of force oc 1/r"), where in the last equation we have » »*V T Wl (the force between two molecules at distance r being Knm2/?-"), and X" (n) is a number depending only on n, having two values for each one of n, according as the force is attractive or repulsive. The first equation shows that if the molecules are simple elastic spheres the viscosity varies as the square root of the absolute temperature (this law is due to MAXWELL) ; the second shows that when the spheres also attract one another the VOL. ccxi. — A. 3 P 474 MU. S. CHAPMAN ON THE KINETIC THEORY OF A (IAS viscosity varies as 8l!*(l+C/6)~l, as was first discovered by SITHKIM, AM>X ; while the List equation shows that, if the molecules attract or repel one another according to the inverse nth power law, then /* is proportional to the (n + 3)/2 (n— !)"' power of 0. It is interesting to notice that Lord RAYLEiGHt predicted this law of variation from a consideration of dimensions alone ; the present formula is complete, with an exact analytical expression for the numerical constant. If we write ,u0 for the viscosity at 0° C., the three laws just indicated may be written n+3 MO ' i+c/0' MO \e When the molecules repel one another according to the inverse fifth power of the distance, the last equation becomes which was obtained by MAXWELL. Experiment shows that the second formula (generally known as SUTHERLAND'S formula) agrees far better with the actual facts than the others. MAXWELL'S two hypothetical laws, /* oc 01/2 and n TT \ spheres ; where after substitution from equations (45), (48), and (52) we have put the formulae into a more convenient form as above ; iv and it/ are, as before, the specific gravities relative to a standard gas whose density at pressure pa and temperature 8a is p^ ; v0 is AVOGADRO'S constant, at the same pressure and temperature. It may be noticed that MAXWELL,* by applying the methods of his fifth-power-law theory to the case of a gas whose molecules were elastic spheres, obtained an expression J times as great as (but in other respects identical with) the first of the equations just given ; as he did not give any details, it is impossible to say how his error arose — probably by a numerical slip. The following equations simply express the law of variation of Du with temperature, as shown by the above three equations :— Du/(Dw)o = (0/00)Va (rigid elastic spheres), "^ (point-centres of force), { 1 + C«/0«}/{ 1 + C>^} (attracting spheres). So far as I am aware, the variation of the coefficient of diffusion with the temperature has never been properly examined experimentally ; the values of Du are generally found only at one temperature, or two at most, and this is insufficient to decide between the second and third of the laws just given. From the analogous behaviour of viscosity, as affected by temperature, however, the third law is probably most nearly true, and the values of C18 have been worked out by SUTHERLAND on this hypothesis.! In view of the importance of the constant C,» it is desirable that further experiments be made on the variation of the coefficient of diffusion with temperature. * ' Nature,' vol. viii. ; ' Scientific Papers,' vol. ii., p. 343. t The present theory shows that Cr>, the temperature constant for diffusion, is a different multiple of the mutual potential of two molecules in contact from that in the case of Cn, the temperature constant for viscosity. This fact was not indicated in SUTHERLAND'S investigation; see ' Phil. Mag.' (5), 38, p. 1, 1894. 480 MR. S. CHAPMAN ON THE KINETIC THEORY OF A GAS As in the case of the coefficient of viscosity, we proceed to give the values of the molecular radii (according to the two theories of elastic spherical molecules, with and without attraction) calculated from the coefficient of diffusion. From the latter we get n!ml;i' in this paper are correct except as regards a factor which in general is a function of the temperature (only), and in particular cases is a numerical constant. The latter cases include the case of elastic spherical molecules (as pointed out by ENSKOG), and also of point centres of force varying inversely as the nth power of the distance. Thus Lord RAYLEIGH'S theorem (p. 474) can be rigorously established. I have endeavoured to form a numerical estimate of the correction factors to be applied to the formulae, but the analytical difficulties involved have thus far proved unsurmountable. The expressions obtained do not converge rapidly, and the calculations are very laborious ; but the problem is now perfectly definite, and improved analysis may remove the difficulties. The present theory must therefore be regarded as approximate only. In conjunction with ENSKOG'S work, however, it provides rigorous proofs of the relations connecting viscosity and conductivity with the pressure and molecular diameter and mass, and in special cases also with temperature; the relations themselves are well known in the case of simple gases, but not in that of compound gases. The numerical constants are not rigorously determined, and are subject to correction ; but the agreement with experiment seems to show that the approximation is a good one. Lastly, it should be remarked that the theory of diffusion is unaffected by the terms omitted, so that the above statements do not apply to it. INDEX PHILOSOPHICAL TRANSACTIONS SEKIKS A, VOL. 211. Alloys of aluminium and zinc, constitution of (RonKNHAlN and ABCHHUTT), 3ir>. Are spectra, effect of pressure upon— silver \ 4000 to \ 4000, gold (DUFFIKLD), 33. AKCHBUTT (S. I..). See ROSKNHAIN und ABCHBUTT. Asymptotic series, theory of ( WATSON), 279. Atmosphere, vertical temperature distribution in ( DINKS), 253. B. I ;< .1 sn i, u. (W. R. and W. E.). The Specific lleat of Water and the Mechanical Equivalent of the Caloric at Temperature* from 0° C. to 80" C., 199. C. CALIBNUAK (II. L.) and Moss (H.). On the Absolute Expansion of Mercury, 1. < ' H \ !• M A N- (S.). The Kinetic Theory of a Gas constituted of Spherically Symmetrical Molecule*, 433. D. DAVID (W. T.). Radiation in Explosion* of Conl-gas and Air, 376. DINBS (W. II.). The Vertical Temperature Distribution in the Atmosphere over England, and nome Remark* on the General and Local Circulation, 253. UIXON (A. C.). On the Series of STPBM and LIOUVILLK, a- derived from a Pair of Fundamental Integral Equation. instead of a Differential Equation, 411. DUFFIBLD (W. a.). The Effect of Pressure upon Arc Spectra. No. 3.— Silver, A 4000 to A 4000. No. 4.— Gold, 33. Explosions of coal-gas and air, radiation in (DAVID), 375. I. Integral equations, Sturm-Liouville neries of functions in theory of (MlBCBH), 111. K. Kinetic theory of gag of spherically symmetrical molecule* (CHAPMAN), 433. VOL. 00X1.— A 483. 3 R 30.3.12 1NDKX. M. \l\iMAii.ix (P. A.). Memoir on the Theory of the Partitions of Numbers. — Part V. Partitions in Two-dimnuia&BJ Space, 76 j Part VI. Partitions in Two-dimensional Space, to which is added an Adumbration of the Theory of the Partitions in Three-dimensional Space, 345. MKIU KB (J.). Stunn-Liouville Series of Normal Functions in the Theory of Integral Kqiuiiioiis, 111. Mercury, absolute expansion of (CALIKNDAB and Moss), 1. \l >-- ill..). See I'.M.I.KNPAU and Moss. N. Numbers, theory of partitions of (MACMAHON).— Part V., 75 ; Part VI., 345. P. Partitions of numbers, memoir on theory of (MAcMJluox).— Part V., 75; Part VI., 345. E. ROSE.NUAIX (W.) and ABCUBUTT (S. L.). The Constitution of the Alloys of Aluminium an