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mmimM illi m mm9. 50&A^-. if Cf PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY VOLUME XIX CAMBRIDGE : PRINTED BY J. B. PEACE, M.A., AT THE UNIVERSITY PRESS PROCEEDINGS OF THE CAMBEIDGE PHILOSOPHICAL SOCIETY VOLUME XIX 30 October 1916—24 November 1919 CAMBRIDGE AT THE UNIVERSITY PRESS AND SOLD BY DEIGHTON, BELL & CO. AND BOWES & BOWES, CAMBRIDGE CAMBRIDGE UNIVERSITY PRESS C. F. CLAY, MANAGER, FETTER LANE, LONDON, E.C. 4 1920 CONTENTS. VOL. XIX. PAGE A self-recording electrometer for Atmospheric Electricity. By W. A. Douglas Rudge, M.A., St John's College. (Eleven figs, in Text) . 1 On the expression of a number in the form ax^ + by^ + cz^ + du'^. By S. Ramanujan, B.A., Trinity College. (Communicated by Mr G. H. Hardy) H An Axiom in Symbolic Logic. By C. E. Van Horn, M.A. (Communicated by Mr G. H. Hardy) 22 A Reduction in the number of the Primitive Propositions of Logic. By J. G. P. NicoD, Trinity College. (Communicated by Mr G. H. Hardy) 32 Bessel functions of equal order and argument. By G. N. Watson, M.A., Trinity College 42 The limits of applicability of the Principle of Stationary Phase. By G. N. Watson, M.A., Trinity College 49 On the Functions of the Mouth-Parts of the Common Pravni. By L. A. Borradatlb, M.A., Selwyn College 56 The Direct Solution of the Quadratic and Cubic Binomial Congruences with Prime Moduli. By H. C. Pocklington, M.A., St John's College 57 On a theorem of Mr G. Polya. By G. H. Hardy, M.A.. Trinity College . 60 Submergence and glacial climates during the accumulation of the Cambridge- shire Pleistocene Deposits. By J. E. Marr, Sc.D., E.R.S., St John's College. (Two figs, in Text) 64 071 the Hydrodynamics of Relativity. By C. E. Weathers urn, M.A. (Camb.), D.Sc. (Sydney), Ormond College, Parkville, Melbourne . 72 0>i the convergence of certain multiple series. By G. H. Hardy, M.A., Trinity CoUege 86 Bessel functions of large order. By G. N. Watson, M.A., Trinity College . 96 A particular case of a theorem of Dirichlet. By H. Todd, B.A., Pembroke College. (Communicated, with a prefatory note, by Mr H. T. J. Norton) HI VI Contents On Mr Ramanujari' s Empirical Expansions of Modular Functions. By L. J. MoRDELL, Birkbeck College, London. (Communicated by Mr G. H. Hardy) . . . Proceedings at the Meetings held dm-ing the Session 1916 — 1917 . Extensions of Abel's Theorem and its converses. By Dr A. Kienast, Kiis nacht, Ziirich, Switzerland. (Communicated by Mr G. H. Hardy Sir George Stokes and the concept of uniform convergence. By G. H. Hardy, M.A., Trinity College Shell-deposits formed by the flood of January, 1918. By Philip Lake, M.A. St John's CoUege 75 the Madreporarian Skeleton an Extraptrotoplasmic Secretion of th Polyps? By G. Matthai, M.A., Emmanuel College, Cambridge, (Communicated by Professor Stanley Gardiner) . On Reactions to Stimidi iii Corals. By G. Matthai, M.A., Emmanuel Col- lege, Cambridge. (Communicated by Professor Stanley Gardiner) Notes on certain parasites, food, and capture by birds of the Common Earwig (Forficula auricularia). By H. H. Brindley, M.A., St John's College Reciprocal Relations in the Them-y of Integral Equations. By ]\Iajor P. A. MacMahon and H. B. C. Darling Fish-freezing. By Professor Stanley Gardiner and Professor Nuttall On the branching of the Zygopteridean Leaf, and its relation to the probable Pinna-nature of Gp'opteris sinuosa, Goeppert. By B. Sahni, M.A. Emmanuel CoUege. (Communicated by Professor Seward) The Structure of Tmesipteris Vieillardi Dang. By B. Sahni, M.A. Emmanuel CoUege. (Communicated by Professor Seward) , On Acmopyle, a Monotypic New Caledonian Podocarp. By B. Sahni, M.A., Emmanuel CoUege. (Communicated by Professor Seward) Proceedings at the Meetings held during the Session 1917 — 1918 . On Certai7i Trigonometrical Series which have a Necessary and Sufficient Condition for Uniform Convergence. By A. E. Jolliffe. (Com- municated by Mr G. H. Hardy) Some Geometrical Interpretations of the Concomitants of Two Quadrics. By H. W. TxjRNBULL, M.A. (Communicated by Mr G. H. Hardy) . Some properties of p (n), the number of partitions of n. By S. Ramanujan, B.A., Trinity CoUege Proof of certain identities in combinatory analysis: (1) by Professor L. J. Rogers; (2) by S. Ramanujan, B.A., Trinity CoUege. (Communi- cated, with a prefatory note, by JVIr G. H. Hardy) .... 117 125 129 148 157 160 164 167 178 185 186 186 186 187 191 196 207 211 Contents vii On Mr Ramanujan'' s congruence properties of p [n). By H. B. C. Darling (Communicated by Mr G. H. Habdy) On the exponentiation of well-ordered series. By Miss Dorothy Wrinch, (Communicated by Mr G. H. Hardy) . . . - . The Gauss-Bonnet Theorem for Multiply -Connected Regions of a Surface By Eric H. Neville, M.A., Trinity College .... 217 219 234 On an empirical formula connected with Goldbach's Theorem. By N. M. Shah, Trinity College, and B. M. Wilson, Trinity College. (Com- municated by Mr G. H. Hakdy) 238 Note on Messrs Shah and Wilson's paper entitled: 'On an empirical formula connected ivith Goldbach's Theorem'. By G. H. Hardy, M.A., Trinity College, and J. E. Littlewood, M.A., Trinity College . . . 245 The distribution of Electric Force between tivo Electrodes, one of which is covered ivith Radioactive Matter. By W. J. Harrison, M.A., Fellow of Clare College. (One fig. in Text) 255 The conversion of saw-dust into sugar. By J. E. Purvis, M.A. . . 259 Bracken as a source of potash. By J. E. Purvis, M.A 261 The action of electrolytes on the electrical condtictivity of the bacterial cell and their effect on the rate of migratioii of these cells in an electric field. By C. Shearer, Sc.D., F.R.S., Clare College 263 The bionomics of Aphis grossulariae Kalt., and Aphis viburni Schr. By Maud D. Haviland, Bathurst Student of Newnham College. (Com- municated by Mr H. H. Brindley) 266 Note on an exp)eriment dealing with mutation in bacteria. By L. Don- caster, Sc.D., King's College. (Abstract) 269 Golourimeter Design. By H. Hartridge, M.D., Fellow of King's College, Cambridge. (One fig. in Text) 271 The Natural History of the Island of Rodrigues. By H. J. Snell (Eastern Telegraph Company) and W. H. T. Tams. (Communicated by Professor Stanley Gardiner) 283 Preliminary Note on the Life History of Lygocerus {Proctotrypidae), hyperparasite of Aphid i us. By Maud D. Haviland, Fellow of Newnham College. (Communicated by Mr H. H. Brindley) . . 293 Note on the solitary wasp, Crabro cephalotes. By Cecil Warburton, M.A., Christ's College 296 Neon Lamps for Stroboscopic Work. By F. W. Aston, M.A., Trinity College (D.Sc., Birmingham), Clerk-Maxwell Student of the Uni- versity of Cambridge. (One fig. in Text) ..... 300 viii Contents PAGE The pressure in a viscous liquid moving throiigh a channel ivith diverging boundaries. By W. J. Harrison, M.A., Fellow of Clare College, Cambridge. (One fig. in Text) 307 The Effect of Ions on Ciliary Motion. By J. Gray, M.A., Fellow of King's College, Cambridge ........ 313 A Note on Photosynthesis and Hydrogen Ion Concentration. By J. T. Saunders, M.A., Christ's College 315 The distribution of intensity along the loosiiive ray parabolas of atoms and molecules of hydrogen and its possible explanation. By F. W. Aston, M.A., Trinity College (D.Sc., Birmingham), Clerk-Maxwell Student of the University of Cambridge. (Three figs, in Text) . . . 317 Gravitation and Light. By Sir Joseph Larmor, St John's College, Lucasian Professor 324 On a Micro-voltameter. By C. T. R. Wilson, M.A., Sidney Sussex College 345 The self-oscillations of a Thermionic Valve. By R. Whiddington, M.A., St John's College 346 Proceedings at the Meetings held during the Session 1918 — 1919 . . 347 Index to the Proceedings with references to the Transactions . . 350 PEOCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY VOL. XIX. PART I. [Michaelmas Term 1916.] ODamijrilJgc: AT THE UNIVERSITY PRESS AND SOLD BY DEIGHTON, BELL & CO., LIMITED, AND BOWES & BOWES, CAMBEIDGE. CAMBBIDGE UNIVEESITY PRESS, C. F. CLAY, MANAGER, FETTER LANE, LONDON, E.G. 1917 Price Ttuo Shillings and Sixpence Net January 1917. NOTICE^. 1. Applications for complete sets of the first Seventeen Yolumes (in Parts) of the Transactions should be made to the Secretaries of the Society. 2. Separate copies of certain parts of Volumes i.— xi. of the Transactions may be had on application to Messrs BowES & Bowes or Messrs Deighton, Bell & Co., Limited, Cambridge. 3. Other volumes of the Transactions may be obtained at the following prices: Vol. xn. £1. 10s. Qd.; Vol. xm. £1. 2s. 6d. Vol. XIV. £1. i7s. 6d. ; Vol. xv. £1. 12s. 6d. ; Vol. xvl £1. 10s. Od. Vol. xvn. £1. 2s. 6d. ; Vol. xvin. £1. Is. Od. ; Vol. xix. £1. 5s. Od. Vol. XX. £1. 10s. Od.\ Vol. XXL £1. 14s. Od.; Vol. xxn. No. 1, Is. 6d. No. ^, 2s.; No. 8, Is. 6d; No. 4, Is. 6d.; No. 5, 2s.; No. 6, Is. Qd. No. 7, 2s. ; No. 8, 2s. ; No. 9, 2s. 4. Complete sets of the Proceedings, Volumes i. — xviil., may also be obtained on application to the Secretaries of the Society. 5. Letters and Communications for the Society should be addressed to one of the Secretaries, " Mr G. H. Hardy, Trinity College. [Mathematical.] Mr A. Wood, Emmanuel College. [Physical.] Mr H. H. Beindley, St John's College. [Biological] 6. Presents for the Library of the Society should be ad- dressed to The Philosophical Library, Neiu Museums, Cambridge. 7. Authors of papers are informed that the Illustrations and* Diagrams are executed as far as possible by photographic "process" work, so drawings should be on a large scale and on smooth white Bristol board in Indian ink. 8. Members of the Society are requested to inform the Secretaries of any change of address. PROCEEDINGS OF THE A self-recording electrometer for Atmospheric Electricity. By W. A. Douglas Rudge, M.A., St John's College. [Received 18 October 1915.] In the course of the writer's work on the local variations of the atmospheric potential gradient, the need was felt for a simple self- recording electrometer. Most of those in use are costly and at the same time rather elaborate in construction. A new arrangement has therefore been devised which has answered the purpose in view, and as the apparatus may be useful in other directions a description is now given. It has been shown* that very considerable variations of the normal potential gradient are produced by clouds of dust raised by the wind, etc. ; and also by clouds of steam escaping under pressure from steam boilers f. These variations are very sudden and do not last for a long time, so that an instrument used for recording them must be fairly quick acting. After a considerable amount of preliminary work, the type of instrument adopted was a modified form of the quadrant electrometer, the record being photographed upon a piece of bromide paper attached to a revolving cylinder. One special use to which the electrometer was to have been applied was to find the relation between the potential gradient and the altitude of the place of observation, and for this purpose it was proposed to construct ten or more instruments, so that a number of observations could have been carried out simultaneously. Some work of this kind has already been done in South Africa from which it appears that the potential gradient near to the ground diminishes with the height of the place of observation above sea level;]:. In order to get satisfactory results it is necessary for the * Proc. Roy. Sor. A, Vol. 90. t Proc. Pflij. Soc. A, Vol. 90. J Trans. Poy. Soc. SnntJi Afrirn, Vol. vi, pai-t 5. VOL. XIX. PT. I. 1 2 Mr Ri(d[/e, A self- recording electrometer stations to be chosen as far removed as possible from the disturbing influence of manufacturing operations, and of railways, and it was intended to have taken a set of observations in the neighbourhood of the Dead Sea, as in that district, stations for the instruments might have been chosen with altitudes varying from 1400 ft below sea level to 3000 ft above, and in an open country. As a number of instruments were required it was necessary to keep the cost of construction low, and this has been achieved in the instrument to be described, so that the cost of material is less than ten shillings and a moderate amount only of mechanical skill is required in the construction. The complete apparatus consists of: (1) The Electrometer. (2) The recording cylinder. (3) The illuminating arrangement. (4) The charging battery. (5) The collecting system. (1) The Electrometer. This consists of four curved pieces of brass cut from a tube of 3 cm. diameter, and attached to a block of ebonite. The alternate pieces were connected together in the usual way. Each conductor subtended an angle at the centre of the mirror of about 60°, and the adjacent conductors were about 1 mm. apart. The " needle " was formed from a piece of silvered paper, 2*5 x 1"5 cm. carrying a small mirror, or a piece of silvered thin "cover" glass could be used for both needle and mirror. A fine wire was attached to the needle to support a piece of wire gauze which was immersed in a small bottle containing paraffin oil, for damping the motion of the needle. The system was suspended by a fine phosphor bronze wire by means of which the needle could be charged, Fig. 1. The whole was enclosed in a thin wooden case having a small window in front, and ebonite plugs to allow of connection being made to the quadrants. (2) The recording cylinder. This is the most novel feature of the instrument, and is constructed from one of those small round clocks which may be bought from a shilling upwards. Two sizes of clock-case are common, of diameters 10 cm. and 6 cm., and both of these sizes have been used. A brass tube is substituted for the hour hand at the fi'ont of the clock, and a similar piece of brass tubing is attached to the arbor at the back of the clock, which is attached to the minute hand and used for setting the clock. These two tubes are in the same straight line and furnish a convenient axis about which the clock as a whole can rotate. If the tube attached to the hour hand is fixed, the clock-case will turn round once in twelve hours, whilst if the minute hand is fixed, the clock for Atmospheric Electriciti/ 3 rotates once in one hour. Two scales of measurement are thus possible and both have been employed. No difficulty was found in taking twenty-four, or two hour records, for although the records overlapjjed it was (|uite easy to distinguish one part from the other. A light zinc tube was slipped over the clock-case to give a good support for the bromide paper which was wrapped round outside. The whole clock was made to balance by fastening small pieces of lead to the inside of the case, but during the working a little Fig. 2. hour hand arbor fixed by the pin P. Fig. 1. The electrometer. irregularity occurs as a consequence of the unwinding of the spring; this however is not very great and a number of clocks could be made to keep time together. The recording cylinder was enclosed in a light tight case with a long narrow slit in front, Fig. 2. (3) Illuminating system. As the apparatus was used out of doors, a lamp was unsuitable as a means of illumination, so that daylight was used and found to be very suitable. The electrometer and recording cylinder were placed at the opposite ends of a light tight box measuring 20 x 17 x 14 cm. A hole was made in the 1—2 j\[r Rudge, A self-recording electrometer J L I , v- \ I \ I u for AtniospJterio Electrlcitij 5 top of one of the ends of the box, and covered over with a piece of silvered glass, upon which a fine vertical scratch — to serve as a slit — had been made. A lens, which could slide upon a rod inside the box, was employed to project the light upon the electrometer mirror, whence after reflection it was returned to the same end of the box as the slit, but at a lower level, and fell upon the horizontal slit in the case of the recording cylinder. By this means a point of light impinged upon the bromide paper, and as the latter rotated, traced out the curve which appears after developing the paper in the usual manner. Fig. 3. To Collector Fig. 4. E electrometer. R recording apparatus. .S' slit. (4) Charging battery. In using the electrometer the opposite pairs of quadrants were kept charged to a fixed potential by means of a battery of the small Leclanche cell used for " flash " lamps. These cells are sold in sets of three and a batch of eight, giving about 35 volts, is quite sufficient for atmospheric observations. The centre of the battery was earthed. The complete apparatus is shown in Fig. 4. (5) The Collector. This consisted of a small plate of brass coated with a radioactive preparation. The plate was fixed in the centre of a very short piece of brass tubing and the open ends of the 6 Mr Rudge, A self-recordiiKj electi'uineter, etc. tubing covered over with wire gauze, so as to prevent loss of radium by rubbing, etc.; Avhilst allowing it to take up the potential of the air. The collecting plate was supported at the end of an insulated wire, and at such a height above the ground as would give a deflection suitable to the sensibility of the electrometer. Up to the present time the apparatus has been used for the purpose of taking records of the variations in the potential gradient, due to the presence of clouds of dust raised by traffic on the roads, or to the variation caused by the steam escaping from passing trains. A number of representative curves are given. No. 1. This is a twelve hour record, taken at a station on the Gog Magog Hill about four miles fi'om Cambridge, and so far from the railway and roadway that traffic had no disturbing influence. No. 2 is a simultaneous record taken in Hills Road at a distance of less than a quarter of a mile from the railway, so that every passing train shows its influence in increasing the positive potential. Nos. 3 and 4 are a pair of simultaneous hour records, three being taken at Cherryhinton reservoir, and four at about 300 yards from the railway. The " peaks " in the latter indicate the passing of a train. No. 5 is a one hour record taken on Hills Road and shows the remarkable influence of the dust raised by passing vehicles. Every vehicle, even an ordinary bicycle, if it raises dust, disturbs the normal electrification. Nos. 6 and 7 are simultaneous records taken at some little distance from the road. Nos. 8 and 9 were taken near the "Long" road railway crossing and show the influence of passing trains. Nos. 10 and 11 are a pair of simul- taneous records, 10 being taken in the Railway yard, and showing the effect of passing train and "shunting " operations ; 11 was taken about a mile away from the line. All the potentials indicated are positive and the records are reduced in reproduction, but an equal range of negative potentials could be recorded, as only one half of the width of the photo- graphic paper was used in the records given. The sensibility of the instrument may be changed by varying the number of cells of the charging battery. 220 volts > 220 voU^^ :z; ^)vi^f^4^ r\/^^(^ /V*— *— " 1 P.M. 2 P.M. ^0. 5. Variation in positive potential due to the clouds of dust raised by traffic on the roads. 1 P.M. No. 6. Taken simultaneously with 5. 2 P.M. 1 P.M. No. 7. Taken simultaneously with 5. V 2 P.M. 1.30 A.M. 10.30 A.M. No. 8. Variation in potential due to steam from passing trains. 2.15 P.M. 3.15 P.M. No. 9. Variation in potential due to steam from passing trains. W^%#»jiw No. 10. Variation in positive potential due to "shunting" of trains. No. 11. Taken simultaneously with No. 10, but at a distance of more than a mile from the railway. Mr Rainanujau, On t/ie ej^pressiun of a nv/inber, etc. 11 On the expression of a number in the form ax- + by- + cz- + dv-. By S. Ramanujan, B.A., Trinity College. (Communicated by Mr G. H. Hardy.) [Received 19 September 1916 ; read October 30, 1916.] 1. It is well known that all positive integers can be expressed as the sum of four squares. This naturally suggests the question : For what positive integral values of a, b, c, d can all positive integers be expressed in the form ax- + by- + cz- + du" ? (ll) I prove in this paper that there are only 55 sets of values of a, b, c, d for which this is true. The more general problem of finding all sets of values of a, b, c, d, for which all integers luith a finite number of exceptions can be expressed in the form (II), is much more difficult and interesting. I have considered only very special cases of this problem, with two variables instead of four ; namely, the cases in which (I'l) has one of the special forms a{x^ + y'- + z^) + bu- (1-2), and a{x'^ +y-)-\-b{z''-\-u?) (1-3). These two cases are comparatively easy to discuss. In this paper I give the discussion of (1"2) only, reserving that of (1"3) for another paper. 2. Let us begin with the first problem. We can suppose, without loss of generality, that a^b^c^d (2'1). If a > 1, then 1 cannot be expressed in the form (I'l) ; and so a = l (2-2). If b> 2, then 2 is an exception ; and so 1<6^2 (2-3). We have therefore only to consider the two cases in which (1"1) has one or other of the forms i X- + y- + cz^ + du-, X- + 2y'- + cz- + dii-. In the first case, if c> 3, then 3 is an exception ; and so l^c^3 (2-31). In the second case, if c > 5, then 5 is an exception ; and so 2^c$5 (2-32). We can now distinguish 7 possible cases. (2-41) x"- -\- y- + z'- + du?. If rf > 7, 7 is an exception ; and so X^d^l (2-411). (2-42) X- + y- + 2z- + du'. 12 Mr Raniaiiujan, On tlie expreasiuii of a namber If c? > 14, 14 is an exception ; and so 2^cZ^14 (2-421). (2-43) x" + y' + 3^2 -1- dii\ If c? > 6, 6 is an exception ; and so '^^d^Q (2-431). (2-44) X' + ly"" + ^z^ + du-. li d >*1, 7 is an exception ; and so 2^d^1 (2-441). (2-45) x'' + 2?/- + '2>z' + diC'. If (Z > 10, 10 is an exception ; and so 3^c^^l0 (2-451). (2-46) X- -r '±y'' + ^z- + diJb^. If c? > 14; 14 is an exception ; and so 4^c^^l4 (2-461). (2-47) x'-^'ly-^-hz^^diC-. If c?> 10, 10 is an exception ; and so o^cZ^lO : (2-471). We have thus eliminated all possible sets of values of a, b, c, d, except the following 55 : 1, 1 2 3, 5 1, 2 2 4, 5 2, 2 2' 5, 5 2, 2 li 1, 6 1, 3 1, 2 6 2, 3 2, 2' 6 2, 3 1', 3, 6 3, 3 2, 3, 6 *3, 3 2 4, 6 1, 4 2' 5, 6 2, 4 1, 1, 7 2, 4 1, 2, 7 3, 4 2, 2, 7 3, 4 2' 3, 7 4, 4 2, 4, 7 1, 5 2 5, 7 2, 5 1^ 2, S 2, 5 2, 3, 8 3, 5 , 2, 4, 9 5, , 1, 2, , 2, 3, , 2, 4, , 2, 5, , 1, 2, 2, 3, , 2, 4, 2 5, , 1, 9 , 2, 4, , 1, 2, 2 4, , 1, 2, , 2, 4, , 1, 2, , 2, 4, 8 8 9 9 9 9 10 10 10 10 11 11 12 12 13 13 14 14 in the for in ax- + by" + cz" 4- da- 13 Of these 55 forms, the 12 forms 1, 1, 1, 2 1, 1, 2, 4 1, 2, 4, 8 1, 1, 2, 2 1, 2, 2, 4 1, 1, 3, 8 1, 2, 2, 2 1, 2, 4, 4 1, 2, 3, 6 1, 1, 1, 4 1, 1, 2, 8 1, 2, 5, 10 have been ah-eady considered by Liouville and Pepin*. 8. I shall now prove that all integers can be expressed in each of the 55 forms. In order to prove this we shall consider the seven cases (2*41) — (2*47) of the previous section separately. We shall require the following results concerning ternary quadratic arith- metical forms. The necessary and sufficient condition that a number cannot be expressed in the form x^-Vy--\-z' (3-1) is that it should be of the form 4^(8ya + 7), (\ = 0, 1,2..., /x = 0, 1,2, ...) (3-11). Similarly the necessary and sufficient conditions that a numbei- cannot be expressed in the forms x'+ tf+2z- : (3-2), .T^+ f + '^z' (3-3), .r 2 + 2 2/- + 2^'^ ( 3 • 4 ) , a;-^ + 23/- + 3^2 (3-5), x- -^ "lif + ^z- (3-6), ^2+ 2y^+ bz- ...(3-7), are that it should be of the forms 4^(16/i + 14) (3-21), 9M 9/*+ <)) (3-31), 4^( 8/.+ 7) (3-41), 4^(16ya + 10) (3-51), 4^(16/^+14) (3-61), 25^(25/x+10) or 25^(25/* + 15)t (3-71). " There are a large number of short notes by Liouville in vols, v-viii of the second series of his journal. See also Pepin, ibid., ser. 4, vol. vi, pp. 1-67. The object of the work of Liouville and Pepin is rather different from mine, viz. to determine, in a number of special cases, explicit formulae for the number of representations, in terms of other arithmetical functions. t Results (3-11)— (3-71) may tempt us to suppose that there are similar simple results for the form ax- + hy- + cz-, whatever are the values of a, b, c. It appears, however, that in most cases there are no such simple results. For instance. 14 Mr Raiitavujan, On tJie exjyression. of a nninher The result concerning ./- + y^ + z- is due to Cauchy : for a proof see Landau, Handhuch der LeJtre von der Verteilung der Prim- zahlen, p. 550. The other results can be proved in an analogous manner. The form x- + y" + ^z"- has been considered by Lebesgue, and the form x'^-\-y'^-{-'^Z' by Dirichlet. For references see Bach- mann, Zahlentheorie, vol. iv, p. 149. 4. We proceed to consider the seven cases (2'41) — (2*47). In the first case we have to show that any number N can be expressed in the form N' = x- + y- + z- + du" (4- 1 ), d being any integer between 1 and 7 inclusive. If JSf is not of the form 4^(8yLt + 7), we can satisfy (4-1) with u = 0. We may therefore suppose that iV^= 4^ (S/jl + 7). First, sup]3ose that d has one of the values 1, 2, 4, 5, 6. Take u = 2\ Then the number N-du' = ^^(8fM+7-d) is plainly not of the form 4^(8/4 + 7), and is therefore expressible in .the form x^ + y^ + z^. Next, let d = S. If /i = 0, take u = 2\ Then N - dii- = 4^+1. the numbers which are not of the form .r- + 2?/- + 10^- are those belonging to one or other of the four classes 25^(8^ + 7), 25^(25^ + 5), 25^ (25/x + 15) , 25^ (25/^+20). Here some of the numbers of the first class belong also to one of the next three classes. Again, the even numbers which are not of the form x'^ + ij- + lOz- ai'e the numbers 4^(16^ + 6), while the odd numbers that are not of that form, viz. 3, 7, 21, 31, 33, 43, 67, 79, 87, 133, 217, 219, 223, 253, 307, 391, ... do not seem to obey any simple law. I have succeeded in finding a law in the following six simple cases: •«■''+ ?/2 + 4,--, X-+ y- + 5z-, x^+ y' + 6z', x^+ y^ + 8z-, x^ + 2y-^ + Qz\ .r2 + 2y- + 8^2. The numbers which are not of these forms are the numbers 4^(8^ + 7) or (8^1 + 3), 4^(8^ + 3), 9^(9ya + 3), 4^(16/x+14), (16m + 6), or (4^ + 3), 4^ (8m +5), 4^^(8^4-7) or (8^ + 5). in the form (ur- + hy" + cz" + du? 15 If ya^l, take« = 2^+\ Then i\r-rf«;^ = 4^(8/^-5). In neither of these cases is TV — d}f- of the form 4^ (8/i + 7), and therefore in either case it can be expressed in the form X- + 2/2 4- z". Finally, let d=l. If ^u, is equal to 0, 1, or 2, take « = 2\ Then N - d^ir is equal to 0, 2 . 4^+\ or 4^+-. If /a^S, take M. = 2^+1. Then iV-*/,^=4^(8/x-21). Therefore in either case N — du- can be expressed in the form a;2 + 2/2 4- Z-. Thus in all cases N is expressible in the form (4'1). Similarly we can dispose of the remaining cases, with the help of the results stated in § 3. Thus in discussing (2-42) we use the theorem that every number not of the form (3'21) can be expressed in the form (3*2). The proofs differ only in detail, and it is not worth while to state them at length. 5. We have seen that all integers without any exception can be expressed in the form m. {x^ +'if + z^) + mi'^ (5-1), when m = \, li^n^l, and m= 2, n = 1. We shall now consider the values of m and n for which all integers with a finite number of exceptions can be expressed in the form (5'1), In the first place 7?? must be 1 or 2. For, if m > 2, we can choose an integer v so that 7iu' ^ V (mod 7)1) for all values of u. Then (nifx + v) — mi^ m where fi is any positive integer, is not an integer ; and so 7nfj, + v can certainly not be expressed in the form (5'1). We have therefore only to consider the two cases in which m is 1 or 2. First let us consider the form cc- + 2/- + z- + nit^ (5'2). I shall show that, when n has any of the values 1, 4, 9, 17, 25, 36, 68, 100 (5-21), 16 Mr lia/uanujan, On tJie expression of a nwniber or is of any of the forms 4yt+2, 4yb + 3, 8^•^-o, 16A; + 12, 32/.^ + 20 ...(5-22), then all integers save a finite number, and in fact all integers from 4?i onwards at any rate, can be expressed in the form (5*2) ; but that for the remaining values of n there is an infinity of integers which cannot be expressed in the form required. In proving the first result we need obviously only consider numbers of the form 4*^ (8yu, + 7) greater than n, since otherwise we may take w = 0. The numbers of this form less than n are plainly among the exceptions. 6. I shall consider the various cases which may arise in order of simplicity. (6-1) 7^ = (mod 8). There are an infinity of exceptions. For suppose that N = Sfju + 1. Then the number N - nu- = 7 (mod 8) cannot be expressed in the form cc- + y- + z'^. (6-2) n=2 (mod 4). There is only a finite number of exceptions. In proving this we may suppose that iV=4^(8/i + 7). Take u=l. Then the number ]SF - oiu^ = 4'" (S/M + 1) - n is congruent to 1, 2, 5, or 6 to modulus 8, and so can be expressed in the form x^ + y^ + z^. Hence the only numbers which camiot be expressed in the form (5"2) in this case are the numbers of the form 4^(8/i+ 7) not exceeding n. (6-3) n=h (mod 8). There is only a finite number of exceptions. We may suppose again that i\r = 4^ (8/i + 7). First, let X =|= 1 . Take u=\. Then N - nu- = 4^ (8/i + 7) - n = 2 or 3 (mod 8). If X = 1 we cannot take u = l, since N - n = 7 (mod 8) ; so we take u = 2. Then JSf- nu- = V (8/i + 7) - 4n = 8 (mod 32). In either of these cases N — nu^ is of the form cc'^ -\-y'^ + z". Hence the only numbers which cannot be expressed in the form (5*2) are those of the form 4^ (8/a + 7) not exceeding ??, and those of the form 4 (8/ti + 7) lying between n and 4?l in the funii aw- + Inf + cz- + da- 1'7 (6-4) /i= 3 (mod 4). There is only a finite number of exceptions. Take iV^=4M8/A + 7). If X^l, take ti=l. Then N — nil- = 1 or 5 (mod 8). If X = 0, take n = 2. Then N - nu? = 3 (mod 8). In either case the proof is completed as before. In order to determine precisely which are the exceptional numbers, we must consider more particularly the numbei'S between n and 4» for which X = 0. For these \i must be 1, and N -nu-= (mod 4). But the numbers which are multiples of 4 and which cannot be expressed in the form .x- + y ' + z' are the numbers 4''(8i. + 7), (/c = l, 2, 3, ..., v^O,l, 2, 3, ...)• The exceptions required are therefore those of the numbers n + ¥{^v + 1) (6-41) which lie between n and 4» and are of the form 8;i + 7 (6-42). Now in order that (6'41) may be of the form (6'42), k must be 1 if 11 is of the form 8A- 4- 3 and k may have any of the values 2, 3, 4, ... if n is of the form 8A;+7. Thus the only numbers which cannot be expressed in the form (5'2), in this case, are those of the form 4^ (8/i + 7) less than n and those of the form ?i + 4''(8i/+7), (y-0, 1, 2, 3, ...), lying between n and 4?i, where k=\ if n is of the form 8A; + 3, and K>\ if ?r is of the form 8A; + 7. (6-5) n = 1 (mod 8). In this case we have to prove that (i) if n ^ 33, there is an infinity of integers which cannot be expressed in the form (5"2) ; (ii) if n is 1, 9, 17, or 25, there is only a finite number of exceptions. In order to prove (i) suppose that iV = 7 . 4^. Then obviously u cannot be zero. But if u is not zero n^ is always of the form 4''(8t/+l). Hence N - nu^ = 7 . 4^ - ?i . 4" (8v + 1). Since n ^33, X must be greater than or equal to k + 2, to ensure that the right-hand side shall not be negative. Hence N - jui^ = ^^ (Sk + 7), VOL. XIX. PT. I. 2 18 31r Ramanujan, On the ex'pression of a number where k = 14 . V-''-- - nv - ^ {n + 7) is an integer ; and so N — nu- is not of the form x- +y'^ + z\ In order to prove (ii) we may suppose, as usual, that N = 4^ (Sfi + 7). IfX = 0, take w=l. Then iV - nil'' = 8// + 7 - 7? = 6 (mod 8). If X^l, take w= 2^-1. Then where k = 4< (fji + l) -^{n + 7). In either case the proof may be completed as before. Thus the only numbers which cannot be expressed in the form (5'2), in this case, are those of the form 8/u. + 7 not exceeding n. In other words, there is no exception when n = 1 ; 7 is the only exception when n = 9; 7 and 15 are the only exceptions when n = 17 ; 7, 15 and 23 are the only exceptions when n = 25. (6-Q) n = 4 (mod 32). By arguments similar to those used in (6'5), Ave can show that (i) if w ^ 132, there is an infinity of integers which cannot be expressed in the form (5*2) ; (ii) if n is equal to 4, 36, 68, or 100, there is only a finite number of exceptions, namely the numbers of the form 4'^ (8yu, + 7) not exceeding n. (6-7) ?i = 20 (mod 32). By arguments similar to those used in (6'3), we can show that the only numbers which cannot be expressed in the form (5'2) are those of the form 4^ (8yLi +7) not exceeding n, and those of the form 4^(8/A + 7) lying between n and 4n. (6-8) n= 12 (mod 16). By arguments similar to those used in (6'4), we can show that the only numbers which cannot be expressed in the form (5"2) are those of the form 4^ (Sfi + 7) less than n, and those of the form n + 4>^(8v + 1), (i/ = 0, 1, 2, 3, ...), lying between n and 4>i, where /c = 2 if n is of the form 4 (8k + 3) and AC > 2 if w is of the form 4 (8A; +7). We have thus completed the discussion of the form (5 2), and determined the exceptional values of iV precisely whenever they are finite in number. 7. We shall proceed to consider the form 2 (ic^ +y" + z') + mir .(7-1). in the fornt (i.r- + bij'^ -\- cz- + (hr ID In the first place n must be odd ; otherwise the odd iiuinbers cannot be expressed in this form. Suppose then that n is odd. I shall show that all integers save a finite number can be expressed in the form (7"1): and that the numbers which cannot be so expressed are (i) the odd numbers less than n, (ii) the numbers of the form 4^^ (16yu. + 14) less than 4n, (iii) the numbers of the form n-\- 4^(16ya+ 14) greater than n and less than 9w, (iv) the numbers of the form cn-\-¥{l(w+U\ (i^ = 0, 1, 2, 3, ...), greater than 9/i and less than 25w, where c = 1 if n = \ (mod 4), c = 9 if w = 3 (mod 4), « = 2 if n^= 1 (mod 16); and /c > 2 if /(-=9 (mod 16). First, let us suppose N even. Then, since n is odd and N is even, it is clear that u must be even. Suppose then that We have to show that M can be expressed in the form x"->r y- -\- Z' + 27?,?'- (7-2). Since %i = 2 (mod 4), it follows from (6'2) that all integers except those which are less than 2n and of the form 4-^ (8//. + 7) can be expressed in the form (7*2). Hence the only even integers which cannot be expressed in the form (7'1) are those of the form 4^(16/* + 14) less than 4n. This completes the discussion of the case in which N is even. If N is odd the discussion is more difficult. In the first place, all odd numbers less than n are plainl37^ among the exceptions. Secondly, since n and N are both odd, u must also be odd. We can therefore suppose that iV = ?i + 2il/, xv" = 1 + 8A, where A is an integer of the form |^•(^• + l), so that A may assume the values 0, 1, 3, 6, .... And we have to consider whether n + 2if can be expressed in the form 2 0r2 + 2/'^ + ^'^) + w(l +8A), or M in the form «- + 2/- + ^- + 4nA (7-3). If M is not of the form 4^ (8/a + 7), we can take A = 0. If it is of this form, and less than 4?i, it is plainly an exception. These numbers give rise to the exceptions specified in (iii) of section 7. We may therefore suppose that M is of the form 4^ (8^* + 7) and greater than 4/?. 2—2 20 Mr Ramamtjan, On the eo;pression of a number 8. In order to complete the discussion, we must consider the three cases in which n = 1 (mod 8), n = 5 (mod 8), and n = S (mod 4) separately. (8-1) /I- 1 (mod 8). If X is equal to 0, 1, or 2, take A = 1. Then M - 4?iA = 4^ (8/i + 7) - 4?i is of one of the forms 8z; + 3, 4 (Sv + 3), 4 {8v + 6). If A. ^ 3 we cannot take A = 1, since if — 4?iA assumes the form 4 (8i/ + 7) ; so we take A = 3. Then M - 4n A = 4^ (SyLt + 7) - 12n is of the form 4 {8v + 5). In either of these cases M — 4nA is of the form x^ + y'^ -\- z^. Hence the only values of M, other than those already specified, which cannot be expressed in the form (7*3). are those of the form 4«(8i; + 7), (z/ = 0, 1,2, ...,«>2), lying between 4?i and 12??. In other words, the only numbers greater than 9'n which cannot be expressed in the form (71), in this case, are the numbers of the form n+4«(8j; + 7), (i^ = 0, 1, 2, ..., /c> 2), lying between 9?i and 25?l (8-2) ?i = 5 (mod 8). If X 4= 2, take A =1. Then ili - 4wA = 4^ (8/i + 7) - 4?i is of one of the forms 8i/ + 3, 4 (8z/ + 2), 4 (8z/ + 3). If \ = 2, we cannot take A = l, since ilf— 4?iA assumes the form 4 (8v + 7) ; so we take A = 3. Then M- 4mA = 4^ (8/A + 7) - \%i is of the form 4 (8y + 5). In either of these cases M — 4/? A is of the form (x? ■\-y'^ A- z^. Hence the only values of M, other than those already specified, which cannot be expressed in the form (7-3), are those of the form 16 (8^t + 7) lying between 4n and 12?i. In other words, the only numbers greater than 2n which cannot be expressed in the form (7"1), in this case, are the numbers of the form n + 4"^(16/ti + 14) lying between 9?? and 2.5». in the form ax~ + by- + oz- -f dti^ 21 (8-3) n = 3 (mud 4). If \ =1=1, take A = 1. Then M - 4»,A - 4^ (8ya + 7) - hi is of one of the forms Si. + 3, 4(4i/+l). If \ = 1, take A = 3. Then M - 4mA = 4 (8/* + 7) - 12/i is of the form 4(4^- + 2). In either of these eases M — 4?iA is of the form j/-^ + -tf + z-. This completes the proof that there is only a finite number of exceptions. In order to determine what they are in this case, we have to consider the values of M, between 4?i and 12w, for which A = 1 and M - 4h A = 4 (8;i + 7 - n) = (mod 16). But the numbers which are multiples of 16 and which cannot be expressed in the form x- + y^ -\- z- are the numbers 4''(8z/+7), (/c = 2, 8, 4, ..., y=0, 1, 2, ...)• The exceptional values of M required are therefore those of the numbers 47« + 4*^ (8i. + 7) (8-31) which lie between 4fi and \2n and are of the form 4(8/iA + 7) (8-32). But in order that (831) may be of the form (8"32), k must be 2 if n is of the form 8/^' + 3, and k may have any of the values 3, 4, 5, ... if n is of the form 8A; + 7. It follows that the only numbers greater than 9« which cannot be expressed in the form (7"1), in this case, are the numbers of the form 9w + 4« (16i^ + 14), {v = 0, 1, 2, . . .), lying between 9w and 25?i, where k=2 if n is of the form 8/v + 3, and /c > 2 if ?i is of the form 8^- + 7. This completes the proof of the results stated in section 7. 22 Mr Van Horn, An Axiom in Symbolic Logic. An Axiom in Symbolic Logic. By C. E. Van Horn, M.A. (Commimicated by Mr Q. H. Hardy.) [Received 30 August 1916: read 30 October 1916.] Philosophy's task is a search for the primal and fundamental elements of the world. Its face is turned in the opposite direction to that of science and mathematics. Philosophy hands back to them its results, and they as best they can construct systematic bodies of doctrine that purport to show us what the world may bo on the one hand (science) and what the world might be on the other (mathematics). As philosophy advances in the pursuit of its task it is continually vacating old ground to science and mathe- matics. The history of this change of boundary can be traced in the changes in the nomenclature of human knowledge : Natural Philosophy has become Physics ; Mental Philosophy has become Psychology ; Moral Philosophy is becoming the inductive science of Ethics. Thus (paradoxically speaking) philosophy's advance is to be marked by the retreat of her boundaries. It is interesting to Avatch this retreat in a field occupied b}' philosophy from its very beginning, and until recently supposed to be its permanent possession. I refer to the field of the foundations of mathematics. Here large areas once occupied by philosophy by sovereign right of long control are slowly passing into the possession of pure mathematics; and by the way both are gainers by the transfer*. To facilitate the mathematical treatment of these new areas a new instrument of investigation had to be invented, namely, Mathe- matical, or Symbolic, Logic. This new logic, which is infinitely more powerful than the traditional logic, and which embraces all that is really self-consistent in the old logic, makes possible a precise and easy handling of all the highly abstract and complex ideas occurring in the noAv fields. For example, both philosophy and the old logic found themselves involved in many a tangle on questions concerning classes and relations because neither possessed the requisite instruments of analysis. Again, philosophy had wandered into a veritable labyrinth of difficulties concerning infinity, quantity, continuity, and so on. Here too the secret of the trouble lay in the inadequacy of the instruments of analysis afforded by the traditional logic. * Much valuable light is thrown upon the details of this process in the writings of Bertrand Russell, especially in the preface and introductory chapters of the Frincipia Mathematica, Vol. i. 1910; and more recently in his Scientific Method in Philosophy, 1914. Ml' Van Horn, An Axiom in Symbolic Logic 28 Nuw however the matter is all changed. Philosophy, equipped with the latest instruments of mathematical logic, is able to deal successfidly with the problems of these fields. In fact so fully have these ideas been analysed that at last philosophy as such has relinquished these fields to pure mathematics. Even more, the whole field of deduction has now become the foundation-branch of mathematics and has developed into a precise Calculus of Pro- positions. Out of it grow by easy stages the Calculus of Classes and the Calculus of Relations, and these in turn grow by equally easy stages into all the manifold branches of pure mathematics as more commonly known. It is in these and similar ways that philosophy and pure mathematics are both gainers by the transfer of the fields recently acquired by mathematics from philosophy. It is now easy to understand why the axioms of mathematical logic (and so of all pure mathematics) lie in the borderland between philosophy and mathematics, and are thus the concern of the philosopher equally with the mathematician. To depart entirely from our figures and adopt others, the rootage of mathematics is in philosophy. It is here too that we meet the innovations of mathe- matical logic that appear so fantastic to the philosopher trained only in the old logic. Its definitions and treatment of some of the common terms of language seem so at variance with what the traditional logician is familiar with that he often views the new logic as the victim of some delusion. It appears however from the nature of the case itself that many of those peculiarities, which from the view-point of traditional logic would be described as abnormal, do not deserve to be so described ; that in fact it is in the theories of the traditional logician and philosopher that the abnormalities really occur*. In order to indicate what seems to me a possible simplification of the axiomatic basis of mathematical logic I wish to introduce in a new form an idea advocated by Shelfer. Its importance lies in the fact that in terms of it Sheffer was able to define the four fundamental operations of logic, namely. Negation, Disjunction, Implication, and Conjunction or Joint Assertion. It is a familiar fact that Kronecker found the use of certain auxiliary quantities (let us call them ' parameters ') of great value in his algebraic investigations, the chief value lying in the fact that their dis- appearance led to desired relations among numbers essential to his investigations. It is a precisely similar use of Sheffer's idea that I desire to make in the field of the philosophy of logic. In terms of it I define, after him, the four fundamental operations of logic. Then, unlike him, I work by means of an axiom to eliminate that idea from the formulae, and in so doing to arrive at the desired * Cf. Russell, Scientific Method in Philosophij, chap. i. 24 Mr Van Horn, An Axiom in Si/ntholic Logic properties and relations of the four fundamental operations. The chief excellence of my method seems to reside in the fact that proceeding as indicated above I have been able to prove as pro- positions of mathematical logic some of the axioms hitherto laid down at the basis of this logic. In its most satisfactory form the axiomatic basis of mathe- matical logic has been stated by Bertrand Russell in the first volume of the Principia Mathematical. In *1 of Vol. i., pp. 98-101 , of the Principia will be found the primitive propositions required for the theory of deduction as applied to elementary propositions. I confine myself to these purposely, for it is here that I have succeeded, I believe, in simplifying the axiomatic basis of mathematical logic. Let p and q be any two elementary propositions. The four fundamental operations give us (1) ~ p {not-p), (2) pv q (either p or q), (3) j9 D q (p implies q), and {4<) p • q (both p and q). After Sheffer, I define these four results in terms of a single undefinable operation. I will call this undefinable operation Deltation. The result of performing this operation upon two elementary propositions p and q is symbolized, after Sheffer, 'pAq' (read " j) deltas q'). The four fundamental operations of logic can be expressed as logical functions of this parameter thus : Negation: ~p. = .jjAj9 D£ Disjunction : /;vg. = .~^jA~g Df Implication : pD q . = .p A <^ q Di Conjunction : p • q . = . '^ (p A q) Df. These definitions of the four fundamental operations of logic as functions of the one undefined parameter, Deltation, are made relevant to our discussion by means of the following axiom. Axiom. If p and q are of the same truth-value, then ' p A q ' is of the opposite truth-value ; but if j) and q are of ojjposite truth - values, then ' p A q' is true. For convenience of reference it might be well for me to state at this point Russell's primitive propositions concerning elementary propositions as he enunciates them in *1 of the first volume of the Principia. *1.1 Anything implied by a true elementary proposition is true. Pp|. t Whitehead and Russell, Princiina Mathematica, Vol. i. 1910, Vol. ii. 191'2, Vol. III. 1913 (Cambridge University Press). X Eussell uses the letters "Pp" to stand for " primitive proposition, " as does Peano. 3fr Van Horn, An Axiom in Sipnbolic Lo(jio 25 *1.H When <^x can be asserted, where x is a real variable, and ' (fjxD yfr x ' can be asserted, where x is a real variable/then yjrx can be asserted, where x is a real variable. Pp. *1.2 h : pvp.D .p Pp. *1.3 \- : q.D .pv q Pp. * 1.4 [■ : pv q .0 .qv p Pp. * 1 .5 \- : py/ (qv r).D .qv {pv r) Pp. *1.6 h: .q'^r.D-.pvq.D.pyr Pp. *1.7 If j9 is an elementary proposition, ~ p is an elementary proposition. Pp. *1.71 If p and (/ are elementary propositions 'pvq' is an elementary proposition. Pp. *1.72 If ^p and i/rj? are elementary prepositional functions which take elementary propositions as arguments, ' (f) pv -ylrp' is an elementary prepositional function. Pp. These are all the primitive propositions that are needed for the development of the theory of deducti(jn, as applied to elementary propositions, according to Russell's method of treatment. It is my purpose to show that by means of my axiom Russell's primitive propositions *1.2 to *1.7l can be demon- strated. I do this by starting at the very beginning and developing the immediate consequences of three of the axioms which I lay down as the basis of the theory of deduction as applied to elementary propositions. The resulting deductive development at length reaches a point where it includes among its theorems Mr Russell's seven pi'imitive propositions and two others that can take the place of his definitions of Implication and Conjunction. Altogether I prove seventeen theorems. Some of these theorems occur as propositions in the first volume of the Principia. Al- though many more theorems can be proved as simply as the ones given, to economize space I shall stop at the point where my development of Mathematical Logic includes the nine theorems mentioned above. I will now state the three axioms used in this paper. The first is * 1.1 given above, the last is my axiom as already enunciated. Axiom 1. Anything implied by a true elementary proposition is true. Axiom 2. Ifp and q are elernentary propositions, then " p Aq' is an elementary proposition. Axiom 3. If p cund q are of the same truth-value, then ' p Aq' is of the opposite truth -value ; hut if p and, q are of opposite truth- values, then ' p Aq' is true. 26 Mr Van Hum, An Aadoiii in Symholic Logic Theorem 1 If }) is an elementary proposition, ~ p is an elementary pro- position, Deni. Axiom 2 gives us ' p Ap' elementaiy when j) is elementary ; 'pAp' is ~ p, by Definition of Negation. Hence the theorem. This is a proof of Mr Russell's primitive proposition *1.7 given above. Theorem 2 Ifj} and q are elementary ])ropositions, ' pv (j' is an elementary proposition. Dem. By Theorem 1 , if p and q are elementary so also are ~ p and ~ q. Therefore, by Axiom 2, ' -^ p A ~ (/ ' is elementary ; but this, by Definition of Disjunction, is ' pv q'. Hence the theorem. This is Mr Russell's primitive proposition *1.7l quoted above. Theorem 3 The propositions p and ~ p are of opposite truth-values. Dem. Two possibilities can occur : 1°:^ true. By Axiom S, " p A p' is false; but this by Definition of Negation is ^ p; hence in this case p and ~ p are opposite in truth-value. 2° : j9 false. By Axiom 3, 'pAp' is true; but this by Definition of Negation is ~ jo ; hence in this case also ^j and f^ p are opposite in truth-value. Hence the theorem. This theorem states in precise form the information usually given in text-books on logic in more or less vague statements that are called ' definitions ' of negation. Theorem 4 \-. pDp. Dem. [Th. 3] h. p and <^ p of opposite truth-values (1) [(1). Ax. 3] \-. pA r^ p (2) [(2). Def. of Implication] h. theorem. This is proposition *2.08f of the Principia. I 0]}. cit. Vol. I. p. 105. Mr Van Horn, An Axiom in Symholio Loyic 27 Theorem 5 If 2) is false, ' p A q' is always true. Bern. Two possibilities can occur : either q true, or q false. In either case ' p A q' IB true by Ax. 3. Theorem 6 If q Is false, ' 2J A q' is alivays true. Proof similar to that of preceding theorem. Theorem 7 llie jyropositions ' p A q' and ' q A p' Jmve the same triUh-ualue. Deni. li' p and q are of the same truth- value then, by Ax. o, ' p A q' and ' q A p' are both of the opposite truth-value. If p and q are of opposite truth-values then, by Ax. 3, ' p A q' and ' q A p' are both true. Hence the theorem. Theorem 8 The proposition f"^ p A 1^ {(^ q A f^ r) is true if any one or more of the propositions p, q, r are true; but if all of these propositions are false then the proposition ~ jj A '^ {^ q A ~ r) is false. Dem. Eight possibilities can occur : 1° : p, q, r all true. Then (Th. 3) ~ ^j, ~ q, ~ r are all false. Hence (Ax. 3) ' ~ 9 A ~ r ' is true. Hence (Th. 3) ~ (~ g- A ~ r) is false. Hence (Ax. 3) the proposition '~ jo A ~ (~ </ A ~ /•)' is true in this case. 2^ : jj and q true, but r false. By Th. S, r^ p and ~ q are false, while ~ r is true. Hence (Ax. 3) ' ~ r/ A ~ ?■ ' is true. Hence (Th. 3) ~ (~ (/ A ~ r) is false. Hence (Ax. 3) the proposition is true in this case. In a similar manner in the following cases : 3° : j) true, q false, r true ; 4° : ]) false, q, r true ; o" : p true, q, r false ; 6° : J) false, q true, r false ; 7° : p, q false, r true ; we have ' ^' j) A ^ (■-- </ A ^^ /•) ' true. But in 8" : p, q, r false, we have ~ jj, r^ q, ^ r all true, by 28 Jllr Van Horn, An Axiont in H[iinholic Logic Th. 8. Hence (Ax. 3) '~ </ A '^ ?• " is false, making ~ (~ (/ A ~ /•) true (Th. 3). Hence (Ax. 3) in this case the proposition is false. Hence the theorem. Theorem 9 The propositions ' <>•' p A f^ (<^ q A f^ r)', 'r^(/A~(~^:>A~ r) ', always have the same truth-valm. This follows at once from Th. 8. At this point I introduce Mr Russell's definition of Equivalence f as it occurs in the Principia. Equivalence: p = q. = .pDq.qDp Df. Theorem 10 h. p= <^ (^ p). Dem. We first prove h. jj D ~ ( ~ p). Two cases arise : 1°: p true. By Theorem 3, '^ ^ is false, ~ (~p) is true, and f^ [f^ {^ py] is false. Hence [Ax. 3] h. I? A ~ [~ (~ jj)] (1) [(1). Def. Implica.] Kj9D~(~p) (2) 2° : p false. By Th. 3, r^ p is true, ^^ {<^ p) is false, and ,^ [r^ ('^i^)] is true. [Ax. 3] h. j5 A ~ [~ {"^ p)\ (3) [(1). Implica.] V. p D (^ (<^ p) (4) Hence in all cases we have V. p'^r^{<^p) (5) We now prove V. ~(~_p)Di9. [Th. 3] V. q and ~ g of opposite truth-values (6) [(6). Ax. 3] h. r^qAq (7) [(7). /] [(8). Def. Implica.] V. '^ {r^ p) A r^ p (8) V. ~(~p)Dp (9) [(5). (9). Def. Equiv.] V. theorem. This is proposition *"4.13 of the Principia^. It is the Principle of Double Negation, and asserts that any proposition is logically equivalent to the denial of its negation. t Op. cit. Vol. I. p. 120, *4.0l. % Op. cit. Vol. I. p. 122. ^fr Van Horti, An A.iiom, in Si/)iibolic Logic 29 Theorem 11 H: pvp . D .p. Dem. [Ax. 3] ■ h. ~ jj and ' <^ p A f^ p' of opposite truth- values (1) [(1). Ax. 3] |-:~_p A ~^9. A . ~_p (2) [(2). Def. Disjunc. Implica.] I-. theorem. This is Mr Russell's primitive proposition *1.2 given above. Theorem 12 h: q.'^.pwq. Dem. Two cases need only be treated : I'' : q true. Then (Th. 3) ~ q is false. Hence (Th. 6) ' ~ jj A <^ q ' is true. Hence ~ (~ ^j A ~ q) is false, by Th. 3. Hence [Ax. 3] h : (/ . A . <^ ( ~ p A ~ (/) ( 1) 2° : q false. [Th. r, |-~(~pA ~g)j F. , . A . ~ ( ~ ,, A ~ ,/) (2) [(1). (2). Def. Disjunc. Implica.] h. theorem. This is Mr Russell's primitive proposition *l.o given above. Theorem 13 h: pv q .D . qy P' Dem. [Th. 7] h : ' ~ jt) A ~ g ' and ' ~ g A ~ p ' of the same truth- value (1) [(1). Th. 3. Ax. 3] h: ~ jj A ~ (/ . A . ~ ( ~ ry A ~ p) (2) [(2). Def Disjunc. Implica.] h: theorem. This is Mr Russell's primitive proposition *1.4 given above. Theorem 14 V : p y {q y r) ."^ . qy {p M r). Dem. [Th. 9] I-: '~p A ~(~(/ A ~?-)' a-nd '~(/ A ~(~|) A ~?-)' of the same truth-value (1) [(1). Th. 3. Ax. 3] h: ~ ;j A ~ (~ (? A ~ /•) . A . ~ [~ q A '^ {r^ p A ~ r)] (2) [(2). Def. Disjunc. Implica.] h: theorem. This is Mr Russell's primitive proposition *1.5 given abt)ve. 30 Mr Van Horn, An Aj'iunt in Sfjtnbolic Logic Theorem 15 J \-:.qDr.D:pvq.D.2i'v r. Bern. There are three cases to be discussed : 1" : li p is true, or if r is true, or if both p and r are true, q being any elementary proposition. [Th. 8] }-: r^l. A.^i^p A '^r) (1) [(1). ::ii. Th. 10] h: Z. A .~(-2J A ~?-) (2) [(2). ~i^^ ~5j h: ~p A ~(/. A . ~(~2) A ~ r)(3) [(3). Th. 3. Th. 6] h: q A ^ r . A . <^ [^ p A '^ q . A . <^ (r^ p A '^ r)] (4) Taken together with the Definitions of Implication and Disjunction, (4) gives the theorem in this case. 2° : If both p and r are false, but q true. In this case ~ ^j and oo r are true by Th. 3. Hence (Ax. S) ' ^ p A ~ r ' is false. The proof in this case proceeds as folloAvs : [Th. 3] 1-: ~(-2) A ~ r) (5) Since q is true, '^ q is false (Th. 3). [Th. 6] h. - 19 A ~ 5 (6) [(5). (6). Th. 3. Ax. 3] V: ^[^^p A ^q. A.'^i^p A ^r)]{1) By Ax. S, ' q A <^ ?' ' is in this case false. [(7). Ax. 3] h: q A ~r.A.'^[~/jA <^5.A.~(~jjA ~ ?•)] (8) As in the previous case this result gives the theorem. 3° : All three false. Hence ~ p and ~ ?■ true as before. In this case ' ^ p A <-^ q' is false by Ax. 3. The proof in this last case proceeds thus : [Th. 3, as in 2°] h. ~(~pA~?') (9) [(9). Ax. 3] h: ~p A ~ g. A . ~(~p A ~ r) (10) In this case q and f--' r are of opposite truth- values. [Ax. 3] h: f^ A~r (11) [(10). Th. 3. (11). Ax. 3] h: ^A~?'.A.'^[~pA'^^.A.'^ (~i^ ^ ~ ''')] (12) As in the two preceding cases, this result, together with the Definitions of Implication and Disjunction, gives the theorem. No other cases can arise. Hence the theorem. This is Mr Russell's primitive proposition *1.6 given above. It asserts that an alternative may be added to both premise and Mr Vail Horn, An Axiom in Si/nihohc Loffic ol conclusion in any implication without impairing the truth of the implication. This completes the list of Mr Russell's primitive propositions that I proposed for proof by means of my axiom, on the basis of the definitions given in this paper of the four fundamental operations of logic. I now propose to prove two propositions which can take the place of his definitions of Implication f and Conjunction j, or Joint Assertion. Theorem 16 Dem. [Th. 4^^'"'^] h: p A -</.D.;9 A ~g (1) [Th. 10] D . ~ (~ p) A ~ ry (2) [(2). Def. Implica. Disjunc] h: pD q .D . ^^ pw q (3) [(1). Th. 10] h: ~(~j9)A ~(y.D.j)A ~ 7 (4) [(4). Def. Implica, Disjunc] h : '^py/q.D.pDq (5) [(3). (5). Def. Equiv.] h : theorem. Theorem 17 h: p . q . = . ^ { '^ p W ^ q). Dem. [Th. 4 *" ^^ ^ '^^ ] h: ~(jo Ary).D.~(p A (/) (1) [Th. 10] D.~[~(~y/) A ~(~^)](2) [(2). Def. Conjunc. Disjunc] I- : p . q . D . ^^ (^ p v r^ q) {S) [(1). Th. 10] h: ~[~(~p)A ~(~5)].D.~(p Afy)(4) [(4). Def. Conjunc. Disjunc] h : '^ (^ p v ^ q) . "D . p .q (5) [(3). (5). Def. Equiv.] h : theorem. With these theorems established the development of the Principia Mathematica can proceed as given by its authors. All that I have done is to reduce the number of axioms needed for that development. Baptist College, Rangoon, Burma. t Op. cit. Vol. I. p. 98, *1.01. + Ibid. p. 116, *3.01. 32 Mr JSicod, A Reduction in the nimiber A Reduction in the number of the Primitive Propositions of Logic. By J. G. P. NicOD, Trinity College. (Communicated by Mr G. H. Hardy.) [^Received and read 80 October 1916.] Of the four elementary truth-functions needed in logic, only two are taken as indefinables in Principia Mathematica. These two have now been defined by Mr Shefferf in terms of a single new function p | q, " p stroke q." I propose to make use of Mr Sheffer's discovery in order to reduce the number of the primitive propositions needed for the logical calculus. There are two slightly different forms of the new indefinable, for we may treat 2:)\q as meaning the same thing as either ~jj . ~g, or <^p}/ ^qt- The definition of <^p is the same in both cases, namely p \ p, while that of pv q simply changes from p/q \p/q with the AND-form into p/p \ qjq with the 07^-form. However, the best course is for us to define all the four truth- functions directly in terms of the new one. In so doing, we find that, while the definition of ~j9 remains the same, and those of pv q, p . q simply permute, as we pass from the ^iV^D-form to the Oi^-form, the definition of pO q is simpler in the latter form. It is p I qjq, as against j;/j) j q \p/p \ q. The OJ?-form is therefore to be preferred §. Definitions. f^p . = . p\p Df. pvq.^.plpiq/q Df. pO q . = . p\ qjq Df. p . q . — . p/q I p/q Df Remaeks on these Definitions, One ought not to aim at retaining before one's mind the complex translation into the usual system, "-^pv^q" as the "real meaning" of the stroke. For the stroke, in the stroke- system, is simpler than either ~ or v, and fi-om it both of them arise. We may not be able to think otherwise than in terms of the four usual functions ; it will then be more in accordance with the nature of the new system to think of the j , not as some fixed compound of -^ and v, but as a bare structure, out of which, in various ways, ~ and v will grow. + Sheffer, Trans. Amer. Math. Soc. Vol. xiv. pp. 481—488. X Sheffer, loc. cit., footnote f, p. 488. % p\q thus corresponds to what is termed the Disjunctive relation in Mr W. E. Johnson's writincrs. |i of the Primitive Propositions of Logic 33 The above definitions give clear expression to the symmetiy between OR and AND ; and this, notwithstanding the choice that we had to make between an Oi?-forni, and an AND-iorva. This is of some interest, because, in general, the very symmetry forces upon us an arbitrary choice, which, in turn, quite obscures the symmetry. I shall use q for q\q whenever convenient. Observe that p I q, i.e. pD q, forms a natural symbol | for implication, allowing of permutation ~q\ p. We may notice in general that the new system brings the four functions into relations far closer than those in Mr Russell's system. For instance, in p/p\p/p-\.p/p the two propositions pv p .D . p and r^pv p coincide. Every stroke-formula falls into two parts on the right and left of a central stem. It will, therefore, add to clearness to use black type instead of dots to indicate the central symbol. Further, slanting strokes are covered by straight ones : thus p/q j p/q stands for (p\q)\ (pj q). The definition of the two primitive notions of the Principia in terms of a single new one tends to reduce the number of the primitive propositions needed. But how far does this reduction actually occur ? Does it extend beyond the obvious substitution of " If p and q are elementary propositions, p\q is an elementary prop." (Sheffer, p. 488) for *r7 and *1'71, stating the same for ~ p and py q respectively ? The reduction goes, as we shall presently find, very much farther. It has first to be said, in order that we may be as precise as possible, that the tuhole amount gained in applying the stroke- definitions cannot with complete certainty be attributed to them. For Mr Russell's system, as it now stands, has not said its last word in that matter. Incidentally, I found that *1'4, pv q .D . q y p, can be proved by means of the other four, with the unimportant change of *1'3, q . "^ . pv q into q . "^ . q v p. In "Association," *1*5, writers for r : p y {q y p) . 1^ . qv ij) V p). The left-hand side, by the help of q ."D . qvp and " Summation," will be found to be implied in pv q. The right-hand side, like- wise, hy p V p . D r p, and " Summation," will be found to imply qvj). The result then follows by using "Syllogism" (obtained from " Summation " with the transformation — - f) twice. p P V p' t By - or ^-^ I mean (following Mr Russell) the substitution of p for q or p, p' for q, q'. By {e.g.) P~ I mean the result of effecting the substitution in P. VOL. XIX. PT. I, 3 34 Mr Nicod, A Reduction in the number Let us, however, take Mr Russell's eight propositions in the form given in Principia. It is my object to reduce them to three — two non-formal and one formal — by means of the stroke-defi- nitions given above. It can be shown, as a first stage, that two formal propositions are enough, namely : (1) p\l)/p. (2) p\q/q\s/q\^. The first proposition is the form of " Identity " (p D p) in the stroke-system. It would, at first sight, appear more natural to adopt the order q/s \ p/s in the left-hand side of (2), since p\qlq-'^-qls\p/s is the syllogistic principle of the stroke-system, giving " Syllogism," pD q .D : q D s . D .pDs when s | s is written for s. It will however be found that the inverted order, s/q 1 p/s, is much more advantageous than the normal syllogistic order, q/s \p/s. For, owing to this " twist," Identity and (2) yield " Permutation," s/p \ p/s, which now enables us to eliminate the twist in (2), and revert to the normal order. From the three propositions thus obtained, the rest follow. This, by the way, illustrates the following fundamental fact. Which form of a given principle is the most general, and contains the maximum assertion, is a function of the symbolic system used. Thus, for instance, in Mr Russell's system, p .D . qwp (a) is more general than p .0 . qD p (b) since (h) is (a) with <^q for q. In the stroke-system, on the contrary, p \ q/q \ p/p, meaning the same thing as (a), is less general than p\ q \p/p, whose meaning is that of (b), since it is obtained from it by writing q\q for q. A further step has to be made in order to be left with only one formal primitive proposition. It consists in adapting to better advantage the form of the primitive propositions to the properties of the stroke-symbolism where implication is concerned. We had above p'^q . = .p\ q/q Df If we look for the meaning of the general form p \ r/q, we find this to be oo 29 V ~ (~ r V ~ 5'), i.e. p .D . r .q. We thus come to the fundamental property that, in the new system, p"^ q is a case of p .D . s . q, whereas in Principia the contrary relation of course holds, i of the Primitive Propositions of Logic 35 This leads us to substitute p \ r/q for 'p \ q/q in the " left-hand sides " of both the non-formal rule of implication and the syllo- gistic proposition (2) above. The reform may be further extended to the proposition (2) as a whole, which might be given the form P ! S/Q instead of P \ Q/Q, with the proviso, if the proposition is to remain true, that *S' must be implied in P. Now, for S, write the pioposition (1) above, p\p/p ; for (as we at this early stage know " unofficially ") a true proposition will be implied by everything. We then have the three primitive propositions of the stroke- system : ( I. If p is an elementary proposition, and q is an Non- elementary proposition, then p\q is an elementary pro- formal 1 position f. \ II. If J) [ r/q is true, and p is true, then q is true. This is the non-formal rule of implication, *1'1, with the modifi- cation just explained. Formal III. p j q/r \t\t/t.\. s/q [p/s. I shall call II " the Rule," and III " the Prop." Remarks on these Primitive Propositions. Observe p r/q in II, while p | q/r in III. This alternance will prove essential for the working of the calculus. In III, I shall use ir for 1 1 t/t, P for p j q/r, Q for s/q \p/s, and shall speak of III as P \ tt/Q. P I ir/Q, by the Rule, yields the same result as the syllogistic proposition (2) above, when the left-hand side P is a truth of logic. This restriction of the syllogistic form to its categorical use with an asserted premiss is a peculiar character of the first proofs to follow, and is of some philosophical interest. One feels inclined to think that III merely asserts together (1) and (2) above. This view, whatever may be the amount of truth it contains, takes AND too much as a matter of course, and tends to lose sight of (a) the fact that III, as a structui;^^s simpler than (2) alone : for III is (2) with t \ t/t instead of s/q \p/s ; and (y8) the very real step from p .q to q, together with the philo- sophical difference between two assertions and only one. The main steps in the formal deduction are : 1. Proof of " Identity," t \ t/t. 2. Passage from P \ ir/Q to the u sual implicative form P [ Q/Q. 3. Elimination of the twist s/q \p/s in Q, and return to the normal order q/s \p/s. t This is the proposition shown by Sheffer to imply the analogous propositions *1*7 and *1-71 in Principia. 3—2 36 Mr Nicod, A Reduction in the numher 4. Proof of " Association," p \ q/r .D.q [p/s. 5. Theorems equivalent to the definitions of p . q, p q in Principia. Proof of Identity, t\t\t. As this first proof from a single formal premiss stands in a unique position, I shall, without in any way obscuring the precise play of the symbols, expound it after a more heuristic order than is usually followed. We start with the Prop. P | tt | Q, and the Rule enabling us to pass from the truth of P to that of Q ; and we have to prove tt. This can only be reached through some proposition of the form -4 1 5 1 TT, where A is a truth of logic f. The proof will thus consist in passing from P | tt | Q to J. 1 5 | tt by some permutative process. A simple two-terms permutative law s 1 5' | ^ | 5, we do not yet possess. Our Prop, yields only a roundabout three-terms per- mutation, slglpjs, subject to the condition of ^jglr being a truth of logic f. This, however, is enough for our purpose. In the Prop., write t. for p, q, r : (a) 7r|7r!Qi, Qi being s|^|^|s. Write now tt for p, q; Q^ for r: then by (a) and the Rule, (b) S ! TT I TT I s. From (b), in the same manner. (c) u I tt/s I s/tt j u. This enables us to pass, by the Rule, from P | tt | Q to (d) Q|7r|P. In order to complete the proof of tt, we need only find some expression which : (a) can be a value for P, i.e. is a case of p\q\ r, and (/3) is implied in some truth of logic, say T. For, by T'lP | P, the Prop., and the Rule, as above, (e) s\P[T\~s. In (e), write Q | tt for s: first by (d) and the Rule, then by T and the Rule, we obtain T\Q\7r, and so (/) t This use of the Rule by anticipation, with still undetermined P's and Q's, is in truth contrary to the nature of a non-formal rule, which must never be used to build up the structure of an argument. It must always be possible to dispense with all such ' anticipated ' assertions in the final form of a proof. This will be seen to be very easy in the present case. of the Primitive Propositions of Logic 37 Now, Qi I 7r| TT fulfils (a) and {^). For (a) tt being the complex expression t\t\t, i s a case of the form q \ r, and (/3) we have, by (c) above, tt i tt/Qi \ Qi/tt \ tt, and by (a) tt | tt | Qj. To obtain the strictest development of the proof we have only to write Qi/tt tt for P and ir ; tt/Qi for T all through the preceding argument. Permutation, s | p | p I s Gives sv p .1) . py s hy ^ , Dem. : Prop. - — — — - , Id., and Rule. j3 q r Tautology, p/p \ p/p \ pjp i.e. py p . -p Dem.: Id.^, Perm., and Rule. P Addition, s\p\sls Gives s ."D .py s by — . Dem. : By Perm, (twice), p \ s/s\sjs \p (a) By Prop, ^-y qrs ' ^ (")' ^ W. +, p \ s/s \ s By Perm., result. Return froivi Generalised Implication P \ tt/Q to P Q/Q. Lemma, pjp \ s/j) Dem. : By Perm, (twice), s/p \ p/s (a) By Prop. -^ — , I- {a), -^ ^ p q r s u\p\ s/p I It Write p/p for ii : by Id. and Perm, (twice), result. t \- (a) means the use of the Rule to pass from a to b iu a sjl). 38 Mr Nicod, A Reduction in the number Theorem, P\irlQ\QIQ\P Dem. : Prop. -^^^-^^ -^ , r Lemma, reeult. p q, r s Hence, by Perm., P \ Q/Q, i.e. P I 5'/^ I s/q i P/^ (^') Syllogism, i? | 5*/^ I q/s \ p/s o s s Gives _p D (/ . D : fy D s . D . jj D 6' for ^^ — Dem. : In this Dem., Permutation is used to correct the twisting action of S\ much as handwriting has first to be inverted, if it is to be seen right in a mirror. By 8' ~ -^ , I" Perm., and Perm., •^ p q, r s qjs I u I u I sjq {a) •^ p q, r s qjs I u I sjq I u (b) By ^- i^lgA^ ^/glW^ g/HW^ ^ H^', h6, result. Association, p \ q/r | q \pjr The structure of the proof is this : Syll. " Il'' ' • p q, r s gives _p I g/r . D : q/r | ?' . {p/r. We now need only the Lemma q \ q/r | r for our result to follow by Syll. twice. Lemma, q \ q/p | p The proof of this lemma — call it L — is as follows : We prove (a) q I LjL, (b) L/L \ q/q. From this, by Syll. and TautoL, the result follows. Dem. : (a) By Syll. ^ , r, s p\qlq-:^-q/p\plp (1) of the Privative Propositions of Logic 89 By Ackl, SylL, I- (1), q. D:q/p\p/2) (2) The right side of (2) implies, by Syll., plp\p.'^.q/p\p (3) By Id., Perm., Add.^/^'^' -'^ , ^ ' ' p, q' q.D:p/p\p (4) By Syll. twice, h (2), h (3), h (4), qD : q ."^ . q/p \p, i.e. q L/L. (b) By lemma to Syll., q/q\s/q; by Perm, and Syll, q/q Iq/s. Hence, q/q \ L/L ; by Perm., L/L \ q/q. Now, by Syll. : L/L 1 q/q .D:q\L/L.D . L/L | L/L. By 1-6, h a, and Taut. -, result. We can now complete the proof of ' Association.' Association, p \ q/r \ q \ p/r Dem. : By Syll., /) | q/r . D : q/r \r .\. p/r By Syll. twice, h Lemma, result. Summation, qDr .D : pvq .D .pv r Dem. : By Syll., Assoc, q \s .D : p\ q/r . D . p\s (1) ^ . s/s, q, p/p , By (1) — ^-^, result. -^ s, r, p Theorems Equivalent to the Definitions of p Dq, p . q, IN Principia. p"^ q • 3 . ^pv q, and reciprocal theorem. That is, p I q/q . D .p/p \ q/q. Bern. : Taut., and Syll. sis D Reciprocal theorem by Add. -^ — — , and Syll. .9, p p\ q ."D . ^p V ~ q, and reciprocal theorem. That is, j9 1 5 . D . p/p \ q/q. 40 Mr Nicod, A Reduction in the nmnher Devi. : Taut. SylL; then, Perm., Taut., and SylL, or S'. Reciprocal theorem by Add, ^-^ instead of Taut. ^ . g . D . ~ {"^p V '^q) and reciprocal theorem. That is, p .q .D . p/q \ p/q. Dem. : Id,, Def, of ~, preceding theorem, and Syll. Reciprocal theorem in the same manner. Appendix, After the substance of this paper had been written, I was given the opportunity of seeing Mr Van Horn's very interesting and original paper dealing with what is practically the same subject, Mr Van Horn recognises clearly the superiority of what has been called above the Oi^-form over the j4iVD-form chosen in Sheffer's text. This deserves the more notice, as Mr Van Horn, I understand, had not Sheffer's article at hand in the time he was writing his own paper. His A, as will be seen from the definitions he gives, is indistinguishable from |. I was much attracted by the harmonious character of Mr Van Horn's third Axiom. It seems to me therefore all the more desirable that certain objections, which Mr Van Horn's proofs in their present form naturally suggest to the reader, should be dealt M'ith, (a) It is not quite plain to me whether " of the same truth- value " (say S for short), " of opposite truth-values " (say 0), are used as indefinables, or as abbreviations. If the former, we have no right to go, e.g., from p q, and '^p, to q, etc., without some axiom to that effect, connecting and S with A, If, on the other hand, S and are abbreviations — as it seems to me they are — the two parts of Axiom 3 stand for not less than four propositions : 1, If jj and q, '^{pAq). 2. If (^p and ~(/, pAq. 3. If p and ^q, pAq. 4, If ^p and q, pAq. We cannot assert the first two, or the last two, or all four, propositions together, because we should then need p . q . D . p, p . q . D . q, before we could make any use of such a synthetic Axiom, of the Primitive P7'opositions of Logic 41 This uncertainty as to the status of S and is not without its effect upoii the proofs. Consider, for instance, Th. 3. In the proof, "1°: p true. By Axiom 3, pAp false" will be seen to require p Sp, concerning the origin of which, and the relation it has to p D j^ (Th. 4), which it indirectly serves to prove, Mr Van Horn says nothing. (/3) In his extensive use of the Principle of Excluded Middle, Mr Van Horn makes no explicit mention of the last steps, that lead from pOq, ^^ pDq, to q. These steps would seem to require several propositions: (1) those carrying us from ^^pyp to qvq — " Summation," plus " Permutation," presumably — and (2) " Tau- tology " qv q .D . q. As Mr Van Horn uses the principle of Excluded Middle in this particular way in the first formal proof given — that of Th. 3 — both the principle itself and the proposi- tions required for its use ought, I think, to be deduced immediately from Axiom 3 ; and I do not see how this is possible. 42 Mr Watson, Bessel functions Bessel functions of equal order and argument. By G. N. Watson, M.A., Trinity College. [Received 1 November 1916: read 13 November 1916.] A proof of the approximate formula Jn{n)' TT 2» 3« w« (the order and argument of the Bessel function being equal and large) was apparently first published by Graf and Gubler*, although the formula had been stated by Cauchyf many years before. The formula has been discussed more recently by Nicholson J and by Lord Rayleigh§, while Debye|| has given a complete asymptotic expansion of Jn{n) in descending powers of 71 ; this expansion is obtained by the aid of the elaborate and powerful machinery which is provided by the mode of contour integration known as the "Methode der Sattelpunkteir"(Methode du Col, method of steepest descents). The earlier writers, just mentioned, employed Bessel's formula 1 /■'" Jn (*') = — I COS (nO — X sin 6) dd, ttJo valid when n is an integer, and it is by no mea.ns obvious to what extent their methods of approximating are valid**. As the correctness of the approximation can be established without the use of contour integration on the one hand and without appealing to physical arguments ff on the other hand, it seems to be worth while to write out a formal and rigorous proof (based on comparatively elementary reasoning) that, when n is large and real, then * Einleitung in die Theorie der Bessehcheii Funktionen, i. (1898), pp. 96 — 107. + Comptes Rendus, xxxviii. (1854), p. 993; Oeuvres (1), xii. p. 163. J Phil. Mag., August 1908, pp. 273—279. § Phil. Blag., December 1910, pp. 1001—1004. II Mathematische Annalen, lxvii. (1909), pp. 535 — 538. 1[ This method of discussing Je"/W<^(s)(fi; consists in choosing a contour on which If{s) is constant, and so Bf{s) falls away from its maximum as rapidly as possible (/(s) being monogenic); it is to be traced to a posthumous paper by Eiemann, Werke, 1876, p. 405. ** See § 4 below. ft For example Kelvin's "Principle of stationary phase" {Phil. Mag., March 1887, pp. 252—255 ; Math. Paiyers, iv. pp. 303—306) is really based on the theory of interference. See also Stokes, Camh. Phil. Trans, ix. (1850), p. 175, foot-note (Math. Papers, ii. p. 341). of equal 07'der and argument 43 Jnin)=^ 7r2*3-^ 2. In order not to restrict ourselves to the case in which n is a positive integer, we take the Bessel-Schlafli integral*, namely sin IITT J II (^) = - cos {nd — X sin 6) dO — ,-nd -X sinh ^ dO, (which is valid whether n be an integer or not), and, after writing n for x, we integrate by parts. This process gives Jn (it) = ^ d nir j 1 "" cos sin mr -^ -Tj. {sin n (0 — sin 6)] dd + d IT J 1 + cosh 9 dd |g-«(^ + siuh^), ,^ nir sinw(^ — sin^) 1 — cos 6 1 + sm nir -n{0-^sm}\d)-\ ^ 1 + cosh^ + '^smn(^-sm^)^^^^^^^ /, mr] Q (1 — cos 6)" sinnTT p sinh^ ^-n(d + s.\nhe) ^n "^ TT Jo (T+COsh^)^^ ;''^' The integrated parts cancel ; and ^^^^^ -n[6 + ^mh0)^0 ^ f"(l+C0sh6^)«-»(^ + «i"l^^)(/^ (1 + cosh^)- Jo = lln; and so, when ?« is large and real, r y . 1 f '' sin ^ sin w<f> , , „ , ^, ?i7r.lo (1 —cose')* ^ where </> has been written in place of 6 — sin 6. It is obvious that <l> inci-eases steadily from to tt as ^ increases from to tt. When 6 is small, (f)r^^6^ and sin ^ . (1 — cos 0)~^ ^ 80-'^. Hence, as ^ -> 0, • <^^ sin 8 (1 — cos 0y Qi ' Now write | (60)* sin ^ . (1 - cos 0)~' =/ (</>) ; Schlatii, 3Iath. Ann. m. (1871), p. 14«. 44 Ml' Watson, Bessel functions then it is fairly evident* that when ^0 ^ir (i.e. when ^ ^ tt), fi{<f>) is bounded and has only a finite number of maxima and minima (and therefore it has limited total fluctuation). Con- sequently, since f -^jr ■• sim/rc/i/r is convergent, we have;]: . v 77 r ^ Lim n~i ^ (f)"'^ sin (n(f>) . f\ {(f)) d(fi =/i (0) | yjr'" sin yjr d-yjr. Therefore, since /i(0)= 1, we have n-ijy~fsm(ncf)).f,{cf>)d<f> = ^T(V> + o(l), and so Jn(n) = 2~ ^ S~ '"^ ir-^ T (^) n~ ■- + o (n~ ^). To obtain the second approximation to Jnin), we obser^ that, when 6 is small, (6«^)^sin^ (i-i&^+^e^-...){i-^ ^e-^+ ^L^e^- ...f Consequently, if </) " ^' { (1-^0^6'^ ~ efj ~ "■^' ^'^^' we have /o (0) = 6 ~ ^ ^ 35, Also, as in the case of/i (</>), we assume§ for the moment that /2(^) has limited total fluctuation in the range (0, tt). The application of Bromwich's theorem is therefore permissible, and we deduce that Lim ni ["(^ - -^ sin (?i</)) ./, {j>)d<l> = 3^ 2 " ^- T (|)/35, M-s-oo J * A formal proof will be given in §5a that /j (^) is, in fact, monotonic and decreasing (we use the term decreasing to mean non-increasing). t Euler's result that / ip''^^~'^ sin \p cl^p — V [m] sin {\mir) , when -\<m<l, is well known. J Bromwich,I?i/ini<e Series, p. 444, proves that, if f{<p) has limited total Jiuetua- f ^ sin Hd) tion in the range (0, h), where 6>0, and if U,^— I — - — f((p)d(p, then H-^ao «-*oo J W J f but his analysis is equally applicable to the more general integral V^=n'"' i (p'^-'^ sin {7i4>) . f (^) d(p (-l<m<l), and hence Lim F„=Lim I i/^^-i sin i// ./(i///?i)(7i/'=/(0) I f-^ sin xj^ df. rt^-x M-*-Qo Jo J i) % A formal proof will be given in § 5 b that/^ (<p) is monotonic and increasing. that is of equal order and argument ["(/)-« sin (7i(/)) . /; (<^) ^0 = 3* 2 - ^ ?i - * r (|)/35 + o {n " '), 45 and so ^ 8 ( f'^ sini/r r'-^ sin^/r ] (6f) -r ??7r j </> ~ I/2 (</)) sin (7?</)) . f/<^ + (/i-2) secon r such sin-^lr Now, by the second niean-vahie theorem, there exists a number a exceeding nir such that liTT "^^ dyjr 1 (nirf sin^p• d/\jr I < 2(mr) •', and so we have at once that, when n is large and real, ^» (n) = ^T—^ ^ + (n - -^ ), which is the result to be established. T(3 obtain a closer approxi- mation by these methods would necessitate some very tedious integrations by parts. 3. We next consider the approximate formula for Jn (n). It is immediately deduced from the Bessel-Schlafli integral that j:/(7i)=- sin (9 . sin 71 (^ - sin (9) . c?^ TT Jo Now we get, on integrating by parts, rsinhde-^^^ + ^'^'^'^^dd Jo ~ nJo 1 sinh 6 d ^-n(d + sinhd)^^0 1 nj + cosh 6 ' dd ^r e-''Ud = 0{n-% 2njQ 46 Mr Watsoji, Bessel functions Hence /„' (n) = - I , -p^ sin ii6 cl(b + («~-), TT / 1 — cos ^ ^ where ^, as previously, stands for 6 — sin 6. Now, if fs ((}>) = ^^ sin ^ . (1 - cos e)-\ then/: (0) = 2^ 3 ' ^ and /^{(fi) has limited total fluctuation* in the range (0, tt). Hence, applying Bromwich's theorem we have o f'^ sin n(6 . , ,, , , , ,., f'" sin ilr - , ,,, J (p'' Jo T^-- and so J",/ (n) = „ ^^^ + o (?i " *) + (n-'), ^ TTIV- when n is large and real ; and this is equivalent to the result stated in § 1. The approximation could be carried one stage further (as in § 2), but it seems hardly necessary to give the analysis. 4. As an example of the necessity for the caution which has to be taken in approximating to integrals with rapidly oscillating integrands, it may be remarked that some of the earlier writers mentioned in § 1 assumed that when x and n are large and nearly equal [in fact, when \x — n\ = o (n^)], then Airy's integral An («?) = -[ cos [nd - cc {6 - ^6')} cW is an approximation to Bessel's integral for J^ (^). This assumption is correct, and it happens that the first tivo terms in the asym- ptotic expansions of An (;») and Jn {oc) are the same. But Airy's integral for An {not) is not an approximation! to Jn (no) when a is fixed and < a < 1, while n — > x . To establish this statement we use Carlini's formula:}: Jn (na) ~ -_ {1 + ^/(l - a2)}» . (1 _ a')i V(27rw) (valid when < a < 1), and after observing that we ma^- write An (yia) = - I — ) / cos IW (mw + lu^)] dw, 7T\naJ J Q '■^ ^ n ' * A formal proof will be given in § 5 c that /I, [cp) is monotonic and decreasing. t For example, the arguments given in the P/u7. Mag., August 1908, p. 274, to justify the approximation seem to me to be as applicable to the second case as to the first. X A translation of Carlini's memoir (published at Milan, 1817) was given bv Jaeobi, Astr. Nach. xxx. (1850); Qes. IVerl^e, vii. pp. 189—245. See p. 240 for the formula quoted. of equal order and argument 47 2?t(l-a) /37r\^ ( nOL\\: ^ where m — ( — J , w IT \naj X we use Stokes' asymptotic formula* cos IItt (mw + 10^)] dw ~ 2 " ^ (3m) " * exp | - ir (|w)^}, .'0 " valid for large values of m. This process gives exp { - in 2^ a ~ ^ (1 - g)^ | {2a(l-a)}V(27rn) ^" ("«)'- r^ ., Mi ./ Hence ./nOi«) ^ /JgLVe^xM where X («)= V(l - «•-■) + log « - log {1 + V(l - «"^)l +i«"'(2 - 2a)i Since % (i) = -02047, a rough approximation to .7io„o (500)/J.iooo (500) is (f )* e-"'*l 5. We now prove the monotonic properties (valid for $ ^ ^ tt) stated in §§ 2, 3 : (A) To prove that /^ (</>) = i (6<^)* sin ^ . (1 -cos ^)-- is a decreasing function, we have d ... .,.,.., _ (3 + 2cos^)<^^^,(^) ' re ^^^' ^^^/^ ^- - — (i-cos^)3 — ' where g, (6) = [5 sin 0(1- cos ^)/(9 + G cos 9)]- 6 + sin 6, so that (y/ ((9) = - 6 (1 - cos ey/{9 + 6 cos 6)' ^ 0, and ^r (0) = 0. We now see that gi{0)^0, and so f/{<ji)^0, which is the result stated. (B) To prove that /,{<!>) =^-^ [(8/6*) - {(^^ sin 6/(1 - cos Of]] is an increasing function, we first prove two subsidiary theorems, namely : B (i). If c = cos 0, s^ sin 0, then the function g, (0) = (85 + 163c + 84c- + 18c^) (/> - is (1 - c) (149 + 157c + 44c0 is not positive. * Math. Papers, ii. p. 343. The result may also easily be derived from NicholBon's expression of Airy's integral in terms of Bessel functions of order ± J, Phil. Mag., July 1909, pp. 6—17, 48 ilf?' Watson, Bessel functions etc. B (ii). The function g,{d) = 2s (7 + 3c) (/)-^ - 3 (3 + 2c) (1 _ c)^ <^ + f s (1 - c? is not positive. To prove B (i) we observe that ^ [g, (6)1(85 + 163c + 84c-^ + ISc^} = - s- (1 - cy (644 -I- 416c + 60c-)/(85 + 163c + 84c- + 18c--)- The denominator may be written in the form I873 + 3O72 + 497 - 12 where 7 = 1 + c, and so the denominator changes sign once on i when < ^ < TT, say a,t 6 = ^. Hence g, (^)/(85 + 163c + 84c^ + 18c0 decreases from to — czd and then from + 00 to tt as ^ increases from to yS and then from /3 to tt. .Hence gi(6) cannot be positive. To prove B (ii) we observe that ^ [15^5 (0)1 [s (7 + 3c)|] = (1 - c) g, (0)/{2 (1 + c) (7 + dcf} ^ 0, by B (i) ; and so g^ (6) ^ g^ (0) = 0, as was to be proved. To prove the main theorem, we have where g (6) = {(3 + 2c) f ^ - s (1 - c) (f)^ (1 - c)-\ Now g' (6) = - ^a (0) f " (1 - c)- ^ 0, so that g(0)^g (0) = 64/6i and so // (^) ^ 0, as was to be proved. (C) To prove that f(<j>) = (f>i sin 6 . (1 -cos (9)-i is a de- creasing function, we have ^^ = i<^-ni-cos^)-^3(^), where g^ (0) = sin ^ (1 - cos e)-S(d- sin 6). Since g/ ((9) = - 2 (1 - cos ey we may use the arguments of (A) to prove the truth of theorem (C). Mr Watson, The limits of applicability etc. 49 The limits of applicability of the Principle of Stationary Phase. By G. N. Watson, M.A., Trinity College. [Received 22 November 1916.] 1. The method of approximating to the value of the integral ». = ^r- cos [m {oD — tf(m)}] dm, Ztt .' where x and t are large, by considering the contribution to the integral of the range of values of m in the immediate vicinity of the stationary values of m {.r — tf(m)], is due to Kelvin*, though the germ of the idea may be traced in a paper published nearly forty years earlier by Stokes f. Kelvin's result is that, if m{x — tf(m)] has a minimum when m — [x> 0, then, as ^ — > oo , u ~ i^-nt) - * {- ,xf" (/.) - 2/-' (fx)] - * cos [t,x\f' (/x) + iTr} ; and this result has imjDortant applications in connexion with various problems of mathematical physics J. Kelvin, in his analysis of this interesting asymptotic formula, takes for granted, on physical groimds, the validity of a certain passage to the limit. This process requires justification from the purely mathematical point of view ; and the necessary justification is afforded by a convergence theorem due to Bromwich§. This theorem plays the same part in dealing with integrals as an analogous theorem, due to Tannery |j, plays in connexion with series. The special form of Bromwich's theorem, which is required in the rigorous investigation of Kelvin's theorem, may be enunciated as follows : If f{x) be a function of x with limited total fluctuation in the range x ^ 0, and if 7 be a function of n such that ny —^ 00 as n —^ 00 , then, if — 1 <m<l, * Phil. Mag., March 1887, pp. 252—255 {Math, and Physical Papers, iv. pp. 303—306). t Camh. Phil. Trans, ix. (1851), p. 175 (Math, and Physical Papers, 11. p. 341). t See Macdonald, Phil. Trans. 210 a. (1910), pp. 134—145. § Bromwich, Theory of Infinite Series, p. 444. In the special case 7H = 0, which is explicitly considered by Bromwich, the result is important in the investigation of Fourier series by the method of Dirichlet. The theorem given by Bromwich on p. 443 is equally applicable to the more general case. II Fonctions d'une variable, p. 183. VOL. XIX. PT. I, 4 50 3Ir Watson, The limits of applicahility i -co x'"^-^f{x) sin nxdx ->/(+ 0) t"'-' sin tdt .' =/(+ O)r(w) sin I m-TT. [// 0<m< 1, the sines may be replaced throughout by cosines ; and, if ny—^ a as n-^ oo , where a is finite, the infinity in the upper limit of the integral must be replaced by o-.] As the formal analytical proof of a theorem* slightly more general than Kelvin's theorem is quite simple, and as sufficient general restrictions to be satisfied by the function' /(?n) are apparent in the course of the investigation, it seems to be worth while to place the theorem on record. It is applicable to all kinds of stationary points, whereas Kelvin considered only cases of true maxima or minima of the simplest type. 2. The main theorem which will be proved in this paper is as follows f: Let a, /3 be any numbers {infinity not excluded), possibly depending on the variable n, such that the real function bt — tf(t) has only one stationary value in the range a^t^ ^, at t = /m, b being independent of n. Let the first r differential coefficients with regard to t o/i| bt — tf(t), be continuousX in a range of values of t of which t = fxis^^ an interior point, it being supposed that the last of them is the lowest which does not vanish at t= /j,, so that r ^ 2. Let F (t) be a real function, continuous when a<t < /3, except possibly at t = fi, and let Urn F{t).{t- ixY = A, Lim F (t) . {tju - 1)"^ := A„ where A, Ay^ are not zero ; for brevity, let (1 — A,)/?- = m.. Then, if the function F(t).\bt-tf{t)-,ji'^f{,,)Y-^-.\b-tf'{t)-f{t)\-^ has limited total fiuctuation^ in the range a^i^/9, and if I nb^ - n^f(,8) - n/ii'f (/.) | , | nba - noif (a) - ntif (^) j both tend to infinity luith n, the approximate value of the integral / s ^ [ F{t) cos [bnt - ntf(t)} dt, * For the connexion between this theorem and a problem, due to Riemann (Werke, p. 260), which has been discussed by Fej^r {Comptes Rendus, November 30, 1908, and a memoir published by the Academy of Budapest in 1909) and by Hardy (Quarterly Journal, xliv. 1913, pp. 1—40 and 242 — 263), see §4 below. t It is convenient to modify Kelvin's notation. X It is necessary iox f{t) to have a continuous first differential coefficient when § If the fluctuation depends on n, it must be a bounded function of n as 7i-».x . of the Principle of Stationary Phase 51 wlien n is large, is r- 1) !(r !)'"-! r(m)[^ cos {nfx,^ f'(fi)+ ^em-n-] + A, cos {nfj:'f'{fM) + 1 7?m7r}] provided that < 1 — X < r ; wAere e = ± 1 according as bt — tf{t) increasinq , . , , ^ ,. , %s an , . f unction when t > a, ana ?? = + 1 accoratnq as the decreasing n • - same function is . . when t < a. When n—^cc hti only increasing such values that cos [nfM-f {/u.)} is always zero, A, may lie in the extended range —r<l — X< r. And, finally, F{t) and bt — tf(t) inay be infinite at t= a, ^, provided only that the integral converges for all sufficiently large values of n. 3. For brevity, write tf (t) ^ (f) (t). Then ^ is given by the equation b-<f>'(,M) = 0, so that, when t — fi, is sufficiently small, bt - tf(t) - t(}>' (/jl) -(f){t) = {/.f (/.) - </) (/.)} -(t- fxyr' {t')/r !, where, by Taylor's theorem, t' lies between fx, and t. Now define a new variable yjr by the equation bt-tf{t) = fi<l>'{ix)-c},ifi,) + f, and let 7, F be the values of yfr corresponding to t = a, t— /3. Noticing that /x^' (/j,) — (f> (fi) = fi^f (/n). we have J. cos {n/M'f (fi)} r^„,,, , u sin{n/jb^f' (a)} T^et/^x • , 7^ Zir J a i/r being a monotonic function of t when a ^t ^ /x and also when fi^t^^. Now e, 7; have been so chosen that e-yjr and rj^jr are positive when t > /M and ^ < //- respectively ; hence, when ^ —>//, + 0, we have df ef '^ {t- fiY </)<'■' {fi) 1 ^ r !. 52 Mr Watson^ The limits of applicahility It follows that ^^^^^A'^^'"'""'"'"'-^^^' as i — > /Lt + 0, where ^^ -(r-l)!(r!)^ </><'•)(;.) 11 (/,<'•) (/.)!}--' Since j wF ] — > oo with h, by hypothesis, we deduce from Bromwich's theorem that cos r p(-)o (If reoo ^<*> sin «+ St '^ ~ "" '"' .1 . <^'>"'"' -n ^'^- Writing %e s &>, we get re CO rcc I (%^)™~^ COS %<^% = e I &>'"~^ cos ctx^o) = eF (??i) cos -| and similarly (%e)'"~^ sin %c^% = F (m) sin ^7«7r. 7?2-7r, In like manner, when t-^ /jl- 0, dt and so, since 1 717 | — > oo with n, we have /, ^ (*) sm "^ 4 ^'^ ~ (-)•■ -'" ^■^i „ (^'''•""' sin ^<'^- Collecting our results, we see that the first approximation to / is / ~ [{AKe + (— )'■ AiKr]} cos ^iutt cos {nfM-f (fM)] - [AK + (-)'• A^K] sin |m7r sin {^i/^y (/i)}] = A [cos (w^-/' (/u.) + ^emir] + J.i cos {ufx^f (fi) + ■|?;??i7r}] (r - 1) ! (r !)'»-! F (m) ^ 27rw'^{|<^<'-'(;u)j}'" ' and this is the result stated. The formula fails to be effective in the neighbourhood of those values of n for which the expression in [ ] vanishes, as the error in the approximation then becomes comparable with the approxi- mation obtained. [It is evident that if the cosine in the integral defining / may be replaced by a sine, then the cosines in the approximation are replaced by sines.] of the Principle of Stationary Phase 53 Cases of practical importance are those in which A = Ai and t = fj> is a, true minimum or maximum of bt — tf{t), so that e and r) are both + 1 or both — 1. The formula then is A cos {n/M'f'ifM) ± ^mTr} . (r - 1) ! (r !)'»-i T(m) ~ 7r?i"* { I (^e-) (fi) I 1*'^ • If nV or ny tend to finite limits, the gamma functions have to be replaced by incomplete gamma functions ; and if one or other tends to zero, we modify the approximation by writing zero for A or A^ respectively in the general formula. The general result reduces to Kelvin's formula when r = 2, X = 0, 711= h, and e = ?; = 1, provided that (with Kelvin's notation) x/t is constant. In that case, a sufficient condition for the validity of the formula is that ^ [{{ma^/t) - mf{m) - i^-f (;.)}*]-^ should have limited total fluctuation when m ^ 0. If X were a function of t, Bromwich's general theorem {loc. cit., p. 443) would have to be used, and the enunciation of sufficient conditions (even in their simplest form) for the validity of the formula, would be exceedingly laborious. The reason for this is that (with the notation employed in this paper) -^ and F (t) dt/dyfr would both be functions of n. 4. The problem of Riemann (see § 1 above) essentially consists in obtaining an approximation for integrals of the type /■"" / ,x cos sin nt. when n is large and a-' (t)—>oo as t—> 0. These integrals are expressible by integrals of the type t-'pit)^"^^ {nt+ (7(t)]dt, Jo sm ! ' so that the problem is, at first sight, very similar to that discussed in % 2—3. There is however an essential difference, namely that, in the problem we have discussed, ntf{t) owes its large rate of increase (which balances the rate of increase of nbt at the stationary point) to the large factor n, whereas, in the problem attacked by Fejer and Hardy, the function a (t) owes its large rate of increase to the infinity of a (t) at ^ = 0. In our problem fj, is fixed, whereas in the other problem the stationary point of nt — (T{t) tends to zero as ?i — * 00 . It seems to be this difterence which accounts for the somewhat elaborate investigation given by Hardy and which 54 Mr Watson, The limits of applicahility makes the theorems of Fejer and Hardy rather deeper than the theorem of §§ 2 — 3. It should be pointed out that there is one integral which can be regarded as coming under either head, namely*, /•oo / X sm . I X " (nx + ax~n dx, j cos ^ ^ ' where n is large, a, \, and r are positive and X and r are chosen so that the integral converges. [For the sine-integi-al, the conditions for convergence are < X < r + 1.] As the integral stands it is of the type discussed by Fejdr and Hardy, with a variable stationary point where x''+^ = arjn. But if we make the sub- stitution and then write v for n''''<»'+i', it becomes j,(A-i)/r[ t-^^^^'^lvit + at-'y^dt, Jo cos ^ ^ ^•' which is of the type discussed in this paper, having a fixed stationary point where t = {ray/^''+^K The reader will have no difficulty in deducing the approximate formula by either method. 5. As an example of the apparent inapplicability of the methods of this paper consider the integral of Bessel for Jn(x) when n and x are both large and a; — ?i, is (n^). The integral is 1 f'^ Jn (x) =- I COS (n6 — X sin 0) dd, and the stationary point is given by cos = nlx; let the root of this equation be = /j,, and let x = n + an^ where a > ; when n is large we have In considering | cos (n0 - x sin 0) d0, we write X = n0 — xsin0 — (n/j, — x sin /i,), and the last integral is expressible by integrals of the type »(tan^-M) cos d0 , sin ^ dx f Jo * I am indebted to Mr Hardy for suggesting that the integral in which a- (t) = llt can be reduced to an integral of Kelvin's type. of the Principle of Stationary Phase 55 Now tq—'^~ ^ co^ 6 o^x{6 — ix) sin ix, when 6 f^ fM and -^r^^oo^iw fx .{6 — ^y. Hence, as ;)^; — > 0, Lde -1 and /, sin '^ c^x \ /-n'tauM-M) (• . 1 rf^") _1C0S /•n'tanM-M)( . i rft'l _1C0S , Now, as ?i -> CO , ?i (tan /x — ^) -> i (2a)^ and so the limiting range of integration is of finite length, 1 d9 Moreover, \/{2x sin fi) . ^^ -^ > — l as ?i — > go ^vhen % is ^■ero, 1 df) that is, when d ^ fi. But, when 6 -> 0, the limit of \/{2x sin A*) • %- ^ is — {2 sin fi (sin /jl — /xcos At)|"^/(1 — cos /a), a^icZ, a.9 n—>cc, the limit of this is not — 1 hit — 2 \/(^) ; and so we cannot infer that •«(tunM-M)f . if^6l) _icos , f' _icos , |V(2.-sm;.).X-^^~j-% -^i^^X^^X-j^^X \sin^^^' where b is Lim •?? (tan yu, — /x). The evaluation of the approximate formula for Jn{^) in the circumstances under consideration consequently seems to require more elaborate analysis than is afforded by the methods contained in this paper. 56 Mr Borradaile, On the Functions of the Mouth-Parts etc. On the Functions of the Mouth-Parts of the Common Prawn. By L. A. Borradaile, M.A., Selwyn College. [Read 30 October 1916.] The food is seized by either pair of chelipeds, or by the third maxillipeds, and is usually placed by them within the grasp of the second maxillipeds, though sometimes it is passed directly to deeper-lying structures. The second maxillipeds are the most important of the food-grasping organs. They have three principal movements; in one, the broad flaps in which they end open downwards like a pair of doors, and with their stout fringes gather up the food ; in another, they rotate in the horizontal plane to and from the middle line of the body, and thus narrow or widen the gap through which the food passes; in the third, the bent distal part of the limb tends to straighten, so as to brush forward any object which lies between them. Frequently these movements are combined. Owing to the facts that the second maxillij)eds cover the mouth-parts anterior to them, and that if they be removed feeding is not properly performed and usually not attempted, it is difficult to trace the food beyond them, but the following seems to be its fate. If it be small in bulk, or finely divided, or very soft, it is passed to the maxillules, by whose strong, fringed laciniae it is swept forwards, and probably caused to enter through the slit between the paragnatha, into the chamber which is guarded by the upper and lower lips. If it be tough or in large masses, the second maxillipeds and maxillules brush it forwards towards the incisor processes of the mandibles. The action of the latter is, by rotating in a vertical plane, to tuck the food into the gap between the paragnatha and the labrum. If the mass be large, pieces are torn off it by this action. Finally, to enter the gullet, the food must pass between the molar processes and be pounded by them. The mandibular palps, maxillae, and first maxillipeds appear to play parts of little importance in regard to the food. The palps are present and absent in closely related genera, and appear to be disappearing in the higher Carides. The same is true of the lobes of the maxillae, which are in constant regular motion to and from the middle line, and probably serve to restrain the action of the scaphognathite. The large laciniae of the first maxilliped may have as their function the covering of the maxillae and protecting them from the food. The labrum undergoes active movements, whose function is probably to aid in keeping the food under the action of the mandibles. The exopodites of the maxillipeds set up a strong current forwards from the mouth. No doubt this aids in carrying away the exhausted water from the gill chamber and the excreta from the tubercles of the green glands. Into the same current particles which have been taken as food are from time to time rejected by the forward kicking of the second maxillipeds. CONTENTS. PAGE A self-recording electrometer for Atmospheric Electricity. By W. A. Douglas Ritdgb, M.A., St John's College . . . On the expression of a number in the form ax^-\-hy^-^cz--\-dv?. By S. Ramanujan, B.A., Trinity College. (Communicated by Mr G. H. Hardy) . . . . . . . . . • H j An Axiom in Symbolic Logic. By C. E. Van Hokn, M.A. (Com- municated by Mr G. H. Hardy) 22 A Reduction in the number of the Primitive Propositions of Logic. By J. G. P. NicoD, Trinity College. (Communicated by Mr G. H. Hardy) . 32 " Bessel fuiictions of eqiial order and argument. By G. N. Watson, M.A., Trinity College . . . 42 ' The limits of applicability of the Principle of Stationary Phase. By G. N. Watson, M.A., Trinity College . . . . . . 49 On the Fionctions of the Mouth-Parts of the Common Prawn. By L. A. BoRRADAiLE, M.A., Selwyn College . . . . . . .56 PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY VOL. XIX. PARTS IL, III. [Lent and Easter Terms 1917.] AT THE UNIVERSITY PRESS AND SOLD BY DEIGHTON, BELL & CO., LIMITED, AND BOWES & BOWES, CAMBRIDGE. CAMBRIDGE UNIVERSITY PRESS, C. F. CLAY, MANAGER, FETTER LANE, LONDON, E.C. 4 1917 Frice Two Shillings and Sixpence Net October 1917. 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Letters and Communications for the Society should be addressed to one of the Secretaries, Mr G. H. Hardy, Trinity College. [Mathematical.J Mr A. Wood, Emmanuel College. [Physical.] Mr H. H. Brindley, St John's College. [Biological.] 6. Presents for the Library of the Society should be ad- dressed to The Philosophical Library, New Museums, Cambridge. 7. Authors of papers are informed that the Illustrations and Diagrams are executed as far as possible by photographic "process" work, so drawings should be on a large scale and on smooth white Bristol board in Indian ink. 8. Members of the Society are requested to inform the Secretaries of any change of address. PROCEEDINGS OF THE The Direct Solution of the Quadratic and Cubic Binomial Congruences with Prime Moduli. By H. C. PocKLiNGTON, M.A., St John's College. [Received 22 January 1917: read 5 February 1917.] 1. The solution of congruences by exclusion methods, although easy enough when the modulus is moderately large, becomes impracticable for large moduli because the labour varies as the modulus or its square root. In a direct method the labour varies roughly as the cube of the number of digits in the modulus, and so remains moderate for large moduli. The object of this paper is to develop the direct method. We take or = a, mod. p, first, discussing the cases where p = 4<m + 3 and p = Sin + 5 in § 2 and that where p = 8ni + 1 in § 3. We next take ar^ = a and discuss the cases where p = Sm + 2, /> = 9m + 4 and p=i9m + 7 in § 4 and that where p = dm + 1 in § 5. 2. Throughout the paper we suppose the modulus to be p where p is prime*. If p is of the form 4wi + 3 the solution of ^•- = a is X = ± a'^'^\ If p is of the form 8m + 5 the solution is ^ = ± a'"+i provided that a-'^+^ = 1. But if not, a'^"*+i = - 1, and as 2 is a non-residue 4-'"+' b - 1 ; so that (4a)-»*+^ = 1 and we have 2/ = + (4a)'"+^ as the .solution of y- = 4o. Hence x=±y;2 or x=±{p+ y)/2 is the solution of x-=a. These values of x can be calculated without serious difficulty by repeated squaring (followed by division by the modulus to find the remainder) and multiplication of the numbers so found (again followed by division). * Hence if it is composite we must factorize it and solve the congrnence for each of the different prime factors. VOL. XIX. PARTS II., III. 58 Mr Pocklington, The Direct Solution of the Quadratic 3. Put D=—a, so that we have to solve cc^+ D = where D is positive or negative but not divisible by p. Let t^ and Mj be so chosen* that t^ — Du-^- = N is a quadratic non-residue of p, and let tn = [{t, + U, ^Dr + (t, - U, sjDY]l% These numbers are clearly integral. Also by use of* which (at first with m = n) we can find the remainders of tn and Un to our modulus without serious difficulty even when n is large. We also have tn — Dun = N'^. Supposing that p is of the form 4m + 1, we have D a quadratic residue of 'p, and tp = t^P = ^i, Up = u-pD^P~'^''i- = u^ ; and now ti = tp^^t-^ + Dup_^Ui^, tij = tp_iUi + t-^Up_^ give on solution tp_^ = 1, Up_-^ = 0. Let p-l = 2r. Then = l/^_i = 2trUr shows that either t,. or u,. is divisible by p. If it is Ur we put r = 2s and proceed similarly. We cannot have every u divisible by p, for u^ is not. We cannot be stopped by having u,n = with m odd, for we always have 4,,^ - Dum'' = N''\ and this would then give ^,„' congruent to a non-residue. But if m is even we can .proceed further. Hence when we are stopped we must have t,n = 0. This gives - Bii,,^ = N''\ and as - D is a residue m must be even. Putting m = 2n we have = ^^ = tn' + Bun', so that the solution of a^ + D = is got by solving the linear congruence UnX =z + tn- In applying the method, if n is the largest odd number con- tained m p-l we first work to get the suffixes n, and then the suffixes 2n, 4<n, 8n, etc. Thus in the case of cc'' + 2 = 0, mod. 41, we see that ^j = 3, Ut, = 1 is suitable, and we find t. = 11, u., = 6 ; t, = 29, u,= 9; t,= 23, u,= 15; t^o = 36, u,o = 34 ;' t.^ = 0. ' The solution of 34^ = 36 is a; =30; and so the two solutions of *•- + 2 = are .x=± 30, mod. 41. ^4. If p is of the form Sm + 2 the only solution of x' = a is a; = l/a'«. If p is of the form 9m + 4> one solution is cc = !/«'"■ and if of the form 9m + 7 one is .x = a^+\ The other solutions are got frojn this by multiplying by (- 1 + 6)12 and (- 1 - e)/2, where t/- + 3 = 0, a congruence which we have shown how to solve. * We have to do this by trial, using the Law of Quadratic Eeciprocity, which " •! ?f i'" *^J® method. But as for each vakie of n half tire vahies of / are .suitable, there should be no ditlficulty in finding one. and Cubic Binomial Congruences with Prime Moduli 59 5. Let 8 be the arithmetical cube root of a, which we assume* not to be a cube. Findf ti, Ui, Vi such that the norm N = t{^ + au^^ + a'-Vi" — SatiU^i\ of the algebraic number is a cubic non-residue of p. We see that as a is a cubic residue of p we have U^' = ti-\- UiS+ i\8'^, so that if U^'-^ = tp-i + Up-i 8 + %_i 8- we have iip^i = Vp_i = 0. Now taking ?7'" where m is in turn (p— l)/3, (p — l)/9, etc. we see that we cannot always have u,n = v„i = 0. Let f/""^ be the last of this series for which this happens. Then m is divisible by 3, for otherwise the norm of [7"\ which reduces to t^, would be congruent to the non-residue iY'". Putting m = on we have tsn = tn + <Kln + a-^n + QatnUnVn, = Usn = 3 (tn'Un + ^^tnVn' + «",rWn), = V.,n = 3 {taUn' + tuVn + aUnVn'')- The last two give tn{av^i"' — Vn)= 0; so that if tn is not divisible by p we have w = Un/v,,. as one solution of n^ = a, for as Un and v,i are not both divisible by p this shows that neither is. They also give Vn (ait^n — tn) = 0, and so w = tnjihi is a solution. Eliminating a from the same two congruences we see that the ratio A, of the two xs, satisfies V + X + 1 = 0, so that they are distinct. The third solution follows immediately. If however tn is divisible by p the two congruences show that either Un or Vn is divisible by p. We now have rtM,i,^=iA'" or a-Vn = -Y". In either case n must be divisible by 3 as before, and we have as one solution x = N^'jun or x = aVnjN'' respectively, where r = n/S. * Simply because of the way in wliich for the sake of shortness we are stating the method. t This again must be done by trial. In order to use the Law of Cubic Beciprocity we must express p in the form ii' + xiv + v'^, which requires the solution of ^2 + 3 = 0. 5—2 60 M7' Hardy, On a theorem of Mr G. Polya On a theorem of Mr G. Polya. By G. H. Hardy, M.A., Trinity College. [Received and read 5 February 1917.] 1. Mr G. P(51ya has recently discovered a number of very beautiful theorems concerning Taylor's series with integral co- efficients and ' ganzwertige ganze Funktionen '. The latter functions are integral functions which assume integral values for all integral (or for all positive integral) values of the independent variable. One of the most remarkable of these theorems is the following*: • Suppose thatg{x) is an integral function, and M{r) the maodmum of I g {x) I for \x\^r. Suppose further that g(0),g{l),g{2),... are integers, and that lim 2-''^JrM{l') = (1). ?'-»-00 Then g {x) is a polynomial. Mr Polya observes that, if it were possible to get rid of the factor \/r from the equation (1), the theorem could be enunciated in a notably more pregnant form, viz. : Among all transcendental integral functions, which assume integral values for all positive integral values of the variable, that of least increase^ is the function 2^. Mr Polya states, however, that he has not been able to effect this generalisation. And my object in writing this note is to show that the generalisation desired may be obtained by a slight modification of Mr P(51ya's own argument, and without the addition of any essentially new idea to those which he employs. 2. Mr Polyaij: reduces the proof of the theorem to a proof that the integral T( \— ^' r g(x)dx ^'^^~2^ij x(x-l)(x-2)...{x-n)' extended over the circle \x\ = r - 2n, tends to zero when ?i — » oo , *■ G. Polya, ' Uber ganzwertige ganze Funktionen ', Rendiconti del Circolo Matematico di Palermo, vol. 40, 1915, pp. 1—16. See also 'Uber Potenzreihen mit ganzzahligen Koeffizienten ', Mathemathche Annalen, vol. 77, 1916, pp. 497— 513, where Mr Polya refers to a third memoir (' Arithmetische Eigenschaften der Eeihenentwicklungen rationaler Funktionen', Journal flir Mathematik) which. 1 have not been able to consult, t Croissance, Wachstum. X Loc. cit., p. 7. Mr Hardy, On a theorem of Mr 0. Polya 61 This he proves by observing that the modulus of J^^ does not exceed n\ M{r) ^ V{n + \)V{n) (r-l)(r-2)...(r-w) r(2n) ^ '' and by an application of Stirling's Theorem. In order to com- plete the proof in this manner it is necessary to assume the condition (1). If however we suppose only that lim2-'-ilf(r) = (2), or J'f(r) = o(2'-) (2'), the proof may be completed as follows. We have where ^= 2ne'^. Now \x — s\ = V(4?r — 4?2.s cos 6 + 8^)'^ 2n — s cos for 1 <s <.n, so that de .){a)-2)...{a;-n)\\ ' U{x- s) 1 if cos ^ > 0, and > n (2n - s cos 6) = (cos ^f 11 (2« sec O-s) n (x - s) 1 ^ n (2n - s cos ^) = I cos ^ i" n (27* | sec ^ | + s) 1 1 if cos ^ < 0. Hence where Kn = nl2^''j Jn = {Kn) + (X„), ''' T( 2na-n) r {2n(r) a'^dO, i„ = „,2»|'V„_£(2^iV,.«rf«, and a = sec 0. A straightforward application of Stirling's Theorem shows that uniformly in 6, where ^ = <t> (0) = {2a - 1) log (2o- - 1) - 2a- log 2o- + log o" + 2 log 2, ^ = ^ (^) = 2(T log 2o- - (2o- + 1 ) log (2a + 1) + log o" + 2 log 2. 62 Mr Hardy, On a theorem of Mr 0. Polya = 2 log 2cr - 2 log (2o- + 1) + ^ = ^ - 2 log (l + ^M > 0. 3. When 6 increases from towards -|-7r, or decreases towards — hir, (T increases from 1 towards oo . Also ^^ = 21og(2c.-l)-21og2^ + ^=2 1og(l-^^)+-^<0, da- Thus <I> steadily decreases and ^ steadily increases. Moreover ^ (0) = 0, ^ (0) = 4 log 2 - 3 log 3 ; and it is easily verified that both <J» and ^ tend to the limit log 2 - 1 when 6 tends to ^tt. We thus obtain, in the first place, i,.=of-'.<--vnj:;y(,^)</«}=o(i). Secondly, we observe that, if S is any positive number, we have ^{d)<^{h) = -7^<0 for h^e^^ir, -^ir^eK-h. Hence we may replace the limits in Kn by - 8 and h, the re- mainder of the integral being of the form 4. All that remains, then, is to prove that /„ = ,U2»f n^^)w(* = 0(l); j_5 1 (2/lcr) The function ^{6) may now be expanded in powers of ^. We find without difficulty that where A = log 2 - ^ > 0. It follows that and we have h = 0bn\ ^e-^»^HO(»^*)^^ = 0U/n r e--^^''^'de\^0{l). Mr Hardy, On a theorem of Mr G. Polya 63 The proof of the theorem conjectured by Mr Polya is thus completed. 5. Mr Polya has also proved an analogous theorem concerning integral functions which assume integral values for all integral values of x, viz.: If ...,g{-2), g{-l), g{0\ g{\), g{2)... are integers, and lim (^^rVrilf(r)=0 (3), then g {a;) is a polynomial. His proof applies, as it stands, to odd functions only, its appli- cation to a completely general function demanding the more stringent condition lim r-*-co ^4r^l 'rm(r) = (3'). He states that it is possible to replace the index | by ^ in all cases, but that, as he has not been able to reduce the condition to lim (^^^y*'ilf(r) = (3"), he has not thought it worth while to publish the details of his work. A modification of Mr P<51ya's argument, in every way similar to that which I have made in the proof of his first theorem, enables us to replace (3) by (3") when g{x) is odd. The same modification in his unpublished argument would, I presume, be equally effective in general. That the number 3 + ^/5 cannot be replaced by any larger number, and so really is the number which ought to occur in any theorem of this character, is shown by Mr Polya by the example of the function which assumes integral values for all integral values of .r. 64 Dr Marr, Submergence and cjlacial climates (luring the Submergence and glacial climates daring the accumulation of the Cambridgeshire Pleistocene Dejwsits. By J. E. Marr, Sc.D., F.R.S, St John's College. [Read 5 February 1917.] A. Introductory. The sequence of events during palaeolithic times is still a subject surrounded by much uncertainty. The area of the Great Ouse Basin is one in which considerable light has already been thrown on vexed questions, and as the examination of the area is carried out in greater detail, important results will be obtained, for in this area we get evidence of the relationship of the palaeo- lithic deposits to those which were formed during a period of submergence and re-emergence, and also to accumulations which give evidence of the occurrence of more than one cold period. The general distribution of the palaeolithic deposits of the district around Cambridge, and their main characters, have long been known, and an account of the deposits, with references to the previous literature, is given in the Geological Survey Memoir The Geology of the Neighbourhood of Gambridqe, published in 1881. Since that memoir appeared, further light has been thrown on the deposits, especially by Professor Hughes, who has given his latest views in a paper entitled Tlie Gravels of East Anglia (Cambridge University Press, 1916). I have devoted much attention to this subject during the last six years and hope to describe my detailed results elsewhere. The present paper is concerned with a discussion of the main problems involved, in hopes that it may direct the attention of workers to the importance of further observations, for the deposits with which we are concerned are only exposed temporarily during the_ working of gravel-pits and the digging of foundations and drains, and it is desirable that all temporary excavations should be carefully studied, and the objects obtained rendered available for study by deposit in Museums, for isolated specimens in private collections are usually mere objects of curiosity devoid of scientific value. B. Submergence and its effects. The actual sequence of deposits. In the fenland and on its borders we meet with marine deposits above sea-level, which have long been known around March and Narborough. They occur above and below fen-level at March accumulation of the Gamhridyeshire Pleistocene Deposits G5 and undoubted marine deposits containing sea-shells are found to a height of at least 50 feet above sea-level in the Nar Valley, and deposits up to 80 feet above sea-level have been claimed as marine. Unfortunately no exposure of these Nar Valley beds has been seen for a very long time, and their exact upward limit is a matter which must remain unsettled until new excavations are made. It is held, with good reason, that the beds of March and the Nar Valley are geologically contemporaneous in the sense that they belong to the same period of sea-invasion, which was subsequent to the accumulation of the chalky Boulder Clay ; and as there is good evidence that much of the fenland was low-lying Nar Level Fig. 1. AB. Slope of ground before marine gravels were deposited. CD. ,, „ after ,, ,, ,. a. Tract of marine gravels. b. ,, interdigitating marine and fluviatile gravels. c. ,, fluviatile deltaic deposits. d. ,, erosion in valley towards its head, during period of deposit of a, b, c. 1, 2, 3. Order of formation of deposits in tracts c and d respectively. (1 is oldest.) Vertical scale greatly exaggerated. ground after this boulder-clay was formed, it would appear prob- able that the March gravels are earlier than those of the Nar Valley, and therefore that a gradual silting up of a bay of the sea took place, until the sediments reached a height of at least 50 feet above present sea-level. During this period of silting the rivers Ouse, Cam and others would build delta-deposits along the lower parts of their courses, with interdigitation of marine and fluviatile deposits in an inter- mediate belt of ground as shewn in figure 1. In this delta- material, the chronological sequence of deposit would be from below upward, as shewn by 1, 2 and 3 in the belt c. The upper waters of the rivers would still be eroding, and the sequence would be from above downwards (see figs, in tract d). After submergence had ceased, it would be replaced by re- emergence, as shewn by the erosion of the rivers to their present 66 Dr Marr, Submergence and glacial climates during the levels, and new deposits 4, 5, . . . (not shewn in the diagram) would be banked against or laid down upon those formed during the period of subsidence and general accumulation in tracts c and a. It will be seen therefore that relative height of deposits above the present river- level is not in itself a necessary indication of The geological surveyors gave the following classification of the Cam gravels: r Lowest Terrace Gravels of the Present River System I Intermediate Terrace [Highest Terrace Gravels of the Ancient River System. I shall treat of three of these, leaving out of account the gravels of the Intermediate Terrace, which I have not studied extensively owing to poor and infrequent exposure of recent years. I shall speak of the gravels of the ' Ancient River System ' as the Obser- vatory gravels, those of the highest terrace of the ' present river system ' as the Barnwell village gravels, and those of the lowest terrace as the Barnwell Station gravels. The ages of these deposits will ultimately be accurately determined by an exami- nation of the fossil evidence, including implements of human manufacture. So far, the evidence of this kind points to the Barnwell village deposits being of two ages, the older formed during the period of delta-growth, the newer during the period of re-emergence and erosion. At the end of the period of delta- growth, and therefore of an age intermediate between those of the supposed two Barnwell village deposits, I would place the Observatory gravel, and certain loams, to be referred to later, and after all of these, the Barnwell Station gravel marking the culmination of the period of re-erosion, for there is evidence of a later period of sinking and deposit after this was formed. This succession is represented in Fig. 2, which shews a section across the Cam valley at Cambridge, before the edges of the valley sides had been destroyed leaving the Observatory gravels as a ridge with lower ground on either side. In the figure the terms Upper, Middle and Lower Palaeolithic indicate the ages of the various gravels as inferred by me from the palaeontological evidence. I am using the term Middle Palaeo- lithic in the sense in which it was used by Prof. Sollas in the first edition of Ancient Hunters as equivalent to Mousterian. I believe therefore that the older Barnwell village gravel is pre- Mousterian, that of the Observatory (in part at any rate) Mou- sterian, and the newer Barnwell village gravel and that of Barnwell Station post-Mousterian, the former being of earlier date than the latter. Mr Jukes-Browne, in an essay on the Post Tertiary Deposits of ■I accumulation of the Camhrldgeslnre Pleistocene Deposits 67 Cambridgeshire, advocated a change in the direction of the rivers near Cambridge between the formation of the Observatory gravels, and those which he regarded as belonging to the 'present river system.' That such a change occurred is admitted, but the evi- dence points to all the deposits save those of the Barnwell Station terrace having been formed before the river diversion occurred. I may now pass on to consider briefly the palaeontological evidence in favour of the order of age indicated above, leaving details for a future paper. In the pits of Barnwell village, and of the Milton Road near Chesterton, loams are sometimes exposed at the base of the over- lying gravels. These loams contain Corhiculaflaviinalis, and with it are associated Unio Wtoralis, Belgrandia margiiiata, and Hip- popotamus, On the continent this is recognised as an early Fig. 2. Section acroiss Cam N. of Cambridge, with higher valley-slopes restored. The figures shew the suggested order of formation of the deposits. Cross- hatching represents modern alluvium of Cam. 5. Barnwell Station gravels (Upper Palaeolithic 2). 4. Newer Barnwell village gravels (Upper Palaeolithic 1). 3. Loams of Huntingdon Koad area. 2. Observatory gravels (Middle Palaeolithic). 1. Older Barnwell village gravel and loam (Lower Palaeolithic). Z= Buried channel. Vertical scale greatly exaggerated. palaeolithic fauna of Chellean or pre-Chellean date, and there seems to be no evidence of the reappearance of this fauna at a later date. In the Geological Magazine for 1878 (p. 400) Mr A. F. Griffith described the occurrence of a palaeolithic implement from one of the Barnwell pits. A cast of this is in the Sedgwick Museum, and it appears to be of Chellean type. Further afield, the occurrence of similar implements at or near fen-level in Swaffham and Soham fens, and at West Row near Mildenhall, and at Shrub Hill near Feltwell, indicates that rivers had excavated their channels to fen-level in those times. There are patches of gravel between the higher Chesterton terrace which corresponds to the Barnwell village terrace and the 68 Dr Marr, Suhmerc/ence and glacial climates dnrmg the Observatory level, but no sections are now seen in them, so we may pass on to the Observatory deposits. In these shells and mammalian bones are very rare, though the former have been found in concretions, indicating that they once lay in the gravels, but have since been dissolved. Implements are relatively abun- dant, and I have found a large number during recent years. Many of them are of Chellean type, others probably Acheulean, but there are a large number of Mousterian type, some having the facetted platform which, as shewn by M. Commont, came into use in Northern France in Mousterian times. It may be noted that the implements of Mousterian type are patinated differently to and in a less degree than those of Chellean type, and I regard the two series as of distinct ages. Either the deposits, which are thick and varied in character, are of two dates, or implements of different ages lying upon the surface were washed into the deposits contemporaneously. This can only be settled by finding a number in situ, a work of great difficulty, but the evidence is in favour of the latter view. I may note that when a valley is being deepened implements of one age only are likely to lie in abundance near the spot where the gravels were accumulating, but when there is general aggra- dation, the highest deposits of the delta-growth are likely to receive washings of implements of various ages which have been lying together, at or near the surface. In any case the age of the newest gravel of a terrace will be determined by the implements of latest date. Lying on this gravel in channels are reddish sandy loams, which must have spread over the gravel, but have since been destroyed by erosion except where so preserved. There is also a deposit of somewhat similar loam but of a lighter colour flanking the gravel at a lower level on either side. It is rarely exposed, and only in shallow sections, but I believe it may be of the same general date as that lying on the gravel. No relics have been found in it, though two implements of possible Upper Palaeolithic date were found on the loam when ' draining the Christ's Cricket Grouod, but they may well have ■ been surface finds. Many other surface finds, some of apparent ' palaeolithic type, are found on this loam belt, and will be referred i to later. | Those gravels of the terraces of Barnwell village age, which | I would refer to a date later than that of the Gorhicula gravels, I are now exposed in a pit near the Milton Road and in another | on the Newmarket Road near Elfleda House, 2\ miles from j Cambridge. These contain a fauna differing from the Gorhicida \ fauna, and including the mammoth, woolly rhinoceros, horse and red deer, the horse being abundant. accumulation of the Cambridgeshire Pleistocene Deposits 69 Implements are scarce, but in both pits I have found some suggestion of upper palaeolithic forms, and in each pit a water- worn pot-boiler has been discovered. In the Barnwell Station pit the common mammal is the rein- deer, associated with the mammoth, tichorhine rhinoceros and horse. In the Geological Magazine for 1916 (p. 339), Miss E. W, Gardner and I recorded the occurrence of an arctic flora in this deposit, with abundance of leaves of Betula nana. A long pre- liminary list of the other plants which indicate arctic conditions was made by the late Mr Clement Reid, F.R.S., but has not yet been published. A few worked flints of undeterminable date have been found, but the fauna indicates the late palaeolithic period, and the late date of these deposits seems to be shewn by the fact that whereas all the others are apparently connected with the old drainage line extending from Cambridge to Somers- ham, these are almost certainly parallel 'to the present course of the Cam : they appear indeed to be the upper portion of the deposits filling an old buried channel of the Cam, evidence for the occurrence of which is borne out by certain observations made by Prof Hughes in the paper to which reference has been given. C. Climatic Changes. There is much difference of opinion as regards the occurrence of alternating glacial and interglacial periods in Pleistocene times, and it would seem that some light is thrown upon this question by the Cambridgeshire deposits and those of adjoining counties. I take the prevalent view that the implement-bearing deposits from the beginning of Chellean times post-date the period of the Chalky Boulder Clay, though others hold a different view, but as the local evidence bearing upon this question has already been recorded I need not enlarge upon this point. If the succession as outlined above be correct the following climatic changes seem to have occurred after the cold period marked by the accumulation of the Boulder Clay : (a) A warm period during the formation of the Corbicula- bearing strata. Arguments in favour of this are well known. (6) A cold period during the accumulation of the Observatory gravels(?) and the newer loams. No evidence of this has been advanced in this area, and a few remarks are necessary. The fauna of the Observatory gravels tells us nothing, and the loams have hitherto furnished no organic remains, but a widespread development of loam marks the Mousterian period, and N.W. Europe is believed to have been subjected to a cold climate during part of the period. 70 Dr Marr, Submergence and glacial climates during the The sections recently seen near Cambridge tell us little, but a brickpit in stratified loam with 'race' nodules similar to those found in the Cambridge sections has long been worked near the railway between Longstanton and Swavesey. It contains boulders, and is actually mapped as boulder-clay. A somewhat similar loam with boulders at High Lodge near Mildenhall has long been known for its implements of Mousterian type. These deposits are at an elevation just below that of the highest palaeolithic gravels, as are those of Cambridge. Further afield there is the very significant section at Hoxne, described in detail in a paper drawn up by the late Clement Reid, F.R.S.. and published in the Report of the British Association for 1896. ' At that locality we have a stratigraphical sequence. Above the boulder-clay lies an aquatic deposit marked by a temperate fauna. It is succeeded by loams with an arctic flora, and above that are loams with palaeolithic implements. They have been usually regarded as Acheulean, but there is one specimen in the Sedgwick Museum which is of a distinct Mousterian type. Taking these facts into consideration, a period of cold climate in this country in Mousterian times seems probable. In any case, the evidence points to a difference of date of the arctic plant-beds of Hoxne and Barnwell Station. (c) The fauna of the beds of the Barnwell village terrace claimed here as of newer date than those containing CoriicM^ajj suggests an amelioration of the climate, but in the absence of a '' well preserved flora, this is doubtful. (d) The Barnwell Station flora, as before observed, is distinctly arctic, and when this flora lived here, we can hardly suppose that our higher hills escaped glaciation. The same remark may be made of the Hoxne flora. This series of changes would accord with the classification of the beds on the continent thus : European Continent Cambridgeshire Pleistocene Wiirm glaciattoxi Barnwell Station beds. Waiiii period Newer Barnwell village deposits. Riss glaciation Observatory gravels and loams. Warm period Corhicula gravels. Mindel glaciation Chalky Boulder Clay. Pliocene Warm period Cromer 'Forest' series. Giinz glaciation Chillesford beds. I merely put this forward tentatively, claiming however that we have in Cambridge proofs of two if not three Pleistocene cold periods. accumulation of the Cambridgeshire Pleistocene Deposits 71 D. Surface Implements. Implements of all ages from earlier palaeolithic to recent times are found lying together on the surftice. Some no doubt have got there from the erosion of deposits which contained them, others belong to the surfece. My object is to insist on their careful collection, with exact records of their localities, even to the particular position in a field where thej^ lay. If they can be shewn to be limited to heights above those of a particular deposit, they may yield valuable information as to geological changes. Two areas in which surface implements are abundant are found very near Cambridge, one on the tract between Castle End and Girton on either side of the Huntingdon Road, on the ground occupied by the Observatory gravels and loams, the other a little south of Fen Ditton, between the railway and the river, and at no great height above the latter. They have not been yet sufficiently studied to enable one to draw definite conclusions, but the former group does not seem to occur below the level of the Barnwell village terrace, which suggests that the river may have eroded its valley below that level to its present position since those implements were made. The other set marks the position of a site on a terrace, which is I believe the terrace of the Barnwell Station deposits, and would indicate the formation of that terrace before this set of implements was manufactured. As the above is merely a preliminary account of these deposits, I have not burdened it with references, nor have I acknowledged the many friends who have helped in the collection of implements and other objects. The bulk of the implements on which my conclusions are based were collected by myself, and the rest by friends chiefly under my supervision, and in no case has any implement been purchased from workmen, so that the collection, which will be deposited in the Sedgwick Museum, is of value, inasmuch as each implement is known to have been obtained from the locality assigned to it. 72 Mr Weatherburn, On the Hydrodynamics of Relativity On the Hydrodynamics of Relativity. By C. E. Weather- burn, M.A. (Camb.), D.Sc. (Sydney), Ormond College, Parkville, Melbourne. {Received 15 December 1916 : read 5 February 1917.] I. The Equations of Motion. I 1. Relativistic equations for the adiabatic motion of a frictionless fluid have been found by Lamia* and Lauef in the form dt^ ■' dx^ ■^ dy^ dz^ ' y dx ^ 7\ 7) f) ^ 7) P — (kv) + 11;^ (kv) + V ;^- (kv) + iv X- (kv) + - ^ = Fy...(l); where m, y, w are the components of velocity at the point {x, y, z) relative to a definite system of reference 8 ; X, Y, Z those of the impressed force per unit of normal rest-mass ; and ry= ^ (2), Vc^ — {u^ •\-v^ + id^) c being the constant velocity of light. The significance of the symbols P and k is as follows. Since the motion is adiabatic the rest-mass of an element of fluid is determined by one variable only, say the pressure p. If we choose some definite pressure p^ as the normal or standard pressure, the element has a definite constant normal rest-mass hm^. If the element occupies a volume hV relative to the system of reference 8, the density k relative to that system is ^| defined by , _8mo * Ann. der Physik, Vol. 37, p. 772 (1912). •|- Das Relativitdtsprinzip, § 36 (2nd ed. 1913). For a more general discussion of the mechanics of deformable bodies from the standpoint of Relativity, cf. Herglotz, Ann. der Physik, Vol. 36, p. 493 (1911); also a paper by Igndtowsky, FIn/s. Zeit., Vol. 12, p. 441 (1911). I Mr Weathei'burii, On the Hydrodynamics of Relativity 73 Using a dash t(j refer in every case to the rest-systeui tS", we have for the rest-density 37»o _ hm^ _ k ~BV'.~y8V y ^"^• The function F is defined by the integral i'=rt (4), •' Pa f^ and in terms of this function k is given by -^=^(1 + ^) <'''■ For the rest-system *S" the quantity y has the value unity, while K becomes k'=1+-^ (5'). The constancy of normal i-est-mass leads, as in the classical theory, to an equation of continuity ¥ + fl.<'-«> + a^<''"> + 3i*^^''> = ° <">■ § 2. Using F and v for the force and velocity vectors, we may write the equations of motion more conveniently |(«v) + v.V(/cv)-[--VP=F (7). ot y Then because the gradient of the scalar product of two vectors is given by V (a • b) = b • Va + a • Vb -f- b x curl a + a x curl b, the second term of (7) is equivalent to - V (k-v") - V X curl (kv), Ik while, in virtue of (5'), VP = c'-'^/c'. Hence the equation may be exjjressed in the form r) 1 ^^ («v) -I- „ V {k-y- -I- cV'-) - V X curl («v) = F. ot 'Ik But again the second term is equal to 1 VOL. XIX. PARTS II., m. 6 2.^ 74 ifr Weatherburn, On the Hydrodynamics of Relativity and the equation of motion takes the ver}^ convenient form ^(a:v) + c-V/c + 2w X v= F (8), where we have written 2w = curl (kv). In cases where the impressed force F admits a potential, so that F = — V F, our equation reduces to ^(«v) + V(t;-/c+F) + 2wxv = (8'). § 3. Glehsch's transformation^. The equation of motion may be expressed in terms of functions analogous to those of Clebsch if we write KV = V(f) + A.V/X (9), (ji, \, fi being three independent functions of x, ?/, z and t. Taking the curl of both members we find immediately that 2w = VXx V;^ ^0). The function w = i curl (kv) plays the same part in the present analysis as |- curl v in classical hydrodynamics. It will therefore, by analogy, be called the vorticity ; and a line whose direction at any point IS the direction of w at that point, a vertex line. Since by (10) w is perpendicular to both VX and V/x it is clear that the vortex lines are the intersections of the surfaces A, = const., /J, = const. Using then dots to denote partial differentiation with respect to t, and assuming the existence of a force potential, we may write (8') as -^ - V ( F+ c-a:) = V(j) + X/i) + XV/i - ^Vx + (v . VX) VyLi - (V . V/x) VX which may be neatly expressed in the form §v,-|vx + Vi.= o (u), where the function H is given by the equation If= (f) + \jiL + V+C"K (12). _ * ^f; Basset, Treatise on Hydrodytiamics, Vol. 1, p. 28 ; also Silberstcin, Vectorial Meciianics, p. 146. Mr Weatherburn, On the Hydro(hjnamics of Relativity 75 On scalar multiplication of (11) by w, it follows in virtue of (10) that w.V// = 0, showing- that H is constant along a vortex line. It can also be shown that H is independent of x, y, z and is therefore a function of i only. For taking the curl of (11) we deduce On scalar multiplication by Vx it follows, by (10), that and similarly that w.Vl'^fl^O, From these we deduce as in the old theory* that ^ = ^ = ...(13). dt dt ^ Thus the first two terms disappeai- from (11), which becomes simply VH = 0, showing that H is constant in space and is therefore a function of t only ; or (j) + \/l + V + c^fc = H (t) (14). From (13) it is clear that the surfaces \ = const, and /x = const., and therefore also the vortex lines which are their lines of inter- section, are always composed of the same particles of fluid. § 4. Steady motion. When the motion is steady partial derivatives with respect to t are zero. If then the impressed force is derivable from a potential V, (8') becomes 2v X w = V ( K + c-k) (15), and the equation of continuity div(/.v) = (16). If we multiply (15) scalarly by v the first member vanishes, showing that vV( l^ + c-/^) = 0. Thus the function V + c-k is constant along a line of flow. Similarly scalar multiplication of (15) by w gives w.V(F+c-^/c) = 0, * Cf. Biis-et, lor. cit. p. 29. 76 Mr Weatherhurn, On the Hijdrodynamics of Relativity and therefore V + C'k is constant also along a vortex line. This is a particular case of the more general theorem, proved in the preceding section, that H is constant along a vortex line. Thus the surface V + c'-V = const, is composed of a double system of vortex lines and lines of flow. II. Irrotational Motion. § 5. When the vorticity i curl («v) is zero the motion will be termed irrotational or non-vortical, being analogous to the motion of that name in the older theory. In this case «v can be expressed as the gradient of a scalar function 0,, which may be called the velocity potential : i.e. 'cv = V(f} : (17). The lines of flow are orthogonal to the surfaces of equal velocity potential. The equation of motion can always be integrated when a force and a velocity potential exist. For (8') then becomes V (<^ + c'k +V) = 0. The function in brackets is therefore constant throughout the liquid, and will be a function of t only; i.e. 4> + c'K+V=f(t) (18). This is the required integral of the equation of motion. An arbitrary function of t may, however, be incorporated in the velocity potential cf), and this equation then written without loss of generality (f) + c-K+ F=0 (18'y When the irrotational motion is steady (c'^k + V) is constant throughout the liquid, and is also invariable in time. In the preceding section, where w was not assumed to be zero this function was only proved constant along vortex lines and lines of flow. The equation of continuity (6), or as it may be written dk , -. ^ + ^-divv = 0, may be expressed in terms of 0, if we write kvJk for v, and expand the divergence of the quotient. The equation then becomes |logA- + v(l).Vc/, + lv^0=.O (19). Mr Weatherhurn, On the Hydrodynamics of Relativity 77 This form is not so short as in the ordinary theory, nor can we obtain Laplace's equation, as there, by assuming the Huid incom- pressible, for such an assumption is inconsistent with the theory of relativity*. § 6. Steadily rotating fluid. Suppose that the Huid is in a state of steady rotation about the ^^-axis, and that the angular velocity of rotation O is a function of the distance r from that axis. We shall now determine what must be the form of this function in order that a velocity potential may existf- If i, J: k are unit vectors in the directions of the coordinate axes V = rn. For irrotational motion this velocity must satisfy the equation curl {kv) = 0, that is IkH + r V ('<:^) = 0, dr the integral of which is /tfir- = const. = fjb, say, so that /cH — ^ (A). The velocity potential <^ is then given by dd) . 1 d<h it dr r do r showing that (f) = fid + const (B), which is an example of a cyclic velocity potential. The integral of the equation of motion is by (18') c"k + V=0 (C). But K involves v" and therefore O, which is itself expressed in terms of k by (A). This equation however gives K' K-C^ whence 12- = -„ , „ „ ,„ , r-{/jb' + r''c-K-) * Cf. § 10 below. It will be shown, howeyer, in § 11 that V-</) = is the equation of continuity for the steady irrotational motion of a Huid of minimum compressibility. t Cf. Lamb, Hydrodynamics, § 28 (1st ed.). 78 Mr Weatherhurn, On the Hydrodjjnarnics of Relatiinty K being given by (5'). On substitution of this vabio in (C) the integral of the equation of motion, viz. becomes V + ~ \l uC^ + r-c-K.'- = (D). § 7. FloLV and circulation. We define the flow from a point P to another Q, along a path of which ds denotes an element, as the quantity [Q kV • ds. J p Whenever a velocity potential exists this is equal to <^y — (j)^. The circulation round a closed curve is the line integral /= /cv.f/s (20) taken round that closed curve. This, by Stokes' theorem, is equal to the surface integral 7 = 1" curl («v).nrf;S' (20') taken over any surface drawn in the region and bounded by the closed curve. When the motion is irrotational the integrand is zero, and the circulation round the closed curve vanishes. It follows that, for a simply-connected region, the velocity potential is single- valued. III. Vortex Motion. I 8. When the vorticity w is not zero the motion will be called vortical or vortex motion. A vortex tube is one bounded bj vortex lines. Considering the portion of a vortex tube betAveei any two cross sections, we find as usual on equating the volume and surface integrals = div curl (/cv) dr = I 2w • nd,S, that the moment of the vortex tube 1 vj'XidS, Avhere the inte gration is extended over the cross section, is the same for all sections. And hence, as in the classical theory, the vortex lines either form closed lines, or else end in the surface of the fluid. Mr Weatherbarn, On the Hydrodijnaniics of Relativitij 79 I shall now show that, on the assumption of a force potential, Kelvin's theorem* of the constancy of the circulation in a closed filament moving with the fluid is true in the present case also. Consider a closed filament consisting always of the same particles, and let ds be a vector element of its length and ds the correspond- ing scalar. Then the circulation round it is / = kv •ds. The time rate of change of this is dl dt ' |(.v).r/s+.v.(;jj f/s VF--VP + /cv.(f/s.V)v 7 --F+/CV. _- OS OS ds ds "lv" .(21). Now the last integral is ds c- c , V/c — (c"- ^. die K 8y^ , , ''hs'-^.Ts^^'- T 27 Vc2 - V- On substitution of this value in (21) that equation reduces to dr dt 9«: 9^ 9 / n' 96' ds ds ds. Hence, since the path of integration is closed and k, V, and kv^ are single-valued functions, the integral vanishes, showing that f- .(22). Thus the circulation does not alter with the tiipe. Corollary. If / is zero at any instant it will remain zero. In particular, if the motion is irrotational at any instant it will remain so, provided that the impressed forces have a potential. § 9. Helmholtz's theoremsf. That these theorems are true in the present theory also follows without difficulty from the form (8') of the equation of motion. For taking the curl of both members we have dw 'dt + curl (w X v) = 0. * Of. Silberstein, loc. cit. p. 161, for the proof of the ordinary theorem. t Ibid. Y>p. 163 — 65. 80 Mr Weatherburn, On the Hydrodynamics of Relativity Expanding the second term and using the equation of con- tinuity, we find rfw T^ dk _ dt k dt ~ ' which, after division by k, may be written d fVT\ w ^ s(l)=I-^^ ■; (23). Differentiation with respect to t gives cZ- /w\ fd Mv\ -. w ;'d _ If then w vanishes at any instant it follows from (23) that the first derivative of w/Z; also vanishes, and from the next equation likewise the second derivative at that instant. Similarly all the derivatives with respect to t vanish at that instant, and the quantity -wjk remains permanently zero, so that the motion con- tinues irrotationa.l. Further, the moment of a vortex filament does not vary with the time. For if ds is an element of such a filament moving with the fluid ds = wds/w, and -J- (ds) = ds* Vv = — w • Vv, at w so that (23) is equivalent to d /wX w d , -, ^ <sUJ = arf(('^'> (24). Now if jj, is the moment of the filament, dm^ the constant normal rest-mass of the element considered, and a the cross- sectional area fx = aw, dniQ = kads, ,1 , w ds ^°"'''* k = ^d^. (25)- Substituting this value in (24), and remembering that dm^ is constant, we have |(/.^s) = ;x|(rfs), and therefore -^ = 0, dt showing that the moment of the filament remains constant. 3h^ Weatherhurn, On the Hydrodynamics of Relativity 81 It has been proved already that a mrtex filament consists alvmys of the same particles of fluid, though this can also be now deduced from (24) and (25), using the invariability of yu.. IV. Fluid of Minimum Compressibility*. § 10. According to the theory of Relativity no velocity can exceed that of light. Hence there is no such thing as an incom- pressible fluid ; for such a fluid would admit a wave propagation with infinite velocity. A fluid of minimum compressibility is one in which a wave can attain a velocity equal to that of light ; and for such a fluid the quantity k is directly proportional to the densityf K = k/kJ, K^k'jk; (27), where A:,,' is a constant representing the normal rest-density, i.e. the rest-density corresponding to the normal pressure p^. For a fluid of minimum compressibility the equations of motion, energy and continuity may by (27) be expressed in terms of the velocity v and the rest-density k'. The equation of motion, viz. becomes on substitution , dv dk c'-„,, J ,„ ^' -ZtT + V ,^ + - V^' = k,'F. dt at y Dividing by 7 and using the equation of continuity to transform the second term, we have at once k' (^ - V div v) + (c^ - v-^) Vk' = A-o'F/7 (28), which is the equation of motion in the required form. Multiplying this equation scalarly by v, and transfortning V • V/t', we obtain ,,/lrfv^ , ,. \ ,, ,, fdk' dk'\ /co'F.v .11.' .1 //. a/^J _ ^,2\ Now dk' _d fk Vc- - -"') dt ~ dt\ c _ dk Vc^ - V-' k dv^' dt c 2c "Jc^ - v" dt = — k' div V - 1 y-k' dv- 2 c- dt ' * Latnla, loc. cit. p. 788 ; Laue, loc. cit. § 37. f Laue, loc. cit. p. 241. 82 Mr Weatherburn, On the Hydrodijnaribics of Relativity in virtue of the equation of continuit}-. On substitution of this value in the last equation it becomes simply 7/1- C" V" U/C Wn __ //-»/,N //divv+ ^ ~- = --^F.v (29), c- ot C^J which is the energy equation in terms of k' and v. These equations (28) and (29) are identical with those found otherwise by Lamia* and Lauef. The equation of continuity is as before ^" + ^divv-0 (30), which takes the required form if k is replaced by jk'. § 11. Steady irrotational motion. In virtue of (27) the equation of continuity may also be written — + div(/cv) = (81), Ob and therefore when the motion is irrotational | + ^^^ = (31'). If it is also steady the iirst term is zero, and we have (as in the older theory for the case of an incompressible fluid) ^^(/) = (31"). Thus for steady irrotational motion of a fluid of niinimimi com- pressibility the velocity jjotential satisfies Laplace s equation. It follows immediately that for such a fluid, filling a simply- connected region within a hollow shell, which is fixed relative to some system of reference S, steady irrotational motion relative to that system is impossible. For by Green's theorem .K'V'dT = I {^(py^dr = — (f)/<:v • ndS — I (f)V-(f)dT. Now the last integral vanishes by the equation of continuity. So also does the last but one : for v • n is zero, being the normal velocity at the surface of the fluid. Hence / fc-v'-dr — 0, showing that v must vanish identically throughout the fluid. * Loc. cit. p. 792. • t Loc. cit. p. 244. Mr WeatJierbuni, On the Hydrodij navvies of Relativity 83 In the present case* the integral of the equation of motion found in § 5, viz. takes the form c^/, + /,;'F=0, or, in terms of the rest-densit}^ //, cH-' + Vk\;\/c--v' = 0. §12. l^^teady motion in two dimensions. Supposing the fluid of minimum compressibility, let its steady motion be parallel to one plane — the plane xy. Introduce a function y^ satisfying the relations dy\ dyjr KV = ^ on; •(32), u, V being, as in § 1, the components of velocity parallel to the axes of a; and y resjjectively. Such a function -v/r exists, the equation of continuity div («v) = being satisfied identically. The function yfr is proportional to the flux of matter across a line AP drawn from a fixed point A to the variable point P (x, y). For owing to an infinitesimal displacement Sx of F the increment in the flux of matter is kvSx = ku'icvSx = k,' ~ Sx. ox Thus if ^ denote the flux -,^ o.i: = /in : - dx. ox ox Similarly Jy ^^ ^ ^'" Jy ^^' showing that '^P = kj-yjr, as stated. The part played by this function yjr is exactly similar to that of the stream function in the two-dimensional motion of a liquid in the classical theory. The present function also is a true stream function. Its value is independent of the path chosen from -4 to P provided the region is simply-connected. For, if ^4PP and * Lamia considers only the case of free motion (F= const.) ; loc. cit. p. 71*5. 84 Mr Weatherbarn, On the Hydrodyncmdcs of Relativity ACP are two different paths, the flux across the complete boundary ACPBA is h-v . ndfi — I div (7bV) dr = 0, as is also obvious because the motion is steady. The lines ■y^r = const, are the actual stream lines : for if P moves subject to this condition there is no flux across the path traced out by that point. The above is true whether the motion is irrotational or vortical. The vorticity w is equal to ,(34). and therefore for irrotational motion -^ must satisfy Laplace's equation V^f = (33). If this relation is satisfied there is a velocity potential (f), and (32) may then be expressed in the form • dcf) _ d-\fr I dx dy I dy dx ) These are identical with the relations subsisting between the stream function and the velocity potential in the classical theory of the two-dimensional irrotational motion of a liquid. They are the conditions that (jy + iyfr should be a function of the complex variable x + iy. The theory of such functions may then be used as in the theory referred to*, to give various possible forms of stream lines and lines of equal velocity potential. § 13. Source, sink and doublet. Similarly the irrotational motion of a fluid of minimum compressibility defined by the velocity potential /-^■l («^>' where r is the distance from a fixed point 0, corresponds to the assumption of a continual creation of matter at the point 0, of amount 4<7rm per unit time. For so that ^•v = — . r^ * Cf. Lamb, loc. cit. chap. iv. i Mr Weatherbarn, On the Hydrodynamics of Relativity 85 The velocity is therefore radial from 0, and kv is inversely proportional to ?'^. The flow of matter per unit time across the surface of a sphere of radius r is ^irm, equal to the rate of creation of matter at 0. Such a motion is then that due to a source of strength m at the point 0. If the negative sign in (o5) were replaced by a positive one, we should have the motion due to a sink at of strength ni. And finally the velocity potential representing a doublet at of moment M and with its axis along the unit vector n is *=""-(;) 86 Mr Hardy, On the convergence On the convergence of certain multiple series. By G. H. Hardy, M.A., Trinity College. [Received 15 May 1917.] 1. In a paper published in 1903 in the Proceedings of the London Mathematical Society*, and bearing the same title as this one, I proved a theorem concerning the convergence of multiple series, of the type which is given (with an improvement in the conditions) on p. 89 of Dr Bromwich's Theory of infinite series. This theorem is one of a class of some importance ; and I propose now to state and prove the leading theorems of this class in a form more systematic and general than has been given to them before. I shall begin by recapitulating, with certain changes of form, some known theorems concerning simply infinite series ; and I shall then obtain the corresponding theorems for double series in a form as closely analogous as possible. The generalisation from double series to multiple series of any order may well be left to the reader. • Simply infinite series, 2. I shall say that a function a.,„,, real or complex, of a positive integral variable m is of hounded variation if ■^ i (^m ~ f'-m+l I 1 is convergent. It is plain that this condition involves the existence of a = lim a^n- Theorem 1. The necessai^y and sufficient condition that a^n shoidd be of bounded variation is that its real and imaghuiry ptarts should be of bounded variation. This follows at once from the inequalities I ^m ^m+l I ^ j Cini (^m+l \ ; j Pm Pm+l i ^ ! ^^ in ^^9n+l |> I (^m C^7n+i I ^ I ^m ^m+i \ "i' [ Pm Pm+i \ > where a,,,, = «,„ + i^,„. * Ser. 2, vol. 1, pp. 124 — 128. See also 'Note in addition to a former paper on conditionally convergent multiple series', ibid., vol, 2, 1904, pp. 190 — 191. of certain multiple series 87 Theorem 2. The necessary and sujjicient cunditiuu that a real function a,„ shoidd be of bounded variation is that it should he of the form A^n—AJ, where A.^. and A J are 'positive and decrease steadily as m increases. The sufficiency of the condition follows at once from the •inequality I Clm ~ ftjn+l ! ^ \A,n, — -4,„ + i) + {Ajn — A ,„,+]). In order to prove that it is necessary, let us suppose that a,„ is of bounded variation, and let us write P„i = ! flm - (I'm+l I (an, " ««<+! > 0), p,„ = (a,,,, - a,„+, < 0), Pm'^ ! "m - (Im+l \ (('in " «/»+! < 0), j)„/ = (a,;, - d ,„^.^ > 0), B = ^' i) 7? ' = S « ' -'J??; — —' /'n> -"m -^ J'n • II! VI Then B,,, and i?„/ are positive and decrease steadily as m in- creases ; and B,n - BJ = 2 ((f„ - a„,+i) = ff,,, - a. in We may therefore take A,„,= B„,, + G and ^j,/= 5,,,' + C, where C and C" are suitably chosen constants. Theorem 3. // a,,; is of bounded variation, and Su,,,, is con- vergent, then 2rt,„M„,, is convei^gent. Theorem 1 shews that it is enough to prove this theorem when a„, is real. Theorem 2 shews that it is enough to prove it when «„, is positive and steadily decreasing. In this form the theorem is classical*. Lemma a. If 2c,„ is a divergent series of positive terms, we can find a, sequence of positive numbers e,„, tending steadily to the limit zero, such that 2e,„c,„ is divei'gent. Lemma /3. If 2c,„, is a divergent sei'ies of positive terms, we can find a sequence of integers m^ such that the series Sc,„' , where Cm' = if m = nil and c,,' = c,,, otherwise, is divergent. Lemma a is due to Abelf. Lemma (3 is quite trivial, and the proof may be left to the reader. * See Bromwieb, Infinite Series, p. 48. Theorem 3 is given by Dedekind in bia editions of Diiicblet's Vorlesungcn iiber Zahlcntheorie : see e.g. p. 255 of the tbird edition. Tbe central idea of all sucb tbeorems is of course Abel's. Tbe line of argument followed bere is due substantially to Hadamard, 'Deux theoremes d'Abel sur la convergence des series'. Acta Matheinatica, vol. 27, 1903, pp. 177 — 184. t 'Sur les series', (Euvves, vol, 2, pp. iy7-~20o. 88 ilf?' Hardy, On the convergeyice Theorem 4. If ^a,f,u,n ■'*' convergent whenever 'lii,,, is cuii- vergent, then, a,„, is of hounded variation. This theorem is due to Hadamard*. We have to shew that, if" X I ^m — f'?n+i j is divergent, v^„ can be so chosen that -n,n is convergent and 1a,nUr,n is not. By Lemma a, we can choose a sequence of positive and steadily decreasing numbers e„, so that e„, — * and Sc,„, where is divergent. By Lemma /3, we can then choose the sequence m,- so that %c„j' is divergent. We take u^ = U,, tL,n = L\n - ?7,„_, (m > 1), where Um,. = 0, and U,n = em if m^mi, the last expression being interpreted as meaning e„, if a,„ = a^+i • We have then mi mi—1 mi-1 1 1 1 which tends to infinity with i. Thus l.amUm is not convergent, while ^Um. converges to zero. We may call a,,;, a convergence factor if 2o„,m,„ is convergent whenever %Um. is so. Theorems 3 and 4 may then be combined concisely in Theorem 5. The necessary and suffi,cient condition that a^n shoidd he a convergence factor is that it should he of hounded variation. Double series. 3. The convergence of a double series, in Pringsheim's sense •]-, does not necessarily involve the convergence of any of its rows or columns |. In this paper I shall confine my attention to con- vergent series whose rows and columns are convergent separately : in this case I shall say that the series is regularly conve7'gent. A regularly convergent double series is also convergent when summed by rows or by columns, and its sum by rows or by colunms is equal to its sum as a double series §. Similarly I shall say that a^.^n tends regidarly to a limit if lim a,„,, n = «n , lim « „,,, „ = «,„ , * I.e. supra. t Bromwich, Infinite Series, p. 72. + Bromwich, ibid., p. 74. § Bromwich, ibid., p. 75. of certain multiple seines 89 and the double limit lim a^n., n= CL, all exist. In this case 0^ a.nd «„ tend to a when m and n tend to infinity. Lemma 7. // SSum^n is regularly convergent, to the sum s, and m n 1 1 then, given any positive number e, we can find co so that I *m,M S I < 6 if either m or n is greater than co. We may suppose s = without loss of generality. Since the double limit exists, we can choose w^ so that | 6',„,^„ \< e if ni and n are both greater than ^i. When &)i is fixed we can choose o)., and «»3 so that the inequality is satisfied for 1 ^m ^(o^, n> Wn and for m > &)3, 1 •^ n ^Wj. We can then take to to be the greatest of <wi, (On, and 0)3. , . Lemma S. In the same circumstances, we can choose w so that pq m n < e if p^m, q^ n, and either m or n is greater than w. This follows at once from Lemma 7 and the identity 1^ _ , m n 4. I shall say that a^n^n is of bounded variation in (m, 71) if (1) a^n^n is, for every fixed value of m or n, of bounded variation in ?i or m, (2) the series is convergent. And I shall say that ar,i^n is a convergence factor if SSa„i_,j,Mm_,i is regularly convergent whenever SSw,„,,„ is regularly convergent. My main object is to prove the analogue of Theorem 5 for double series, i.e. to establish the equivalence of these two notions. VOL. XIX. PARTS II., III. 7 90 Mr Hardy, On the convergence It will be convenient to write The condition that a^n,n should be of bounded variation is then that the series i | A„, a,„,,,i|, S|A„a^^,,j|, and iS ] A„i^,i«,„,„ j should all be convergent. It is clear that these conditions in- volve the regular convergence of a^^n to a limit a. Theorem 6. If the condition (2) is satisfied, and a„,,i and «!_„ are of bounded variation in ni and n respectively, then a,n,n is of bounded variation in (m, n). n — l For A^a^,„ = A^a^_i- 1, A^_^«.^,^, v = l m—\ m— 1 ni-\n-\ S |A^a^,«!^ 2 |A^a^,i|+ S S |A^_^a^,J, /u. = l /u. = l n = l 1^ = 1 so that 2 I A^,a^,M j is convergent. Theorem 7. If a^n,n is of bounded variation in {m, n), then a,n= lim am.^n, ««= lim cv,n are of bounded variation in m and n respectively. For a^= ai 1,— % A ^t=i /xfi^Ai, »»' W-1 71-1 ao 11-1 S I «;, — a^+i| ^ 2 I A^tti,^ j + 2 2) I Aj^,^a^,„|, and so 2 ] a^ — a^+j j is convergent. Theorem 8. The necessary and sufiicient condition that a,n,n should be of bounded variation is that its real and imaginary parts should be of bounded variation. This follows from Theorem 1 and the inequalities I ^m,n(^'m,n \ ^ j ^m, ?i^m,9i I "r | ^in,n Hm.,n |j where am,n = '^m„n + i^m,n- of certain multi'ple series 91 Theorem 9. The necessary and sufficient condition that a real function a.m^n should he of bounded va,riation is that it should he of the fo7'm Am,n — A\n,n, whcrc and A\n^n satisfies similar conditions. Suppose first that a-m^n is of tlie form indicated. It is plain that the series ^^7n-^m,n> ■^^7i-^7n,nt ■"-^^m, ?i-^?n., n> m n and the corresponding series formed from ^',„„>i, are all convergent. Further we have and similar inequalities for A„o,„_,i and \n,n(^m,n- Hence «,„,„ is of bounded variation. Next suppose that A,n,n is of bounded variation, and let Pm,n = I ^m,n(^7n,n \ \^7n,7i^m>7i ^ ")> Pm, 7i =" " \^rn, 7i^7n,7i < ^)) P 7)1,71 ~ I ^m,7iO-7n,7i I ( ^m, Ji C^m, ?i ?$ ^/i P m,7i ~ ^ \^m,n(^7n,7i > ^)- Suppose also that C» 00 00 00 ^-'711, Jl — -^ — JJlJL, VI ^^ 771, 71 — -^ — ' Z' M, I' • 771 ji m rt Then it is plain that ^7n-Drn,7i^ ^> ^n^in, 7i-^^i ^7n,7i-t^7n,n^^^ and that B'^^n satisfies similar conditions. Also 00 00 J^7n, n — -O 7JJ , jj = -^ .i ^fi, V Ojh, V ^^ ^m, ?!. ^m ^7i "r '^j m w (^771,71 ^^ -ttnijTi -O 7ji, n + ftjn ~r ttjj ft. But, by Theorems 7 and 2, we have where 0^., CJ, D,,, and D,/ are positive and steadily decreasing functions. Thus ^7n, >i ^^^ ■^7n,7i -^ on, 7i > where Am,7i — Bm^n "H ^m + Dn + Jif, A ,n,7i— B rn.^n + ^m + -L'oi + -^ > £^ and E' being suitably chosen constants; and it is clear that Am,7i and A',n,7i will satisfy the conditions of the theorem if E and E are sufficiently large. 7—2 92 Mr Hardy, On the convergence Theorem 10. // a.m.,n *'* of bounded variation, and 2St/,„,,j, is regularly convergent, then SSa,„,,„M,„,„ is regidarly convergent. In virtue of Theorem 8, it is enough to prove this when o.,n,n is real. In virtue of Theorem 9, it is enough to prove it when am.,n > 0, A„,a,rt,,i ^ 0, Ana,„,„ > 0, ^m.,na.m,n > 0. In the first place, by Theorem 3, every row and column of the series S2a,„^„iA,«,,i is convergent. In the second place, we have m n m n m n p-l («. q q-l P V pg + 2 A^ a^, g 2 2 %i, j + , 2 A^ a^^ ^ 22 Uij + a^^ ^ 2 2 Uij * . m in n ii m n in n It follows that, if _p ^ m, q^n,we have p q <a m,n-'^'m,,n> 2^2^ a^l y ^fji, V mn where Hm^n is the upper bound of 1 '^ " i '^'%Ui,j (/-t ^ m, n ^ v). \ mn I Now p q mn f P 1 p n m q^ 22 a^, ^iV,"- 22 «M,^ '?''/",>'= 2 2 + 22+22 1 1 ' 11 \in+l n+1 m+1 1 1 w+l II and so p q 21 1 1 1 1 ' ' 11 where h^^n is the upper bound of 22«t,j for all values of k, I, /x, and i/ such that fi'^k, v^l, and Tom or l>n. * See pp. 124—125 of my paper quoted in § 1, where the general form of this identity, for multiple series of any order, is given. Similar transformations of double series were given independently by M. Krause, ' Uber Mittelwertsatze im Gebiete der Doppelsummen und Doppelintegrale', Leipziger Berichte, vol. 55, 1903, pp. 240 — 263. See also Bromwich, 'Various extensions of Abel's Lemma', Proc. London Math. Soc, ser. 2, vol. 6, 1907, pp. 58—76, where further interesting applications of the identity are made, of certain multiple series ' 93 Hence, by Lemma 8, we can choose co so that p q m n 11 11 if m and n are greater than eo. Thus the double series is con- vergent, and, since its rows and cokimns are convergent, it is regularly convergent. When a^n,^n and its various differences are positive, this theorem is nearly the same as that referred to in § 1. It is related to the latter theorem, in fact, as what Dr Bromwich calls 'Abel's test' for ordinary convergence is related to 'Dirichlet's test'.* The more direct generalisation is as follows. Theorem 11. If a,n^n is of bounded variation and tends regularly to zero, and m n 2^ 2^ U/j,^ „ 1 1 is bounded, then S'Zam,n%n,7i ^■^ regularly convergent. The proof is similar to that of Theorem 10, and I need hardly write it out at length. The theorem shews, for example, that the series 2S cos (md + n^) (a + mco + nw'f ' where 6 and <^ are real, w'jui is positive or complex, and the real part of s is positive, is regularly convergent except for certain special values of 6, </>, and a; or again that the series ^^ cos (md + n(f)) (arn^ + 2bmn + cn'-y ' * Theorem 10 itself does uot seem to have been enunciated before, even in the specialised form. The nearest theorem which I have been able to find is one given by G. N. Moore, ' On convergence factors in double series and the double Fourier's series', Tia)is. Amer. Math. >Soc., Vol. 14, 1913, pp. 73 — 101. Moore's theorem (a particular case of a theorem concerning Cesaro summability) is as follows : if (1) 23«„j, ,j is convergent as a double series in Priugsheim's sense, VI n (2) |2S«^^J<:7i, (3) ^m, » ^ 0, 00 00 (4) lim 2 |«„,,,J =0, lim S|a„^,„|=0, m-^cc »=1 w-^-a)m = l (5) 22 I M^, „ a,„, „ I is convergent, then 22a„j „ Um,n *® convergent. I 94 Mr Hardy, On the convergence H 11 where 6, (f), a, h, and c are real, and a, ac - ¥, and the real part of s are positive, is regularly convergent except for certain special values of 9 and 0. In either of these series, of course, the cosine \ may be replaced by a sine. In order to prove the converse of Theorem 10 we require two lemmas analogous to Lemmas a and yS. Lemma e. If SSc^.n is a divergent series of positive terms, we catt find €„i^n so that (1) e^^^n decreases luhen ni or n increases, (2) €m,n tends regularly to zero, and (3) the series Sl,e^n,nGm,n is divergent. (1) Suppose first that at least one row or column of the original series, say the vth row 2c„i, ^, is divergent. By Lemma a, we can choose a steadily decreasing sequence 97^, with limit zero, so that XvmCm,p is divergent. We take €vi, n = Vm (n ^ v), €,n, n = {u > v), and it is plain that the conditions of the lemma are satisfied (2) Suppose that every row and column is convergent; and let (m) (w) Then 'S,j„^ is divergent. We choose a steadily decreasing sequence r]m so that 'S^Tjmjvi is divergent. Then SSc'„,,,„, where is divergent; and so Xjn, where yn ^ ^ Vm Cm, n j (m) is divergent. We now choose a steadily decreasing sequence ^„, with limit zero, so that ^^nln is divergent. It is clear that, if we write // y (^ m,n — Vmbn^^m.jn — ^?w, n C^/i,, ^ , all the conditions of the lemma will be satisfied. Lemma ^. If SSc^.w is a divergent series of positive terms, we can choose a sequence of pairs of integers (nii, Ui), tending to infinity ! luith i, so that the series X%c\n,n, where c'm,n = if 771 = nit, n^ni or m^nii, n = ni, and c,n,n = c,,,,,,,, otherwise, is divergent. The modification to be made in the series is effected by drawing perpendiculars on to the axes from the points {mi, n^), and annulling all terms which correspond to points on these perpendiculars. Let a^ denote the sum of the terms whose of certain multiple series 95 representative points lie on the perpendiculars from (m, m) on to the axes. Then So"„i is divergent. Applying Lemma to this series we obtain the construction required, nii being in fact always equal to Ui. Theorem 12. If S2a,„.^,i,i<„,,,j, is regularly convergent ivhen- ever SSif.„i.,,i is regularly convergent, then a,n,n "^^ ^f hounded variation. In the first place it follows from Theorem 4 that ■u,„,^n is, for every value of n (or m), of bounded variation in m (or n). It remains only to shew that SS | A,„^„a,„,_,i | is convergent. Suppose, on the contrary, that it is divergent. By Lemma e, we can choose a sequence of positive numbers €,„,„, tending regularly to zero, so that 2Sc,rt,n, where is divergent. We can then modify this series as in Lemma ^ without destroying its divergence. Now let m n U —^l.u 1 1 and suppose that if m = mi, n ^ ni or m ^ mi, n = ni, and that otherwise ^ in n, — this last formula being interpreted as meaning e^yi^n if These equations define u,n,^n uniquely for all values of m and n, and it is plain that U'm,n tends regularly to zero, so that 2S^/,„,,t is regularly convergent. On the other hand ^ -^ '■hn,n ^'■m,n "i ■^ '^m,n ^i')n,n ^ m,n — -^ ■^ ^ w,n> 11 11 11 which tends to infinity with i. Thus ^'%a„r,nUm,n is not convergent. This proves Theorem 12. Combining it with Theorem 10 we obtain the analogue of Theorem 5, viz. Theorem 13. The necessary and sufficient condition that am^n should be a convergence factor is tJiat it should be of bounded variation. 96 Mr Watson, Bessel functions of large order * Bessel functions of large order. By G. N. Watson, M.A., Trinity College. \^Received 14 June 1917.] 1. When the order of a Bessel function is large, the asymp- totic expansion of the function assumes various forms depending on the values of the ratio of the argument to the order of the function. The dominant terms of the asymptotic expansions are given by the formulae : (i) When n is large, x is fixed and ^x< 1, then Jn {nx) ~ (27rw)"* (1 - a^y^x"" [1 + V(l - a?)]"' exp [n V(l - x% (ii) When n is large, x is fixed and x>l, then Jn{nx)<^{^'jrny^{x'^- l)~^cos [?i V(*'^ - l)-n^e(r^x- lir]. _ 2. (iii) When n is large and € = 0{n ^), then J^{n + ne)^T(i)/{7r2is^n^]. The corresponding complete asymptotic expansions, valid for general complex values of n and x, have been given by Debye*. Accounts of the history of the approximate formulae are to be found in Debye's memoirs and also in two papers f which I have published recently. It is evident that there are transition stages between the domains of validity of the three formulae quoted ; and not much is known about the behaviour of J„ (jix) in these transition stages. Consequently I propose to establish approximate formulae (involv- ing Bessel functions of orders'] + ^) which exhibit the behaviour of the Bessel function right through the transition stages. These formulae are more exact forms of some approximations which Nicholson § obtained some years ago without estimating the margin of error or the precise ranges in which the results were valid. * Math. Ann., lxvii. (1909), pp. 535—558. Milnchen. Sitzungsberichte [5], 1910. t Proceedings, xis. (1916), pp. 42—48. Proc. London Math. Soc. (2), xvi. (1917), pp. 150—174. J These functions have been tabulated by Dinnik, Archiv der Math, und Phys., XVIII. (1911), p. 337. § Phil. Mag., Feb. 1910, pp. 228—249. Mr Watson, Bessel functions of large order 97 The approximations which I shall obtain are derived by shewing that certain integrals of Airy's type* are effective approximations to the integrals which occur in Debye's analysis. It will be assumed that the reader is familiar with Debye's memoirs, although it seems desirable to modify the notation to a considerable extent. The formula for Jn (nx), when x^l, is of importance in connexion with the maxima of the Bessel function f. The two formulae which will be obtained in this paper are as follows : (I) When a ^ 0, Jn ( n sech a) ~ 27r~^ 3 " ^ tanh a X exp [n (tanh ol + ^ tanlr' a — a)} . if ^ {^n tanh* a), where the error is less than 37i"^ exp [?i(tanh a — a)|, and K,,,{2) denotes the Bessel function of Basset's type (see § 6). (II) When ^ /3 ^ \7r, Jn {>i sec /3) ~ ^ tan /3 cos [n (tan ^ — ^ tan^ yS — /3)} X [J_ I (-3-*^ tan=^ ^) + /, (i/i tan* /3)] + 3" ^ tan yS sin [n (tan ^- ^ tan* j3 - /3)} X [/_ , (iw tan* /S) - J^ O tan* y8)], where the error is less than 24/?i. Part I. The value of Jn(n.v) ivhen O^^-^l. 2. We take Sommerfeld's integral The stationary points of x sinh w — w, qua function of w, are given by cosh w — l/x ; accordingly we replace x by sech a, where a ^ ; and then, putting w^ot + t, we have jiTTl J 00 -ni + n (sinh t — t)] dt. The exponent has a stationary point at ^ = 0, and the method of steepest descents provides us with the contour whose equation is / {tanh a (cosh t-\)-\- (sinh t-t)] = 0. * These integrals have been expressed in terms of Bessel functions by Nicbolsou, Phil. Mag., July 1909, pp. 6—17, and by Hardy, Quarterly Journal, xli. (1910), pp. 226—240. + Proc. London Math. Soc. (2), xvi. (1917), p. 169. 98 Mr' Watson, Bessel functions of lar-ge oi'der The portion of this curve which is suitable for our purposes consists of an arc* on the right of the imaginary axis in the ^-plane with its vertex at the origin and with the lines I {t)= ±tt as asymptotes. If we write t = u + iv, where tt, v are real, the equation of the curve becomes cosh (a + u) = V cosec v cosh a. We shall put tanh a (cosh t - 1) + (sihh t — t) = — r, so that as t traverses the contour t diminishes from + x to 0, and then increases to + oo ; and therefore Jn (^'^) = o-^ e«(t^'ii^*-«) I j + \ e""^ (dt/dr) dr ; in the first integral v ^0 and in the second integral v ^ 0. Now define T by the equation j- 1^2 tanh a + ^T' = -T. A contour in the T-plane on which t is real is a semi-hyperbola touching the imaginary axis at the origin and going off to infinity in directions inclined ± ^tt to the real axis. If we write T=U+iV, where U, V are real, the equation of the hyperbola becomes Utanha + ^U'' = iV-\ Taking the semi-hyperbola as the T-contour, we shall shew that an approximation to foD+iri /• 00 exp ( Jtt?') e-'^-'dt is e-''^dT. J CO - Tii ■' 00 exp ( - JttO It is easy to see that the difference of these integrals is D idt dT) , {] ^ Jo ] (dr dr] and so the problem before us is reduced to the determination of an upper bound for \d{t— T)/dT |. * This curve is derived from the curve shewn in fig. 4 (p. 541) of Debye's first! paper by turning it through a right angle and talcing the origin at the vertex. The! degenerate case when a is zero is shewn in fig. 5. \ t Since T = ^t^ ta,nh a + lt^ + (t*) when | 1 1 is small, the curve in the T-plaue closely resembles the curve in the i-plane near the origin ; and, the parts of the curves near the origin being the most important when n is large, we are obviously able to anticipate that the integrals under consideration are approximately equal. Mr Watson, Bessel functions of large order 99 3. We shall now shew that, whenever t ^ and when, corre- sponding t(^ any given value of t, we choose V to have the same sign as v, we have the inequality \d{t- T)}dT I ^ Stt. Since, corresponding to any value of t, the two values of t are conjugate complex numbers (and similarly for T), it is evidently sufficient to prove this inequality when v and V are both positive. On comparing the values for r in terms of t and T, we perceive that ' ( 7' - {H^ + tanh a + 1 (T' + Tt + f')} tanh a (cosh ^ - 1 - ^t-) + (sinh t-t- ^t-'). Also d(t- T)jdT = { jTtanh a + ^T''}~' - {sinh t tanh a + (cosh t - 1)}-^ ^ t-T T {sinh t tanh a + (cosh ^ — 1 )] ^^(^-r ) + (sin h t - i) ta nh ajl^(cosh t-l-^f) ^ TXta'nh a + iT) {sinh « tanli^+ (coshT-nf)} " Now I sinh t tanh a + (cosh t—l)\ — sech a v/[(cosh u — cos v) {cosh (2a + w) — cos vW ; and since {cosh (2a + (t) — cos v] — cosh- a (cosh ii — cos v) = sinh- a (cosh m -t- cos y) + sinh 2a sinh a and {cosh (2a + u) — cos v] — sinh- a (cosh a + cos y) = cosh^ a (cosh <( — cos v) + sinh 2a sinh a we 6'ee that \ sinh ^ tanh a + (cosh ^ — 1) | exceeds both cosh ^^ — cos V = I cosh i — 1 1 and also tanh a \/(cosh- u — cos- y) = tanh a | sinh ^ j . We now divide the range of integration into two parts, namely T ^ 1 and :^ T ^ 1. 4. Consider first what happens when t^ 1. If I T| ^ 1, we have (on the T'-contour) T = I IT- tanh a + 1^=* | < ^ + i < 1. 100 Mr Watson, Bessel functions of large order Also, if I ^ I ^ 1, we have (on the ^-contour) T = I (cosh ^ - 1) tanh a + (sinh t-t)\^ S \t i '"/''* ! ^ e - 2 < 1. Hence, when r^l, we must have both \T\^1 and also \t\^l. But, when | T| ^ 1, since U^O, we have I (dr/dT) I = I Ttanh a + ^-T- 1 ^ ] tanh a + ^T\>^. Also, when | i | ^ 1, we have u or v (or both) greater than l/\/2, and we always have v less than vr. Hence, by the result of § 3, I (dr/dt) I = I sinh t tanh a + (cosh t — 1)\^2 (sinh'^ ^u + sin^ ^y), and this exceeds the smaller of 2sinhni/V8), 2sin^(l/v'S). Consequently i (dt/dr) I ^ 1 cosec^ (l/\/8) = 4-14 < 27r - 2. Therefore, when t ^ 1, we have \d(t — T)/dT \ < ^ir. We shall make use of this inequality in § 6. 5. Consider next what happens when ^ t ^ 1. If I T| ^ 2, we have (on the I^-contour) T = I i-r^ tanh a + iTM ^ 4 ! 1 tanh a + ^T\> l\ Tl^^. Also noting that u and v increase together, when v "^ ^tt, we have (on the ^-contour) T = u + tanh a — cos v sech a sinh (a + it) ^ M + tanh a ^ tanh G — a + log {-^-TT cosh a + \/(l7r" cosh" a — 1)}, on expressing u in terms of v and noting that v cosec v exceeds -^tt. This function of a increases with a and so it exceeds log{j7r + V(i7r^-l)} >1- Hence, tuhen r ^ 1, we ??iwsi have both \T\^2 and also v^^tt. Further, when v^I-tt, we have cosh u ^ sech a cosh (a + it) = v cosec w ^ I-tt < cosh I'l, so that \t\" -^ ^TT' -h (I'ly < 4, a7id therefore \t\<2. That is to say, when r^ 1, neither \ T\ nor j t j exceeds 2. Also, for all values of t, du/dv = {1 —V cot v)/i\/(v" — sin^ v sech^ a) ^ (1 — ?; cot «;)/«; ^ ^-v, and so u'^^v'^. Mr Watson, Bessel functions of large order 101 Further, when v ^ ^tt, we have* 1 — v cot v ^ ^J{v- — sin- v), and so dujdv ^ 1 , i.e. v ^ w, whence at once we have v \/2 '^\t\. Next, when | ^ | $; 2, we have \^m\xt\^\t\[l - ^\t\^ - -,^^,\t\* - ...] In like manner we may prove that, when | ^ | ^ 2, I cosh i - 1 I ^ 4 I ^ J-/18 ^l\t\\ I cosh t-\-\t'\^\t IV20, i sinh t-t\^^\t IV24, | sinh t-t-^,f\^\t I7IO8. We are now in a position to obtain an upper bound for | 7^- ^ |. It is first evident that \\{T+t) tanh a + 1 (^' + ^« + t') I ^ / {i ( 2^ + tanh « + 1 ( r^ + r^ + 01 ^ I V tanh OL-^ ^uv ^ |-w(tanha + i-y-) ^|i|(tanha + yVl^l')/\/8 ^U|2(itanha + T'H|^I)/V8. Hence, by the result stated in § 3, \l-t\^d> \t\ -ytanha+ 1^1/18 ^^^^^l^l ^i|«l. To obtain a stronger inequality, we write the equation of § 8 in the modified form {T-t){T + t) {1 tanh a + ^{T-\-t)] = -{T- tfl24> + tanh a (cosh t-l-^f-) + (sinh t-t- ^t^). The expression on the right does not numerically exceed I ^ 17192 + tanh a.\t \'/20 + \ t IVIOS ^ tanh a . 1 1\'/20 + | ^1748 ; and since \{T + t) [^tixnh.a + ^(T + t)]\ exceeds both -H^ltanha and also ^|^1", we see that \T-t\^{j^^+i)\t\'=i\tm5. If we now f mother restrict t so that | ^ | ^ 1, the last inequality- gives \{-(T- i)V24 + tanh a (cosh t-1- ^f-) + (sinh t-t- |^«)} | ^ 4-^ I « 1 7(24 . 150 + tanh a.\t |720 + | ^ J7108 ^tanha. 1^1720 + 1^17104, * Since -j~ {siii'^j' (1 - v cot v)^ - sin-i; {v- - sin- c) j- = -- 2 sin 2v . (v^ - sin^ u) < when < u < Itt. 102 Mr Watson, Bessel functions of large order and this inequality, combined with the modified form of the equation of § 3, gives I T- ^1^(1/10 + 1/13) jip'^i^lVS. Hence, when |^!^1, \T-t\^^\t\, and so \T\^^\t\; and therefore, when U | ^ 1, \T\ exceeds the value which it has when 1^1 = 1, and so, a fortiori, | T\ ^ |. It now follows that, when both r and \t\ do not exceed 1, we have d^_dT ^ i 1 1 dr dr II ^ I . I (sinh t tanh a + (cosh i — 1)} | 1 1 [yiO + 5 tanh a \ t |V24 + 1 1 |V20 "^ (8 1^ IV25) I {sinh t tanh a + (cosh t-l)]\ = (23 i ^ 1732 + 125 tanh a 1 1 1/192} -^ I {sinh t tanh a + (cosh t— 1)} j . Since the denominator exceeds both 1 1 ^ | lanh a and i | ^ |", we see that \d{t- Tydr I ^ (23/8) + (125/128) < 27r. If T ^ 1 and 1 ^ j ^ I ^ 2 we use the second expression of § 3 for d{t— T)ldr. Replacing \T—t\ in the numerator by 4 | ^1^15 and \T\ in the denominator by |, we get in a similar manner \d{t- T)ldr \^[\t IV3 + 125 tanh a . | i |7192 + 11 1 ^ | V60| -^ j {sinh t tanh a + (cosh t — 1 )} | ^ 4 {I ^ i/3 + 1 1 I ^ I V60}/13 + 125 i « I-/128 <37r. 6. It is obvious, from the results of §§ 4, 5, that, luhenever T ^ 0, we have \d{t- T)ldT I < Stt ; and from this result we have 27r r] „ idt dT\ ,1 ., /•" „ , ^, The evaluation of I e""**^(ZT presents no special points J CO exp( — -J-irt) of interest ; the simplest procedure is to modify the contour into two rays, starting from th^ point at which T = — tanh a and making angles + -J^tt with the real axis. Mr Watson, Bessel functions of large order 103 If we write T = — i&xih. a ■\- ^e*>' on the respective rays, the integral becomes /•OO e'-'^*' exp (i?i tanh-' a) A exp { - i?ip - ^n^e^""' tanh^ a] d^ JO - g-i'^' exp (1??, tanh' a) . exp { - i?^p - i^j^^-i'^'' tanh- aj d^. These are integrals of Airy's type ; on expanding exp (— |n^e* ■•'^* tanh- a) in powers of tanh a and integrating term-by-term — a procedure which is easily justified — we get on reduction Itti tanh a . exp (i?i tanh^ a ) . [/_ j^ (^n tanh' a) — /j^ (J-?i tanh" a)], where, in accordance with the ordinary notation, On introducing Basset's function K^Az), defined as ^ TT cot niTT [/_^ (z) - /,„ (z)], we obtain the final formula 2 t/^ (n sech a) = ~ [tanh a exp {?? (tanh a + i^ tanh' a - a)] TT V " '^ X Kx Qn tanh' a)] -|- SdiU-^ exp {w (tanh a — a)], where | ^i j < 1. When w is large the ratio of the error term to the dominant term is of order n~^\/tanha, n~^, vT^, according as n tanh'' a is large, finite or small. The formulae (i) and (iii) of § 1 agree with this result when a is finite and when n tanh' a is small, respectively. Part II. The value of Jn(nx) when x^l. 7. It is convenient to regard Hankel's solutions of Bessel's equation, iT^'^' and Hn^^^ as fundamental. The ordinary solutions are expressed in terms of these functions by the equations Jn (nx) = i {^,,w (nx) + HJ'^ {nw)], The integral formulae of Sommerfeld's type are 1 roo+ni Hn^'^ (nx) = -—. e« (xsinhw-tv) dyj^ TT'l J _oo 2 roo—iri 104 Mr Watson, Bessel functions of large order The stationary points of xi^m\iw — iv, qua function of w, are given by cosh ?<; = !/«. As <\lx^l, we put x = sec/3 where ^ /3 < I^TT ; and two stationary points are given by w= ± ^i. Now it has been shewn by Debye that a branch of the curve * I{x sinh lu - w) — I {x sinh ijS — i/3) is a suitable contour for HJ^\ and the reflexion of this contour in the real axisf is a suitable contour for HJ-K On making a change of variable by writing w = t + i^, we have 1 rQO+7r?-(/3 where i tan /3 (cosh t — 1) + sinh t — t = — T. If we put t = 'u + iv, where u, v are real, the equation of the contour is cosh w = (sin ^ + v cos /3) cosec (v + /3), and, on the contour, r = u — sec /S sinh u cos (v + /3). When V is given, cosh u is given and the sign of u is ambiguous ; we take u to have the same sign as v, in order that the contour may be of the requisite type. Next define T by the equation ^TH tan /3 + ^T' = - r. We write T^U+iV, where U, V are real; a contour in the T- plane on which t is positive is that branch of the cubic :|:, whose equation is (t/^ _ T/2) tan ^ + 1 F(3[7- - V) = 0, which passes from — cc — i tan ^ through the origin to x exp (^Tri). Taking this curve as the contour, we shall shew that an approximation to r<x+Tri—ip rco exp (Jn-i) e-''^dt is e-'^^dT. .'-cxj-i/S J -00— i tan/3 * This curve is derived from the curve shewu in fig. 2 (p. 540) of Debye's first paper by turning it through a right angle and taking the origin at tlie node. The reader will observe that tlie character of the contour has changed with the passage of X through the value unity. t Since iJ^*^), HJ") are conjugate complex numbers when n and x are real, it will be sufficient to confine our attention to HJ^^. X Of course t is real on the whole cubic ; as T traverses the specified portion of it, T decreases from + oo to and then increases to + oo . Mr Watson, Bessel functions of large order 105 8. Before proceeding further, we shall shew that the slopes of the contours in the t-plane and in the T-plane never * exceed s/S. If we write (sin ^ + V cos /3) cosec (v + /3)^ ^|r (v), we have dv ^sinhu^ ^ [{^(^)}2 _ i]l ^ du '\Jr' (v) ~ ■\lr'(v) Now yjr' (v) = cosec (/3 + v) {cos yS — cot (^ + v) (sin l3 + v cos /8)}, and so -yjr' (v) is positive when /3 + v is an obtuse angle. When ^/3 + v^^TT, however, we find that cos /3 tan (/3 + v) — (sin 13 -\- v cos /3) is an increasing function which vanishes with v. Hence yjr' (v) has the same sign as v (and therefore the same sign as u), and consequently dv ^ [{ylr(v)\'-lf du I yjr' (v) I It is therefore necessary to prove that i.e. that x (^) = '^ li"' W' " if (^)}' +1^0. Now % (0) = 0, and it is consequently sufficient to shew that X {^) h'^s the same sign as v. Since X{v) = 2y{r\v){Sf"(v)-ylr{v)} and yfr' {v) has the same sign as v, it is sufficient to prove that Srfr" (v) - f (v) ^ 0. Since yjr (v) sin (v + /8) reduces to a linear function of v, its second derivate vanishes, and so the inequality to be proved reduces to f {v) - 3-f' (v) cot (v + yS) ^ 0, i.e. to (sin ^+v cos /3) {1 + 3 cof^ {v + /3)} - 3 cos ^ cot (v + /3)^0. But sin yS + V cos yS - 3 cos ^ cot {v + /3)/{l + 3 cot^ {v + /3)} has the positive derivate 4cos yS (1 + 3 cot"(w + /3)}~^ and is positive when v = — ^; hence it is positive throughout the range — ^^v-^TT-^. And this is the result which had to be proved. * In the limiting case when ^ = 0, the ^-contour has slope ^3 immediately on the right of theorigin, and the T-contour consists of the rays arg r = 0, arg T — ^tt ; so there is no better inequality of the form stated. VOL. XIX. PARTS II., III. 8 106 Mr Watson, Bessel functions of large order In like manner, we find that dV _ (tan;e + F)^(tan/8 + ^7)^ ^dU~ tan'/S + Ftan/S + iV- ' and it may be proved by quite simple algebra that the square of this last fraction does not exceed 3. From the results just proved it follows on integration that \v\^\u\^/S, |F|^|i7!V3, and hence h'!^iUI> \v\^^\t\^3, \U\^^\T\, |Fi$i|TlV3. 9. We now return to the integrals of § 7. As in the corre- sponding work of §§ 2 — 3, we have to obtain an upper bound for \d{T — t)ldr i ; we shall in fact shew that this function does not exceed 127r. We notice that formulae corresponding to those given in | 3 are {T-t){hAT+t)i\>?.u^-^^{T^^Tt + 1?)] = i tan yQ (cosh t-l-hP) + sinh t-t- ^t\ d{t- T)/dT = {iT tan /3 + ^T']-^ - {{ sinh nan /3 + (cosh t - 1 )]-^ t-T " T [i sinh t tan ^ + (cosh ^ - 1)} ^t{t-T) + i (sinh t - t) tan ^ +_(cosliJ^- 1 - ^i') "^ T {% tan y8 + ^ T) (TsmhTtan yS + (coshl -1)} Now I i sinh t tan /3 + cosh t — 1\ = sec 13 V[(cosh u — cos v) {cosh v — cos (2/3 + w)|], and since (cosh u — cos (2/3 + v)] — cos- B (cosh ii — cos v) = sin^ /3 (cosh u + cos v) + sin 2/3 sin v ^ (1 + cos v) {sin- /S + sin 2/3 tan -^ vj ^ (1 + cos w) {sin- 13 - sin 2y3 tan ^^] we have . I i sinh t tan /3 + (cosh ^ — 1) | ^ cosh w — cos « = | cosh t — l\. Also cosh w - cos (2/3 + ?;) ^ 2 sin= (^8 + -iwi ^ 2 sin- 1/3, Mr Watson, Bessel functions of lar^ge order 107 and so I i sinh t tan /3 + (cosh t — l)\^sm^^ sec /S V[2 (cosh it — cos v)} ^ tan/9 I sinh ^t\. That is to say \ i sinh t tan /3 + (cosh i — 1 ) j exceeds botJt, j cosh ^ — 1 j and also tan /3 | sinh ht\. In order to simplify the subsequent anafysis, it is convenient to place a restriction on /S. We shall coiisequently assume in futui^e that O^/S^Itt, so that tan^^^l. This restriction is not of importance so far as the final result is concerned, because Debye's formula, quoted in | 1 (ii), is effective whenever sec/3^1 + S, where 8 is any positive constant ; and so it is certainly effective when sec /3 ^ \/2. The importance of the analysis in the present investigation is due to the fact that it is valid for small values of ;8. 10. Consider what happens when r ^ ^, whether v, V are both positive or both negative. When I 2'| ^f , we have (on the T-contour) T = I iiTHanyS + irT' \ ^\T'\{^ + ^\T\)< h, and if I i I < I, we have (on the ^-contour) T = I [i tan /3 (cosh ^ — 1) + (sinh t — t)]\ 00 $ t |«|'"//?i!^e*-l -1 = 2-12 -1-75 <^. Hence, when t ^ |-, we must have both | ^j ^ f and \t\ ^ f . But, when | T | ^ | , we have I (dr/dT) \ = \T\.\itan/3+iT\^\T\.\^R(T)\^^\T\"-^^\. Also (as in § 4) when | ^ | ^ f , we have j (dr/dt) \ = \i sinh t tan /3 4- (cosh t— 1)\ ^ I cosh ^ — 1 I = cosh u — cos v^2 sin- (fjr ^2) = 0-137, and so j (dt/dr) | ^ 7-3. From these results we see that, when r^^, \ d (t -T)/dT\< 15 <57r. We shall make use of this inequality in § 12. 11. Consider next what happens when ^t ^^, whether v, V are both positive or both negative. When \T\^2, we have (on the IT-contour) Also, when \t\ ^2 and v + /3 ^ ^ir, we have u ^ ^tt \/3, and then T = u — sec /3 sinh u cos (v + /3)^u^ 1. H_-2 108 Mr Watson, Bessel functions of large order Next, when | ^ | ^ 2 and ^ <^r + jS -^^Tr, v/e have cosh w = (sin + v cos yS) cosec (v + 0) ^sm/3 + (Itt — /3)cos/3 < ^7r< cosh 1*1, since sin yS + (^tt — ^) cos /3 is a decreasing function of y8. This gives 2 ^ | i | < \/{(-^7r)- + (1"1)^} < \/3'7, which is impossible; so that, when \t\^2, we cannot have /3 ^v + /3 -^ ^tt. Lastly, when \t\'^2 and ^ v ^ - /3, we have w :$ 0, and so - w ^ V(4 - /S'O ^ V{4 - (iTT)}^ > 1-8, and T = sec /3 sinh (— u) cos (?; + /3) — (— it) ^ sinh (- w) - (- u) > ^ ( l-8)» > i . Therefore, whenever | ^ | ^ 2, we have r^^. Hence, when ^ t ^ |- , we must have both \t\^2 and | T | ^ 2. Next we shall shew that R[^t+ T'^/{T + 1)} has the same sign as u and U. The function under consideration is equal to l^u {{U + uf + (F+ vy\ + (U"^-V')(U+u) + 2UV{V + v)]^[{U +uy + (V + vf]. Taking U, V, u, v positive for the sake of definiteness, we see that the numerator of this fraction exceeds lu{U' +V') + u{U-'- F^) = ^u(SU' - V) ^ 0. Similarly we can prove that the numerator is negative when U, V, u, V are all negative. It follows from this result that \R{t+Ty(T + t)}\^^\R{t)\^l\tl We ewe now in a position to obtain an upper bound for \1' — t\ when 1 1 1 and \ T \ are both less than 2. First suppose that | ^ | ^ | . Then, from the formula quoted at the beginning of § 9, j {T- t) I .\{T+t)\.\ {laan /3 + i^ + iTy{T+t)} | = I i tan y8 (cosh t-1- lt~) + (sinh t~t-lt^)\ 00 < 2 \t\'"'lm\^D\t\*lllQ. 1)1 = 4: But \T+t\^\t\and \{^its.n^ + it + iTy{T+t)}\>i\R{t + ri(T+t)}\^^^^\t\. Hence, when | i | ^ l, we have | (T - ^ | ^ 120 | ^ |7119. Next, keeping | ^ | ^ | , we take the formula i2iT-t)(T+t){iUnl3 + i{T+t)} = -^\(T-ty + i tan 13 (cosh ^ - 1 - i t~),+ (sinh t-f-^f) 3Ir Watson, Besael functions of large order 109 and observe that and also, in view of the fact that, as t varies through positive vahies, t + T traces out in the Argand diagram a curve, through the origin, whose slope obviously never exceeds V3, the distance of all points of this curve from — 4^ tan $ must exceed 2 tan /3. Hence | i tan /9 + ^ (T + ^) | ^ ^ tan /3. Using these two inequalities, combined with the fact that j(T— i) j ^ 120|^|-/119, and the obvious inequalities \T+t\^\t\, I cosh t-l-^t'\^\t 1 V23, |sinhi-^-i^-|^|^i-Yll9, we deduce from the last equation for T— ^ that \T-t\^l\t\ {120 1 1 \ll\^Y + 4 1 ^ IV23 + 16 U IV119 < U I'- Using now the inequality \T —t\i^\t\^ in place of I T- ^1^120 1^17119, we get \T-t\^l\t\' + ^\t IV23 + 16 I ^ IV119 < (1/24 + 4/23 + 16/119) \tY ^ ^\t\K Using now the inequality \T —t\^\\t\^,yNQ get, in place of the last result, \T-t\% (1/192 + 4/23 + 16/119) | ^ 1=^ ^ ^ | ^ p'. From this result it follows that, when l^j^^, \T —t\% ^\t\, and so iri^l^l^l. Consequently, from the formula for d{t - T)ldT given at the beginning of § 9, we see that, when | ^ | $ |^, \dt^_dJ dr dr i\t\ {i|«|.|(cosh«-l)| + i (ii I ^ l)N 1* ^i^h ^ ^^^ /^ + (cosh t — 1)| Now, when | ^ | < 2, 1 po«?h /_l|>i|^|2ri_ 4 _ 16 _ 1>1|/|2 and \smh.t\^\t\[l-l;-^- ...]>^\t\; and so, using the results of § 9, we get \d(t- T)ldr \ ^ 16/11 + (576/121) [4 (1/6 + 1/23) + 6/5] <12, when 1^1 ■S^. . lio Mr Watson, Bessel functions of large order Lastly, when |:^|^|^2, we have l^^j^ 11/24, and so, by the method of § 10, we get \d{t- T)ldr I <S 4 (24/11)^ + \ cosec^ (i ^2) <35-3<127r. 12. It follows from the results of §§ 10, 11 that, for all positive values of t, \d{t-T)ldT\<12Tr, and consequently + di < 24<7rjn, ; [dr dr] '^ so that ir,,« (n sec y8) = A e''^ (t=i»^-^) g-"- dT + 24^6.,/ n, TJ"* J-Qo-itan/3 where | ^al < 1- To evaluate this integral, where — r = ^T'^ i tan /8 + ^T'^, we take the contour to consist of the two rays arg(T+ itan/8) = 7r, ^ir ; on writing T= — i tan /3 — ^, —i tan j3 + ^e^"' on the respective rays, expanding the integrand in jDowers of ^ and integrating term by term we find that /■ooexp(47ri) e-''"dT J -00 — ?!tan/3 = §771 tan ^ exp (— l^vn tan^ /3) X [e" *''' J_ 1 (i?^ tan^* ,8) + e*" /i (iw tan^ ^8)] = 3" -7ri tan /5 exp (|■7^^ — |-7w tan^/3) i/^.'^' {^n tan" y8). Since J„ {n sec ^) = R [^w"' ('^ sec /3)], /_„ (w sec ^) = R [e"'^^' i^,,"' (" sec ^8)] , it follows at once that, when ^ /3 ^ ^tt, Jn {n sec /9) = 3~^ tan /3 cos {m (tan /3 — ^ tan^ /3 — /3)} . [JL i + t/i] + 3" Han /5 sin {n (tan /3 - i tan^* /Q - /3)} . [/_ .^ - J^J + 24(9/7^, J"_,i (?i sec /3) = 3~^ tan /3cos [ji (tt + tan /3 — -^ tan'* y8 — /S)} . [/_ x + Ji] + 3" ^ tan /3 sin {n (tt + tan /3 - i tan=* ^ - /3)} . [J_ j - ^^] + 245'77i, where the arguments of the Bessel functions J±x on the right are all equal to ^ntan^/3, and | ^ I, \6'\ are both less than 1. It is easy to see that, except near the zeros of the dominant terms on the right, the ratios of the error terms to the dominant terms are of orders Vl^^^^an^), n~'^, ?i~*, according as 7i tan^ /3 is large, finite or small. Mr Todd, A particular case of a theorem of Dirichlet 111 A particular case of a theore^n of Dirichlet. By H. Todd, B. A., Pembroke College. (Communicated, with a prefatory note, by Mr H. T. J. Norton.) [Received 14 June 1917.] [The following note is an extract from an essay submitted to the Smith's Prize Examiners. It will, perhaps, be convenient if I preface Mr Todd's argument by explaining its relation to the theory of algebraical numbers. The principal theorem is a famous one of Dirichlet's on the unities of an algebraic corpus or order. It will be remembered that if ^ is a root of an irreducible equation of the nth degree, the coefficients of which are integers, then, if the coefficient of the nth. power of the unknown is 1, ^ is an algebraic integer, and if in addition the absolute term is + 1, ^ is a unity ; and further, that if ^ is an integer of the ?ith degree, then the order of '^ is the aggregate of numbers w of the form JUq ~r~ ^1 ^j *T • • • *^"ii 1 'J ) where x^... x^-i are rational whole numbers, every member of the order of ^ being an integer of the wth or some loAver degree. Dirichlet's theorem*, as modified by Dedekind and others, asserts that if the irreducible equation satisfied by ^ has r real and 2s imaginary roots, then the order of ^ contains r + s — 1 fundamental unities, e^, .... e,.+^.._i , which are such that every unity contained in the order is expressible in one and only one way as a product ' ^ r+s-l' M'here t; is a root of unity contained in the order and m^, ... , m,.+.,_i are rational integers ; and that, conversely, every such product is a unity and a member of the order. The simplest cases of this theorem are those in which the equation satisfied by "^ is (i) a quadratic with two imaginary roots, (ii) a quadratic with two real roots, (iii) a cubic with one real and two imaginary roots and (iv) a quartic of which all the roots are imaginary. In the first case, and in this alone, there are only a finite number of unities in the order, and they are all roots of unity ; in the other cases * The theorem, when stated completely, has a wider scope, corresponding to a wider definition of an ' order ' than is given above : what is there defined is more properly called a 'regular order'. A general statement and proofs are given in Bachmann, Zahlentheorie, vol. v., eh. 8. 112 Mr Todd, A 'particular case of mentioned there is one and only one fundamental nnit}'^ and in cases (ii) and (iii) + 1 are the only roots of unity which the order contains. In case (i) the theorem is easy to prove. In case (ii), if P + 2bt + c = is the equation satisfied by "^j the unities of the order are essentially the same as the solutions of the Pellian Equation x"" - (b- - c) y^ = ± 1, and Dirichlet's results can be deduced from the theory of this equation. In other cases the proof of the theorem is much more difficult. Mr Todd is concerned with the case in which ^ is the cube root of an integer — which comes under the heading (iii) above. If ^ = n, the general theorem ass( -rts (a) that the order of ^ contains an infinity of unities, (b) that they are all expressible in the form where 7 is a particular one among them and m is a positive or negative whole number, and (c) that every number of this form is a unity of the order. Mr Todd's essay contained an elementary proof of (6) and (c) ; the proof of (c) does not essentially differ from that given in text-books, though this was not known to him at the time, but the proof of (6) appears to be new and forms the subject of the following note. — H. T. J. N.] If ^^ — - n, and T =x + y^ + 2'^^ is a member of the order of ^, then r^ = nz + x"^ + 2/^2^ SO that r satisfies the cubic equation \x — t, y, z \ nz, x — t, y =0 ; I ny, nz, x — t *4 hence it follows that F is a unity of the order if and only if x, y, z satisfy the Diophantine equation = af^- ny' + n^z^ — Snxyz = ±1 (i). It will be the object of this short note to give a simple elemen- tary proof of the fact that, if the existence of unities is assumed, then every unity of the order of ^ can be expressed in the form X, y, z nz, X, y ny> nz, X a theorem of Dirichlet 113 where V is one particular unity of the order, and m is a positive or negative integer or zero. In what follows we shall restrict ourselves to the positive sign on the right-hand side of equation (i), since the negative sign merely replaces {x, y, z) by (— x, — y, — z). Also when x, y, z are all positive, we shall refer to {x + 2/'^ -f s^-) as a " unity of positive integers ". Suppose that r = *• + 2/^ + z"^" is any unity of the order of '^ : we shall first prove the following inequalities, viz. : |a--2/^|, JT/^-^^-j, |^^-^-a;!^2/V(3r) (ii). For, if we write a-^- x — y*^, ^ = y^-z^\ and 7 = z^- — X, we see that the equation satisfied by x, y, z can be thrown into the form r(a^-h/3-^ + 7^)=2: so that we have a-^+/3^ + 7' = 2/ri and a + /3 + 7 = ]' From these two equations, assuming F to be constant, we find that the maxima and minima for each of a, /8, 7 are ± 2/V(3r) ; from which the truth of the statement (ii) follows immediately. Further, we have the fact that if F = a; + 2/^ + z^" is any unity of the order and F> 1, then x, y, z will be positive. For, since F> 1, we have the inequalities \x-y'^\,\y'^-z'h^\,\z'^^-x\< 2/V3 < US. But, r being positive, the only possibilities of negative signs occurring amongst x, y, z are either (a) one negative and two positive or (6) two negative and one positive ; and in each case two of the inequalities given would take the form I X, + yU<^ I < 115, where \ and fi are positive integers and <^ ^ \/2, which is obviously impossible, except in the trivial case of one or more of the quantities x, y, z vanishing : it will be seen, on examining the inequalities, that the only possibility is x = l, y=0, z = 0, which gives r = l and so is excluded. Hence x, y, z must be positive. From this 114 Mr Todd, A particular case of we can easily shew that if there exists an}^ unity in the order other than + 1, then there exists a unity of positive integers other than + 1 of which any other unity of positive integers is- a positive integral power. For suppose that T is any unity of the order other than + 1 : then by definition of a unity it follows that the three numbers -r, i/r, -i/r, will be unities of the order also : and of these four it is plain that one will be positive and greater than 1, i.e. it will be a unity of positive integers. Now take any number k> 1; then there will be only a finite number of F's for which k >T >1, since for any such F we must have kXoO, K>y>0, k> z>0. Hence there must be a unity of positive integers which is greater than + 1 and less than any other ; let this one be 7. Suppose that F is any unity of positive integers which is, if possible, not a positive integral power of 7. Then we shall have F >,7, so that we can assume that F is intermediate in magnitude between 7^ and 7^+S where p is some positive integer. But by the last part of Dirichlet's Theorem we know that F/7^ - is also a unity of the order, i.e. we have found a unity of the order which is less than 7 and greater than + 1, which contradicts the assumption that 7 was the least unity greater than + 1. Hence F must be a positive integral power of 7. Finally Ave have the result that, if F is any unity of the order, it can be expressed in the form where 7 has its previous significance and jo is any positive or negative integer or zero. For if F is any unity of the order, other than + 1, the numbers -F, 1/F, -1/F also will be unities, and one of these will be positive and greater than 1, and so will be expressible in the form where g- is a positive integer. Hence F can be expressed in the form where p is some positive or negative integer or zero. The result obtained can be put into an interesting geometrical form as we shall proceed to shew. a theorem of Dirichlet 115 It is evident that any rational point (*•, y, z) in space of three dimensions can be regarded as being determined by its affix V^x-]ry^ + z^'\ where ^ is the real root of the equation ^^ = n : also the affix of any point determines a plane through that point and parallel to the asymptotic plane of the surface whose equation is i\=(jc? -\- ny"^ + n"z^ — Snxyz = 1 ; such a plane we shall call a " r-plane ". We shall now prove the following proposition : The V-planes of any two consecutive integj^al points on the surface A = 1, together with the surface itself, enclose a space of constant volume. The equation A = 1 can be written in the form [x + y^ + 2^-| [{x - y'^y + (2/^ - z^^-'f + {z"^^ - xy^ = 2 ; so that the section by the F-plane of the point (^, rj, f) will be given by the equations x" + 2/-^2 ^ ,^^^.^2 _ ,^^y^ _ ^2^^. _ ^^.y ^ ijY (i) and a; + 2/^ + z"^'- = T. Evidently the quadric (i) and the surftxce A = 1 are cut in a common section by the F-plane of the point (^, 77, ^). It is this quadric that we shall now examine. If by any rotation of axes it becomes ax- + hy^ + 6'^^= 1, we shall have (from the usual properties of invariants) a4-6 + c=r(l+ 7i^ + ^-), \ ah + bc + ca = ^ P^- (1 + m^ + ^-), I abc = ; j so that the quadric is evidently a cylinder, and the direction of its axis is the line x = y^ = z^". Suppose that c = ; then the area of a right section of the cylinder will be -rr/^/iah) = |^/V3 (1 + w^ + ^^). But the angle between the normals to the right section and the F-plane is the same as the angle between the two lines and x = y/'^ = zl'^^; i.e., is cos-i {3^7(1 + n^ + ^-)} : 116 il/r Todd, A partictdar case of a theorem of Dirichlet hence the area of the section made by the F-plane will be 27r V(l + «^ + ^-)/3nV3 F. Now the perpendicular distance between two near F-planes, r and r + ST, is 8r/V(l + >i^ + ^-), and so the element of volume enclosed by these two planes and the surface A = 1 will be, to the first order, 27r ST 3nV3'^' Integrating this between the limits F = 7^+^ and F = 7^' (i.e. the F-planes of any two consecutive integral points), we find that the volume of the space enclosed is 27r log 7/3/1^3 ; and since this is independent of the integer p, our proposition is proved. a Mr Mordell, On Mr Ramanujans Empirical Expansions, etc. 117 On Mr Ravianujan's Empirical Expansions of Modular Functions. By L. J. Mordell, Birkbeck College, London. (Com- municated by Mr G. H. Hardy.) [Received 14 June 1917.] In his paper* "On Certain Arithmetical Functions" Mr Ramanujan has found empirically some very interesting results as to the expansions of functions which are practically modular functions. Thus putting (^X^{<o„ CO,) = r [(1 - r) (1 - rO (1 - r^) . ..? = S T {n) r-, he finds that T{mn) = T(m)T{n) (1) if m and n are prime to each other ; and also that 2 ^^ = Ul/(l-T(p)p-^+p^^-) (2), n=l "' where the product refers to the primes 2, 3, 5, 7 He also gives many other results similar to (2). My attention was directed to these results by Mr Hardy, and I have found that results of this kind are a simple consequence of the properties of modular functions. In the case above A (&)i, Wa) (r — e'"'^"", (o = coi/coj) is the well-known modular invariant (^f dimensions — 12 in co^, 0)2, which is unaltered by the substitutions of the homogeneous modular group defined by ft)/ = aoii + bco.2, (o./ = Cftji + dw2, where a, b, c, d are integers satisfying the condition ad —bc = l. Theorems such as T {mn) = T {m) T {n) had already been investigated by Dr Glaisher f for other functions ; but the theorems typified by equation (2) seem to be of a new type, and it is very remarkable that they should have been discovered empirically. The proof of Mr Ramanujan's formulae is as follows. Let f{(Oi, 6)2) be a modular^ form of dimensions ~ k in coj, co,, which is a relative invariant of the homogeneous modular gi'oup, so that /(fw/, w.^)l f{(o^, 0)2) is a constant independent of «i, (Oo,. * Transactions of the Cambridge Philosophical Society, vol. xxii. , no. ix. , 1916. t See, for example, his paper " The Arithmetical Functions P (m), Q (in), fi (;n) ", Quarterlij Journal of Mathematics, vol. xxxvii., p. 36. + For an elementary introduction to the modular functions, see Hurwitz, Mathematische Annalen, vol. 18, p. 520, 118 Mr Mordell, On M^- Ramanujan's Let also p be any prime number; then we may take (&)i, pa>^, («! + CDa, p(o<^ . . . (ft)i + ( jO — 1) fUo, J9&)y), ( ptWi , &),.) as the reduced substitutions of order j)- Then for many modular forms* it is well known that unities ^, fu, ^i, ••■, |>-i can be found so that is also a relative invariant of the modular group. This is also true of the quotient Q = (fi/fioy^Jwo), which is a modular function of co. Q is really an automorphic function whose fundamental polygon (putting (o = x + Ly)is that part of the upper CO plane bounded by the lines x = ±^ and external to the circle a;2 + 2/2= 1, but we reckon only half the boundary as belonging to the fundamental polygon. The only infinities of Q are given by the zeros of /(&)i, 0^2) = 0, and if these zeros are also zeros of the numerator of at least the same order as of the denominator, it follows that Q has no infinities in the fundamental polygon. Hence Q is a constant, so that (f)~Qf((Oi, &>.,). Suppose now that where ^1 = 1. Then becomes (^)" S's' |.^,r^/^e^-'VP and in the examples with which we are concerned all the terms will vanish, because of the summation in X, except those for which s = (mod p), and then the sum will become ^ pA, Hence we have Equating coefficients, we find, if s is prime to p, pAsp = Qp^Ag. * This fact is intimately connected with the transformation equations in the theory of the modular functions. We may note that it is often more convenient to select the reduced substitutions in different ways. Empirical Expansions of Modular Functions 119 Taking .9=1, pAp = Qp", so that -Agp^ AsAp... (3). If no restrictions are placed on s we find, by equating coefficients of rP', ^^s+^,Asp.= QAsp. From this Asp.-ApA,p + ^p''-'A. = (4). From equations (3) and (4), we can prove that A^n = A^An if m and n are prime to each other. For all we really have to shew is that, if 2? is a prime and s is prime to j), then ^,,^a = AifAp\. But from equation (4), we have Asp\+2 - ApAspK-hi + ^p"-^ Agp\ = 0, and Ap\+2 — ApAp\+i + ^j)''-^ A^k = (4a). Hence the theorem follows by induction, for if it is true for \ and X + 1 it is true for X, + 2. But it is true for X = and for X= 1 (equation 3): hence it holds universally. We notice also that equation (4a) is a linear difference equation of the second order with constant coefficients*. Hence, since A,= l, 1 + ApX + Ap-iX" 4 Ap^x^ + . . . = 1/(1 - ApX + ^p^-Kx-), from which, by putting x = l/jf, pS p2S p3S '^ ••■ / ^ p' p^^ J' Putting for p in succession the primes 2, 3, 5 ..., multiplying together the corresponding equations, and remembering that A,nn = A^nAn if vi and /; are prime to each other, we have « where the product refers to the primes 2, 3, 5 — The simplest application of these results is given by the function fa {co„ &).3) = A ^— ft),, (Ooj This is obvious if we put fx^= ApK. 120 My^ Mordell, On Mr Ramanujan's where a is a divisor of 12. Its expansion in powers of r involves only positive integral powers of r and starts with I — j r. /a(&)i, 6)2) is not however an invariant of the modular group. We can avoid this difficulty by taking /(co,, eo,) = [A(ft)i, w^)]"'^'^. In this case* Kp-ni ^f(pco^,a),) =L(-1) ^ iyA(p&)i, &)2)J , provided we exclude p = 2 and p = S. Putting for the moment /m \" / "= —4-'! we find (—:) [iyA(a,„a,,)]- = S^^,ri-^^^ CO p-i OKpni ( a \ 2K7ri /■ «■ , „N 1 2 K = = 22 e"~6 S = 0/<: = But since ^ 4" ^ or 3, p' —1=0 (mod 12). Hence 2 eP i 12 +*;=o, K = unless a{\ —p-)/l2 + s = O(mod_p), that is a 4- 12s = (mod jd), and is then equal to p. Hence ^ is a power series in r^^^^ (really of the form ?-'^/i2(^ + Br+ Gr" ...)), starting with r'<"'+'^^s)i\ip^ where s is the smallest positive integer for which a + 12s = (mod p). Now the only zeros of /(wi, oa^) = in the fundamental polygon are at ft) = iX) or r = 0, and /(o)i, «2) = (^y '■'"' (1 + ^^' +^^ ...). But putting a = 1 2/6, so that h is an integer, a + ] 2s 1 + 6s 1 a 12p bp "6^12' since 1 + 6s = (mod p). Hence <^lf{ui^, Wa) is a constant, and equations (8), (4), (5) apply to the function V a We note also that |= (- l)«(^-i)/2. * Hurwitz, I.e., p. 572, or Weber, Lelirhuch der Algebra, vol. 3, p. 252 i «/12 Empirical Expansions of Modular Functions 121 When p = 2, these theorems hold if a = 4 or 12. For the functions ^\f\* are selected as before, and it is clear that the argument above applies, as « (1 —p^)l\2 is an integer. Lastly, when p = S, these theorems hold if a = 3, 6, 12, and the functions ^k/k* are selected as before. Hence, altering our notation, we have the following theorems. If a is a divisor of 12 and p 12 24 36 -j,,^ 00 r [(l -r") (l-r«) (l -r") .-..J^" = S fa(n)r'\ n = \ then fa {m)fa (n) =fa (mn) (6), if m and n are prime to each other ; and ^ />- (^) ^ n 1 /(i -^^ ^^^ I ^~ ^K^^''~' ] (7). n=i n' / V p' P~ The product refers to the primes 2, 3, 5, etc., except that p = 2 is excluded except when « = 4, 12, and jt) = 3 is excluded except when a = .3, 6, 12. We notice that when a — 1, 2, 3, or 6, jt? = 2 is not excluded as a factor of say m in (6), as in this case fai'ni) and fa (mn) are both zero. Similarly for jo = 3 when a = 1, 2, 4. The result (6) is given by Mr Ramanujan wh6n a = 12, as are most of the cases of (7). We shall now shew how in many cases we can find simple expressions for fa{p). If a = 1, it is known that, by a result due to Kulerf, r[(l-?-i2)(l-r'^^)...]2 = [ 2 (-If ?• 2 J — 00 (6ot + 1)2+(6w + 1)2 = SS(-l)'"+"r 2 = tt(-l)^r^'^^^\ where ^ = 3 (m + ?i) + 1, r) = n — m, so that ^, ij take all integer values satisfying ^ = 1 (mod 3), ^ + 77 = 1 (mod 2). Hence f(p) — 2(— 1)'' if p = ^^ + 9r}^ and we take both ^ and 77 to be positive. If ^ = — 1 or ±5 (mod 12), f{p) is obviously zero. This is Mr Ramanujan's result (118). If a = 2, it is known (Klein-Fricke, vol. 2, page 374) that r [(1 - r«) (1 - r'^) ...]* = ^2 (- 1)^ ^r^'+^^^+^r,^, where f , r) take all integer values satisfying 1=2 (mod 3), 7; = 1 (mod 2). * Hurwitz, I.e., vol. 18. t See also Klein-Fricke, Modulfunktionen, vol. 2, p. 374. VOL. XIX. PARTS II., III. 9 122 Mr Mordell, On Mr Ramanujans Hence /.(jt>) = 2 i(- 1)^| extended to the solutions of p= ^ + S^rj + Srj- for which ^ = 2 (mod 3), 17 = 1 (mod 2). This * can be written as /2 ( p) = ^v, where p = Su^ + v'-, u is positive and V = 1 (mod 3). Also f\ {p)=0 if p = - I (mod 3). This is Mr Ramanujans result (127). If a = 3 we have, from Klein-Fricke, vol. 2, page 377, r [(1 - r^) (1 - r«) ...]« = - 12 (p - 7)') rf=+''^ where ^ takes all even values and rj all odd values. Hence if p = ^^ + 7)"^, I is even, 97 is odd, and both f and ■?; are positive. Also /3( j9)= if ^ = 3 (mod 4). This is Mr Ramanujans result (123). If a = 4, then by Klein-Fricke, vol. 2, page 373, r \(l — r^) (1 — r^) . . .? = +2p r^""*"^^*''+^''', where ^, rj take all values for which |^ = 2 (mod 3). Hence /4 (^) = ^Sp extended to all the solutions of ^ = p + 3^77 + '67j\ where | = 2 (mod 3). Thisf can be written as /4 ( jd) = 2 (v" — 9vu-), where p = 3w'^ + ^^ tt is positive, and y = 1 (mod 3). This is Mr Ramanujans result (128). When a = 6,/6 {n) is known by means of the representations of n as a sum of four squares. Mr Ramanujan has overlooked the fact that in his result (159) 2c^ is —J\{p)' The theorem /b (wO/e in) =/6 (mn), is due to Dr Glaisher. When a = 12, we have Mr Ramanujan's results given as equations (1) and (2) in this paper. He also gives results when a = i , |. * When ^ is even put ^ = 2v, 7] = u-v, and when | is odd put ^ = du-v,'r) = v-7i. Both these cases are admissible, and we find that p=v^ + Bu" and v = l (mod 3). Also S {-1)^ ^=2v + 2v - {Su -v) - ( -3u-v) = Gv, where now w is taken as positive. t See the last footnote. In addition to the two cases there considered, 7/ even is admissible. Put then 7] = 2u, ^— -v -3ii, from which p = v^ + 3ii'^ and v = l (mod 3). Empirical Expansions of Modular Functions 123 where [- j and ( ] are symbols of quadratic reciprocity, so that (:-^) =(-l)V , (I) = 1 if i, EE ± 1 (mod 12), and (|) = - 1 if jj = ± 5 (mod 12). If /; = -S, ('^\ = 0. These are particular cases of Euler's theorem that s-^^^^=ni/(i--^^^) if the function y" satisfies the condition f{m7i)=f(m)f(n), the product refers to any group of primes, and the summation to all numbers whose prime factors are included in the group. Thus r (1 _ r2J)(l _ r'') ... = i (_i)»y.(««+i)^= v f'}\ ,,.«^ . -c»' 1,3, 5... V''^/ and r [(1 - ?•») (1 - r^^) ...]'= t (- 1) ^ wr^' = 5 f ^ nr"\ 1,3,5... 1,3,5... \ n J Finally, Mr Raman ujan gives two results, equations (155) and (162), of which the first is 5 -^ = lT9i=^ n 1/(1 - 2c, p-' + (- IV P'-n where Cp = u^ - (4w)- and u and y are the positive integers satis- fying u^ H- {^v)'=}f: But if JO = 3 (mod 4), Cp is taken to be zero, /lo (w) is defined * by =^r[(l-r^)(l-r^)(l-r«)...]"/[(l+r)(l-r-^)(l-|-r^)(i-r^)...h and this is equal tof ii 2(a,- + i2/)^r^'+^'. — 00 —00 The second result is I'^^^lT^^W-Sc^p-^+i'^-'X (1^ = 3,5...), " The functions /io(?i),/i6 (n) arise in iinding the number of representations of n as a sum of 10 and 16 squares respectively and the series 2 S (a; + i?/)-*r^""'"*' is well known in this connection. t From this, it follows that the result can be also proved as a particular case of Euler's product. 124 Mr Mordell, On Mr Ramanujan's Empirical Expansions, etc. where /le (?i) is defined* by 1 = r [(1 + r)(l-r2)(l +rO(l -r^) . ..]'/[(! -r"){l-7^){\-r') ...]«. Mr Ramanujan overlooks the fact that Cy = ^fu{p)- These results can be proved by aid of the principles used in finding equations (3) and (4). We should however have to consider now invariants of a sub-group of the modular group, and it seems hardly worth while to go into details. * The functions /lo ('O'/ieC**) arise in finding the number of representations of n as a sum of 10 and 16 squares respectively and the series 2 S [x + njY r* "^^^ is well known in this connection. PROCEEDINGS AT THE MEETINGS HELD DURING THE SESSION 1916—1917. ANNUAL GENERAL MEETING. October 30, 1916. In the Comparative Anatomy Lecture Room. Professor Newall, President, in the Chair. The following were elected Officers for the ensuing year : President: Dr Marr. Vice-Presidents : Dr Fenton. Prof. Eddington. Prof. Newall. Treasurer : Prof. Hobson. tSecretanes : Mr A. Wood. Mr G. H. Hardy. Mr H. H. Brindley. Other Members oj the Council : Dr Duckworth. Dr Crowther. Dr Bromwich. Dr Doncaster. Mr C. G. Lamb. Mr J. E. Purvis. Dr Shipley. Dr Arber. Prof. Biffen. Mr L. A. Borradaile. Mr W. H. Mills. Mr F. F. Blackman. 126 Proceedings at the Meetings. The following was elected an Associate of the Society : W. Morris Jones, Emmanuel College, The following Communications were made : 1. Methods of investigation in atmospheric electricity. By C. T. R. Wilson, M.A., Sidney Sussex College. 2. On the functions of the mouth parts of the Common Prawn, By L. A. BoRRADAiLE, M.A., Selwyn College. 3. On the growth of Daphne. By J. T. Saunders, M.A., Christ's College. 4. A self-recording electrometer for Atmospheric Electricity. By W. A. D. Budge, M.A., St John's College, 5. An axiom in Symbolic Logic. By C. E. Van Horn. (Com- municated by Mr G. H. Hardy.) 6. On the expression of a number in the form aar -i- hy- + cz^ 4- du". By S. Ramanujan, Trinity College. (Communicated by Mr G. H. Hardy.) 7. A reduction in the number of primitive propositions of Logic. By J. G. P. NicoD, Trinity College. (Communicated by Mr G. H. Hardy.) November 13, 1916. In the School of Agriculture. ^ Dr Marr, President, in the Chair. The following were elected Fellows of the Society : F. W, Green, M,A,, Jesus College, R, I, Lynch, M.A. The following was elected an Associate of the Society : N. Yamaga, Fitzwilliam Hall, The following Communications were made : 1. The surface law of heat loss in animals. By Professor Wood. 2. Inheritance of henny plumage in cocks. By Professor Punnett and Capt. P. G. Bailey. Proceedings at the Meetings. 127 3. On extra mammary glands and the reabsorption of milk sugar. By Dr Marshall and K. J. J. Mackenzie, M.A., Christ's College. 4. Experimental work on clover sickness. By A. Amos, M.A., Downing College. (Communicated by Professor BifFen.) 5. Bessel's functions of equal order and argument. By G. N. Watson, M.A., Trinity College. February 5, 1917. In the Sedgwick Museum. Dr Marr, President, in the Chair. The following was elected a Fellow of the Society : F. W. H. Oldham, B.A., Trinity College. The following Communications were made : 1. Submergence and glacial climates during the accumulation of the Cambridgeshire Pleistocene Deposits. By Dr Marr. 2. Glacial Phenomena near Bangor, North Wales. By P. Lake, M.A., St John's College. 3. The Cretaceous Faunas of New Zealand. By H. Woods, M.A., St John's College. 4. Exhibition of the Fruit of Chocho Sechium edule : remarkable in the Nat. Order Cucurbitaceae, native of the West Indies and culti- vated also in Madeira as a vegetable. By R. I. Lynch, M.A. 5. The limits of applicability of the Principle of Stationary Phase. By G. N. Watson, M.A., Trinity College. 6. The Direct Solution of the Quadratic and Cubic Binomial Congruences with Piime Moduli. By H. C. Pocklington, M.A., St John's College. 7. On the Hydrodynamics of Relativity. By C. E. Weather- burn, M.A., Trinity j(^ollege. 8. The Character of the Kinetic Potential in Electromagnetics. By R. Hargreaves, M.A., St John's, College. 9. On the Fifth Book of Euclid's Elements. (Fourth Paper.) By Dr M. J. M. Hill. 10. On a theorem of Mr G. Polya. By G. H. Hardy, M.A. Trinity College. 128 Proceedings at the Meetings. February 19, 1917. In the Botany School. Dr Marr, President, in the Chair. The followiug Communications were made : 1. (1) On an Australian specimen of Clepsydropsis. (2) Observations on the Evolution of Branching in the Ferns. By B. Sahni, B.A., Emmanuel College. (Communicated by Professor Seward.) 2. On some anatomical characters of coniferous wood and their value in classification. By C. P. Dutt, B.A., Queens' College. (Com- municated by Professor Seward.) CONTENTS. PAGE The Direct Solution of the Quadratic and Cubic Binomial Congruences tuith Prime Moduli. By H. C. Pocklington, M.A., St John's College .57 On a theorem of Mr G. Polya. By G. H. Hardy, M.A., Trinity College . 60 Submergence and glacial climates during the accumulation of the Cam- bridgeshire Pleistocene Deposits. By J. E. Maer, Sc.D., F.R.S., St John's College . . . . . . . . . .64 On the Hydrodynamics of Relativity. By G. E. Weatherburn, M.A. (Camb.), D.Sc. (Sydney), Ormond College, Parkville, Melbom'ne . 72 On the convergence of certain multiple series. By G. H. Hardy, M.A., Trinity College . . .86 Bessel functions of large order. By G. N. Watson, M.A., Trinity College 96 A particular case of a theorem of Dirichlet. By H. Todd, B.A., Pembroke College. (Communicated, with a prefatory note, by Mr H. T. J. Norton) Ill Oil Mr Ramanujan's Empirical Expansions of Modular Functions. Bj^ L. J. MoRDELL, Birkbeck College, London. (Communicated by Mr G. H. Hardy) . . .117 Proceedings at the Meetings held during the Session 1916 — 1917 . , 125 PKOCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY VOL. XIX. PART IV. [Michaelmas Term 1917— Easter Term 1918.] (iDambtitrgc : AT THE UNIVERSITY PRESS AND SOLD BY DEIGHTON, BELL & CO., LIMITED, AND BOWES & BOWES, CAMBRIDGE. CAMBRIDGE UNIVERSITY PRESS, C. F. CLAY, MANAGER, FETTER LANE, LONDON, E.G. 4 1918 Price Two Shillings and Sixpence Net July 1918. NOTICES. 1. Applications for complete sets of the first Seventeen Tolumes (in Parts) of the Transactions should be made to the Secretaries of the Society. 2. Separate copies of certain parts of Volumes I. — xi. of the Transactions may be had on application to Messrs BowES & Bowes or Messrs Deighton, Bell & Co., Limited, Cambridge. 3. Other volumes of the Transactions may be obtained at the following prices: Vol. xn. £1. 10s. 6d.; Vol. xm. £1. 2s. Qd.; Vol. XIV. £1. 17s. 6d ; Vol. xv. £1. 12s. ed. ; Vol. xvl £1. 10s. Od. ; Vol. xvn. £1. 2s. 6d.; Vol. xvin. £1. Is. Od.; Vol. xix. £1. 5s. Od. ; Vol. XX. £1. 10s. Od.; Vol. xxL £1. 14s. Od.; Vol. xxn. No. 1, Is. 6d.; No. 2, 2s.; No. 8, Is. 6d.; No. 4, Is. 6d.; No. 5, 2s.; No. 6, Is. 6d.; No, 7, 2s.; No. 8, 2s.; No. 9, 2s.; No. 10, Is.; No. 11, 2s.; No. 12, 3s. 6d. 4. Complete sets of the Proceedings, Volumes L— xviii., may also be obtained on application to the Secretaries of the Society. 5. Letters and Communications for the Society should be addressed to one of the Secretaries, Mr G. H. Hardy, Trinity College. [Mathematical.] Mr A. Wood, Emmanuel College. [Physical.] Mr H. H. Brindley, St John's College. [Biological.] 6. Presents for the Library of the Society should be ad- dressed to The Philosophical Library, New Museums, Cambridge. 7. Authors of papers are informed that the Illustrations and Diagrams are executed as far as possible by photographic "process " work, so drawings should be on a large scale and on smooth white Bristol board in Indian ink. 8. Members of the Society are requested to inform the Secretaries of any change of address. PROCEEDINGS OF THE Extensioits of Abel's 'Theorem and its converses. By Dr A. KiENAST, Kiisnacht, Zurich, Switzerland. (Communicated by Mr G. H. Hardy.) [Received 26 September 1917.] Introduction. Abel pruved in 1826 the theorem : n ''If liui !£ (^ ejists and is finite, then lim % a^x" = lim ^ a„." Let us write Sn 1 (Ik t""' _1 S6V n n ' ! ^(A + 1) _1 i (« n 1 / Then H(3lder* proved in 1882 Theorem 1. If lim t^^ exists and is finite, then .(1). lim S ai^x" = lim t]^ . * Bromwifh, Iiijinite scries, p. 313. VOL. XIX. PART IV. 10 130 Dr Kienast, Extensions of In 1897 Mr Tauber, and in 1900 Mr Pringshciui, published the following converse of Abel's theorem : Theorem 2. The two conditions lim 2 ««.'■" = I ( finite), 1 " hm - 2 KCi^ = n-*'Oo ^^ 1 are each necessary for the convergence ofX a^, i.e. for the existence of n lim ^a^ — l; n-*-ao 1 and, taken together, they are sufficient'^. In the present paper I replace the means (1) by ^ (2.) 1 « ?l 1 {n = 2, 3, . («. = X + 2, 0^ = 1,2,. {n = 2, 3, . •■) Defining r\'^ by ■) •) r;'+^^ = V - 7-(^) Ol = X + 1, X + 2, . . . ) I prove, in Part I, Theorem 3. Tlie tivo conditions •(3), lim S a^x" = I (finite), x-*l 1 Broniwjch, IntinlU series, p. 251. Abel's TJieurcDi and its converses 131 are each necessary for the existence of lim s^^^ = / ; (tnd, taken together, they are suMcient. This theorem includes the analogue of 1 : Theorem 4. If lim s\^ =1 exists and is finite, then lim 2 aK*" = I- x^\ 1 It is easy to verify that lim ij,^* = lim 6-J^* if \ = 1 or 2 : for higher values of X this relation certainly holds if both limits exist, as follows from Theorems 1 and 4. In Part II, I propose to extend Theorem 3 to certain other mean values; and Part III contains some general remarks about the converse of Abel's theorem. Part I. ]. In the researches which follow I have to make use of the following theorems. n Theorem 5. // lim 2 a« = lim s,,. = I (finite), and Uk is positive and n, lim 2 6k = lim t^ = ^ , 1 " then lim ~^b^s^= lim s„ = / (-1). This theorem is due to Stolz*. Theorem 6. Suppose that b^ is positive and 26« divergent; a)id let D be the region defined by p<2cosi|^ {\ylr\^\lr^<^'Tr), where i — x = pe'''^. Further suppose that tb,\x''\/\^b^x''\<G, where (f is a finite constant, for all values of x inside the region D. * Bromwich, Injinitc series, p. 378. 10—2 132 iJr Kienast, Extensions of Finally suppose that a^lbn tends to tlie lint it I luhen n tends to infinity. Then lim (S a^afjt b^,/f) = I (o), tuhen X approaches 1 along any path inside D. This theorem is due to Pringsheim*. It is to be supposed throughout this paper that, when x tends to 1, its approach to 1 is along some path inside B. Theorem 7. If the radius of convergence of F {x) = '!E a^x" is r, then lim anX'"' =0 (\x\< r). If the radius of convergence of Q(x) = Xa^x" is unity, it will I remain unchanged if Q(x) be transformed in any of the following ways : (i) by suppressing a limited number of terms, (ii) by multiplying by ./;°", a being an integer, (iii) by multiplying by ^ _ ^ , = ^ «'\ (iv) by integrating term by term, (v) by differentiating a limited number of times. Using in succession one or other of these operations, there result the following power-series, all with radius of convergence unity : xF' (x) — S Ka^x", \(x)= —A^F'{x)] = lrl'>x, X — X I F,(xy= ^ ^ 1 ~l '^ ) - Pj (x) dx \ =X r,^ x", 1 — X P.^ (x) dx Pg (x) dx = ir^^x'^\ * 4Qta Mathematica, vol. 28, p. 7. Abel's Theorem and its converses 133 Thus the series S r*^' , ,.r'^"^^~" converges if Lt I < 1 ; the same is the case with ^^ or i r(\+;\«,-'^^^-l J) + A — 1 Differentiating the last series (X — 2) times, we obtain SO^ + 1 )0; + 2) ... (^9 + X - 2)7-<^+;':;,a''^ ; which gives 00 Theorem 8. // the radius of convergence of P (a-) = 2 a^.r" is 1 unity, then for every \x\<\ r (A + k) n+A-2 r. hm ?■ , , , ./• = 0, /l+A — 1 ' hm - r ic = 0, lim (>i + 1 ) (7? + 2) . . . (n + \ - 2) r^+^^l^ «^" = 0. 2. The demonstration of Theorem 8 depends on certain identities. The formula ' n " leads, by successive summation, to the series of equations (6). (2) (1) 1 (2) n I) j^ /( ^(A) _ _(A-1I _ J- ^.(A) If lim s =1 exists, then, bv Theorem 5, lira s . also exists and is equal to /, and therefore one of these identities gives r 1 (A+i) r\ Inn - ?' = 0. Theorem 9. If lim .s^^' = I exists and is finite, then n-*-x r 1 (A + l( n. lim 7" = 0. 7l-»-X " Thus the second condition of Theorem 3 is necessaiy. 134 Dr Kievast, E.vtensioiis of 3. I proceed to prove some other identities. We have ,(1) i KQ K 1 a« = - r -/ U<i>_-,.(i' ! ,(^) ,(A-|-1) 1 K " M) Jl + 1 M .(7); and by successive substitution we find 1 n ' " (3) Q,„.,(3) , O/., 1\^(3) /„, 0\ ,,(3) " ' - n—l> n + l n + /2) +1|,.C2)_,,(2) } = (n + 1) C - 3m'- , + 3 (.^ - 1) r^' - {n - 2) ,-- ^ fl '■ n n-l> Writing , . (n + X - yu, - 2) r (\+k) ^ ^ M+A — M — 1 K.K.m .(8). we can easily verify that 6r, =tT>.i — (X — l)0. ,, \,K,n. \ + l,K,n ^ ■' A,K+l,w Moreover Developing a^ in this way we obtain, after a finite number p of steps, the formula cin= S A+K = P A, K A, K, n \^ iUp)_, .(p) .(10). The upper index of all the 7''s is the same throughout this expres- sion. For the present purpose it is not necessary to determine the coefiicients c^ ^, which are integers. In consequence of the definitions of s and r we have 7>) = (n>vi). Abel's Theorem and 'its converses 135 But it is not difficult to see that the recurrence formulae (8) and (9) still hold, if the number p of steps exceeds the index n. It is only necessary to put ?-^" = whenever n > m. The form of the relations (7), viz. is the cause why the coefficients of the remaining terms are not influenced by the fact that some terms disappear. Thus 'I'' A, K A, K, n 1 A+K=p \n=\ + S -a;^{7>'_7» I (11). n ' » n-V V ^ n=p "• To evaluate the first of these sums we have (A+k) (1 - xf s (« + 1 ) . . . {n + X - 2) r;;;;:, ^" n = \ in ,»H+1' A, (C, 7J \,K,Vl + V ' say. Each of the i\(\ + l) terms contained in the second sum has the form K{p + i)(p + 2)...{p + x-2)rl^;;!y+', {fi= V, V + 1, ...\; V = 1, 2, ...X; p = m — v + 1). Therefore, by Theorem 8, we have, for every j x \ < 1 and any finite X, limS G,^^^^.r"^(l-a^^{n + l){n+2)...{n+X-2)rl^^;;^^x'\ «4-*-» 1 1 The second sum in (11) gives '"- 1 . N , \ U-1 -\ vif,.(p)_^,.W }./= S r^' (p) n + 1 ;i -'^ ^ (p) m m «' = {\-x) S - - 7>' A'" + S ~ , 7>* X'' + i 7>^ .^'" p II + \ " p II (v + 1) " m '« (12); and again, by Theorem 8, we have, for every , x j < 1 and any finite p. lim % x' {r^'^ - 9>\} = (1 ^ X) 2 — ^ 7>\*'" + ^—^ m-» p *'' " ""' p» + l " ,n(n- + 1) " ,.(p) _.,."_ 136 Dr Kienast, Extensions of Thus we have established Theorem lO. //' l.a^x^ has unity as radius of convergence, then S ancc" = t c. 1 A+K=p ' 00 M = l .(f^) ^," 4. V 1 ,,(P) ,..« (13) 4. Equation (13) has now to be considered when .r->l. To the first terms on the right-hand side we apply Theorem 6, which gives 00 2(n + l)...(n + x-2)r;;)^_^^-'^ ^» hm . = lim -Jl±h=l_ , T^^— jyi 2 (/i + 1) . . . (h +X- 1)^'* Again, by Theorem 6, CO 1 -. lim (1 - ^;) 2 -4t ^'I^^'" = lim - r^"* : and finally Theorem 8 gives Theorem 11. If ta^x" has unity as radius of convergence, and if \im~rl^^ = 0, J » 00 1 then hm 2 a^ x" = lim 2 - j-^"' x" x^l 1 0,-^! p «(/?+!) " 5. Furthermore, equations (7) and (12) lead to «'' 1 , , , , m-l n '"•"1 1 m—l 1 1 P w + 1 " p 7z(w+l) ■" ■ ^m ^'^ •'' • Putting X = 1, it follows that -2 ^ 7^^'"^= 2 L^,>+i» , 1 ,.(p+i) Abel's Theorem and its converses 1:17 Hence Theorem 12. //' Xok-"''" Iki-'^ <inity as f(((Iiiis of convergence, and if 1 then .(p) '^^ 1 r " = lini 2, ,(p+i) I Another identity is acquired by developing s\f (;? = p + l^ /) + 2, ...) in the form P + 1 P+2 >)_JP) 1 j(P-l) pVi p+2 1 ^(p-1) 1 V.*"-^' V/i(?W — 1) I "-1 1 ( \ VI (m + 1) (p) til en Theorem 13. If .s"' and r ''' are defined as in (2) and (3) S^'^= S .(P) If lim ,9''^* = I exists and is finite, then, by Theorem 5, II ^x r ('^+1) 7. hm s = I : II ll->-orj and by Theorem 13 S 1 .(A+i) = /. .(^+1) Therefore by Abel's theorem TV -'- (A + 1) III V ■'- inn 2. - "Tvr ,r — z — 7 Ts x^i A+i ?H (?/i + 1 ) '" A+i m (m + \) '" On the same assumption, Theorem 9 gives lim -■^•^^"^ ■ =0; and therefore Theorem 1 1 gives = /. lim 2 n.^x" — lim 2 ^,(A+l) « 1X+1«(" + 1) " " 7 1^8 Dr Kienast, Extensions of Thus we obtain Theorem 4. If Km s\';^ = I exists and is finite, th en Mni^a^x" = I. X-»-l 1 The first condition of Theorem 3 is therefore necessary. 6. To demonstrate the rest of the assertion in Theorem 8, it follows from the hypothesis lim \/^^'^ = that Theorem 11 is applicable. Thus the assumptions are transformed into lim 2 - ^rl^+'\-=.i lim-7^;^+^'=0. 71-*- 00 ^i From this last equation follows I '' 1 <>.^^^ .. 1 .(A+l) lim:^S -^r(^+^'=lim^^^ = ,,-^00 nx+lK + l « ,,^^ 71 + 1 Hence Theorem 2 can be applied to the series 2 - x'' • 11 , • • n{n + \) ' and the conclusion is that lim t -r~~^, = I. Theorems 12 and 13 now yield lim 4^^=/, with which the proof of Theorem 3 is completed. 7. The foregoing deductions are valid for X = l, 2,.... For X = they still hold, except those in § 6. This case requires the proof of the following special case of theorem 2 : Theorem 14. // limS^-i--r^V = /, ^^1 1 n(n + 1) « lim ~r^'^ = I Abel's Theorem and its converses 180 then r'''=l. 1 n(n+l) This proof is actually given by Mr Tanber, and is therefore the basis of the theorems of this paper. Part II. 8. Let 6k denote the terms of an infinite sequence of positive real numbers, which have the properties (1) lim S b^ = lim tn = oo .(14), (2) n 1 tends to a limit or oscillates between finite limits. Then Theorem 16. The two conditions QO lim S «'««"' = / (finite), x^l 1 1 » hm --l^AaA = 0, n-*-cc I'll 1 are each necessary for the convergence of S a«, i.e. for the existence of II lim %aK= I : and, taken together, they are sufficient. Abel's theorem states that the first of these conditions is necessary. If lim Sn = I, then lim a„ = 0, and by Theorem -5 1 " lim - S 6a.?a-i = lim ii^X ^'11 1 n 1 n --2&ASA--S6AfO '« 1 f»i. 1 /. The identity 1 " 1 " f ft 2 f« 1 n(^w gives, as a consequence of lim ,<?„ = /, 1 Therefore the second condition is necessary too. nm ^ uxu\ ^^^ Dr Kienast, Extensions of 9. To prove the converse, we require two identities. If J^ hi n we have X a.af =f ^ + 2 - |^« - p,_,] x^ / — 1 ''K + l '«+r ^« a' w-1 1 ^K+l 1 t^ ^K + i Putting a; = 1, this gives the identity .(15). _\^t<±i-t^ Pk . p L t ^f ''K+i hi •(16). If we suppose lim ^ = 0, it follows that lim^.r'»=0 for every I .t | < 1 ; and, by Theorem 6, lim(l-^)i^,,- = 0. Now passing in (15) to the limit (first n -^ oo and then a?— » 1 we find that if lim -^'=0, then lim 2 o„ a;" = lim 2 ""^^ -^JL.^" n^j) Theorem 15 starts from the assumptions lim ^ = 0, n-^-x I'll CO lim S Ui^x" = I. x^\ 1 The first assumption shows that (17) is available ; and this equation gives, with the second assumption, hm z ~ -i-^ a* = /. x-^\ 1 t. L ,, Abels Theorem and its converses 141 NoAv Theorem 2 can be applied to the series S --^ — - j-^ m^", provided that hiu -1,K — , — J = (lo). Assuming for a moment that this condition is satisfied, Theorem 2 leads to lim ^i<±}-zi<l^ = i. Il^-X I ix *K+l and (16) gives finally lim Sn = I, proving the theorem, which is the analogue to Theorem 2. Condition (18) depends on the 6's as well as on the as; but since lim J-^ = 0. it will certainly be fulfilled when -Zk -— n I Ik tends to a limit or oscillates finitely. For, e being given, we can choose K so that I ^ ^A+, -t^ Pk ^1 "^^ ^ tK+i -tK pK , e V ^ ^A+i - tK — ^ A. < - Zi /^ 7 i — A, - . n 1 Ik t\+i n \ Ik t^+i " < t^\ We may suppose, for example, that tn = ri''; \ogn; log log 7i;... 10. Adding to the notations used hitherto 1^7 (1) / .1 A A-l 71 ' ^lbj^, = sf, '"n 3 S Ox ^ = CJn , 2 ^A-l and restricting the choice of the numbers b^ not only as done in 8, but further by supposing that the two limits lin, L ^1 , lim h+ip^h+^ ^1 (19) 142 Br Kienufit, Eidenaioiiii of shall exist, or at any rate that the functions under the limit sign shall oscillate finitely, I procffcd to prove Theorem 16. TJie two cunditions CO lim 2 ««*■" = I (Jinite), x^l 1 M-S-OG hi 2 fA— 1 are each necessary for the existence of the limit lim s^'^ = l: n mid, taken together, they are sufficient. It is not possible to demonstrate this theorem for every set of numbers 6^. The following example shows this. Mr Riesz has pointed out* that 1 'i, 1 hm ^ - Sk WH-Qo log n 1 K exists and is finite in the case of However, Abel's limit lim 2 Kr''^~'^^x'^ x^\ 1 does not exist, as the function behaves like r(«)(iogi)"' when *■— »1. ].l. The demonstration depends on some identities analogous to those employed in the case of the arithmetic means, viz. n ^n - 1 Pi ! (2) H 11 - 1 tn n t 2 h. tK- * See G. H. Hardy, ' Slowly oscillating series', Proc. London Math. Soc, ser. 2, vol. 8 (1910), p. 310. Abel's Theui'em cuid its converses 143 which scries of relations might be continued. They show (in con- junction with Theorem 5) that lim s' = / whenever lim s — /, from which we deduce Theorem 17. //" lim s|J' = I exists <incl is finite, then lim^ii,^ = 0. Thus the second condition of Theorem 16 is necessary. 12. We have also 7 Pn—1 '^n—i and thus V ,, ...K _ ^ ' h + ^ 'i" ,.K _ ^ ^« + l '/« " ^A-l „.K 1 1 Ok+1 2 fx f^K II -\ In. 1 i«- = (1-^) V^!^±LZLil,,,K_i^ ^K + l ^«.+ l ^, #« - ^«_i ^« - (y«_i + Z ^ — T-?— w'' ■1 t^ b^ + 'Jn+i'^ ^n-^ ^n+\ bn+i [ 1 K+2 I t>«+l &«+2 ^J + 1 h ^ ^ i h * • "n+1 "n+i 2 f/f Ox Now the series has a radius of convergence at least as great as 1, since lim -=^ = f?t— 1 and =■ -^ tends to a limit or oscillates finitely. Thus n-1 ba •' lira p X'' = 144 Dr Kieiiad, E.dennwns of for every x < 1 , arul therefore i a.,- = {i-xfi p" ^« + (1 - ^) V ^±LZL^J ^ ^ Taking account of the conditions (19), it follows from Theorem 6 that lim(l -xf%p'a:- = 0, x^-\ 1 Ok +2 and Iini (1 - x)t ^tir_^«+> f+1 ,.. = 0, X^\ 1 Ok-I-1 0^+2 SO that lim % a^x" = lim S ^ "^^ ~ ^^ a;'^^^ (20). 13. Lastly we have the identity 14 If lim 6-|/^ = ^ exists and is finite, then, by (21), i ^"^' ""^^^ 1 ^K + l converges to the sum I. Therefore by Abel's theorem and since (Theorem 17) lim ^ = 0, equation (20) is valid, and thus lim 1a^x'' = l. x^l 1 We have therefore Theorem 18. Let the coefficients h^ be chosen so as to satisfy the conditions (19). Then, if lim 6'J/* = I exists and is finite. lira 2 a^x" = I. x^\ 1 The hrst condition of Theorem 16 is consequently necessary. Abel's Theorem and its converses 145 15. The proof of the converse begins with equation (20), which is valid since lini ~ = 0. Therefore X^-l 1 f/C + 1 This is equivalent to the first condition of Theorem 15. But the second is satisfied too, viz. lini yit. ^A^i^i ^ n„, 1 [-^^^^ -())+...+ (fy, - r/„_,)] «-*-x hi 2 tie M-*-c» f)i = lim '^ = 0. Thus lim S '^'^^'~'^'' = ^. and, by equation (21), lim s[y = I, which completes the demonstration. The conditions (14) and (19) imposed on the numbers b^ are not necessary but only sufficient. The conditions necessary and sufficient would depend also on the coefficients «« of the power series considered, so that for a given series 'S^a^x" a given set 6^ )nay be admitted which must be excluded for other series ^CkX". Part III. 16. Theorem 2 is in a sense a perfect converse of Abel's theorem, from which all these researches originated. Series for which Abel's limit exists may be divided into two classes, those which are convergent and those which are divergent, series for which the limit does not exist being excluded. Theorem 2 shows that the first class consists of those, and those only, which satisfy the condition lim -i«ct, = (22). The second class consists of those, and those only, which do not satisfy the condition. The condition (22) is satisfied, in particular, if lim nau = (23) VOL. XIX. PAliT IV. 11 146 Dr Kienast, Extensions of But this condition, unlike (22), is not a necessary condition for convergence. Recent investigators have generalised the condition (23) in a different manner. Thus Mr J. E. Littlewood proved* the theorem : " 2 a^ IS convergent, provided lim 2 a^x^" = A and 1 na„ \ < K." 1 x^l 1 And still more recently Mr G. H. Hardy and Mr J. E. Littlewood f proved Theorem 19. If lim 2 a,,x'^ = A , and a„ >--K, then 2 a„ x^i n converges to the sum A. But however interesting in themselves these two theorems and their proofs may be they are less perfect than Theorem 2. For the conditions j na,, j < K and na,, > - ^ are neither necessary for convergence nor is either, together with \imta,x'' = A, necessary, nor do they characterise the non-converging series for which Abel's limit exists. Their interest is in fact of a quite different character from that of Theorem 2. It is not difficult to state similar theorems which are open to the same objection but which give information in cases where the last two theorems fail. 17. The terms a^ of any sequence can be written in the form «« = ^ » where t^ is subject to the same conditions as in Theorem 15. This theorem then shows that CO ^ 00 ■ T " ^ -^ is convergent, provided lim 2 ^- a;" = ^ and lim -2c = " 1 '''' x^l 1 ^K ,i-*oo tn 1 " Now the second condition is certainly satisfied if lim 2 c« tends to a limit or oscillates finitely. The only limitation'thus imposed upon the order of magnitude of a« is that \c^\<K, i.e. that the order of /c | a« | does not exceed that of ^ . Instead of the condition T */' ,^- ^ittlewood, 'The converse of Abel's Theorem on power-series', Proc London Math. Sac, ser. 2, vol. 9 (1911), p. 438. . -^ '<^c. _t G.H^ Hardy and J. E. Littlewood, ' Tauberian theorems concerning, power- series and Dirichlet s series whose coeflicients are ijositive ', Proc. London Math '^nr ser. 2, vol. 13 (1914) p. 188. See also E. LanJau, I)^.^./..,;"^ ^^'^^^^ ciniger neuerer ErgebniHse der Funktionentheorie. (Berlin, 1916) pn 45 etsea ■ ihl actual theorem is stated in § 9 and finally proved in § 10 (Die HardyiLittlewoodsche Umkehrung des Abelschen Stetigkeitssatzes). Abel's Theorem and its converses 147 ««K > — A" of Theorem 19 we have \X t^a,, < K, a condition which I 1 allows Kti^ to tend to infinity in either direction. That such cases exist, in which S «« is convergent, is shown by the fact that t-— (0<e<</)<27r-e) is convergent if t^ is any function of k which tends steadily to infinity with k. 18. A similar result can be obtained from another theorem of Messrs Hardy and Littlewood, viz. : Theorem 20. If f{x) = S a^af is a potver series with positive coefficients, luul f{x)^ ^ as x^\, then n 2 a« ~ /I. * 1 From this theorem it is possible to deduce Theorem 19 (see above) of the same authors. Now the hypothesis is equivalent to CO lim (1 — x) f{x) = lim % (a^ — a^-i) x" = 1, and the conclusion is lim -^a^= hm - S J 2 (a^ - «a-i) h = 1. M-*M '>l 1 M-*.0O ^ 1 ( 1 j Thus Theorem 20 is equivalent to CO n Theorem 21. If limXb^x" = 1, and if the sums «„ = ^ 6« x-*l 1 1 are alt positive, then 1 'i lim - 2 6'k = 1. »-*-oo 11 1 Here again is a condition which, in case the series converges, does not prevent the real numbers Kb^ from tending to infinity in both directions. * G. H. Hardy and J. E. Littlewood, I.e. See also E. Landau, I.e., § 9. 11 — 2 148 Mr Hardy, Sir George Stokes and the Sir George Stokes and the concept of uniform, convergence. Bv G. H. Hardy, M.A., Trinity College. [Received 1 Jan. 1918. Read 4 Feb. 1918.] 1. The discovery of the notion of uniform convergence is generally and rightly attributed to Weierstrass, Stokes, and Seidel. The idea is present implicitly in Abel's proof of his celebrated theorem on the continuity of power series ; but the three mathe- maticians mentioned were the first to recognise it explicitly and formulate it in general terms*. Their work was quite independent, and it would be generally agreed that the debt which mathematics owes to each of them is in no way diminished by any anticipation on the part of the others. Each, as it happens, has some special claim to recognition. Weierstrass's discovery was the earliest, and he alone fully realised its far-reaching importance as one of the fundamental ideas of analysis. Stokes has the actual priority of publication ; and Seidel's work is but a year later and, while narrower in its scope than that of Stokes, is even sharper and clearer. My object in writing this note is to call attention to and, so far as I can, explain tw^o puzzling features in the justly famous memoir-f- in which Stokes announces his discovery. The memoir is remarkable in many respects, containing a general discussion of the possible modes of convergence, both of series and of integi'als, far in advance of the current ideas of the time. It contains also two serious mistakes, mistakes which seem at first sight almost inexplicable on the part of a mathematician of so much originality and penetration. The first mistake is one of omission. It does not seem to have occurred to Stokes that his discovery had any bearing whatever on the question of term by term integration of an infinite series. The same criticism, it is true, may be made of Seidel's paper. But Seidel is merely silent on the subject. Stokes, on the other hand, quotes the false theorem that a convergent series may always be integrated term by term, and refers, apparently with approval, to the erroneous proof offered by Cauchy and Moignoj. Of this there is, I think, a fairly simple and indeed a double * The idea was rediscovered by Cauchy, five or six years after tlie publication of the work of Stokes and Seidel. See Pringsheim, ' Grundlageu der allgemeineu Funktionenlehre ', Encyld. der Math. Wiss., II A 1, §17, p. 35. t ' On the critical values of the sums of periodic series', Trans. Canib. Phil. Soc, vol. 8, 1847, pp. 533-583 (Mathematical and physical papers, vol. 1, pp. 236-313). X See p. 2-42 of Stokes's memoir (as printed in the collected papers). concept of uniform convergence • 149 explanation. In the first place it must be remembered that Stokes was primarily a mathematical physicist. He was also a most acute pure mathematician ; but he approached pure mathematics in the spirit in which a physicist approaches natural phenomena, not looking for difficulties, but trying to explain those which forced themselves upon his attention. The difficulties connected with continuity and discontinuity are of this character. The theorem that a convergent series of continuous functions has necessarily a continuous sum is one whose falsity is open and aggressive : examples to the contrary obtrude themselves on analyst and physicist alike. The falsity of this theorem Stokes therefore observed and corrected. The falsity of the corresponding theorem concerning integration lies somewhat deeper. It is easy enough, when one's attention has been called to it, to see that the proof of Cauchy and Moigno is invalid. But there are no particularly obvious examples to the contrary : simple and natural examples are indeed somewhat difficult to construct*. And Stokes, his suspicions never having been excited, seems to have accepted the false theorem without examination or reflection. This is half the explanation. The second half, I think, lies in the distinctions between different modes of uniform convergence which I shall consider in a moment. Stokes's second mistake is more obvious and striking. He proves, quite accurately, that uniform convergence implies con- tinuity f. He then enunciates and otfers a proof ;J; of the converse theorem, which is false. The error is not one merely of haste or inattention. The argument is as explicit and as clearly stated in one case as in the other ; and, up to the last sentence, it is perfectly correct. He proves that continuity involves something, and then states, without further argument, that this something is what he has just defined as uniform convergence. It is merely this last statement that is false. Stokes's mistake seems at first sight so palpable that I was for some time quite at a loss to imagine how he could have made it. A closer examination of his memoir, and a comparison of his work with other work of a very much later date, has made the lapse a good deal more intelligible to me ; and my attempts to understand it have led me to a number of remarks which, although they contain very little that is really novel, are, I think, of some historical and intrinsic interest. 2. There are no less than seven different senses, all important, in which a series may be said to be uniformly convergent. * See Bromwich, Infinite sfrlea, pp. 110-118; Hardy, ' Notes on some points in the integral calculus', XL, Messentjer of Matlieniatica, vol. 44, 1915, pp. 145-149. t p. 282. I use ' uniform ' instead of Stokes's ' not infinitely slow '. X p. 283. 150 Afr Hardy, Sir George Stokes and the I shall write the series in the form 00 S tin (^f^) ; 1 and I shall suppose, for simplicity, that every term of the series is continuous, and the series convergent, for every x of the interval a^x ^b. I shall denote the sum of the series by s (x) ; and I shall write Sn (OC) = II, (x) + Uo {x)+ ...■\- iln (.«), S (x) = Sn (x) + r,, (cc). The fundamental inequality in all my definitions will be of the t3^pe \rn(a!)'\^e (A), I shall refer to this inequality simply as (A). When we define uniform convergence, in one sense or another, we have to choose various numbers in a definite logical order, those which are chosen later being, in general, functions of those which are chosen before. I shall write each number in a form in which all the arguments of which it is a function appear explicitly : thus no (^, e) is a function of ^ and e, Uo (e) one of e alone. It will sometimes happen that one of the later numbers depends upon several earlier numbers already connected hy functional rela- tions, so that it is really a function of a selection of these numbers only. Thus h may have been determined as a function of e ; and ??o niay have to be determined as a function of ^, e, and h, so that it is in reality a function of ^ and e only. I shall express this by writing «o = ^2o(^, e, S) = no(|^, e); and I shall use a similar notation in other cases of the same kind. 3. The first three senses of uniform convergence are as follows. A 1 : Uniform convergence throughout an interval. The series is said to he uniformly convergent throughout the interval (a, b) if to every positive e corresponds a.n no (e) such that (A) is true^for n ^ Wo (e) and a^x^^b. This is the ordinary or ' classical ', and most important, sense, the sense in which uniform convergence is defined in every treatise on the theory of series. A 2 : Uniform convergence in the neighbourhood of a point. The series is said to be uniformly convergent in the neighbourhood of the point ^ of the interval (a, b) if an interval (f — 8 (I), ^ -\-B (^))* can be found throughout luhich it is uniformly convergent ; that is to say %f a positive 8{^) exists such that (A) is true for every positive e, for n ^ n^ (^, S, e) = ??o (?> e), and for * A trivial change is of course required in the definition if t = « or ^ = b. The same point naturally arises in the later definitions. concept of mil form convergence 151 A3: Uniform convergence at a point. The series is said to be uniformly convergent at the point x = f {or for x = ^) if to every positive e correspond a positive S (^, e) and an »o(|, €, B) = nQ(^, e) such that (A) is true for n ^n^{^, e) and for ^-S(^,e)^x^^+S(^,e). 4. Before proceeding further it will be well to make a few remarks concerning these definitions and their relations to one another. The idea of uniform convergence in the neighbourhood of a particular point (Definition A 2) is substantially that defined by Seidel in 1848*. It is clear, however, that definitions A 1 and A 2 were both familiar to Weierstrass as early as 1841 or 1842f. It is obvious that a series uniformly convergent throughout an interval is uniformly convergent in the neighbourhood of every point of the interval. The converse theorem is important and by no means obvious, and was first proved by Weierstrass | in a memoir published in 1880. This theorem would now be proved by a simple application of the ' Heine-Borel Theorem ', and is a par- ticular case of a theorem which will be referred to in a moment. Definition A3 appears first, in the form in which I state it, in a paper of W. H. Young published in 1903§; but the idea is present in an earlier paper of Osgood ||. The essential difference between definitions A 2 and A 3 is that in the latter S is chosen after e and is a function of ^ and e, while in the former it is chosen before e and is a function of f alone. In each case n^ is a function of two independent variables, ^ and e. It is plain that uniform convergence in the neighbourhood of ^ involves uniform conver- gence at ^, and at (and indeed in the neighbourhood of) all points sufficiently near to ^. But uniform convergence at ^ does not involve uniform convergence in the neighbourhood of |. It is important, however, to observe that uniforni convergence at every point of an interval involves uniform convergence throughout tJie interval. This important theorem is proved very simply by * ' Note iiber eine Eigensehaft der Reihen, welche discontinuirliche Functionen darstellen', Munchener Ahliandlungen, vol. 7, 1848, pp. 381-394. This memoir has been reprinted in Ostwald's Klassiker der e.vakten Wisscmchaften, no. IK!. The reference tliere given to vol. 5, 1847, is incorrect. (■ For detailed references bearing on this and similar historical points, see Pringsheim's article already qnoted. X See the memoir 'Zur Functionenlehre ' {Ahliandlungen aus der Funktionen- lehre, pp. 69-104 (pp. 71-72)). § 'On non-uniform convergence and term-by-term integration of series', Proc. London Math. Soc, ser. 2, vol. 1, pp. 89-102. II 'Non-uniform convergence and the integration of series', American Journal of Math., vol. 19, 1897, pp. 155-190. See Prof. Young's remarks on this point at the beginning of his later paper ' On uniform and non-uniform convergence of a series of continuous functions and the distinction of right and left ', Proc. London Math. Soc, ser. 2, vol. 6, 1907, pp. 29-51. 152 Mr Hardy, Sir George Stokes and the Young, in his paper already quoted, by means of the Heine-Borel Theorem * ; and it plainly includes, as a particular case, Weierstrass's theorem referred to above. 5. It seems to me that the definition given by Stokes is not any one of A 1 , A 2, A 3 ; and that, if we are to understand him rightly, we must consider another parallel group of definitions. These definitions differ from those given above in that (A) is supposed to be satisfied, not for all sufficiently large values of n, but only for an infinity o/ values. B 1 : Quasi-uniform convergence throughout an interval. The series is said to he quasi-uniformly convergent tlvroughout (a, h) if to every positive e and every N corresponds an n^ (e, N) greater than N and such that (A) is true for n = n^ (e, N) and a^x^b. B 2 : Quasi-uniform convergence in the neighbourhood of a point. The series is said to be quasi-uniformly convergent in the neighbourhood of f if an interval (^ — 8(f), | + S(f)) can be found throughout which it is quasi-uniformly convergent ; i.e., if a positive 8(f) exists such that (A) is true for every positive e, every N, an «o (f . 8, e, iV) = ?io (f , e. ^) greater than N, and f — 8 (f ) ^ .'c ^ f + 5 (f ). B3: Quasi-uniform convergence at a point, llie series is said to be quasi-uniformly convergent for iV = ^ if to every positive € and every N correspond a positive S (f, e, N) and an no{^,e,8,N) = n,(^,e,N), greater than N, such that (A) is true for n = ??o (!> f> N') and for Definition B 1 is to be attributed to Dini or to Darboux+. Another form of it has been given by Hobson|. As Arzela and Hobson§ have pointed out, a series is quasi-uniformly convergent throughout an interval if, and only if, it can be made uniformly convergent by an appropriate bracketing of its terms. Definition B 2 is for us at the moment of peculiar interest, for (as I shall show in a moment) it is really this definition that is given by Stokes. . Definition B 3 is also of great interest, both in itself and in * Choose € and determine 5 (^, e) and n^ (|, e), as in definition A3, for every f of the interval. Every point of {a, b) is included in an interval {^-d, ^ + o). By the Heine-Borel Theorem, every point of (a, b) is included in one or other of a finite sub-set of these intervals. If N (e) is the largest of the Hq's corresponding to each of the intervals of this finite sub-set, then (A) is true for n^N and a ^ .r ^ 6. This is the essence of the proof, though, like all proofs of the same character, it requires a somewhat more careful statement if all apj^earance of dfpendence upon Zermelo's AusicahUprinzip is to be avoided. t See Pringsheim, I. c. + ' On modes of convergence of an infinite series of functions of a real variable', Proc. London Math. Sac, ser. 2, vol. 1, 1903, pp. 373-387. Hobson (following Dini) uses the expression ' simply uniformly'. § L. c, p. 375. concept of uniform convergence 153 relation to Stokes's memoir. For the necessary and sufficient con- dition that s (x) should he continuous for x=^ is that the series should be quasi-uniformly convergent for x — ^. This theorem is in substance due to Dini*. I give the proof, as it is essential for the criticism of Stokes's memoir. (1) The condition is siificient. For I s {x) - s (I) I ^ { Sn {x) - Sn (|) ' + | r„ {x) j + | r^ (f) |. Choose e, N, S (^, e, N), and n = ??o {^, e, N) as in definition B 3. Then [ r„ {x) I < e for ^—Z^x^^ + h. Now that n is fixed we can choose Si less than 8 and such that [ s,i {x) — «» (^) j < e for ^ — Si ^ a" ^ ^ + Sj. And thus |s(.«)-s(f)i<3e for ^ — §1 ^ .« ^ ^ + Si , so that 5 {x) is continuous for a? = f . It is plain that this argument proves, a fortiori, that A 2, A 3, and B 2 all furnish sufficient conditions for continuity at a point, and A 1 and B 1 sufficient conditions for continuity throughout an interval. (2) The condition is necessary. For I rn {x) \^\S {x) - S (^) I + ! Vn (f ) | + ! S„ {x) - 5„ (|) |. Suppose that e and N are given. Then we can choose S (^, e) so that \s{x) — s{^)\<e for f — S ^ « ^ ^ + S, and n^ (|, e, iV ) so that Vq > N and j r^^ (|) | < e. And, when n^ has thus been fixed, we can choose S] (^, e, n^) = Sj (^, e, N) so that Si < S and 1 6'„^ {x) — Sn^ (f ) | < e for I — Si ^ .^■ ^ I + Si . Thus | r,i (^) j < 3e for n = no> N and ^ — Bi^X'^^ + 8i, so that the series is quasi-uniformly convergent for x=^. 6. If a series is uniformly convergent at every point ^ of an interval, it is (as we saw in § 4) uniformly convergent throughout the interval : definition A 3 (and a fortiori definition A 2) passes over, in virtue of the Heine-Borel Theorem, into definition A 1. It is important to observe that this relation does not hold between B 3 (or B 2) and B 1 : a series quasi-uniformly convergent at every point of an interval (or in the neighbourhood of every such point) is not necessarily quasi-uniformly convergent throughout the interval. We can apply the Heine-Borel Theorem in the manner indicated in the first sentences of the footnote * to p. 152 ; but the last stage of the argument, in which every one of a finite number of difterent integers is replaced by the largest of them, fails. What we obtain is the necessary and sufficient condition that s {x) shoidd he continuous throughout the interval ; and this is not ^' Foiulaiiii')iti..., p. 107 ((jerinan translation, GruiuUa(ii'ii...,p\). 143-145). 154 Mr Hardy, Sir Georr/e Stokes and the the condition B 1 but a condition first foi-mulated by Arzela*, VIZ. C: Quasi-uniform convergence by intervals {convergenza uniforme a tratti). ^ The series is said to he quasi- uniformly con- vergent by intervals if to every positive e and every N correspond a division of (a, h) into a finite number v (e, N) of intervals 8,. (e, N), and a corresponding number of numbers n,(e, Nj, all greater than A^, and such that (A) is true for ?? = 7?,.(?- = 1, % ...,v) and all values of X which belong to 8,.. The deduction of Arzela's criterion from B 3, in the manner sketched above, was first made by Hobsonf. There is one further point which seems worth noticing here, although it is not directly connected with Stokes's memoir. Dini J proved that if u^ (x) ^ for all values of n and x, and s (x) is con- tinuous throughout {a, b), then the series is uniformly convergent throughout (a, b). This theorem is now almost intuitive. For it is obvious that, for series of positive terms, quasi-uniform conver- gence in any one of the senses B 1, B 2, or B 3 involves uniform convergence in the corresponding sense A 1, A 2, or A 3. If then s {x) is continuous throughout (a, b) it is continuous for every f of (a, b) ; and therefore the series is quasi- uniformly convergent for every f ; and therefore uniformly convergent for every |; and therefore uniformly convergent throughout (a, b). 7. Let us now consider Stokes's definitions and proofs in the light of the preceding discussion. It is clear, in the first place, that Stokes has in his mind some phenomenon characteristic of a small, hit fixed, neighbourhood of a point. ' Let u^ -{-U.+ ... (66)', he says§, ' be a convergent infinite series havmg U for its sum. Let v, + v, ■]-... (Q7) be another infinite series of which the general term v.,, is a function of the positive variable h and becomes equal to Un when h vanishes. Suppose that for a sufiiciently small value of h and all inferior values the series (67) is convergent, and has V for its sum. It might at first sight be supposed that the limit of V for A=0 was necessarily equal to U. This however is not true.... ' Theorem. The limit of V can never differ from U unless the convergency of the series (67) becomes infinitely slow when h vanishes. * ' Sulle serie di funzioni', Memorie dl Bologna, ser. 5, vol. 8, 1900 up 131-186 701-744. ' t L. c, pp. 380-382. J L.c. (German edition), pp. 148-149. See also Bromwich, Infinite series, p 125 (Ex. 6). ■ '^' § p. 279. concept of II inform convergence 155 ' The convergency of the series is here said to become infinitely slow when, if n be the number of terms which must be taken in order to render the sum of the neglected series numerically less than a given quantity e, which may be as small as we please, n increases beyond all limit as h decreases beyond all limit. 'Demonstration. If the convergency do not become in- finitely slow it will be possible to find a number n, so great that for the value of h tue begin with and for all inferior values greater than zero the sum of the neglected terms shall be numerically less than e....' Stokes's words, and in particular those which I have italicised, seem to me to make two things perfectly clear. (1) Stokes is considering neither a property of an interval (a, b) im Grossen (such as is contemplated in A 1 or B 1), nor a property of a single point which (as in A 3 or B 3) need not be shared by any neighbouring point, but a property of an interval im Kleinen, that is to say a small but fixed interval chosen to in- clude a particular point. His definition is therefore one of the type of A 2 or B 2. Stokes's failure to perceive the bearing of his discovery on problems of integration is made much more natural when we realise that he is considering throughout a neighbourhood of a point and not an interval im Grossen. And this remark applies to Seidel as well. (2) Stokes is considering an inequality satisfied for a special value of n, or at most an infinite sequence of values of oi, and not necessarily for all values of n from a certain point onwards. In this respect there is a quite sharp distinction between Stokes's work and Seidel's. What Stokes defines is (to use the language of this note) a mode of quasi-unifo7'ni convergence and not one of strictly uniform convergence. It seems to me, then, that what Stokes defines is what I have called quasi-uniform convergence in the neighbourhood of a, point (B2). 8. If we adopt this view, Stokes's mistake becomes very much more intelligible. He proves, quite correctly, that uniform con- vergence in his sense implies continuit}^ : his proof, stated quite formally and by means of inequalities, is substantially that given in 1 5, under (1). He then continues* as follows. ' Conversely, if (66) is convergent, and if U= Vof, the con- vergency of the series (67) cannot become infinitely slow when h * p. 282. Tbe italics are mine. t Ffl is what Stokes calls 'the value of V for h = 0', by which he means, of course, its limit when h tends to 0. 156 Mr Hardy, Sir George Stokes and uniform, convergence vanishes. For if Un, V^ represent the sums of the terms after the nth in the series {QQ), (67) respectively, we have V^Vn + V,:, U=U^+Un'; whence v,:=Y-u-{v,,-u,,)^-u,:. Now V-U, Yn- Un vanish with h, and Ua vanishes when n becomes infinite. Hence for a sufficiently small value of h and all inferior values, together with a value of n sufficiently large and independent of h, the value of F,/ may be made numerically less than ^ any given quantity e however small ; and therefore, by definition, the convergency of the series (67) does not become in- finitely sloiv when h vanishes.' Now this argument is, until we reach the last sentence, perfectly accurate, and indeed, if we translate it into inequalities, substantially identical with that given in § 5, under (2). Stokes proves, in fact, that continuity at | involves quasi-uniform convergence at |. Where he falls into error is simply in his final assertion that this property is that which he has previously defined, the mistake being due to a failure to observe that his intervals of values of h depend upon a prior choice of e. In a word, he confuses, momentarily, B 2 and B 3. The ordinary view that Stokes defined uniform convergence in the same sense as Weierstrass compels us to suppose that he confused B 3 with A 1 , or at any rate with A 2 : and this is hardly credible. I add one final remark. If we could identify Stokes's idea with B_3, instead of with B 2, we could acquit him of having made any mistake at all, since B 3 really is a necessary and sufiicient con- dition for continuity. We could then regard Stokes as having anticipated Dini's theorem. This view, however, does not seem to me to be tenable. • Mr Lake, Shell-deposits fornied by the flood of January/, 1918 157 Shell-deposits formed by the flood of January, 1918. B}- Philip Lake, M.A., St John's College. [Read 18 February 1918.] The heavy snow of the third week in January 1918 was followed by a very rapid thaw and a considerable fall of rain, and the Cam, in consequence, rose to an exceptional height. In the neighbour- hood of Cambridge the floods were the most extensive of recent years, the water reaching its highest level on Sunday, Jan. 20. The traces of the flood remained visible for several weeks, its limits being marked in most places by straws, twigs, silt, etc., with a sprinkling of land and fresh-water shells. But below the town, near the railway-bridge, the shells were so abundant as to form a remarkable deposit, which seems to deserve a special record. It was not till the 25th Jan. that I saw it, and the following notes are drawn up from the observations made on that day and on two or three subsequent visits. The deposit lay partly upon the tow-path and partly in the shallow ditch on the iimer side of the path, and it extended with little interruption from the immediate neighbourhood of the 'Pike and Eel ' to a point about 850 yards below the railway-bridge, a total distance of approximately 850 yards. Occasional patches occurred still farther down, and scattered shells even as far as Ditton Corner. Beyond Ditton the tow-path was in several places covered with a thick layer of silt, but I saw no more shells until within sight of the lock at Baitsbite. The deposit was somewhat irregular and it was difficult to form an estimate of its average width, but this can hardly have been less than a foot, and was probably much more. Above the railway-bridge the shells were mixed with silt, especially in the ditch on the inner side of the path ; but even here the proportion of shells was large, and in places they formed the bulk of the deposit. Below the railway-bridge the deposit was free from silt and consisted entirely of shells. In the shallow hollows formed by the irregularities of the surface, it was often an inch or two deep, so that it was possible to scoop up the shells by the handful. Owing to its colour it showed conspicuously as light streaks upon the slightly darker path. By far the greater part of the deposit consisted of Limnaea, L. stagnalis and L. peregra being the most abundant species ; but other fresh-water shells also occurred and land-snails were by no 158 Mr Lake, Skell-depu'sits foniied bij the means rare. Mr C. E. Gray, of the Sedgwick Museum, went down shortly after my first visit, and in a very short time obtained most of the following species, but a few names have been added to the list from specimens collected subsequently : Sphaerium corneum (L.), Bithynia tentaculata (L.), Vivipara contecta (Millet), Valvata piscinalis (Miiller), Limnaea stagnalis (L.), „ peregra (Mtiller), „ auricularia (L.), Pkuiorhis corneus (L.), „ umbilicatus Miiller, „ caiinatits Miiller, „ vortex (L.), „ contortus (L.), Pliysa fontinalis (L.), Helix nemoralis L., Theba cantiana (Mont.), Hygromia striolata (Pfr.), Vitrea draparnaldi (Beck), „ cellaria (Miiller). Even now the list is probably far from complete, and a closer examination would no doubt reveal the presence of many other forms. The last five species are land-shells, and, with the exception of Vitrea cellaria, they occurred in Mr Gray's first collection and were identified by Mr Hugh Watson. Vitrea draparnaldi does not appear to be a native of the county, but is found in and near green- houses ; for instance, in the Botanical Gardens. In Mr Gray's first collection, which was made below the railway-bridge, it was repre- sented only by a single specimen, which we supposed to have come from the florist's greenhouses close by. But at a later date he found it to occur abundantly at the beginning of the tow-j)ath, some five or six hundred yards above the greenhouses. In order to make sure that the specimens really belong to this species they were sent to Mr Watson, who agreed with the identification. Since there were so many specimens of Vitrea draparnaldi at the beginning of the tow-path, and so few (at least comparatively) below the railway-bridge, it seems clear that they cannot have been carried far, for otherwise they would have been more evenly dis- tributed. It is most probable indeed that there was a colony of this species in the immediate neighbourhood. The nearest greenhouse that I have been able to find above the locality where the species was so abundant is five or six hundred yards off, and stands well I flood (if Jcumuvij, 1918 159 away from the river. The specimens can hardly have come from there, and it is more likely that the colony lived out of doors and nearer to the river. Nevertheless its progenitors may have been 'escapes'. The greenhouses below the railway-bridge have now been out of use for some time, and the snails that were in them must have been forced to seek new quarters. Most of the shells, both land and fresh-water, were perfect or nearly so, and all of them were empty. Neither Mr Gray nor myself found a single specimen with any remains of its former inhabitant. The greater number were very fresh in appearance, but some of the land-shells had evidently been exposed to the weather for some time, and some of the fresh-water shells had lain in the mud long- enough to become discoloured or incrusted as if the process of fossilization had begun. The specimens of Vitrea draparnaldi, it may be noted, were all fresh-looking. Apart from the extent of the shelly deposit, its freedom from silt below the railway-bridge was perhaps its most important feature, for it shows that even a muddy river like the Cam may produce a purely calcareous deposit. The fact that the shells were all empty indicates that those belonging to the river must have lain in its bed for some time; and in this connection an observation made by Mr Gray is of interest. Some years ago at Bottisham, when dredging operations were going on, he noticed that the mud brought up by the dredger was full of fresh-water shells. During floods the river digs up its bed and, as on the occasion here described, it may deposit the shells in one place and the silt in another. In the case of an artificially controlled stream like the Cam, floods are comparatively rare ; but in an unrestrained river we may reasonably expect them to be both more numerous and more extensive. It seems quite possible therefore that neither the clayey fresh-water limestones of the Wealden nor the purer fresh- water limestones of the Purbeck series required lagunary conditions for their formation. 160 Mr Matthui, la the Madveporarian Skeleton Is the Madveporarian Skeleton an Extraprotoplasmic Secretion of the Polyps ? By G. Matthai, M.A., Emmanuel College, Cam- bridge. (Communicated by Professor Stanley Gardiner.) [Read 18 February 1918.] In 1881 von Heider (5) suggested that the calcareous skeleton of the Madreporaria is formed by the deposition of carbonate of lime within certain specialised ectodermal cells (calicoblasts*) consti- tuting an outer layer, and repeated this conclusion in a subsequent paper (6). In 1882 von Koch (8) inferred from embryological obser- vations that the skeleton is deposited outside the living tissues, i.e. is extraprotoplasmic in origin. In 1896 Ogilvie (9) supported von Heider's view and argued that, by repeated calcification of "cells" of the calicoblastic layer of ectoderm, successive strata of calcareous " scales " are formed, and slightly modified her opinion in 1906 (10). Fowler (4) had previously accepted von Koch's view. In 1899 Bourne (2), from his studies on the Anthozoan skeleton, supported von Koch's conclusions and entirely disagreed with von Heider and Ogilvie. He further held that, whilst in Heliopora and the Madreporaria the corallum is formed outside the living calicoblastic layer, the spicules of the Alcyonaria are formed within certain ectodermal cells or scleroblasts which either remain in the ectoderm or wander into the mesoglaea (2, p. 506). Following von Koch and Bourne, it is noAv generally believed that the Madreporarian skeleton is an extraprotoplasmic formation and that Alcyonarian spicules are entoplastic products. After a ground-down section of an Astrgeid corallite has been slowly decalcified on a slide, somewhat homogeneous organic remains (distinguishable from algal filaments penetrating the skeleton) are left which react to any of the common stains. This is clear indication that the calcareous matter has been deposited in an organic matrix. Bourne regards this matrix as due to the "disintegration of calicoblasts" (2, pp. 520 and 521, fig. 21), assuming that the organic basis was not part of the living calico- blastic ectoderm. His view is that carbonate of lime is secreted by the calicoblastic layer and is passed through its outer border (the " limiting membrane ") into the decaying part outside, exactly as the Alcyonarian spicule is " from its early origin, separated from the protoplasm which elaborated the material necessary for its further growth by a layer of some cuticular material" (2, p. 537), * Von Heider's original rendering of this word is chalicoblast, of which the first half, I am informed, is derived from the Greek x'^^'li which in Eomau characters should be spelt clialix. Subsequently, Fowler changed the spelling to calycoblast, and in 1888 both this author and Bourne adopted the present form calicob/ast. an Extraprotoplasmic Secretion of the Polyps! 161 viz., the spicule-sheath. At the same time, Bourne contends that the spicule is entoplastic in formation whilst the Madreporarian coralkim is exoplastic. To be consistent, both the spicule and the corallum would have to be regarded as formed either within living protoplasm or outside it, but spicules could not be viewed as intra- protoplasmic products whilst assuming the extraprotoplasmic origin of the corallum. Duerden (3) held that the organic basis of the corallum of Siderastrea yadians is a " secretion " of the calicoblastic layer of ectoderm to which it is closely adherent (pi. 8, fig. 45) and is "a homogeneous, mesoglaea-like matrix within which the minute cal- careous crystals forming the skeleton are laid down " (p. 34). Since he refers to the skeleton as " ectoplastic " in origin (p. 113), it is evident that he agi-ees with Bourne in the view that the organic matrix was not part of the living tissues when calcareous matter began to be deposited in it. But in the account of these authors there is no more evidence to show that, in the Madre- poraria, the organic ground substance or "colloid matrix " (2, p. 539) was non-living at every phase of skeleton formation than that the areas of the scleroblasts of the Alcyonaria in which the deposition of spicular matter took place had not, at least at the initial stages of this process, formed part of the living protoplasm. Further if, in the Madreporaria, the calcareous matter were deposited outside the living calicoblastic ectoderm, it is difficult to understand how the manifold patterns of eoralla so charac- teristic of this gi'oup of organisms can have been built up*. But if the matrix in which carbonate of lime is laid down is part of the living calicoblastic sheet, it follows that the protoplasm must regulate the arrangement of the calcareous matter into the various skeletal types which, in large measure, maintain their re- spective form independent of changes in environmental conditions. Similarly, the formation of the various kinds of spicules of the Alcyonaria can be adequately explained only if calcareous deposition takes place within living protoplasm, and indeed. Bourne has drawn attention to the phenomenon that " the spicules of the Alcyonaria show a definite and complex crystalline structure, the details of which are, indeed, moulded upon and dominated by an equally complex organic matrix..." (2, p. 517). The intraprotoplasmic origin of spicules in the Alcyonaria might, without difficulty, be ascertained since sections can be made with- out decalcification, whereas in Heliopora and the Madreporaria possessing massive eoralla, satisfactory sections are possible only after decalcification, and in this condition the skeleton may appear * In explanation of this phenomenon, Bourne suggests that "the general arrangement of the fasciculi of crystals is dominated, in some manner of which we are ignorant, by the living tissues which clothe the corallum " (2, p. 539). VOL. XIX. PART IV. 12 162 Mr Matthai, Is the Madrejjorarian Skeleton as though formed outside the living tissues. A further difficulty with regard to the Madreporaria is that, except perhaps at the growing points, the skeleton would secondarily lose its intraproto- plasmic character and appear to be external to the living tissues by having displaced most of the protoplasm in which it was deposited, just as the discrete condition of fully developed Alcyonarian spicules is due to the increase of calcareous matter at the expense of the protoplasm in which it was formed. From the above considerations it would appear to be highly probable that von Heicler was right in regarding the Madreporarian skeleton as formed within the calicoblastic protoplasm. Bourne directs much of his criticism to von Heider's suggestion that the striae in the calicoblastic layer (i.e., in the processes of attachment) are calcareous fibres, but it is not improbable that, in the unde- calcified condition, some of these processes of attachment might be partially calcified. When thin sections of Astrseid coralla are examined under a microscope, they frequently appear to consist of calcareous pieces united by sutures resembling the " laminae " or " trabecules " of the skeleton of Heliopora (1, p. 463, pi. 11, figs. 7 and 8) and the " tra- becular parts " of the Madreporarian skeleton as figured by Ogilvie (9, p. 124, figs. 13, 19, etc.). Each piece is composed of calcareous strands radiating from a dark centre or line which, as Ogilvie sug- gested, appears to be the organic remains of the protoplasm in which the calcareous needles were laid down. There is some similarity between these elements and the spicules of Tuhipora (7, figs. 9 and 10) which, according to Hickson, are not fused together but dovetailed into one another as in the membrane bones of Mammals (p. 562). The resemblance is also marked in the case of the scale- like spicules of Plumarella (2, figs. 6 and 7) containing dark centres from which calcareous fibres or rods radiate. It is difficult to gather from Bourne's account what he considers to be the unit of skeletal structure in the Alcyonaria. Are spicules such units* ? But spicules are not all homologous elements since they are formed in protoplasmic areas containing one or more nuclei and no limit can be set to their size in the various genera (2, pp. 508-517), an extreme case being the scale-like spicules of Primnoa and Plumarella, each of which is " formed by several cells, or at least by a comparatively large coenocytial investment containing many nuclei " (p. 510). Or, is a spicule a calcareous piece which behaves like a single crystal when examined under crossed Nicols? The same confusion prevails with regard to ske- letal units in the Madreporaria — whether they are represented by " fibro-crystals " (Bourne), "crystalline sjjhgeroids" (von Koch) or * Bourne applies the term spicule to "an entoplastic product of a single cell or of a ccenocyte " (2, p. 504). The italics are mine. an. Extra protoplasmic Hecretiou of the Polyps ? 163 " calcareous scales " (Ogilvie). The latter are not calcified calico- blastic "cells" as Ogilvie contended since the calicoblastic ectoderm is now found to be a multinucleated sheet of protoplasm devoid of cell-limits, i.e., a syncytium. In fact, there is hardly any evidence to show that the skeleton of the Anthozoa is made up of homologous units just as it is highly doubtful if their soft parts are composed of uninucleated units or cells. The significance of the Anthozoan skeleton would consist in its probable formation within syncytial protoplasm according to physical laws under the presiding activity of the living protoplasm which would direct the complex skeletal architecture. The cal- careous deposit further appears to be differentiated into elements which remain separate as spicules in most Alcyonarians but are united to form a compact skeleton in certain Alcyonarians, e.g., Tuhipora, Corallium, Heliopora, and in all the Madreporaria (in which the calcareous matter may undergo subsequent rearrange- ment). From this point of view, a separate calcareous piece of an Alcyonarian might be regarded as a diminutive corallum, and the corallum of a Madreporarian as a massive spicule, and finally, the formation of the Anthozoan skeleton would be essentially similar to the formation of membrane bone in Vertebrates*. References. 1. Bourne, G. C. " On the Structure and Affinities of Heliopora ccerulea, Pallas. With some observations on the Sti'ucture of A'euia and Hetero xenia." Phil. Trans.., CLXXXVi, p. 455, 1895. 2. Bourne, G. C. " Studies on the Structure and Formation of the Calca- reous Skeleton of the Anthozoa." Quart. Jour. Micr. Sci., xli, p. 499, 1899. 3. DuERDEN, J. E. "The Coral Siderastrea radians and its Postlarval Development." Carnegie Institution, No. 20, Washington, U.S.A., 1904. 4. Fowler, G. H. " The Anatomy of the Madreporaria : I, Flabelhcm, Rhodopsanimia." Quart. Jour. Micr. Sci., xxv, p. 577, 1885; and Stud. Owens Coll., I, p. 243, 1886. 5. Heider, a. R. von. "Die Gattung Cladocora, Ehrb." Sitzb. Akad. Wis- sensch. Wien, Lxxxiv, p. 634, 1881. 6. Heider, A. R. von. " Korallenstudien : Astroides calycidaris, Blainv., u. Dendrophyllia ramea, Linn." Arbeit. Zool. Inst. Graz. i, No. 3, p. 153, 1886 ; and Zeitsch. Wiss. Zool., XLiv, p. 507, 1886. 7. Hickson, Sydney J. " The Structure and Relationships of Tuhipora.^ Quart. Jour. Micr. Sci., xxiir, p. 556, 1883. 8. Koch, G. von. " Ueber die Entwicklung des Kalkskeletes von Asteroides Cali/cularis und dessen morphologischer Bedeutung." Mitth. Stat. Neapel, III, p. 284, 1882. 9. Ogilvie, Maria M. "Microscopic and Systematic Study of Madreporarian Types of Corals." Phil. Trans., clxxxvii, p. 83, 1896. 10. Ogilvie, Maria M. "The Lime-forming Layer of the Madreporarian Polyp." Quart. Jour. Micr. Sci., XLix, p. 203, 1906. * It is interesting to note that structures analogous to fibrous connective tissue, tendon and bone of Vertebrates, occur in the Madreporaria, viz., the middle lamina ( = mesoglfea), processes of attachment and the calcareous coraUum, a matter which will be discussed in a future communication. 12—2 164 Mr Matthai, On Reactions On Reactions to Stimuli in Corals. By G. Matthai, M.A., Emmanuel College, Cambridge. (Communicated by Professor Stanley Gardiner.) [Read 18 February 1918.] The following is a brief record of feeding-experiments made on living Astrseid colonies during a short stay at the Carnegie Bio- logical Station at Tortugas (July 16 — Aug. 2) and at the Bermuda Biological Station on Agar's Island (Aug. 20 — Sep. 14) in the summer of 1915, which, though necessarily incomplete as they had to be undertaken in the midst of other work, gave some indication of the nature of reactions to stimuli in the Madreporaria. In order to watch the behaviour of living Corals, colonies of most of the recent species recorded from those localities were kept in aquaria of running sea- water, viz. : Mceandra lahyrinthifo7^mis (Linn.), Moeandra strigosa (Dana), McBandra clivosa (Ell. and Sol), Manicina areolata (Linn.), Colpo- phyllia gyrosa (Ell. and Sol), Isophyllia dipsacea (Dana), Isophyllia fragilis (Dana), Dichocoenia Stokesi, Ed. and H., Easrnilia, aspera (Dana), Favia fragum (Esp.), Orhicella cavernosa (Linn.), Orbicella annidaris (Ell. and Sol.), Stephanocoenia intersepta (Esp.), Ocidina diffusa, Lam., Mycetophyllia lamarckana, Ed. and H., Siderastrcea radians (Pallas), Siderastrcea siderea (Ell. and Sol.), Agaricia purpurea, Les., Porites astreoides, Lam., Porites furcata, Lam., Porites clavaria, Lam., Madracis decactis (Ly.), and Acropova muricata (Linn.). In Isophyllia dipsacea (Dana), when a particle of meat was placed on the oral disc with contracted mouths, the oral lip was slowly directed towards the particle and the mouth became dilated, to an extent depending on the size of the food-particle. The latter was, in the meantime, slowly moved into the oral open- ing by ciliary action. To facilitate this event, the periphery of the oral disc was drawn over towards the dilated mouth and the disc itself was somewhat depressed, thus deepening the peristomial cavity. During distention of the mouth, the stomodgeum was everted and, consequently, the coelenteric cavity Avith its convolutions of mesenteries became exposed.^ After the food-particle had passed into the coelenteric cavity, it was caught in the mesenterial coils. If the fragment of meat was large, the mouth remained widely open till the former had been reduced in size by the digestive action of the mesenterial filaments. The stomodtEum was subsequently with- drawn and the mouth opening gradually narrowed. But if, before this, the oral lip was touched with a glass needle, it did not contract as it would do instantaneously if no food-particle had previously to Stimuli in Corals 165 been swallowed. Every mouth that was tested could thus take in particles of meat. The touch of the food-particle on the oral disc was also a stimulus for the expansion of the tentacles around the mouth and of those around the neighbouring oral openings. When a particle of meat was placed on the tentacles of a colony of Mceandra labyrinthiformis (Linn.), it was slowly passed on to the oral disc, but the tentacles did not show any sign of contraction. At the same time, the oral disc was depressed and arched over the mouth opening till finally its margin closed over the peristome. In the meantime, the tentacles were fully distended, the entocoelic ones were directed obliquely towards the oral opening, those of one side passing between those of the opposite side. The food- particle was now hidden from view. After it had passed into the ccjelenteric cavity and had presumably undergone partial digestion, the periphery of the oral disc gradually moved outwards carrying the tentacles with it, thus again exposing the peristomial cavity. The principal movements in these two cases are: (1) Ciliary movement passing the food-particle into the nearest oral aperture. (2) The direction of the oral lip towards the food-particle pari passu with the dilatation of the mouth. (3) The narrowing and deepening of the peristomial cavity, which help to roll the food-particle into the oral opening. (4) The expansion of the tentacles of the affected oral disc and of those of adjacent oral discs. (5) The eversion of the stomodeeum and consequent exposure of the coelenteric cavity and mesenterial coils. (6) The return of the soft parts to their original condition by the retraction of the stomodseum into the coelenteric cavity, recoil of the oral lip to its normal extent, shortening of the tentacles, flattening of the oral disc and withdrawal of its periphery carrying the tentacles outwards. When a drop of meat-juice was gently placed on a colony of Favia frag am (Esp.), the oral apertures in the neighbourhood were slowly distended after a short pause. The inner or entocoelic row of tentacles was then extended and directed over the oral disc, meeting or intercrossing over the mouth as had been noticed in the case of Mceandra labi/rinthiformis (Linn.), thus hiding the oral region, whilst the exocoelic tentacles were arched outwards. Similar move- ments were observed in Mceandra strigosa (Dana). When meat-juice was spurted by a pipette on sea- water con- taining a colony of Orhicella cavernosa (Linn.), strong contraction of the soft parts was set up in the neighbourhood, the polyps en- tirely closing up. This was followed by the protrusion of convolutions of mesenteries through mouth openings, oral discs and especially through edge-zones, combined with secretion of mucus over the polyps, the former obviously to paralyse prey and the latter to 166 Mr Matthai, On Reactions to Stimuli in Corals entangle food-particles. Shortly afterwards, the oral apertures were widely distended to let in the meat-juice but the process was un- accompanied by eversion of stomodsea. Similar events were observed in Manicina aj-eolata (Linn.). When finely powdered carmine was scattered in sea-water con- taining a colony of Manicina areolata (Linn.), it was partly taken into the stomoda^a, the oral lips becoming conspicuously stained. The carmine was, however, subsequently passed out of the stomodaea, showing thereby, that the mouth openings could function as in- halent and exhalent apertures. When a tentacle of any of the Astraiid colonies was touched with a fine glass needle, it was suddenly withdrawn in a manner resembling pseudopodial movement and the neighbouring tentacles were also retracted. In Porites and Madracis, whose soft parts are composed of small polyps, the instantaneous contraction of a polyp due to mechanical stimulation caused the contraction of its neigh- bours as well. In all these cases, the wave of contraction started from a centre, viz., the point of stimulation, but remained local and did not spread over the entire colony. Series of movements such as the above, made in response to chemical and tactile stimuli, are reminiscent of amoeboid or stream- ing movement of protoplasm, the soft parts of the colonies appearing to serve as the medium for the transmission of stimuli*. If the initial stimulus be too strong, the sudden contraction of the soft parts, due to the mechanical impact, is followed by slow purposive movements. The amoeboid character of the movements of the soft parts of Astrseid Corals is in conformity with their histological structure which, on examination, revealed neither a muscular nor a nervous system, although a neuro-muscular apparatus has been supposed by most authors to exist in Madreporaria. The so-called muscular fibres at the base of the ectoderm and endoderm seem to be of the nature of specialised connective tissue fibres, for in both teased preparations and in sections of 4/z — 10/i thicknesses these are found to be without nuclei and to form part of the middle lamina (= meso- glsea) which is itself composed of fine fibres cemented together by a homogeneous matrix containing a few scattered nucleated cells. Fibrils pass into the middle lamina through the granular stratum present at the base of the ectoderm (and less frequently at the base of the endoderm), but these fibrils do not show any histological differentiation which would justify us in regarding them as belong- ing to nerve elements f. * Carpenter I'egarded the feeding reactions of Isophyllia as muscular in nature and as brought about by the transmission of impulses of a " nervoid character," but he had not investigated the histological structure of its soft parts {vide Con- tributions Bermuda Biol. Station, No. 20, Cambridge, Mass., U.S.A., p. 149, 1910). t For a detailed account of the minute structure of coral polyps vide "The Histology of tlie Soft Parts of Astraeid Corals " to be published shorth'. Mr Brindley, Notes on certain parasites, food, etc. 167 Notes on certain parasites, food, and capture hy birds of the Common Earwig (Foi-ficula aiiricularia). By H. H. Brindley, M.A., St John's College. [Read 18 February 1918.] (rt) Effects of pa7'asitism. In a paper entitled " The effects of Parasitic and other kinds of castration in Insects " (Jour. Exper. Zool. viii. Philadelphia, 1910) Wheeler expresses the opinion (p. 419) that Giard has given good reasons for supposing that the dimorphism exhibited by the forcipes of male earwigs from the Farn Islands, Northumberland (Bateson and Brindley, " On some cases of variation in secondary sexual characters statistically examined," Proc. Zool. Soc. Lond. 1892, p. 585), is due to "differences in the number of gregarines they harbour in their alimentary tract." The reference to Giard is C.R. Acad. Sci. cxviii. 1894, p. 872, where he writes " J'ai tout lieu de croire qu'une interpretation du meme genre (referring to the changes evoked in Carcinus by the action of parasites) pent s'appliquer pour la distribution des longueurs des pinces des Foi'ficules males. II est possible, en effet, d'apres la longueur de la pince, de prevoir qu'une Forficule male possede des Gregarines et qu'elle en possede une plus ou moins grande quantite." In criticism of the above statements Capt. F. A. Potts and myself published a letter in Science, Philadelphia, Dec. 9, 1910, p. 836, in which we gave reasons for disagreeing with Wheeler's conclusion : viz., (i) that in the absence of any further account by Giard the above passage could not be taken as direct evidence that he had examined the intestine of Forficula for gregarines and found a correspondence between their presence and the condition of the male forcipes ; (ii) that out of several thousand earwigs collected by us on the Farn Islands in 1907 over 50 males of different forceps lengths were carefully dissected with the results that the gregarine Clepsydrina ovata was found to occur commonly in the alimentary canal, that it occurred indifferently and was absent indifferently in " low " and " high " males, and that no correlation could be traced between the number of parasites and the length of its forcipes. Moreover, no difference in the development of the testes or other internal sexual organs could be detected in low and high males respectively. Since the above was written I have (August 1917) examined the alimentary canal of 51 earwigs out of a large batch obtained at Porthcressa, St Mary's, Isles of Scilly, where the males exhibit 168 Mr Brindley, Notes on certain parasites, food, and capture well-marked dimorphism (Camb. Phil. Soc. Proc. xvii. part 4, 1914, p. 831). The results summarised are as follows: Infection by Clepsydrina ovata. Number examined Not infected Infected Number of gregarines found Average number of gregarines in the infected individuals Low males High males Females 23 23 5 12 11 1 11 12 4 323 238 53 29 20 13 Thus the evidence so far obtained is that the dimorphism of the forcipes in F. auricularia </ is not a result of or influenced by gregarine infection — though in view of the well-established effects of such parasitism on the secondary sexual characters of another arthropod in Geoffrey Smith's case o^ Inachus dorsettensis modified by the gregarine Aggregata {Mitt. Zool. Stat. Neap. xvii. 1905, p. 406), the absence of positive evidence to the contrary at the time Wheeler wrote, but now obtained, certainly afforded ground for his support of Giard. In this connection I may quote a letter from Geoffrey Smith, whose recent death at the battle front brings us into common mourning with Oxford zoologists for a friend and colleague. Writing to me about 1907 he said, " Have you noticed that Giard attributes all cases of High and Low Dimorphism to parasitic castration ? I am sure this is not right, but there is no doubt that parasitic castration is a much more frequent occurrence than is commonly supposed." These words, and a footnote to the same effect in his paper " High and Low Dimorphism " (Mitt. Zool. Stat. Neap. XVII. 1005, p. 321), are typical of the writer's insight and balanced judgment. It may be stated that the gregarines in the Porthcressa earwigs fell roughly into categories of small, medium, and large, but they all seemed to be C. ovata. Rather more than half were small individuals, and those of medium size were slightly in excess of the large, but the sizes were not recorded in the case of the first few earwigs examined. Very large numbers were found in syzygy, and such associated individuals were of all three sizes. One instance of syzygy of a large with quite a small individual was observed. There was no noteworthy difference between the hy birds of the Common Earwig (Forficula auriculana) 169 numbers of gregarines of different sizes or between the proportion of free gregarines to those in syzygy in their low and high male hosts respectively. During our stay on the Scilly Islands in 1912 Capt. Potts and myself, in company with Capt. J. T. Saunders, found in St Martin's several earwigs parasitised by a gordiid larva {sp. incert.), the coils of which, though projecting between the terga of the abdomen, seemed to have no effect on the health and activity of their hosts. The same apparent absence of deleterious effects was noticed in three of the Porthcressa batch of 1917 which were found to be similarly infected. In one, a low male, a large gordiid occupied most of the body, and no portion of the alimentary canal posterior to the crop could be found ; in a high male similarly infested by a large gordiid there was very little of the hind gut left ; and an adult female contained three or four gordiids of various sizes, the gut in this case being intact and apparently healthy. A fourth individual, a low male, was not parasitised when examined, but as the gut was partially atrophied, it had probably been recently deserted by a gordiid. All these infected individuals seemed as active and healthy and to possess fat bodies as large as those not infected ; the earwig's resistance to such extensive destruction of internal organs is very noteworthy. As Clepsydrina ovata inhabits the chylific ventricle and hind gut and as the presence of gordiids evidently often results in destruction of these portions of the alimentary tract, the latter parasite is likely to be exclusive of gregarines, and these were absent in all three of the males mentioned above (including that with the hind gut intact), while only two were found in the female. That the presence of parasitic worms has sometimes serious effects on the insect's health is suggested by the recent observations of Jones recorded in " The European Earwig and its control," a report on the invasion of Newport, R.I., in 1911 by Forficida auricularia and its subsequent spread ( f/. >§. Dept. Agric. Bidl. 566, Washington, June, 1917), from which it appears that 10 per cent, of earwigs kept in the laboratory were killed by the infection of a worm identified as Filaria locustae, whose average length is given as 83 mm. This however is a size exceeding considerably that of the gordiids in the Scilly earwigs, which I have called " large " when attaining a length of 50 mm. In southern Russia Forficula tomis, Kolenati, is parasitised by the tachinid fly, Rhacodineura antiqua (Pantel, Bull. Soc. Entom. France, No. 8, Paris, 1916, p. 150), but I do not know if it attacks the common earwig. The paper quoted mentions the capture of the adult fly in Holland and Portugal. Lucas {Entom. XXXVII. 1904, p. 213) reports F. auricularia (or ? lesnei) attacked by scarlet acarine mites. 170 Mr Brindley, Notes on certain parasites, food, and capture Among fungoid parasites, EntomopMhora forficulae diminishes the number of earwigs (Picard, Bidl. Soc. Etude Vulg. Zool. Agric. Bordeaux, Jan. — April, 1914, pp. 1, 25, 37, 62). It is possibly this species which has caused heavy mortality among the earwigs which I have kept in captivity in the Zoological Laboratory during recent years. Infection by the above or other fungus is a very frequent result of damp in the soil or in the plaster of Paris cells bedded with coco fibre which I have employed. The most effective preventive of fungus has so far been keeping the earwigs in roomy glass dishes lined with virtuall}^ dry sand and supplj^ing water only by wetting the vegetable food given. (6) Food. In " The Wild Fauna and Flora of the Royal Botanic Gardens, Kew," 1906 {Kew Bull. Add. Series V), Lucas writes (p. 23) of the Common Earwig, " It is an animal feeder. Does it do. as much damage as is supposed ? " And Ealand in " Insects and Man," 1915, p. 266, states "most gardeners would assert that the insect is destructive to cultivated plants. Careful observation and experiment, however, show that it is carnivorous and that it devours caterpillars, snails, slugs, etc.... its habit of hiding in such flowers as the sunflower and dahlia have earned it an undeserved reputation for evil." I find that seven out of nine recent and more or less compre- hensive manuals of Economic Entomology do not mention earwigs at all, which is fair evidence for considerable doubt as to their being harmful insects. Of the two works in which earwigs are mentioned one speaks of them as destructive to mangolds, turnips, cabbage crops, and plant blossoms, while the other states dahlias as attacked, " but nearly all plants suffer." Virtually every fruit grower and horticulturist of whom we make enquiry assures us that earwigs are most destructive pests, but is the general belief thus expressed really well founded ? Recent literature leaves the impression that in certain localities earwigs may be specially harmful to plants of economic value, though an explanation of this capriciousness is wanting. Theobald (Rep. on Econ. Zool., South-Eastern Agric. Coll., Wye, April 1914) gives hops as attacked by F. auricidaria. Lind and others in a summary of the diseases of agricultural plants in 1918 (79 Be- retning fra Staiens Forsogsvirksamded i Plantekidtur, no. 30, Copenhagen, 1914) state that in one locality in Denmark cauli- flowers were completely destroyed by the Common Earwig, which seems a very exceptional event. Sch^^iyen in Beretning om skadein- sekter og plantesygdommer i land og havchruket 1915 (Report on the injurious insects and fungi of the field and the orchard in 1916), hy birds of the Common Earivig (Forficula auricnlaria) 171 Kristiania, 1916, mentions that in many parts of Norway different vegetables, cabbage in particular, were extensively damaged by F. auricularia. Tullgren, in a report on injurious animals in Sweden during 1912 — 1916 (Aleddelande frdn Centrcdanstalten for Jorshruksforsok, no. 152 ; Entomologiska Avdelningen, no. 27, p. 104), records damage by F. auricidaria to ornamental plants, barley, wheat, and cabbage. In the case of the invasion of New- port, R.I., by the Common Earwig, Jones {op. cit.) reports that the quite young individuals eat tender shoots of clover and grass, and possibly grass roots ; while later on shoots of Lima Bean and dahlia and blossoms of Sweet William and early roses are attacked, with a general preference for the bases of petals and stamens rather than for green shoots. Adults are recorded as feeding almost wholly on petals and stamens, though clover, grass and terminal buds of chrysanthemums and other "fall flowers" are also devoured. Sopp, "The Callipers of Earwigs" {Lanes, and dies. Entom. Soc. Proc. 1904, p. 42), records having seen a female earwig using her forcipes to repeatedly pierce damp decaying seaweed on which she was apparently feeding. Ltistner {Centralhl. Bakt. Parnsit. u. Infektions- krankheiten, XL. nos. 19-21, Jena, April 1914, p. 482) has summa- rised the work of over thirty observers of the contents of the crop of the Common Earwig. Altogether 162 individuals were thus examined, and the conclusion was arrived at that earwigs normally feed on dead portions of plants and on fungi such as Gapnodium, living leaves and flowers being attacked when circumstances favoured the change. Dahlia leaves and petals were very readily devoured. How far earwigs are a pest to ripe fruit seems not to have been investigated, but it was concluded that as a rule they may be regarded as harmless save in special cases. It was admitted however that the further the enquiry went the less definite were the results. In view of the diversity of reports as to the favourite food plants of earwigs and the general want of exact information as to the damage likely to be done by earwigs in a flower or kitchen garden I carried out a small series of observations on the earwigs obtained last August from St Mary's, Isles of Scilly, which were kept in captivity in the Zoological Laboratory for some weeks, primarily for the purpose oj" examining their alimentary canal for parasites. These earwigs, several dozen in number, were kept in a large glass dish bedded with sand slightly damped occasionally. They had no animal food save that afforded by those which died. In order to obtain information as to preference for one kind of plant above another they were given three different species, taken haphazard, at a time for a period of two days or more. A summary of the results is as follows : — Aug. 20 and 21. Vegetable marrow leaves were ver\' much 172 ilf?' Brindley, Notes on certain parasites, food, and capture eaten ; horse-radish leaves very little touched ; Michaelmas Daisy leaves and flowers hardly, if at all, touched. Aug. 22 and 23. Beetroot leaves were much eaten, the leaf stalks m particular, these being opened out and the pith taken : white phlox leaves and flowers, the petals much gnawed and pollen grains were found in the gut : dwarf bean leaves, little touched. Aug. 24 to 26. Blue Anchusa leaves and flowers, the petals were much eaten but the leaves neglected : white rose leaves and flowers, petals devoured but leaves untouched: golden rod (Solidago) leaves and flowers, leaves nibbled at sides here and there but flowers apparently neglected. Aug. 27 to 29. Yellow Oenothera flowers and pods, the petals were much eaten but the pods remained untouched : white Japanese anemone leaves and flowers, petals eaten to some extent, leaves neglected : raspberry foliage, the leaves were not nibbled, but the earwigs congregated in numbers on their hairy undersides, an action much more pronounced than in the case of any of the other plants given throughout the observations. Aug. 30 and 31. Cabbage leaves were destroyed by the blade bemg gnawed down between the veins to the midrib while the ends of the veins were shorn off: rhubarb leaves, eaten a good dea : scarlet runner leaves, flowers, and pods, apparently quite Sept. 1 to 3. Plum fruit unskinned was much attacked- potato tuber and rather unripe apple, both unskinned, were not touched at all. Sept 4 to 10. On the 4th the plum was removed, but the apple and potato were not attacked during the seven days. Sept 11 to 15. On the 11th the apple was cut across, with the result that it was slightly gnawed during the five days : the potato remained untouched. Sept. 16 to 2:l On the 16th the potato was cut across, which was followed by its being very thoroughly attacked, though the apple was not entirely deserted. Of the 51 earwigs whose alimentary canals were examined for gregarine 7 contained spores of Fuccinea graminis (one had as many as 180 and another 100), while the food of another individual included numerous unidentified enjiomophilous pollen grains Both spores and pollen grains appeared to be very slightly if at all" digested. It is hoped to extend the observations in the coming summer, as those recorded above were limited to only a few of the possible food plants and only adult earwigs were kept. It may well be that there are differences in the preferences of nymphs and adults, and as the former are in the majority till about the end of July, it is possible that they may be harmful to certain plants m particular, as Jones's observations (o;j>. cit.) suggest. hy birds of tJie Covinion Eariuig (Forficula auricularia) 173 It seems established that a large number of ordinary garden species are liable to serious attack by earwigs, and that the latter can continue healthy on a purely vegetable diet. But much further information of a detailed kind is required befoi'e we can explain why in a given locality a particular kind of plant is attacked while in another it is neglected. Does it mean that the presence or absence of suitable animal food is a factor ? As regards animal food, there is a considerable amount of evidence that earwigs are often carnivorous by choice, very possibly they are so usually (cf Rlihl, M.T. Schweiz. Ges. vii. 1887, p. 310). In respect of eating dead animal matter I have found that when kept in captivity they devour the soft parts of their fellows who have died even when fresh vegetable food is available. In this necrophagous habit they resemble cock- roaches. Jones {op. cit.) states that dead flies and dead or dying comrades are devoured. Lustner (op. cit.) finds that only dead animal matter is taken. This conclusion points to too limited an inquiry and want of taking into account the possible presence of food plants which were more attractive than available living prey. In any case his opinion that earwigs should not be regarded as beneficial is traversed by the records of their killing certain insect pests of plants. Round Island, the northernmost islet of the Scilly group, is swarming with earwigs, and they congregate in vast numbers in the light-keepers' midden inside the discarded pressed beef tins. If, as seems probable, they reached the islet before the lighthouse was built a change of diet seems to have occurred, as the indigenous vegetation is chiefly Armeria maritima, Cochlearia officinalis and Mesembryanthemum edide. There is no turf It is of course possible that they seek the potato peelings also thrown into the midden and that their numbers inside the discarded tins mean that the latter are frequented partly for shelter. If the Round Island earwigs have really turned during comparatively recent years from a herbivorous to an extensively carnivorous diet, Rosevear, another islet of the Scilly group may, in a sense, be a converse case. It is the other locality in the Scilly group in which (as far as I know) the earwig population is densest. Like Round Island, it is very small, but differs from it in being uninhabited. But from 1850 to 1858 it was occupied by the builders of the Bishop Rock Lighthouse, so is it possible that the abundance of earwigs is due to the animal food available in the past ? However this may be the present diet of the Rosevear earwigs appears likely to be vegetarian in the main, unless the islet harbours some insect or other small arthropod suitable for food. The commonest plants are Armeria maritima and Lavatera arborea, the latter growing luxuriously. But before the abundance of earwigs on Rosevear 17-i Mr Brindley, Notes on certain jKtrasites, food, and cajytare can be discussed adequately something must be known of the con- ditions obtaining on Rosevean and Gorregan, its small and only immediate neighbours. Of these islets I possess no information at present. Also, there are other peculiarities as regards the earwigs of Rosevear and Round Island which are beyond the scope of the present paper. There is no doubt that earwigs sometimes kill and devour other insects larger than themselves, though the event is probably somewhat exceptional. Chapman ("Notes on Early Stages and Life History of the Earwig," Entom. Record, xxix. no. 2, Jan. 1917) states that "animal food, such as dead insects, seemed always acceptable " to earwigs in captivity. Sopp {op. cit. p. 42) regards earwigs as probably "omnivorous feeders, largely carnivorous by choice, but often phytophagous, frugivorous, or even necrophagous of necessity." Whether attack on living animals as prey is common I cannot say, I have no observations of my own to record ; ^ but it appears that occasionally the forcipes, organs of much disputed function, are used for this purpose. Sopp (op. cit.) has seen them employed to seize and crush large flies which were . subsequently devoured and quotes an instance of a larva similarly attacked from the records of another observer. Burr (Entom. Record, Sept. 1903) saw a blue-bottle seized by the forcipes of a male Labidura riparia kept in captivity. Lucas {Entom. xxxviii. 1905, p. 267) records a female of this species as using the forcipes to capture a cinnabar moth larva, which was afterwards devoured. Jones {op. cit.) records that the Newport, R.I., earwigs attack and devour " certain sluggish unprotected larvae." There are many observations which show that earwigs in some localities prey upon small insect larvae, and in certain instances they have been recommended as a means of diminishing plant pests. Thus the following references, as also others quoted in this paper, have appeared in issues of The Review of Applied Entomo- logy, 1913—1918. Bernard {Technique des traitements contre les Insectes de la Vigne, Paris, 1914) states that they devour the pupae of one or more of Clysia amhiguella, Polychrosis botrana, and Sparanothis pilleriana {v. also 'Kirkaldy, "^^i^o??*. xxxiii. 1900, p. 87). Dobrodeev {Mem. Bur. Entom. of Gent Board of Land Administration and Agric, Petrograd, XL no. 5, 1915) makes a similar report as regards the destruction of the first tAvo Tortricidae named above by earwigs. Molz {Zeits. Angeiuandte Ghemie, Leipzig, xxvi. nos. 77, 79, 1913, pp. 533, 587) speaks of earwigs as natural enemies of the vine moth. Feytaud {Bull. Soc. Etude Vidg. Zool. Agric. Bordeaux, xv. nos. 1—8, Jan.— Aug 1916, pp. 1, 21, 43, 52, 65, 88) states that earwigs destroy the eggs and larvae of the coccid vine pests Eidecanium persica and (probably) Pulvinaria vitis. Harrison in "An unusual parsnip hy birds of the Coiiiiiwn Earwig (Forficula auricuLiria) 175 pest" {Entomologist, XLVI. Feb. 1913, p. 59) reports them as most effective in killing and eating Depressaria heradicwa, the "parsnip web-worm." Brittain and Gooderham (Canad. Entorn., London, Ont., XLVii. no. 2, Feb. 1916, p. 37) make a similar state- ment. There is no doubt that our knowledge of the bionomics of the earwig is at present very imperfect. As in the case of other very common animals far too much has been taken for granted. The earwig's nocturnal habit, its tendency to assemble in great numbers between two closely apposed surfaces, and its "frightening attitude " of flexing its abdomen dorsalwards with opened forcipes all tend to give it a reputation for evil which very probably is but partially deserved. We all know how the habit of entering crevices is responsible for the belief that it gnaws through the tympanic membrane with the result of mania or even death. Perce-oreille speaks for itself. It seems fairly established that its universally bad reputation among gardeners is founded on tradition and want of judgment combined with neglect of the increasing evidence that its presence is sometimes beneficial by its destructiveness to more harmful insects than itself That it eats the petals of dahlias and chrysanthemums to some extent is true, but as far as my own observations go the outlay of time and material devoted to the traditional protection of the flowers by inverted flower pots stuffed with straw seems hardly worth while. The great attraction which the flowers have for earwigs seems to be the closeness and number of their petals, which provide a daytime shelter whence nightly excursions for feeding are made. Anyone possessing a garden may greatly add to our knowledge of favourite foods; observation at night is particularly needed. As regards garden varieties of roses the case against earwigs is probably more severe. (c) Capture by birds. During the last decade systematic investigation of the contents of the alimentary canal of British wild birds by several observers has resulted in most useful information as to which should be regarded as harmful and which as neutral or beneficial to agri- culture. It is manifest from the laborious and painstaking work now at our disposal that many of the reputations, good or evil, which certain common birds have in the eyes of farmers and gardeners need considerable revision, in some cases even reversal. As regards the capture of earwigs by birds, it appears that they are not a favourite food when we bear in mind how numerous they are sometimes and that they are large enough to be easily seized. No doubt their nocturnal habit affords much protection from capture. 176 Mr Brindley, Notes on certain parasites, food, and caj^ture Collinge in " The Food of some British Wild Birds " (London, 1913) reports on the contents of the crop, etc., of 29 of the com- monest species, among which only four contained earwigs, and these were very few in number. Thus in 404 House Sparrows 2 earwigs were found, 1 in each of 2 birds; in 721 Rooks 2 ear- wigs were found, 1 in each of 2 birds ; in 40 Skylarks 3 earwigs were found among 2 birds; in 64 Song Thrushes 7 earwigs were found among 2 birds. Newstead in " The Food of some British Birds " (Sapp. to Journ. of Board of Agric. no. 9, Dec. 1908) records observations on the swallowed food of 128 species, the outcome of 871 post-mortem and pellet examinations carried out in various years from 1894 to 1908. He finds that 10 sj)ecies had eaten earwigs, the numbers of birds examined and the numbers of earwigs found being : 1 Whimbrel, 40 earwigs ; 2 Green Woodpeckers, 24 earwigs ; 2 Starlings, 3 ear- wigs; 1 Nuthatch, 3 earwigs; 1 Chaffinch, 1 Great Titmouse, 1 Redbreast, 1 Song Thrush, 1 Whinchat, 1 Woodcock, 1 earwig each. Theobald and McGowan in '•' The Food of the Rook, Chaffinch and Starling" {Sapix to Journ. of Board of Agric. no. 15, May 1916) put on record a particularly valuable and interesting series of observations, as they examined the food month by month during nearly 2^ years, viz., from Jan. 1912 to May 1914, the inquiry covering 277 Rooks, 748 Starlings, and 527 Chaffinches. An analysis of their results as regards earwigs for the 2^ years is as follows: 3K ^c^ Birds examined Earwigs found Average number of earwigs taken by each bird Starling Chaffinch Rook 372 277 121 154 7 3 •41 •025 •025 Starling Chaffinch Rook 376 248 156 199 5 3 ■ -53 •020 •019 I have divided the year into two j)eriods of six months con- formably with the seasonal presence or absence of earwigs on the surface of the ground. From October to March most male earwigs hy birds of the Common Earwig (Forficula auriciilaria) 177 die and the females are hibernating. In view of this it is curions that earwigs should be taken as numerously during this period as during the six months when both nymphs and adults can be found easily. The numbers recorded for Rook and Chaffinch are small, though a large number of birds were examined. The Starling is a great insect eater; is it possible that it habitually searches for buried insects during the colder months and devours earwigs found with the rest ? This action may be true for the other two birds also. The figures for all three are certainly curious. So we find only 13 species of birds reported as having captured earwigs, and most of them as very sparingly. The Starling is not recorded by Collinge as an earwig eater. The above quoted reports certainly suggest that wild birds cannot be relierl upon to diminish earwigs in a garden. Many of the most insectivorous are not reported as feeding upon earwigs at all. They may be distasteful, and a large number together emit a well-defined odour, and the same is true of a number preserved in alcohol. Be this as it may, domestic fowls always eat them readily, a fact which is noted by Jones {op. cit.) in the case of the invasion of Newport, R.I. He also mentions that toads will eat them. Miss Maud D. Haviland, Hon. Mem. B.O.U., to whom I am indebted for assistance with regard to the literature of the subject and for kind advice in the preparation of these notes, informs me that she has noticed a Redbreast take earwigs in preference to earthworms. ADDENDA. Under (b). Mr H. Ling Roth informs me that he has found earwigs very destructive to iris pods, with resulting premature fall of seeds, in a garden at Halifax, Yorks. Under (c). Gurney, in "Ornithological Notes from Norfolk for 1916" {British Birds, x. 1917, p. 242j, records that his father in October, 1843, found several earwigs in a Stone Curlew. VOL. XIX. PART IV. 13 178 M(ijo7^ MacMahov and Mr Darling, Reciprocal Relations Reciprocal Relations in the Theorij of Integral Equations. By Major P. A. MacMahon and H. B. C. Darling. [Received 1 February 1918. Read 4 February 1918.] 1. Let f{oc)K{ajt)dx = ylr,{t) J a, and f2{cc)/c(a;t)da; = yjr^{t); J a., then, if we suppose the functions f^,f and k to be such that the order of integration is indifferent, we have fbi rbo rb, / /i (•^) fa (^t) da; = dy \ f {x)/., {y) k {xyt) dx = \\Uy)i^i{yt)dy, or, as it may be written, / A(oo)yjr,(xt)dx= f,(x)yfr,(xt)dx (1). *i J a.2 In the Messenger of Mathematics, May 1914, p. 13 Mr Rama- nujan has employed this result to deduce a number of ' interesting relations between definite integrals. The method is very suggestive and appears capable of considerable extension. For example, if f{x)K[e{x,t)\dx = ^lr,{t)\ [b. \ (2), and / fM'c{e{x,t)]dx = ^lrM *^^n \j^ (•^) ts [0 {a; 01 dx = ^J, {x) f, {0 (x, t)} dx . . .(3), provided that {x, 6 {y, t)]=- d [y, 0{x, t)\ (4). The functional equation (4) is satisfied by 0{^,t) = cl,-^f(x) + cl>(t)\ (5), where / and are arbitrary functions ; which is a general form of solution and includes among others such solutions as H^>t) = c}>-^{f(x).cl>{t)} (6), ^ ^ \f(^) + cP{t)\ ^'>' ^{^,t) = cf^~^f(x) + cf,{t)+f(x)cf,(t)] (8). in the Tlieory of Integral Equations 179 Thus, to derive (7) from (5) let f{x) = coth-i [P{oc)], (/> (0 = coth-i [(^1 {t)\ : then (5) becomes 0-' [coth-' [F{x)] + coth-i {(^1 {t) W Now let (^"^ (^) = u, then ir = <^(m) = coth"' 1^1 («)}, whence ^i {u) = coth 0, and u = 4>r^ (coth 2) ; that is (/)-' (^) = 01-1 (coth ^), and therefore (5) reduces to _^\ F{x)4>,(t)+l ] F{x) + (ji,(t) 0r which is of the form (7 ). As an example of the use of (2) and (3) in the determination of relations between integrals, let /i (•'^) = sin X, /, {x) = cos X, and, using the form (6) for 0, let 0{x, «) = e»'-'o.'^', K (x) = X. bi = b.2 = a, Ui = «y = 0, and Then, putting we have from (2) yfr^ (t) = sin x . e^iog'' J dx (log t . sin a — cos a) e^^^st 4. 1 ^ l + (logO' ~ and yfr..it)=\ cos.'r.e-'"'°s'rf.« Jo (log t . cos a + sin a) e"^^^ — log ^ l+(logO-^ Substituting these values in (3), and then putting log^ = l/r for brevity, we obtain '*" [x sin {x — a) + 7' cos {a; — a)} e'**'"' , r' + «' ' -' it' sin X -\- r cos x i ■-' X SI •'0 y.2 _j_ ,^2 f/./ = 0: 13—: 180 Major MacMalion and Mr Darling, Reciprocal Relations so that, provided r is not zero, we have X sin {x — a) + r cos {x — a)\ e"'^'^ J r- + x^ X sm X -{-r cos .r , r^ + ic- ^ an identity which may be verified by differentiation with respect to a. Putting x = r tan ^ and then replacing ^ by a;, (9) becomes •*^'^~' «/'• cos(^ + a- r tan .^■) „ tan x '^-^"^'^ ' "^ ■_::^-J g a tan x ^^ J cos X .tan 1 a/;- ^^g (^ _ ^ ^^^ ^A = ^ ax (10), J cos X which admits of ready verification by differentiation with respect to a. The identities (9) and (10) hold generally, provided that the constants are finite; we have seen that r must not be zero. It will be noticed that both (9) and (10) are of the form Jo Jo where the upper limits of integration involve a. 2. As another illustration of how the method admits of genera- lisation, let fAx)'c{0{x,t)]dx = y\r,{t). J «, rb, and f2{x)K{d {x, t)] dx = yjr, (t) : J 0.2 fbi . fb, then I /i (x) v/^a {\ (x, t)} dx=\ fo {x) f, {\ (x, t)} dx J a, J a, when \ {x, t) = 4>^-^ {/(x) + (j>, (t)} and e(x,t)=g{f(x) + cl>,it)}, f, g, (pi and (f>2 being any functions. It should be observed that A. becomes 6 when (f)^ = ^2 and g = 02~^. Other corresponding pairs of functions are \(^,O = </>rM/(^')-0i(O), 0(x,t)^g{f(x).cj>,{t)}> and M^^0 = <^r^J4^r\^4^|, 'f(x)cf>,{t) + l] e{x,t)=g f{x) + (f>,{t) so that in tJie Theory of Integral Equations 181 8. A further extension is obtained when the kernel k includes more than one parameter t; thus let /i (x) K [6 {x, ti , Q} dx = -f, (^1 , t,), /„ (x) K [d {x, t„ t^] dx = yfr. (t, , t.^, J a.2 \ fi (!/) « [^ {!/> f^ (^> ii> Q> V {x, ti , t)}] di/ J a, = -v/tj \/j, {x, ti, t^, V {x, ti, t.^\ and f Vi (//) « [^ y^ f^ (•'•' ^i> Q, V {x, t, , f,)}] dy = \/ro [^ {x, t,, ti), V {X, t,, Q). Now consider /i (•^-'O -^/^a [/A (*', ^1, t;), V {x, ti, t^)} dx = f V"i (^) ( I ' /3 (i/)/^ [0 [y, ti (,*•, ^x, t^}, V {x, t, , t,)}] dy) dx. ■J tti ^ ■ fl2 ' This double integral is equal to if ^ {y/, /x (.r, t„ t,), V {x, t„ t.^} = e [x, iM (y, t„ t.;), V (y, t„ L)}. Now suppose fl (X, t„ «,) = <^,-' [f{x) + (/>! {t,) + (/>! (^2)}. ^ {x, t„ L) = </)-! \2f(x) -f- (/>! (^0 + </), (t,)} ; then 6^ {y, /j.{x, ti, t,), v{x, t^, t,)} = </>-! {2/(2/) + 2/ (*■) + (/), (t,) + 01 (t,) + (/), (^0 + 0. (g). This is symmetrical in x and y, so that we may write /^(^■, ^1, ^2) = 0rM/3(*')+ 03(^1, 4)j, /. C^-, t„ t,) = 0,-^ 1/4 (^0 + 04 (^1, ^2)}, (a^, t„ t^ = g [f, (x) +f, (x) + 01 {t,) + 0, (QK leading to 5'{/3(^)+/4(i/)+/;(*')+/4(^O+03(^l, ^ + 04(^1, t.^\, 182 Major MacMahon and Mr Darling, Reciprocal Relations which is symmetrical in x and v/ ; and hence it follows that I J[(x)ylrn {/m(x, ti, L), v{x. t^, ig)} dx ■ ' a, = I fo,{x)'\^i {/"-(*■> t\, ^2)) v{^> ^1) ^2)) dx. As a particular case we may write ^L {x, t, , L) = cf^r' (./X^O + ^1 (A) + ^1 (^2)}, V (X, t, , t.;) = (/),-' {/(*■) + 02 (^1) + C^2 (4)}, (x, t„ Q = g {2/(*0 + 4>, (t,) + 4>, (01, and again ^l{x, t„ Q = ct>-' {^,f{x) + (i>(Q + cj)(t,)], {x, t„ Q - ct>-' {/(x) + (^0 + (/) (t,)}, the case where /x i^ y and each resembles as much as possible. It is evident that the case in which the kernel includes any number of parameters may be treated in the same manner and presents little difficulty. 4. The method may also be extended to double integrals. Thus let /i {^> y) K^ [^ (*'. y> ii> 4)1 dxdy = f^ (t,, to), J »! J a,' /■2 (^-^ y) « [^ (^, 2/. ii, 4)1 dxdy = -f . (^i, 4) ; [b, rb,' then / /i {cc, y) i/r^ {/x (^■, y, t^, 4), ^ (a-', y, 4, 4)} c^^'f^^/ J «, >/ a,' 6/ /2(^, 2/)'fi l/^(«> y. 4, 4)> ^(^S ^» 4, 4)1 dxdy if ^ {^r, w, yu, (a;, y, ^j, 4), v {x, y, t^, t^)] = [x, y, IX {z, w, ti, t.^, V (z, IV, ti, 4)1- If A, B, G, D, E be functional symbols, one solution is ix{x, y, t„ O = A-' [B{x, y) + C {t„ t,)] v{x, y, t„ t,) = D-^[B{x, y) + E(t„ t,)} (x, y, t„ t,) = B (x, y) + kA (t,) + ^D (t,). in the Theory of Integral Equations 183 5. Let US next consider the case of three integral equations !'\f\{x)K{dU;t)}dx = f,{t), J «, /; (a:) K [e {x, t)] dx = ^/r, (t), J a,, r\f,(x)K{d(x,t)]dx=f,(t). We have 'yA''^)f.{OOr,t)\ir,{d{x,t)}dx\ = r Mx)^}r,{e{x, t)}f,{e(x, t)] dx\ (11), = fV;cr) ti {^(^'> 01 ir,{0(x, t)} dx ] if certain conditions are satisfied. For '"' f\{x)y^,[e{x,t)]ylr,[e{x^t)]dx = I '' ./; i-'^) f ' /. (z/) '^ [^ (>/> t)\ dy f V; {z) K\e{z, t)\ dzdx, and the equalities (11) will hold good if, for example, k (x) = x'^ and d[y,d{x,t)\.e[z,e{x,t)] is unaltered by the circular substitution {xyz). Now suppose that ^ e{x,t)^f{x)t-^ (12;; then [y, {x, t)] . 6 {z, d (x, t)\ =f(y)f(z) {x, t) Hence if k(x) — x'^ the relation (12) satisfies the conditions. The generalisation to the equality of n integrals is apparent, and in that case 0(x, t)=f{x)t''-'^ is a solution. We have also /i (.'•) f; 1^ (*•, 0} ^3 {^ (.'<-■, t)] dx fb, = f, (.'/;) yjfs {\ (x, t)} ^fr, {X (x, t)] dx J rta 184 Major MacMahon and Mr Darlwcf if X {x, t) =f{x) t'\ e (x, t) = { f{x)Y'-'t''', and in particular if X {x, t) =f{x) t^, e {x, t) = [\ {x, t)Y-''. A solution may also be obtained when k (x) = [f, in which case K[e[y,e{x,t)\'\.K[d{z, 6'(^, 0}] = e^'•^'^^''■'*^^'^^'''^^''•*^^• Putting d{x,t)=f{a^ + lt, we have e [y, d (x, t)] + e{z,e (x, t)] =/(y/) +f(z) + f(x) + ^t, which is of the symmetrical form required. 6. In the cases investigated above the kernels of the several integral equations have been functions of the same form. It is, however, easy to extend the method to the case where the kernels are functions of different form. Thus if /i (x) /ci {0 (x, t)\ dx = -v/tj (t), bo /a (x) K. {6 (x, t)} dx = ylr.2 (t), we are led to the condition K, [6 [y, \ (x, t)]] = K, [6 [x, \ (y, t)]]. Case 1. Let Ki(z)—z, k2{1/z) = z; then the condition becomes 0{y,X(x,t)].d{x,\(y,t)}^l; a solution of which is d(x, t)^xWi^\ (A (OK % {0(0, F{^)i where \ {x, t) = (jr^ F {x), and ;j^ is any function. Case 2. Let k^ (z) = z, k.,(-z)^z; then the condition is 0{y,X(x,t)} + e{x,\{y, 01=0; a solution of which is d (x, t) = x {F{x)., c/, (01 -x\<^ (0. F{x)\. Case 3. Let k^ {z) = z\ k,_ (s) = (1 - ^0'" ; then the condition is \0[y,\{x,t)\J^ld{xMy^m=^\ a solution of which is e {x, t) = x[F{x), </>(0} ^ lixWia^), 4>{i)]y + (%{</> (0, F{x)]yr- Prof. Stanley Gardiner and Prof. Nuttall, Fislt-freezing 185 Fish-freeznuj. By Professor Stanley Gardiner and Professor Nuttall. [Read 18 February 1918.] Fish-freezing commenced in 1888, in connection with Western American sahnon. It was started to preserve the excess of fish caught during the runs for canning in the shick season. The busi- ness proved so profitable that fish began to be distributed all over North America and exported to Europe, the chief market in the latter being Germany. The fish are, as soon as possible after catch- ing, brought to the refrigerator, frozen dry on trays at about 10° F., this process taking about 36 hours. The fish then are drawn into a room at 20" F., where they are dipped into fresh watei', their sur- faces being thus covered with a glaze of ice. They are then packed in parchment paper in strong wooden cases and exported to Europe by refrigerator cars and cold storage steamers. The process is also applied to halibut, haddock, cod, pollack and various flat fish in America. It succeeds in preserving the fish for an indefinite period of time, but the product breaks up in cooking, tending to become rather woolly and loses flavour and aroma. To meet this a fresh process has now been developed, freezing the fish in brine consisting of about 18 per cent, of salt at a tem- perature of 5° to 20" F. The brine is an excellent conductor of heat and cold. A large fish freezes thoroughly in three hours, a herring in twenty minutes. After freezing, the fish returns to the same condition as it was when placed into the brine; there is no woolliness, no loss of flavour or aroma. The difference is due to the fact that, whereas in dry freezing there is a breaking up of the actual muscular fibres, due to the formation of ice crystals, in brine freezing the ice crystals are so small that the muscular fibres are entirely unaffected and on thawing return to the normal. In neither form of freezing is there danger from moulds or putrefaction if the fish is stored below 20^ F. The authors advocate the creation of a vast store of frozen her- rings against time of scarcity, instead of the herrings being pickled and exported. The value of fish as food is weight for weight about the same as meat, containing the same constituents. If the excess of the herring catch were stored in this way, there would be, on pre-war figures, a store of herrings in this country to meet the necessity for -albuminous food in the British Isles for at least eight weeks. 186 Mr Sahni, On the branching of the Zygopteridean Leaf, etc. On the branching of the Zygopteridean Leaf, and its relation to the probable Pinna-nature of Gyropteris sinuosa, Goeppert. By B. Sahni, M.A., Emmanuel College. (Communicated by Professor Seward.) [Read 20 May 1918.] ( 1 ) The supposed quadriseriate " pinnae " of forms like Staurop- teris and Metaclepsydi^opsis are tertiary raches, the vascular strands of the secondary raches (pinna-trace-bar, Gordon) being completely embedded in the cortex of the primary rachis. All Zygopterideae therefore have a single row of pinnae on each side of the leaf. (2) This revives the suggestion that Gyropteris sinuosa Goepp. is a free secondary rachis of a form like Metaclepsydropsis. (3) The genus Glepsydropsis should include Ankyropteris because: a. A fossil described in 1915 (Mrs Osborn, Brit. Ass. Rep., p. 727) com- bines the leaf-trace of Glepsydropsis with the stem of Ankyropteris, the leaf-trace in both arising as a closed ring. h. In G. antiqua Ung. also the leaf-trace arose similarly, as shown by a section figured by Bertrand {Progressus 1912, fig. 21, p. 228) in which a row of small tracheides connecting the inner ends of the peripheral loops represents those lining the ring before it became clepsydroid by median constriction. 3 The Structure o/Tmesipteris Vieillardi i)aw^. By B. Sahni, M.A., Emmanuel College. (Communicated by Professor Seward.) [Read 20 May 1918.] The most primitive (least reduced) of the Psilotales. Specifically distinct from T. tannensis in (1) erect terrestrial habit, (2) distinct vascular supply to scale-leaves, (3) medullary xylem in lower part of aerial stem. On Acmopyle, a Monotypic New Galedonian Podocarp. By B. Sahni, M.A., Emmanuel College. (Commimicated by Professor Seward.) [Read 20 May 1918.] Indistinguishable ivova. Podocar pus in habit, vegetative anatomy, drupaceous seed, megaspore-membrane, young embryo, male cone, stamen, two-winged pollen and probably male gametophyte. Chief differences: (1) seed nearly erect; (2) epimatium nowhere fi"ee from integument, even partaking in formation of micropyle; (3) outer flesh with a continuous tracheal mantle covering the basal two-thirds of the stone. PROCEEDINGS AT THE MEETINGS HELD DURING THE SESSION 1917—11)18. ANNUAL GENERAL MEETING. October 29, 1917. In the Comparative Anatomy Lecture Room. Dr Mark, President, in the Chair. The following were elected Officers for the ensuing year : President: Dr Marr. Vice-PresideiUs : Prof. Newall. Dr Doncaster. Mr W. H. Mills. Treas'itrer : Prof. Hobson, tSecretaries : Mr A. Wood. Mr G. H. Hardy. Mr H. H. Brindley. Other Members of Council : Dr Bromwich. Mr C. G. Lamb. Mr J. E. Purvis. Dr Shipley. Dr Arber. Prof. Bitfen. Mr L. A. Borradaile. Mr F. F. Blackman. Prof. Sir J. Larmor. Prof. Eddington. Dr Marshall. The following Communications were made to the Society : 1. On the convergence of certain multiple series. ByG. H.Hardy, M.A., Trinity College. 2. Bessel functions of large order. By G. N. Watson, M.A., Trinity College. 188 Proceedings at the Meetitujis 3. A particular case of a theorem of Dirichlet. By H. Todd, B.A., Pembroke College. (Communicated by Mr H. T. J. Norton.) 4. On Mr Ramanujan's Empirical Expansions of Modular Functions. By L. J. MoRDELL. (Communicated by Mr G. H. Hardy.) 5. Extensions of Abel's Theorem and its converses. By Dr A. KiENAST. (Communicated by Mr G. H. Hardy.) November 12, 1917. In the Comparative Anatomy Lecture Koom. Professor Marr, President, in the Chair. The following Communications were made to the Society : 1. Some experiments on the inheritance of weight in rabbits. By Professor Punnett and the late Major P. G. Bailey. 2. The Inheritance of Tight and Loose Paleae in Avena nuda crosses. By A. St Clair Caporn. (Communicated by Professor Bitten.) February 4, 1918. In the Comparative Anatomy Lecture Room. Professor Marr, President, in the Chair. The following Communications were made to the Society: 1. On certain integral equations. By Major P. A. MacMahon. 2. (1) Sir George Stokes and the concept of uniform convergence. (2) Note on Mr Ramanujan's Paper entitled : On some definite integrals. By G. H. Hardy, M.A., Trinity College. 3. Asymptotic expansions of hypergeometric functions. By G. N. Watson, M.A., Trinity College. 4. (1) On certain trigonometrical sums and their applications in the theory of numbers. (2) On some definite integrals. By S. Ramanujan, B.A., Trinity College. (Communicated by Mr G. H. Hardy.) Proceedings at the Meetings 189 February 18, 191S. In the Comparative Anatomy Lecture Room. Professor Marr, President, in the Chair. The following were elected Fellows of the Society : E. Lindsay Ince, B.A., Trinity College. S, Ramanujan, B.A., Trinity College. The following Communications were made to the Society : 1. Fish-fi-eezing. By Professor Stanley Gardiner and Professor NUTTALL. 2. Shell deposits formed by the flood of January 1918. By P. Lake, M.A., St John's College. 3. (1) Reactions to Stimuli in Corals. (2) Is the Madreporarian Skeleton an Extraprotoplasmic Secre- tion of the Polyps 1 By G. Matti^ai, M.A., Emmanuel College. (Communicated by Professor Stanley Gardiner.) 4. Notes on certain parasites, food, and capture by birds of Forficuhi cmricnlaria. By H. H. Brindley, M.A., St John's College. May 20, 1918. In the Botany School. Professor Marr, President, in the Chair. The following was elected a Fellow of the Society : C. Stanley Gibson, Sidney Sussex College. The following Communications were made to the Society : 1. (1) On the branching of the Zygopteridean Leaf, and its relation to the probable Pinna-nature of Gyropteris simiosa, Goeppert. (2) The Structure of Tmesipteris Vieillardi Dang. (3) On Acmopyle, a Monotypic New Caledonian Podocarp. By B. Sahni, M.A., Emmanuel College. (Communicated by Professor Seward.) 2. Asymptotic Satellites in the problem of three bodies. By D. Buchanan. (Communicated by Professor Baker.) CONTENTS. PAGE Extensions of Abel's Theorem and its converses. By Dr A. Kienast, Kusnacht, Zurich, Switzerland. (Communicated by Mr G. H. Hardy) 129 Sir Oeorge Stokes and the concept of uniform convergence. By G. H. Hardy, M.A., Trinity CoUege 148 Shell-deposits formed by the flood of Jamtary, 1918. By Philip Lake, M.A., St John's College 157 7s the Madreporarian Skeleton an Extraprotoplasmic Secretion of the Polyps? By G. Matthai, M.A., Emmanuel College, Cambridge. (Communicated by Professor Stanley Gardiner) . . . .160 On Reactions to Stimuli in Corals. By G. Matthai, M.A., Emmanuel College, Cambridge. (Communicated by Professor Stanley Gar- diner) 164 Notes on certain .parasites, food, and captitre by birds of 'the Common Earwig (Forficula auricularia). By H. H. Brindlet, M.A., St John's College 167 Reciprocal Relations in the Theory of Integral Equations. By Major P. A. MacMahon and H. B. C. Darling 178 Fish-freezing. By Professor Stanley Gardiner and Professor Nuttall 185 On the branching of the Zygopteridean Leaf, and its relation to the pro- bable Pinna-nature o/Gyropteris sinuosa, Ooeppert. By B. Sahni, M.A., Emmanuel College. (Communicated by Professor Seward) . 186 The Structure of Tmesipteris Vieillardi Bang. By B. Sahni, M.A., Emmanuel College. (Communicated by Professor Seward) . . 186 On Acmopyle, a Monotypic New Caledonian Podocarp. By B. Sahni, M.A., Emmanuel College. (Communicated by Professor Seward) . 186 Proceedings at the Meetings held during the Session 1917 — 1918 . . 187 (>.r ^-.^ PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY VOL. XIX. PART V. [Michaelmas Term 1918 and Lent Term 1919.] TfiSTi m NOV 2 01946 ^J] AT THE UNIVERSITY PRESS AND SOLD BY DEIGHTON, BELL & CO., LIMITED, AND BOWES & BOWES, CAMBRIDGE. CAMBRIDGE UNIVERSITY PRESS, C. F. CLAY, MANAGER, FETTER LANE, LONDON, E.G. 4 1919 Price Three Shillings net April 1919. NOTICES. 1. Applications for complete sets of the first Seventeen Volumes (in Parts) of the Transactions should be made to the Secretaries of the Society. 2. Separate copies of certain parts of Volumes I.— xi. of the Transactions may be had on application to Messrs BowES & Bowes or Messrs Deighton, Bell & Co., Limited, Cambridge. 3. Other volumes of the Transactions may be obtained at the folloAving prices: Vol. xn. £1. 10s. 6d.; Vol. xm. £1. 2s. Gd. Vol. XIV. £1. 175. 6d. ■ Vol. xv. £1. I2s. 6d. ; Vol. xvl £1. IO5. Od. Vol. xvn. £1. 2s. 6d; Vol. xviii. £1. Is. Od; Vol. xix. £1. 5s.M. Vol. XX. £1. 10s. Od; Vol. xxl £1. 14s. Od.; Vol. xxn. No. 1, Is: 6d. No. 2, 2s.; No. 3, Is. 6d.; No. 4, Is. 6d.; No. 5, 2s.; No. 6, Is. 6d. No. 7, 2s.; No. 8, 2s.; No. 9, 2s.; No. 10, Is.; No. 11,2s.; No. 12,3s. 6d No. 13, 2s.; No. 14, 3s. 6d. 4. Complete sets of the Proceedings, Volumes L— xviii., may also be obtained on application to the Secretaries of the Society. 5. Letters and Communications for the Society should be addressed to one of the Secretaries, Mr G. H.* Hardy, Trinity College. [Mathematical.] Mr A. Wood, Emmanuel College. [Physical] Mr H. ff. Brindley, St John's College. [Biological] 6. Presents for the Library of the Society should be ad- dressed to The Philosophical Library, New Museums, ' ^ ' Cambridge. 7. Authors of papers are informed that the Illustrations and Diagrams are executed as far as possible by photographic ''process" Avork, so drawings should be on a large scale and on smooth white Bristol board in Indian ink. 8. Members- of the Society are requested to inform th Secretaries of any change of address. e PROCEEDINGS OF THE On Certain Trigonometrical Series which have a Necessary and Suffijcient Condition for Uniform Convergence. By A. E. Jolliffe. (Communicated by Mr G. H. Hardy.) [Received 1 June 1918; read 28 October 1918.] 1. The series S^n sin nd, where (a,i) is a sequence decreasing steadily to zero, is convergent for all real values of 6, and it has been proved by Mr T. W. Chaundy and myself* that the series is uniformly convergent throughout any interval if /?a,i-*-0, this con- dition being necessary as well as sufficient. A generalization of this theorem is as follows : If (Xn) is tt sequence increasing steadily to infinity and (an) is a sequence decreasing steadily to zero, then the necessary and suffi- cient condition that the series Sa„+i(cos A,.,i^ — cosX,i_^i^)/^, which is coyiver gent for all real values of 6, shoidd be uniformly convergent, throughoid any interval of values of 6, is Xnan^b. I shall prove rather more than this, viz. that the condition is sufficient for uniform convergence and necessary for continuity. When ^ = 0, it is understood that the value assigned to any term of the series is its limit as 6 tends to zero, so that for ^ = the sum of the series, which I shall denote by Sun, is zero. Since, by Abel's lemma, i Wn+i + ... + Up\< 2an+i/0, it is evident that there is continuity and uniform convergence throughout any interval which does not include ^ = 0, so that it is only intervals which include ^ = that we have to consider. * Proc. London Math. Soc. (2), Vol. 15, p. 214. VOL. XIX. PART V. 14 192 3Ir Jollife, On. Certain Trigonometrical Series which have A very trifling modification of the analysis which follows will show that, so far as an interval which includes ^ = is concerned, the same results hold for the series 2a„+i (cos \n^ — cos \n+i^) cosec 1)9, where h is any fixed number. If either |(X„+i — A,„) or ^(Xn+i + X„) is always an integral multiple of some fixed number b, then X„. differs by a constant from an integral multiple of 26, and the series is periodic with a period Tr/b. In this case the results which are true for an interval which includes ^ = are true for any interval. The particular series 2a«sinn^ corresponds to b = ^, X,,j=» +^. 2. Since the sum of the series when ^ = is zero, it follows that, for continuity at 6 = 0, the sum of the series, when 6 is different from zero, must tend to zero as 6 tends to zero in any manner. In particular, the sum when 6 = 7r/2\„, must tend to zero, as n tends to infinity. When 6 = 7r/2\,j, let m be the integer such that Xm-i^ $ TT < \n6- It should be noticed that we may have m — l = n, and that When m—1 >n, cos \p-i6 — cos \p6 is positive, so long as p is not greater than ?7i — 1, and consequently 0{U^ + Uo+ ... +Um-i) > a-n (cos Xi6 — cos \n^) + «w-i (cos X„^ - cos A,„,_i 0). Also, by Abel's lemma, 6 (Um + U,n+i + ... + U,n+g) > "m (cOS X^O - 1) for all values of q. Hence the sum of the series is greater than [an COS \6 - {a,n-i - a„i) cos Xn-iO - am]IO, which, since a,„,_i ^ a^ and cos Xm-i ^ is negative, is greater than (an cos X^d- a„,)/6 = 2Xn («« - «,»,)/-"■ + bn, where bn denotes a„(l —cosXi6)/d and consequentl}^ tends to zero as n tends to infinity. When m — 1 = ?i, we can divide the series up into (U^ + U2+ ... + Un) + (Um + Um+i +...), and, noticing that cos X^-i 6 = 0, we see that the sum is greater than (an cos Xj^ — am)/ 6, as before. a Necessary and Suficiejit Condition /or Uniform Convergence 193 Hence the sum of the series, when 6 = 7r/2X,i, can in no case tend to zero, as n tends to infinity, unless X,i (a,i — a,n) -^ 0. If X)i (an — a,n) -* 0, then, given any positive number e, we can find V such that X,i (a„ -«,„)< e for n^v. Denote ni by (n, 1) and let (/i, 2) be the integer formed from (n, 1) in the same way that {ii, 1) is formed from n, and so on. Then ((n - «n, 1 < e/\n , (/„, i - ttn, o < e/X^, i , , for n ^ V, and by addition Un < € (1/X,„ + l/Xn,i + ... + lAn,i>) + (hi,p. Now X,i_i > 2\,i,, \,i,^2 > 2X„_i, and so on, so that «„ < 2e/\„ + cin,p. Also when % is fixed we can choose p so that an^p < e/Xn, and we shall have therefore Xnttu < Be (n ^ v). Hence XnCin^O is a necessary condition that the sum of the series should be continuous at ^ = 0, and a fortiori that it should be continuous throughout any interval which includes ^ = 0. 3. To show that this condition is sufficient for uniform con- vergence in any interval, and d fortiori for continuity at any point, it is sufficient to show that I Un+i+ ... +Up\ <AM, for all values of 6, where A is some fixed number and M is the greatest value of Xt-a^ for r^n + 1. Since the value of the series is changed in sign only by changing the sign of 6, it is sufficient to consider positive values of 6 only. By Abel's lemma I Un+i + ... +Up\< 2an+i/d < 2Xn+ian+i/7r, if ^ ^ tt/Xji+j. If ^ ^ T^/Xp, every term of Un+i + . . . + Up is positive ; and, if u,. is one of these terms, Ur ^ M (cos Xr-iO — COS X,.^)/X,.^ ^ 2if sin l(Xr - X,_i) 6 sin h{X, + X,_,) O/X^d < MO {Xr - X,_i), so that Un+\ + . . . + Up < MOXp < ttM. If irjXp < 6 < 7r/X„i+i, let Tr/Xq+i < 6 ^ 7r/Xg, and divide lln+i + ... ^Up up into Un+i + . . . Uq and Uq+i + ... +Up. Then | Un+i + ... + Uq\< ttM, and I Uq+i + ... +Up\< 2aq+^l0 < 2aq+iXq^,/7r < 2ilf/7r. Therefore j Un+i + ... -\- tip\ < {ir + 2/7r) M. U~2 194 Mr Jolliffe, On Certain Trigonometrical Series which have Hence for all values of 6 I Un+^ + ... + ?i^ i < (tt + 2/7r) M, and therefore the condition X,ia„ -*- is sufficient for uniform con-"" vergence and a fortiori for continuity in any interval. il 4. If 'Xn tends to infinity more rapidly than n, the series does not seem to be capable of any modification. If A,^ = ^?2 + B, where A and B are fixed, we obtain practically the series 2a„ sin nO and nothing more. But when X^ tends to infinity more slowly than n, and with a certain measure of regularity, the theorem can be transformed in an interesting manner. We have, in fact, the following theorem : If \n tends steadily to infinity and \n+i — ^n tends steadily to zero, then the necessary and sufficient condition for the uniform convergence of Zft^i (Xji+i ~ Xji) sm A.,jC7 is Xndn -* 0. As before, I prove rather more, viz. that the condition is suffi- cient for uniform convergence and necessary for continuity. This theorem will follow at once from the theorem just proved, if we can show that the series Sa« {(cos \nd - cos Xn+i0)/6 - (Xn+i - Xn) sin XnO] is uniformly convergent throughout any interval. Here the con- dition Xn+ittn ■-* is equivalent to X^an -^ 0, since Xn+i — X^ -^ 0. We can verify immediately that cos y — cos X — sin y sin (sc — y) = sin^ h{x — y) (cos y — cos x) + ^ sin {x — y) (sin x — sin y). It follows by Abel's lemma that, if Xn+i — X^ decreases steadily, so that sm{Xn+i — Xn)d and sin ^(Xn+i — Xn) decrease steadily, then S {cos Xn — cos X,i+i 6 — sin {Xn+^ — Xn) sin Xnd] -rt+1 < 2 sin^ 1 {Xn+i ~Xn)6 + sin {Xn+i - X«) 0- Also, given any e, we can choose v so that Xn+i — X^ < e for n ^ v. Hence, for n'^v, we have p 2 {cos XnO — cos Xn+i6 — sin (Xn+i — Xn) 6 sin XnO] w+1 < 2e2^2+66'<3e^, for any interval of values of 0, if e is sufficiently small. a Necessary and Sufficient Condition for Uniform Convergence 195 It follows also that i -& I I Z sin (X^+i - \,)6 sin X,^^ < 2 + Se^ < 3, |w + l I for n ^ V. Now (X„+i - Xn) 6 cosec (X.,1+1 - X„) ^ - 1 decreases steadily to zero, and is less than Therefore 2 6{\n+i - \i) sin Xu^ — 2 sin (X„+i — X„)^ sin XnO n+l jj+1 Hence < e'6' p % {(cos X,i^ - cos \n+i6)ld - (Xn+i - ^/i) sin \nd} n+l < 3e + e^O < 4e (n ^ i^). Hence the series 2o„ {(cos Xn^ - cos \n+id)l6 - (X„+i - X„) sin X„^} is uniformly convergent throughout any interval, and hence the result enunciated follows. 5. If instead of a sequence (X„) we have a function X (x) such that, as iT-^oo, X{a;) increases steadily to infinity and \'{a;) de- creases steadily to zero, then Xn+i — ^n decreases steadily to zero. The series 2 (X'„ — X^+i + X,i), where \'n denotes the value of X' (x) when x = n, is convergent and is moreover absolutely convergent, since X',^ — \n+i + Xn is positive. Hence, by Weierstrass' M test*, the series Sa,i (X'„ — X,i+i + X,i) sin X„^ is uniformly convergent throughout every interval. It follows then that ajj,X„ ^ is the necessary and sufficient condition that the series Sa^X'„ sinX,j^ should be continuous at every point and uniformly convergent throughout every interval. In particular the series 2a,i?i'~^ sin (n'^), where t is any real number not exceeding 1, is continuous at every point and uniformly convergent throughout every interval if n^a^-^O, this condition being necessary as well as sufficient. * Bi'omwich, Infinite series, p. 113. 196 Mr TurnbuU, Some Geometrical Interpretations Some Geometrical Interpretations of the Concomitants of Tivo Quadrics. By H. W. Turnbull, M.A. (Communicated by Mr G. H. Hardy.) [Received 6 July 1918; read 28 October 1918.] § 1. In the Mathematische Annaleii, Vol. LVi, Gordan has given a system of 580 invariants for two quaternary quadratics. It appears that by carrying out the processes of reduction a little further, the irreducible forms can be shewn to number 123 at most. That is to say, the system is about as complicated as the ternary system for three conies which Ciamberlini* first established. It is therefore worth while to give geometrical interpretations to members of the system for two quadratics. In the following pages about a hundred of them are shewn. The geometrical significance of the residue appears to be remote. Using the classification introduced by Gordan, the numbers of forms of each type J which have not been reduced are shewn in the subjoined Table. The rows of the Table give the numbers of forms of each particular order in the three sets of coordinates x, p, u, which define points, straight lines, and planes respectively. Detailed lists of these forms will be found at the heads of the paragraphs which deal with separate types. References Order in x, p, u J' J2 Type J3 J' J5 Total 1 i 1 §5 Invariants 5 1 5 1 §6 Covariants 4 1 5 ! 5) Con travari ants 4 1 5 1 §§ 7-14 Complexes 6 1 1 4 4 16 ' § 15 Mixed (1, 0, 1) 1 2 3 §21 (1, 0, 3) 4 4 1 5) (3, 0, 1) 4 4 i §22 (2, 0, 2) 1 6 7 ! §§ 17-20 (0, 1, 2) 1 6 1 1 9 ! 5) (2, 1, 0) 1 6 1 1 9 ! § 18 (0, 3, 2) 1 1 i )) (2, 3, 0) 1 1 ; §23 (0, 2, 2) 4 4 J5 (2, 2, 0) 4 4 § 16 (1, 1, 1) 4 12 16 ! §23 (1, 2, 1) 12 6 6 24 1 » (2, 1, 2) 4 4 1 5) (3, 0, 3) 2 2 Totals 21 7 71 12 12 123 Ciamberlini, Giornale di Matematiche, Vol. of the Concomitants of Ttuo Quadrics 197 Notation. § 2. Let Ui, Uo, Ui, u^ be plane coordinates; and let v, w be cogredient with u. We may then typify line coordinates by Pij = (uv)ij = UiVj - vflij, (i, j = 1, 2, 3, 4) ; and X or point coordinates by cc^ = (uvtu)23^ and three similar ex- pressions for X2, x-i, ^4, Then the symbolic system of Gordan can be exhibited as follows. Let the point equations of the quadrics be /=<''^' = «/'=..., and /" = h^- = bj^ = .... Let the line equations be u=iApy = {A'py = ..., U' = {Bpy = (B'pf=.... Let the tangential equations be 2 = uj = uj = ..., X' = M,3" = ?'j3'^ = . . . . Then the connections between the symbols are A = (ta', B = bb', a = aa'a", ^ = bb'b". And all concomitants of the system can be expressed in terms of TO P'l'OT'^ 4, {dd'p), (dd'd%), {dd'd"d"'), where d signifies a or b. But the irreducibles can be shewn to be composed of the following types, (la, b^\ Ucc', b^, (Ap), (Bp), iia, u^, (abp), (Abu), {Ban), (AB), a^, ba, (A/3x), (Bax), (a^p), (AB)', F„ F, ; where (A^x) — a^a^' — a/a^ = «^(V , say, and («/3/>) = UaVp — u^Va = luvp , {AB)' = (Ahu)k', F, = {(ibp) a^' - (abp) a^ = (Abp^). Reciprocation. § 3. By interchanging the symbols (a, a), (b, /S), (u, x) without altering vl, jB or ja, we obtain from any given concomitant the reciprocal form. Thus the bracket factors (A^x) and (Abu) are 198 Mr TurnhuU, Some Geometrical Interpretations reciprocals. So also would be (ahcu) and (a^ya;), the latter being of a type not arising for less than three quadrics. Though the process by which Gordan arrived at such symbols as {A0iv) and (a/3p) was purely analytic, it is interesting to observe that from the geometrical point of view such analytical results were almost inevitable. Below will be found several examples of the use of this principle of duality. The fundamental forms. § 4. A brief investigation would reveal the importance of the following forms, to which special symbols are therefore attached. Let / denote a^-, f denote hx~, S „ u^", %' „ %^ n „ {Ap)\ W „ {Bpf, k „ {AjBxf, k' „ {Baxf, X " {Ahuf, X » {Bauf, 77^2 „ (abpf, Dia „ (ct^pT, and C „ {AB)(Ap)(Bp). Some account of these forms may be found in Salmon, Analytical Geometry of Three Dimensions (revised by Rogers), Yol. i, Ch. ix. There %, %, %', S' are denoted by a, r, r', a' (§ 214) : A^, k' are the T, r of § 215 ; n, 7ri2, O' are the ^, ^^, ^' of § 217. Invariants. § 5. The irreducible invariants are a^, bj, (ABf, a^^, bj or the A, 0, ^, @', A' of Salmon, § 200. In fact, there are no other types, for two quadrics of any dimension n, than the n + 1 co- efficients of X, in the discriminant of The five covariants and contravariants. I 6. The covariants (w + 1 in the case of w-ary forms*) are the four quadrics /, k, k', f and the quartic J defined by apbaaj)x (AB) (A^x) (Bax). This is indeed the jacobian of the four quadrics, and represents the four planes of the self-conjugate tetrahedron (c£ Salmon, § 233). * Of. Turnbull, ' Quadratics in n variables' (pp. 235-238), Camb. Phil. Trans., Vol. XXI, No. viii. H of the Concomitants of Two Quadrics 199 Correlatively, S, %, x, S' are the four quadrics in u which make the system of contra variants together with j = a^haUaU^ i-^B) {Ahu) {Ban). This latter represents the four vertices of the same tetrahedron. In fact, the jacobian of u^, u^, (^Ahuf, (Bau)" is (a/3 Ab Ba) Ua,u^ {Abu) {Bau), where A = a a", say ; expanding the first bracket this becomes iijd^" (bBa) M - d/ b^ {ci"Ba) M + &„d/ {ci"Ba) M, where each term represents two, with a, a" permuted, and M is short for UaU^{Abu){Bau). But the factor Ua is reducible to Ua' (Gordan, il, § 6) ; which means in this case that the symbols u of the factors u^,, (Abu) would be bracketed. Hence the product in- volving cia is zero. Thus the jacobian is equal to baCip'(d"Ba)M, = b^a^ {a"Ba')M+ba.h' {a"ab"a) M (if B = b'b") = — ba a^ (a'a'B) M (as before) A correlative reduction applies to the case of /. The complexes. § 7. A complex is a function of ]), or line coordinates, but not explicitly of ii or x. There are eight quadratic and eight cubic complexes in the system. The quadratics are (ApY or n, (Bjyf or TI', (abpf or tt,.,, (a/3p)' or IIi., {AB){Ap){Bp) or C, {abp){a^p)a^b„., F;' and Fi. Differentiation. I 8. Let p be any symbolic product belonging to the whole dP system; then — - {i=l, 2, 3, 4) would be composed of terms each OXi with one odd S}?mbol Qj or 6,: left over. Thus the four symbols ^:— may be considered as the coordinates of a certain plane. For dxi ^ ^ example the coordinates of the polar plane of a point (x) with regard to a^- are («a;«i, axU.,, a^a^, a^ai). Likewise ^ — would give a set of point coordinates. 200 Mr TurnhuU, Some Geometrical Interpretations Again, ^ — would give six quantities which would symbolise the coordinates of a certain linear complex : and, m some special cases, the coordinates of a straight line. For example, is a useful way of denoting the six quantities (Aj)) A^ {i,j = 1,2, 3, 4), which represent a straight line, since they satisfy the identical re- lation existing between the six ^^-coordinates of a straight line. Li7ie coordinates. § 9. This identity satisfied by line coordinates (p) is ^pijPM=0 (1), which we denote by co (p) = 0. Symbolically, the condition that two lines p and q should intersect is {pq) = 0. If p is the line common to two planes u, v, and q is that common to u', v', then this condition is (iivuv) = 0. If two lines p, q intersect, then Kpij + Xqij represents the co- ordinates of any line of the plane p, q passing through the connnon point of p, q. Since the line p touches the quadric / if {Ap)- = 0, it follows that the line (k, \) touches this quadric if k' (Apf + 2k\ (Ap) (Aq) + X2 (^Aqf = 0. Hence (Ap)(Aq) vanishes if p intersects the conjugate of q in f; for then p and q are harmonic conjugates of the two tangents to / r)TT in this pencil of lines (k, \). This shews that the coordinates ^::— , op i.e. (Ap)Aij, are those of the line conjugate to p in the quadric f. Analytically it is evident that these coordinates represent a line and not a linear complex, since they satisfy the required condition (1). In fact (Ap) (A'p) (A A') = A {AAy <o (p^. But the left member of this equation is the symbolic equivalent of substituting (ApJAy forp in (1): which proves the statement. Complexes and their polars. I 10. Let (Dpy = represent one of the quadratic complexes of § 7. Then (Dp) By gives the coordinates of a linear complex * Cf. Gordan, ii, § 6. of the Concomitants of Two Quadrics 201 pola7' to (p) in (Dpf. If (p) is a member of the complex (Dp)-, the polar is called the tangential linear complex. The complex (Dp)Dij is not usually a special linear complex. The preceding case was exceptional. For in that case the quad- ratic complex was (Apf = 0, and all the rays touched the quadric/. The comptlexes tt,., IIi.,, C, (ahp){oL^p)a^ba. § 11. The principal quadratic complexes which occur are 'jr,, = (abpy, U,,= ial3pY, G^{AB){Ap){Bp). The two former are well known, ttjo being the aggregate of lines cutting the quadrics harmonically, and rTi. being the correlative complex. The third, G, is the complex of lines whose conjugates, in /and /' respectively, intersect. For the conjugate of -p in / is {Ap){A) and in /' is {Bp){B). Again, G is satisfied too by the singular lines of the complex ttj.,. For if p is a line of {ahp)- = 0, its tangent linear complex (§ 10) is {ahp) (abq) = , q representing current coordinates : further, jj is a singular line if this tangent linear complex is special, i.e. if (abp){aba'b')(a'b'p) = 0, which reduces to (AB) (Ap) (Bp) = 0. Correlatively C also contains the singular lines of the complex ITi.,. Again, the singular lines of the complex G belong to the com- plex (abp) {a^p) Ufiba. This follows in the same way as in the above case. But a more direct interpretation of this last form arises from the apolar* condition for two linear complexes ; if the polar linear complexes of a line (p) with regard to ttjo and ITis are apolar, then (abp) (a^p) cipba vanishes. The complexes F{~, Fi. § 12. Besides the original complexes {Apy and {Bpy, and the four complexes of § 11, there remain two more quadratics, i'V and F^. Just as {abpf is the harmonic complex between / and /', so F^^ is the harmonic complex between /' and k, while F^ is that between / and k'. To prove this we build up a form (/', ky from /' and k, in the same way as (/ f'f, i.e. {ahpY, is built from / and /. Then {f',ky^{b,^,{ABxrf = (bx', 2ft^"^aa;'-— la^a^'axaxY = 2ap^ (abpf — 2a^a^' {abp) {abp) = [{abp) a/ - {a'bp) a^J = F,^ (| 2). * The linear complexes {Dp) — 0, {Ep) — are apolar if {DE) - . 202 Mr T'urnbull, Some Geometrical Interpretations The eight cubic complexes : F, (ahp) K (Bp), F, (oL^p) ap (Bp), F, (ahp) K (AB) (Ap), F, {oc^p) a^ (AB) (Ap) ; and four involving F^. § 13. If aj^, hx, Cx are three quadrics, the lines p cutting them in involution are given by the cubic complex {hep) (cap) (ahp) = 0. Let us denote this complex by the symbol {a^, h^', c^)- Then {f,f', k') may be formulated, and we shall have (ax% hx^, k') = {{ahp)axb^, {Baxff = ((abp).axbx, - 2b:bx"ba"bx + "^^'b^'J = - 2 {abp) {ab"p) (bb'p) bjb^' + 2 {abp) (ab'p) (bb'p) b^"\ The second term is zero, since b, b' are interchangeable. The first term is F^ (abp) ba. (Bp) to a constant coefficient. Reciprocally (S, ^',x) represents F.2 (a^p) a^ (Bp) ; and there are two like forms involving Fi. § 14. This leaves four complexes such as F2 (abp) ba (AB) (Ap) to be interpreted, but the geometrical significance is not at all immediate. If however we write (/, /', k') as (Bpf, then the line (p) has a polar linear complex (npf(Dq) = 0. And if q = (Ap)(A), i.e. if q is the conjugate line of p in the quadricy, then (Dpy(DA)(Ap) = 0. This latter form is equivalent to F2 (abp) ba. (AB) (Ap) : and similar results follow for the other three forms, as in § 13. The mixed concomitants. § 1.5. To denote the order of a form, let (i,j, k) mean that the order is i in x, j in p, and k in u. Then there are three linear forms (1, 0, 1) and sixteen linear forms (1, 1, 1). The three linear forms (1, 0, 1) : Up % ax, Uahabx, (AB)(Ab u ) bx'. If (v) is the polar plane of a point (x) in /, then (v) = ax(a). Hence Usa&av. = is the condition that a conjugate plane of u in/' of the Concomitants of Two Quadrics 203 should be the polar of x in f Similarly for Uahahx- Again, {AB){Ahu) h^ vanishes if the polar of x in/' is conjugate to u in ;)^, i.e. in {Ahiif = 0. The sixteen forms (1, 1, 1) : two like ax {Ban) (Bp), two like Ua {Bax) (Bp), „ a3^a^{a^p)ua, „ „ ax(abp)baUa, „ „ ax{Bau)(AB)(Ap), „ „ u^(Bax){AB)(Ap), „ (abp){Abu)(A/3x)ap, „ „ (oL^p)(Abu)(A/3x)ba. § 16. The polar plane of a point (x), with regard to /, meets a plane (u) in a straight line whose coordinates are (au) a^. If ax(aBu){Bp) = 0, this line cuts the conjugate of p in /'. Let us denote this relation by (f^, 11'). The significance of the reciprocal of this, viz. (Stt, n'), is obvious. This accounts for four forms since either /or/' can be employed. Suppose we word this relation differently and say that the plane {u) cuts the polar of (x) in/ in a line which lies in the linear complex polar of (p) in II' : then a like meaning attached to (fx, His) interprets tta;% (^^p) Ua.. So also i-u, 'ir^^ = ax{abp)ba.Ua, (/,, {AB) {Ap) (Bp)) = a, (aBu) (AB) (Ap), with reducible terms, and (tu, (AB) (Ap) (Bp)) = lu (Bav) (AB) (Ap), while (kx, TTia), (Xu> II 12) denote the remaining two forms of the above list. To complete the set of sixteen forms we merely write %' for S, k' for k, and so on. The polar quadrics (0, 1, 2) and (2, 1, 0). I 17. There are nine forms of order (2, 1, 0), any one of which represents a quadric associated with a given line (p) ; or, from another point of view, represents a linear complex associated with a given point (x). The simplest of these is (abp) ajj^. Let this denote the polar quadric of the line (p) with regard to the system /*+ X/'. It is convenient to use the symbol p (ff) for this relation. The equation (abp) aj)x = is the analytical condition required when the polar planes of a point (x) with regard to / and /' meet in a line which intersects (p). For the coordinates of these polar planes of x are denoted by a-^a^, bib^ (* = 1, 2, 3, 4). Hence the coordinates of their line of intersection are aj)x(ah)ij; and this line cuts (p) if (abp) axbx = 0. 204 Ml' Turnhull, Some Geometrical Interpretations ■ Forming the invariant of the polar quadric, we obtain an ex- pression which reduces directly to {{AB)(A2)){Bp)\^. Hence if j9 belongs to the complex G, its polar quadric is a cone. § 18. Again, the tangential equation of the polar quadric (abp) cij)y; = is formed in the same way as Ua~ is formed from a^-. A simple reduction leads to {Ap) (Bp) (abp) (aBu) (bAii). Likewise the point equation of {ajBp) u^ a^ involves the form (Ap) (Bp) (a^p) {AjSx) {Box). This interprets the two forms of orders (0, 3, 2) and (2, 3, 0). 1 19. Again, if we form the polar quadric oi {p) with regard to each pair of quadrics /, /', h, k', we obtain the following results : p (/, k) equivalent to (Ap) (A /3a;) a^a^, with a like form for p (f, k'), p{f,k') „ F,a^(Bax), „ „ p(f',k), p (k, k') „ (A^x) {cc/3p) (Bax) [AB). If, further, (q) is the conjugate line of (p) in {Bpy,i.e. in/', then q (/, k) is equivalent to a^a^ (A/3x) (AB) (Bp), and q' (/', k') „ b^^b, {Bax) (AB) (Ap). All these equivalences are readily verified, but we give a special proof for the case of p (k, ¥). In fact, the polar plane of dk X m. k = 0, i.e. in (A^xy = 0, has coordinates — , which may be OXi symbolised as (A^x)(A/3)*. So also the coordinates of the polar of X in k' are denoted by (Bax) (Ba). Hence the line of inter- section of these polars is denoted by (Aj3x)(Bax) [ABa^], which is equal to (A ^x) (Bax) (AB) (a^)* ; and the line cuts p if (A^x) (Bax) (AB) (aj3p) = 0. § 20. These eight polar quadrics now enumerated, viz. p (f, f), p (f, k), ..., q (f, k'), must be supplemented with one more form, (a^p) cipbaa^bx, to complete the set of nine forms (2, 1, 0) belonging to the irreducible system of two quadrics / and f. The geometrical significance of this last form is as follows : the line joining the two points, Xi and x^, cuts p ; Xj^ being the pole in / of the plane whose pole in /' is x, and x^ being the pole in /' of the plane whose pole in / is x. * {A^) = a a' -a' a, and the combination of (A^) with (Ba), as a transvectant, into [ABa^] is essentially the reduction of Ch. ii, § 15 in the paper of Gordan. of the Concomitants of Two Quadrics 205 Correlatively there are nine forms (0, 1, 2), quadratic in u, exactly parallel with the above, of which (a^p) UaU^ is the simplest. The four forms (3, 0, 1) and then' correlatives: (Abu) (AjSa;) a^axb^, {A^x) (Abu) ba.Ua.u^, (AB) (A/3x) (BoLx) a^a.^Ua, (AB) (Abu) (Bau) baiua^, and four similar forms interchanging f and f, \ 21. If a^', bx", Cx" signify any three quadrics, then (abcu) a^b^Cx vanishes when the common point of the polars of (x) in the three quadrics lies on the plane (u). Applied to the quadrics ff, k, k' taken three at a time, this condition involves the four forms (3, 0, 1) indicated above. The correlative condition, applied to each set of three from among S, S', %, %', gives rise to the four forms (1, 0, 3). For example, if we select /,/', k as the three quadrics, then the condition is (Abu) (A^x) a^axbx = 0. The polars of (x) in all four quadrics/,/', A:, k' meet in a point if (x) lies on any face of the self-conjugate tetrahedron (A^x) (BoLx) a^baaxbx = 0. The remaining forms of the system. I 22. None of the remaining forms appear to have any special geometrical importance : but we give a few examples. First, as to the forms of order (2, 0, 2), we may exhibit them as follows: ax (aBu) (Bax) lu and a similar form, (Abu)(A^x)baa^iL„,ax „ „ (AB) (Abu) (Bau) axbx and a correlative form, and [(AB)J. Suppose (q) to denote the common line of the plane (u) and the polar of (x) in / and (q) to denote the line joining (x) to the pole of (u) in /. Then the condition that q, q should satisfy the harmonic relation (Bq) (Bq') = becomes on substitution Ux (aBu) (Bax) Ua = 0. Thus the first in the above group of forms is interpreted. The second form vanishes if two lines (q), (q) satisfy the harmonic relation (abq) (abq') = 0, where (q) now denotes the intersection of the plane (u) with the polar of (x) in k, while (q^) is the same as before. Again, the third form of the set vanishes if the lines in which 206 Mr Turnhull, Some Geometrical Interpretations, etc. the polars of {x) in f and /' cut the plane {u) satisfy the har- monic relation for the complex C=(AB)(Ap)(Bp). Finally the last form [(ABYY, which is equivalent, except for reducible terms, to (Abu)bx {Ab'u)hx (§ 2). is involved in the con- dition that the line common to (a) and the polar of (x) inf shuuld touch /. § 23. Next there are four forms of order (0, 2, 2), such as (Ap){Abu) (abp)apUp, (Ap)(AB){Bau){al3p)apUa, and four cor- relatives of order (2, 2, 0). All of these have obscure geometrical properties, though they present no difficulty to identify. After this there are twenty-four forms of order (1, 2, 1). The simplest of these is (Abu) b^ (Ap) (Bp), which vanishes Avhen u, X, p satisfy the following conditions : if the polar of (x) in /' meets (p) at a point {y), and if the polar o{(y) in/' cuts the plane (u) in a line (q), then p, q satisfy the harmonic relation {Ap) {Aq) = 0. The remainder of these (1, 2, 1) forms are of like nature. Beyond this there are four forms (2, 1, 2), and two forms (3, 0, 3), none of which present concise geometrical interpretations. i Mr Ramanujan, Some properties o/p(n) 207 Some properties of p (n), the number of partitions of n. By S. Ramanujan, B.A., Trinity College. [Received 3 October 1918 : read 28 October 1918.] § 1. A recent paper by Mr Hardy and myself* contains a table, calculated by Major MacMahon, of the values oip{n), the number of um-estricted partitions of n, for all values of n from 1 to 200. On studying the numbers in this table I observed a number of curious congruence properties, apparently satisfied by p {n). Thus (1) |)(4), p(9), 23(14), p(19), ... = 0(mod. 5), (2) p{^\ P(12), p{n\ i^(26), . . = (mod. 7), (3) P(6), P(17X P(28), i^(39), . . = 0(mod. 11), (4) i^(24), yo(49), ^3(74), p{m, . .. =0(mod. 25), (5) p{^% P (54), ^(89), i>(124),. .. HO(mod. 35), («) i>(47), i9(96), /;(145), i>(194), . .. = 0(mod. 49), (7) p(39), i^(94), ^(149), = (mod. 55), (8) pi^n p(138). = (mod. 77), (9) ^(116), B0(mod. 121), (10) jt^(99),. = 0(mod. 125). From these data I conjectured the truth of the following theorem : If h= b'^mV and 24A, = 1 (mod. S), then p{\), p(\ + 8), p{X + 28),... = 0(mod.B). This theorem is supported by all the available evidence ; but I have not yet been able to find a general proof. I have, however, found quite simple proofs of the theorems expressed by (1) and (2), viz. (1) p {57n + 4) = (mod. 5) and (2) p (7m + 5) = (mod. 7). * G. H. Hardy and S. Ramanujan, 'Asymptotic formulae iu Combinatory Analysis', Proc. London Math. Soc, ser. 2, vol. 17, 1918, pp. 75—115 (Table IV, pp. 114—115). VOL. XIX. PART v. 15 208 Mr Ramamijan, Some properties of p (»), From these (5) p (35m + 19) = (mod. 35) follows at once as a corollary. These proofs I give in § 2 and § 3. I can also prove (4) p {Ton + 24) = (mod. 25) and (6) p (49n + 47) s (mod. 49), but only in a more recondite way, which I sketch in § 3. §2. Proof of {\). We have (11) X {(1 - cc) {l-x-){l- x^). ..Y = X (1 - 3« + haf -1x^^ ...){\-x-x^^ X' +...) = t(- lY+^(2/ji+l)x^+^-'^'i^+'>+i''^^'' + ^\ the summation extending from ^ = to fi = x and from v = — oc to y = 00 . Now if l+i/A(/x+l) + ii^(3i^ + l) = 0(mod. 5) then 8 + 4/x (/x + 1) + 4z^(3!^ + 1) s (mod. 5), and therefore (12) (2ya + l)^ + 2(i; + l)^ = 0(mod. 5). But (2/x + 1)^ is congruent to 0, 1, or 4, and 2(v + 1)- to 0, 2, or 3. Hence it follows from (12) that '2/x + l and v i-1 are both multiples of 5. That is to say, the coefficient of x^^^ in (11) is a multiple of 5. Again, all the coefficients in (1 — x)~^ are multiples of 5, except those of 1, x^, x^°, ..., which are congruent to 1 : that is to say (1 — xf 1 —x" 1 —x^ or -z rr = 1 (mod. 5). Thus all the coefficients in (1 ~ x') (1 - X'') (1 -x'')... 1 {{l-x){l-x"~){l-x') ...Y (except the first) are multiples of 5. Hence the coefficient of x^'^ in ^(l_^..)(l_^ao)... _ (l-a^)(l-x-) ^.^ {l-x)(l-x^){l-a^)...~^^ ^^ ""'^'--^ {(l-x){l-x')...Y is a multiple of 5. And hence, finally, the coefficient of .r"' in (1 -x){l- X') (1 - x^) is a multiple of 5 ; which proves (1). the number of partitions of n 209 § 3. Proof of (2). The proof of (2) is very similar. We have (13) a-\(l-x){l-af)il-x^..:f = x- (1 - 3a; + 5x' - 7a-« +...)- = S (- 1)'^ + " (2//. + 1) (2i/ + 1) a,-+i'^<'^+i> +^v(. + i)^ the summation now extending from to oo for both jx and v. If 2 + l/^(^ + l) + iz/(z; + l) = 0(mod. 7), then 16 + 4/A(ya + l) + 4z/(z7 + l) = 0(mod. 7), (2/A + l)" + (2i^-l-l)- = 0(mod. 7), and 2/x + 1 and 21^ + 1 are both divisible by 7. Thus the coefficient of .^■™ in (13) is divisible by 49. Again, all the coefficients in (l-a;7) (l-a;») (l-a;^!)... [{I - x)J\ -^x^) {V-a?) ... Y (except the first) are multiples of 7. Hence (arguing as in § 2) we see that the coefficient of a'"" in (l-^-)(l-^')(l-^')--- is a multiple of 7 ; which proves (2). As I have already pointed out, (5) is a corollary. § 4. The proofs of (4) and (6) are more intricate, and in order to give them I have to consider a much more difficult problem, viz. that of expressing p {X) + jj (\ + h) X + j) (A, + 2S) X + . . . in terms of Theta-functions, in such a manner as to exhibit ex- plicitly the common factors of the coefficients, if such common factors exist. I shall content myself with sketching the method of proof, reserving any detailed discussion of it for another paper. It can be shown that (14) ^^ " ^'^ ^^ " ^"^ (1 - ^1 . ^. 1 {l-x''){l-x^){l-x'^) ... ^-'-x' -^X' _ |-^ - Zx^ + or (^ -' + 2x^-' ) + ^^(2g-^ - ,rp) + x^{3^-' + xl')+5x^ ~ ^ -' - 1 1 X - x'^' ' where P = ^^ ^ " ^1 ( ^ Z ^(LtI^L • ' ^ {\-x^)(l-x'){l-x''){l-af)...' the indices of the powers of.'?;, in both numerator and denominator 15—2 210 Mr Ramanujan, Some properties of p (n) of |, forming two arithmetical progi'essions with common difl-erence 5. It follows that (15) (l-oc^) (1 - x^°) (1 - X'') ...{p{4<)+p(9)a; + jj(14).7;-+ . ..} 5 . Again, if in (14) we substitute cox% (o^x% (o"x\ and q}*x% where w" = 1, for x^, and multiply the resulting five equations, we obtain \ (l-ay^){l-x^^)(l-x^^)... Y^ 1 ^^^ \(l-x){l-x'^)(l-x')... I ^-'-Ux-x'^-^- From (15) and (16) we deduce (17) ^(4)+_p(9)^'+_p(14)a;-+ ... _ {(1 - x'){l - x'^yj l-x^')...}' . I {(1 -x){l - x')(l-a^) ...Y ' from which it appears directly that ^ (5m + 4) is divisible by 5. The corresponding formula involving 7 is (18) p(5)+p(12)x + p(19)x''+ ... {(1-^)(1-^'^)(1-^^)...}^ - , ^9^ { a-x')(l-x^^)(l-af^)...Y ^ {{l-x)(l-x'){l-x'')...Y ' Avhich shows that p (7m + 5) is divisible by 7. From (16) it follows that p (4) X + p (9) x'- + p (14!) x^ + ... 5{{l-x^){l-x^'>)(l-x'')...Y X {l-x'){l-x'">){l-x'')., {l-x)(l-x'')(l-af)... {(l-x){l-x^){l-x')...Y' As the coefficient of «^'* on the right-hand side is ^ multiple of 5, it follows that p {25m + 24) is divisible by 25. Similarly p(5)x + p (1 2)x'-+p(19)x^+ ... 7{(1-«0(1-^")(1-^'')---}' (l-x')(l-x^')... = x(l-3x+5x^- 7a'« + . . .) Ki-^)(i-^^)-r ^^ja-^)(i-^^) {(1-^)(1-^-) 18 ' "J from which it follows that p {4<9m + 47) is divisible by 49. [Another proof of (1) and (2) has been found by Mr H. B. C. Darhng, to whom my conjecture had been communicated by Major MacMahon. This proof will also be published in these Proceedings. I have since found proofs of (3), (7), and (8).] Prof. Rogers & Mr RamanKJan, Proof of certain identities 211 P roof of certain identities in combinatory analysis: (1) by Prof. L. J. Rogers ; (2) by S. Ramanujan, B.A., Trinity College. (Com- municated, with a prefatory note, by Mr G. H. Hardy.) [Received 3 October 1918 : read 28 October 1918.] [The identities in question are those numbered (10) and (11) in each of the two following notes, viz. q q* q^ + l-l + (I-g)(l-5'=)'^(l-5)(l-f/)(l-9^) + -" = 1 (1) and q^ q^ 512 + n^ + (i -q){i- f) + j\-~-^(y-f) (1 - f) + • • • = ^ ...(2). (1 - r/) (1 - (/) (1 - f) (1 - (/) (1 - q^-^) (1 - f/0 ^ ^ On the left-hand side the indices of the powers of q in the numerators are n^ and n (n + 1 ), while in each of the products on the right hand side the indices of the powers of q form two arith- metical progressions with difference 5. The formulae were first discovered by Prof Rogers, and are contained in a paper published by him in 1894*. In this paper they appear as corollaries of a series of general theorems, and, possibly for this reason, they seem to have escaped notice, in spite of their obvious interest and beauty. They were rediscovered nearly 20 years later by Mr Ramanujan, who communicated them to me in a letter from India in February 1913. Mr Ramanujan had then no proof of the formulae, which he had found by a process of induction. I communicated them in turn to Major MacMahon and to Prof. O. Perron of Tubingen ; but none of us were able to suggest a proof; and they appear, unproved, in Ch. 3, Vol. 2, 1916, of Major MacMahon's Combinatory Analysis'^. Since 1916 three further proofs have been published, one by * L. J. Rogers, ' Second memoir 011 the expansion of certain infinite products', Proc. London Math. Soc, ser. 1, vol. 25, 1894, pi?. 318—343 (§ 5, pp. 328—329, formulae (1) and (2)). t Pp. 33, 35. 212 Prof. Rogers d- Mr Ramanujan, Proof of certain identities Prof. Rogers* and two by Prof. I. Schur of Strassburgf, who appears to have rediscovered the formulae once more. The proofs which follow are very much simpler than any pub- lished hitherto. The first is extracted from a letter written by Prof Rogers to Major MacMahon in October 1917 ; the second fi-om a letter written by Mr Ramanujan to me in A.pril of this year. They are in principle the same, though the details differ :|:. It seemed to me most desirable that the simplest and most elegant proofs of such very beautiful formulae should be made public with- out delay, and I have therefore obtained the consent of the authors to their insertion here. It should be observed that the transformation of the infinite products on the right-hand sides of (1) and (2) into quotients of Theta-series, and the expression of the quotient of the series on the left-hand sides as a continued fraction, exhibited explicitly in Prof Rogers' original paper and in Mr Raman ujan's present note, offer no serious difficulty. All the difficulty lies in the expression of these series as products, or as quotients of Theta-series. — G. H. H.] 1. {By L. J. Rogers.) Suppose that \q\<l, and let F,,^ denote the convergent series (1 - ^»^) - ^«(^«+i-'« (1 - x'"^q"''') C\ where ., _ (1 - ^) (1 - xq) (1 - iC(f) ... (1 - xq'-^) ^'- (i_5)(i_5-.)(i_23^...(l_5.) -■ the general term being Then V,n - Fm-i = ^*"~' (!-«;)- x"q''+'-'^ {(1 -q) + x'>''~Hf"-' (1 - Ar^)} d _,_ ^in^m+z-im |(^1 _ ^^2) ^ ^m-i^im-2 (^l _ ^^2^J (J^^ (• 2^_ Suppose now that the symbol 77 is defined by the equation vf{«^) =f(xq)^ Then ( 1 - f) C, = (1 - x) rj C,-! , (1 - xq^) 0,. = (1 - x) rj C,. * L. J. Kogers, ' Ou two theorems of Combinatory Analysis and some allied identities', Proc. London Math. Soc, ser. 2, vol. 16, 1917, pp." 315—336 (pp. 815— 317). t I. Schnr, ' Ein Beitrag zur additiven Zahlentheorie uud zur Theorie der Kettenbriiche', Berliner Sitzungsberichte, 1917, No. 23, pp. 301—321. J I have altered the notation of Mr Eamanujan's letter so as to agree with that of Prof. Rogers. ill combinatory analysis 213 Hence, arranging (2) in terms of t^C'i, 776*2, ..., we obtain V -V ' m ' in—\ \ — X = a.'"^-i {(1 - ^.n-«i+ig»i-"i+i) _ .^H^«+?n (1 _ ^^n-m-H ^jin-sm+i^ r]Ci+ ...} = a;'''-'vVn-n,+i (3). CO If we write v^^ TI (1 — *•(/'■)= F,,^ (4), >- = then (3) becomes v„, — v,n-i = ^''"^~'^ vVn-m+i (o). It should be observed that Fo and Vo vanish identically. In particular take n = 2,m = l, a,nd n = 2, ni = 2. We then obtain Vi = 7]Vo, V2 — v^ = xrjv^ ; and so Vi — 7]Vi = ocqr)'-v^ (6). Now let Vi = l + a^x+a2cc-+ (7). Then from (5) 1 + aiX + a-iO? + . . . — (1 + a^xq + a^xif + . . .) = a-^ (1 + tti^y/ + aojf-(f + ...) ; and so a^=—l~^ (^2 = p. ry^ ^ (8). 1-q (l-q)(l-q') But when x = q, C,- = 1 ; and so V, = {l-q)-q^{l-q^) + f^{l-q^)- (!)). From (4), (6), (7), and (8) it follows that ^l-q^(l-q)(l-q'^)^ ^ a-q)-qH ^- q') + q''('^-q')--- .. ^^ (l-5)(l-2'^)(l-f/)... ^ '^- Similarly we have and, when x — q, and V, = {1- q') - q^ (1 - g«) + 5" (1 - f/«) - ... . \ ^ X X'Q^ 77 1-^ (1- 7) (!-(/-) l-q (1-g) (!-(/) 214 Prof. Roger's S Mr Ramanujan, Proof of certain identities Thus {l-q){\-q'){\-t) ^ 2. {By S. Ramanujan.) Let G{x) = l + t^ ^^^^^ ^' ""i \i-q){l-q^){l~f)...{l-qn If we write 1 -a;g2'' = 1 - (^^ + 5''(1 -^g"), every term in (1) is split up into tAvo parts. Associating the second j)art of each term with the first part of the succeeding term, we obtain 1 —xq G (a;) = {1- xhf) - x'^q^ ( 1 - x-'q') l-q + ^q (1 ^q ) (i_^)(i_^.>) {-)■ G(x) Now consider II(x) = ., ^ - G (xq) (3). \—xq Substituting for the first term from (2) and for the second term from (1), we obtain x^q' a;*g" (1 — xq-) H {x) = xq - ^3^- {(1 -q) + xc^ (1 - xf)] + in combinatory analysis 215 Associating, as before, the second part of each term with the first part of the succeeding term, we obtain H {X) = ,rq (1 - ^Y/) jl- ccY' (1 - ^^q') 1 ^ 'M i^ ^?)(l_^)(l_5.)(l_,/) + ' = xg{l — x(f) G {x(f) (4 ). II now we write K{x) = ^-^--^^^^^- , we obtain, from (3) and (4), andso A-(^)=l+fl^^^ (5). In particular we have 1^A_ t^- 1 _ {\-q)0{q) .... 1+ T + 1 + ... K{\) G{1) ^ " or I q f _ 1 — g — g* + g' + g^^ — (7). 1 + 1 + 1 + . . . 1 - ^2 _ ^3 ^ ^9 + gll This equation may also be written in the form 1 <!_ f__{i^q){i_::q')iX-j')(i^)i^-_thi: 1+1+1 + ... {\-f){\-q^){l-q^){l-cf){\-t'-)... (8)- If we write ;,,._ G J^) ^ ' {\-xq){l- xq') (1 - xf) ...' then (4) becomes F (x) = F (xq) + xq F {xq"), from which it readily follows that 216 Prof. Rogers d' Mr Ramanujan, Proof of certain identities In particular we have 1 - r/ - rf + (f + ry" - . . ^l-q^{l-q){l-(f) (i_,^)(l_f/)(l_,/) {l-q)il-cf){l-f) 1 and 1 + ^^ + ,g' + ... = {l-q)G(q) .(10), 1_^ (l-g)(l-5^)' ••• (l-5)(l-^-^)(l-r/), 1 — q — q^ + q' -\- (f^ — ... (i-^)(i-9^ya-^Yy^-" 1 (l_^.)(l_53)(l_(^7)(l_^8)(l_^l.),.. .(11)- Mr Darling, On Mr Ramanujan's congruence properties of p (n) 217 On Mr Ramanujan's congruence properties of p («). By H. B. C. Darling. (Communicated by Mr G. H. Hardy.) [Received 3 October 1918: read 28 October 1918.] 1. Proof that p {5ni + 4) = (mod 5). Let u ^(l-x){l- aJ") (1 -x')...; then by Jacobi's expansion a^' = "ST (- 1)" (2» + 1) **"^''+'^ n = » so that in d'-ur, where d denotes differentiation with respect to x, the coefficients are of the form i {n - 1) n (n + 1) (n + 2) {2 (n + 3) - 5], and therefore d^u^= (mod 5) (1). Again, in d^u" the coefficients are of the form A^{n' + n-4<){n-2)(n-l)n(n + l)(n + 2)\2(u + 4')-7], and therefore an<3= (mod 7) (2). /1\ 1 2 Now 8- ( - = /ci-u + - (duf ; \Uj U' u also du^ = Zu'du, and d-ii^ = ^u'^dhi + Qu (du)-. Hence a^(-) = -o\9"''' + (r7(9'*')' (3)5 \uj ?>u^ 9«' and thus, by (1), we have ^' ^ - II ^^'''^^ ^" ^ ~ 27I" ^^'''^' ^"^^^ ^^ ' so that 8^ f-] = (mod 5) (4). Again if Iju be expanded in powers of x, and the operator 3* be applied to the resulting sei'ies, it is evident that the coefficients of all powers of x of the forms om, 5m + 1, om + 2 and 5m + 3 will be multiplied by a factor divisible by 5 ; but that the coefficients of the powers of x of the form 5m + 4 will be multiplied by a factor which is not divisible by 5. Hence it follows at once from (4) that p {oin + i) = (mod 5). 2181 3Ir Darling, On Mr Ramanujan's congruence properties of p {n] 2. Proof that p {7m + 5 ) = (mod 7). Differentiating (3), we have d' (-] = - ^, dHi' + ^^ dud-u' + A 9 (9"')' ('iiocl 7 ) OU SU'^ i)'U' = - ~dhi^ + j^^didu^y (mod 7). Similarl}^ having regard to (2), a* (-] = ^- du^d^u^ + ^a- (du'T- (mod 7), d' (^) = ^. d'vPd'w + ~ d' (dtt'f (mod 7) (5), ^0-^,(^-'y^l,^'('"'y('--^'^ ^6). Again a'(-^) = -3'©H-6a.Q = .-.3{..8»Q}; SO that, by (5) and (6), 36 (^] = !_' [43 (x'^d'u'd-u') + 63 {a^«3« (du'f]] (7). Now d{dit'')- = 2dHo'dit', 3^ (8m3)2 = 23^ i(33«.3 + 2 (d-u'^y. Thus, by (2), 3^ (du^Y = Qd'u'^d'-u^ (mod 7) ; and therefore, by (7), we see that that is, by (2), ^x 38 (- j = 3 {w'd'ii'dHi'] (mod 7) ; = ai'dhi^dHi^ + 6w^d^ u^'d' u' (mod 7) uj = d'u^d{£c'd-a') (mod 7) (8). But the coefficients in 3 (x'^a-u^) are of the form i (n - 1) 9i {n + 1) {n + 2) [2 (n - 3) + 7} {(// - 2) (n + 3) + 14}, and are therefore divisible by 7 ; and therefore, by (8), 3« (-) = (mod 7). Hence, by considerations similar to those in the latter part of § 1, we see that p{7m + 5) = (mod 7). Miss Wrinch, On the eicponentiation of well-ordered series 219 On tlie exponentiation of luell-ordered series. By Miss Dorothy Wrinch. (Communicated by Mr G. H. Hardy.) [Read 29 October 1918.] The problem before us in this paper is the investigation of the necessary and sufficient conditions that P'^ should be Dedekindian or semi-Dedekindian when P and Q are well ordered series. The field of P'^ is the class of Cantor's Belegungen and consists of those relations which cover all the members of the field of Q with members of the field of P : several members of the field of Q may be covered with the same member of the field of P, but every member of the field of Q is covered with one member of the field of P and one only. In order to prove that P'^ is Dedekindian it is necessary to prove that every sub-class of the field of P^ has a lower limit or minimum with respect to P^. If there is a last term of the series P'^ it is the lower limit of the null class. Unit sub-classes have their unique members as minima. It remains, then, to con- sider sub-classes with two or more members. Now the relation P*'* orders two relations R and *S' by putting R before S, if R covers the first Q-term, which is not covered with the same P-term by both R and h, with a P-term occurring earlier in the P-series than the term with which 8 covers it. Suppose A, is a sub-class of the field of P'^ with at least two members. We will call Qm'^ the first Q-term which is not covered with the same P-term by all \'s; and Tp^\ that subset of X which consists of those members of \ which cover Q„/A, with that term, in the class of P-terms with which various X's cover Qm'^; which occurs earliest in the P-order. Tp'\ will therefore be contained in \ and not identical with it. It will be seen that P'^-terms belonging to Tp'\ come earlier in the P'^-order than terms of \ not belonging to it. Constructing Tp'T'X we get a smaller subset of \ : members of this subset occur earlier in P''* than other members of X. Continuing this process with X, T/X, T/Tp'X, Tp'Tp'iyx,... /*,... V,..., we obtain smaller and smaller sub-classes of X: if /a precedes v in this order, members of v occur earlier in the P'^-order than members of fM which are not members of v. We take the common part of 220 Miss Wrinch, On the exponentiation of well-ordered series all these subsets of \, i.e. the class of relations which belong to all the sets \, 2 P \, 1 P 1 P \ . . . ', and get a subset of X p'iT'pW^ which, again, consists of members of \ which come earlier in the P'^-order than members of A. not belonging to it. Repeating the original procedure we get Tpy(%h'\ Tp'TpYiT'p)^'\, ..., and so obtain a series of sub-classes of A, ordered by the serial relation A {Tp, \), where A is the relation between /j, and v when v is contained in /i but not identical with it. And this is a well-ordered relation : CQjisequently it will have an end, viz. p%Tp^Ay\. If this is not null, it consists of a single member, which will be the minimum of X in P^. But if it is null we will put PQ'X = s'N {a^ . ^,e (Tp^Ayx . iV= (i^/.) r eQm VI- Then PQ'X is a relation covering a certain section of the Q-terms with P-terms : PQ'\ agrees in the way it covers the Q-spaces with each member yu, of ^ {Tp, \) as far as QniV- PQ''^ will therefore cover Q-spaces up to z, if there is a yu- which is a member of the field of A (Tp, \) such that z precedes QmV in the Q-order. If no member of the field of J. (Tp, X) agrees in the covering of Q-spaces beyond a cer- tain member z of the field of Q, PQ'X covers no spaces beyond z with P-terms and for this reason is not a member of the field of P'^. If P is a P'3-term which agrees with PQ'X in the covering of Q-spaces as far as it goes, R precedes all the members of X, in the P^-order ; further, any member of the field of P^, following R and all relations agreeing with PQ'X as far as it goes, follows at least one member of X. Hence, if there were a maximum in the P'^- order in the class p of members of the field of P*? which agree with PQ'X as far as it goes, this relation would precede all X's and any relation following it would follow at least one member of X. If the class consists of one term R, it will have a maximum, namely R itself: R will then be equal to PQ'X and PQ'X will, therefore, be the lower limit of X. But p is a unit class only when PQ'X covers Miss Wrinck, On the exponentiation of ivell-ordered series 221 the ivhole of the Q-terins with P-terras. When PQ'X does nut cover the whole of the Q-tenns, but covers Q-terms only up to z (say), all p's will agree in their covering of Q-spaces up to z, and the remaining Q-spaces will be covered differently by different members of p. To get a maximum of the p's with respect to P*^, we want a relation S which is a p such that no member of p comes later in the P^-order. Now. if P has no last term, every P-term is followed by other P-terms. However S covers z and the Q-spaces after z, by replacing the term covering any member of the field of Q after ^ by a member of the field of P following it in the P-order, we obtain a relation T which is a p and follows ;Si in the P'^-order. ;S' is, consequently, not the maximum of p in the P^-order. Now if z in the field of Q is covered by PQ'X, the term innnediately following z will also be covered by PQ'X. Therefore, if Q is a finite series or an co, PQ'X will always cover the whole of the Q-terms ; since, as X has at least two members, it will always cover one Q- term. Any X will then have a lower limit or minimum with respect to P*^. In such cases, P*? will certainly be Dedekindian with the addition of a last term, whether P has a last term itself or not. But if Nr'Q is greater than o), it is possible to find a subclass X of the field of P^ which is such that PQ'X does not cover the whole of the field of Q. For, let 1 and 2 represent the first and second terms in the P- series and let (e.g.) i...hi(r)2...h2(aiii represent a relation which covers the first ^ Q-terms with 1, sub- sequent terms up to (but not including) the ^th term with 2, and all remaining terms with 1. Such a relation is clearly a member of the field of P'^. Consider the class of I'elations X which cover all Q-spaces up to z with 1, and all the Q-spaces following z with 2, as z is varied from the second Q-term to the ^th, where ^ is an ordinal number with no immediate predecessor. We will arrange this class of relations in the P^-order. 1... 1-1(^)2. ..h2(0,22.... {^<0 11112 h2(f), 22 11122 f-2(0> 22 11222 ^-2(0> 22 12222 ^-2(0, 22 This class has no minimum in the P'^-'-order, and PQ'X covers all 222 Miss Wrinch, On the exponentiation of well-ordered series the Q-places up to the ^th with 1 and does not cover the subse- quent Q-places at all. It is therefore not a member of the field of P^. But, as we have seen, every relation which agrees with PQ^X as far as it goes, and covers the other Q-places with any P- terms whatever, precedes all X's : and any member of the field of P^ following this relation, and all relations agreeing with PQ'X as far as it goes, follows at least one member of X. Thus, e.g., the relation ll...hl(^), 2111 precedes all X's, and any relation following it and all relations agreeing Avith PQ'X as far as it goes (as e.g. the relation 11211... h 1(0, 21211...) follows at least one relation belonging to X, e.g. the relation 11122... \-2{0, 222... Thus \ will have a lower limit if and only if there is a maximum among the relations covering all places up to the ^th with 1. And this is the case when and only when P has a last term u (say). For then the relation 111 \-l{^)uiiu... will be the lower limit of X. Thus if Nr'Q is greater than o), it will be the case that all existent sub-classes of the field of P'^ will have a lower limit or minimum when and only when P has a last term. A non-existent subclass (i.e. a subclass with no members) will have a lower limit or minimum when and only when P has a last term. If Nr'Q is greater than co, P^ is Dedekindian when P has a last term, and if P has no last term P^ even with the addition of a last term is not Dedekindian. We thus arrive at the following conclusions. When P and Q are well-ordered series, (1) P^ is Dedekindian when and only when P has a last term ; (2) if Nr'Q is greater than co, P*^ with the addition of a last term is Dede- kindian if and only if P has a last term ; (3) if P^ is made Dede- kindian by the addition of a last term when and only when P has a last term, Nr'Q is greater than co. These propositions will now be established. [The symbols used are those o/Principia Mathematica. Among the propositions referred to, those whose nwnhers are greater than 1 are proved in P.M., ivhile the others are established in the course of this paper.'] *01. QjX = mmQ'y{s'X'y^eQKjl) Df *-02. Tp'X = XnM {M'QJX = mmp's'X'Q^,'X) Df Miss Wrinch, On the exponentiation of well-ordered series 223 *-03. A=\fl{^lQ\.^Ji^\) Df *1. I- : P, Q e n . X 6 . D . 5'Cnv'P« = mm'(7^«)'X [*207-l7] *11. V:F,QeQ..\el.\C C'F'^ . D T'X = min (P^^)')^ Dem. h.*l7619. 'D\-:R6a'P^.Dj,.^{RPm) (1) l-.(l).*205-18. Dl-.Prop *'201. \-:P,Qen.\C C'P'i . E ! T/X . D . f{Tp^Ay\ = B'Cnv'A {Tp, X) . A (Tp, X) e O Devi. [•02] h. TpeRl' A nCh-^1 (1) h . (1) . *258-231 . D h . Prop *-202. h : E ! Tp'X . D .p'(Tp^Ayx ^eB'Tp [*-201] *-203. \-:P,Qen.XC C'P'^ .X^eOwl.D.E! Tp'X Dem. \-:.XQ G'P'i .R,S€X.X€ a'R . D^^ . R'w = S'x:D .XeO vjI :. [Transp] D I- :. Hp . D : P, >Sfe A, . D . g^ . i2^« =|= iS^'a- . iceQ^E :. [*250'121] D h :. Hp . D : E ! min,/^ (i-'A,'^ ~ e u 1) :. [rOr02] D I- :. Hp . D : E ! Q,„'X . E ! Tp'X *-2031. (- : E ! T/X . D . E ! Q,/X [*-02] *-204. h : P, Q e O . X C C"P'^ X ~ e D' Tp . D . X e w 1 [*-2()3 . Transp] *-205. f- . Hp r203 . D . p'iTp^Ayx e u 1 [**-202-203-204] r211. h . E ! Q,„*\ . D . s'X'QjX ~ e w 1 [*-01] *-212. h :. E ! Q,,,'X : D : {s'X)[Q'Q,,,'X el -^C\s: ReX . D.P6l->Cls Pewi. [*176-19] hz.ReC'P'i .D'.zeC'Q.D.R'z^eO (1) l-.(l).r02. Dh.Prop VOL. XIX. PART V. 16 224 Miss Wrinch, On the exponentiation of well-ordered series r213. \-:S€l->C\s.R(iS.D.S[a'R = R Dem. V .^u.v . uSv . V € Q.'R . ~ (uRv) . D . g«, v, u' . a 4= w • uSv . u'Rv . <^ (uRv) . V € d'R : D\-:.R(lS.D:'3^u,v. uSv . v e a'R . -- (uRv) . D . g/t, V, n' . u' =j= u . nSv . u'Sv : 0\-:.RQ.S.Sel->Ch.D:-^ {gu, v . iiSv . v e Q'R . ~ (itRv)] D\-:.R(lS.Sel-^Ch.Di nSv . v e a'R . D„,, . uRv : D\-:.R(lS.S€l-^Ch.:>:S\-a'R = R *-2131. ^:.'3^l^.aCa's'uT.{s''!^)\-ael^C\s:0:Re^.D.R[a = {s''ST)[a = (p''S7)[a Dem. h . *40-13 . *41-44 . Dhz.Rezy.D: xRy .D.x (sV) xj : a'RQa\s'^)'.. '^ViaXia'R.Ret;T.':>.R\aQ.{s'zj)\a (1) I- *-213 . (1) . D F : a C iVR .Re^. {s'^) pet e 1 -^ Cls . :^.R\a = {s'7JT)\a (2) h:.g;!tn-.i^ero-.Djj. xRy .yea: D : g[*S' . S e-sT . xSy .yea (3) I- . (3) . D h : a ! OT . D . {p'^) \a G (s'tsr) \a (4) f- . (4) . *-213 . D h :. a: ! w . D : (5^-sr) [^a e 1 -^ Cls . D . (i'OT)Pa = (^^i3-)['a (5) l-.(2).(5). Dh.Prop r214. V :. Hpr20.3 : D : E e \ . D . R[Q'QJX = {s'\)[Q'Q^,'X = {p'X) [Q'QJX [*40-13 . *41-44 . r2131] r215. h :. Hpr203 :D:Re\- T/X . S e Tp'X . D . R [Q'Q^'X = S [Q'Q^'X . {S'QJX) P (R'Q^'X) Dem. [r02] \-:SeTp'X.D.SeX (1) [**-01-02] h : ^ ! Q^,'X . S e T/X .ReX- Tp'X . D . S'Q,,,'X = minp 's'X'QjX . R'Q^'X + min's'X'Q^'X (2) K (1) . (2) . r2l4 . D h . Prop Miss Wrinch, On the expouentiatluit of luell-ordered series 225 *-216. f- : . Hp *-208 :D:Re\- Tp'\ .SeTi^'X.D. SF'^ R [*-2 15] *-217. h : . Hp *-203 : D : /x e ( Tp^A y\ . R eX- /m . S e fi, .D . SP'^'R Bern. h . *40-23 . D h :. p C (Tp^Ayx .'■^l p : fie p . S e /j, . ReX- fi . D . SP'^R :D:Sep'p.R€X-p'p. D^„, . SF'^R (1) h . (1) . -r216.*258-241 . D I- . Prop r218. h: }ii)*-203. '3^1 p'{Tp*Ayx.D.7Y(Tp^Ayx = mill {P'^yx Dem. V . *r217-201 . D I- : Hp . D : ^' e \ -p^Tp^Ayx . Rep'iTp^Ayx.D.RP^S (1) h.(l). Dh.Prop *-31. hi.SeX-.D: k=p'{{Tp^Ayx n^/Sf} . D . ke(Tp*Ayx.S'^eTp'k Dem. [*22-43] h :. SeX. D : h = p'{{Tp^AyXn e'S] . D . '3_p.pC (Tp^Ayx . g! p . p = [{Tp^Ayx n e'S] . k=p'p [*257-125] D h :. A'^eX . D : k=p'{{Tp^Ayx n eSS'J . D . a/3 . It/p e {Tp^Ayx . p = {{Tp^Ayx n e'S] . k =p'p [*258-211] D h :. 6' e A . D . k = p'{{Tp*Ayx n e'S] . D . ^^e(Tp*^)'X (1) l-.(l).*2o7-125. Dl-:.S€X.D.k = p'{{Tp*Ayx n e'^'l . D . ?p'^^e(^p*^)'x (2) [*40-12] D\-:.SeX.D.k^p'{{Tp^AyXne'S].D. ^ e (T^p*^ yx n e'S .D^.kCr^: [Transp] D\-:.SeX.D.k =p'{(Tp^Ayx n e'S] . D . 16—2 226 Miss Wrinch, On the exponentiation of vjell-ordered series l-.*22-43.*-04.(2). D\-:.SeX.D.k =|/{(rp*^)'X n e'S\ : D . ~ (k C Tp'k) . Tp'k e {Tp^A y\ ( 4) l-.(3).(4). D h :. >S'eX . D . k =p'{{Tp^Ay\t^ e'S} . D . r^iSeTp'k) (5) |-.(1).(5). Dh.Prop r32. h : Hp r203 . /. ?p A, . E ! ?p V ■ ::> ■ (Qn/^) Q (<^m» Dem. [*-212] l-:Hpr203.i?eX.D.^Q(Q^/X)D.i^*5el (1) [**-02-203] \- .fiTpX.D.fjiCX (2) h . (1) . (2) . O 1- :. Hp -r203 .f^TpX.Sefx. :>:zQ(QjX).D.S'zel (3) [*-02] f- : Hp *-203 . fjuTpX. S e /j, . D . S'Q,,,'X = minp'P [s'X'y - e u 1 } (4) 1- . (3) . (4) . I- :: Hpr203 .f^TpX.Se/x. D :. a/ : (QJX) Qz' : uQz' . D,„ . ^S^(^ e 1 (5) K (5) . h : Hp*-203 . /. ?p X . El ?pV • ^ • (Qn/^) Q (Qm'/-) r321. h : Hp r203 . ^a (.4 {Tp, X))v.El Tp'v . D . (Q,„V) Q ('Q^'i.) Dem. [**-02-203] \-ipC (Tp^Ayx .'Rlp.'K^.p'p. D.lyl'Q.^Yp'^elyjO (1) [*40-12] h'.Xep.D.'p'pCX (2) K(l).(2). D\-:.pC{Tp^Ayx.'3_lp.'3_lp'p: D-.Xep.O. i'X'Q,,yp'p ~ e 1 u (3) h . (3) . -r02 . D\-:.pC {Tp^Ayx .'3,1 p .'Rlp'p . *• > D : Tp% Xep.D. s'Tp'X'Q,,yx e 1 . i'Tp'X'Q,^'p'pr^e 1 yj : D\- :. pC{Tp^Ayx.'3l p .^Ip'p: :>:Tp%Xep.D.Q,^'X^Qjp'p (4) Miss Wrinch, On the exponentiation of well-ordered series 227 I- . (1) . rOl . (4) . D f- : pC{Tp^A)'\ . g ! p . g ! jo'p . (InP'p ^V !.v'X',y-eO u i; . Q,/\ = miiV.??~e() w 1 (5) I- . (o) ■ D h : p C (rp*^)'\ . a ! /9 . a ! jt)'p . r,/X,\6p.D.(Q„A)Q(Q,nyp) (H) V . (6) . *-82 . *-258-24l . D h . Prop *-33. V : Hp *-203 . D . PQ'X e 1 -^ Cls [*-04] I- : Hp . t!7 = ^ [gyti . yLt 6 (r/,*4)'\ . |-:.Hp(l).D:i¥,iV6t^.Di,,v- [r201.*250-113] D h :. Hp (1) . D : M, Ne^. D^^, v ■ 3/^, ^^ - IJi [A {Tp,\)]v.v.v \A (Tp, \)] /Lt : [**-321-214] D F- :: Hp (1) :. D :. i¥, iYe ^ . D,^, y ■ D\-:.Rp(i):D.M,N€^.DAVMCa'N. M = N \ QM/ . V . a'N C a'i¥ . N = M\-a'N':. D:.Hp(l):D.if,.V6t^. D . 2/ e a'lM n a'N . D . 3f'y = N'y (2) f-.(2). D.Prop *-34. I- : /x e ( '/p*^ )*\ . E ! ?p'X . Hp r203 . [*-04] 1- . Hp *-203 . E ! Tp'tx .fx.€( Tp^*A )'X . D.(/>V)rW„>CPQ'X (1) 1 228 Miss Wrinch, On the exponentiation of luell-oj'dei^ed series [*-33] h . Hp . D . PQ'X € 1 -^ Cls |-.(l).(2).*-213.Dh.Prop r341. h : /x €(Tp*Ayx . Hp *-208 . p'(Tp*Ayx . Dem. [*-201] !-:Hp.D.'B^Cnv'^(Tp,\) = A: D h : Hp . D . E ! ?p V (1) |-.(l).r84.DK.Prop *-35. f- :. i^ [^ a^PQ'X = PQ'X . Hp *-203 . p'(Tp^AYX = A . ReC'P'^::)f>:SeX.Ds-RP'^S Dem. [**-84-31] h . Hp . <Sf e X . /<: = p^K^P*-^ y^ '^'^\ ■ D^.Er'Q'Q^'x=5:r^Qm'x (1) [**-34-02] h . Hp . ^ e \ . k=p'\{Tp^Ayx nV'S] . D^ . gT . T e fp'k . K^e^'X = T'Q^'X . (2''Q„^X)P(,Sf,Q,^X) (2) h . (1) . (2) . D I- :. Hp .DpiSeX. D,, ■ PP^>S^ *-4. h : : Hp *-203 . p'(Tp^A yx = A.ze a'PQ'X :.D:.S ['Q'z = (PQ'X) [~Q'z : (PQ'X'z) P {S'z) .D.'3^U.UeX. UP'^S Dem. [r04] h-.ze a'PQ'X .:>.'^pi.fMe {Tp^AyX . 2 e'q'Q^'iM : [r341] D h : . Hp . D : ayu . ^ = Tp V ■ 3 ■ (PQ'^) [~Q'Qm'i^ [r341] D h : Hp . D . a/., f^ . C/^ 6 ?p ^ ■ (i'Q*^) T Q'Qm'Tp^M' = U['Q'Q^'Tp'v.zQ^iQ^'v): [*176-19] D I- : Hp . D : ,S7 Q'^ = PQ'X [Q'z . (PQ'X'z) P (S'z) . D . a ^ . [/ e X . UP'iS Miss Wrinch, On the exponentiation of well-ordered sei'ies 229 *-41. t- : . Hp *-203 . p\Tp^-A y\ = A . i2 e C'P'? . 7? [ (VPQ'X = PQ'\.D. RP'i V : D : g^S' . ,S' e X . .S'pv V Dem. [*17G-19] h:Hp.D.-(rra'PQ'\ = PQ'A,): [*176-19] D H : Hp . D . 32 , 5 6 (J'PQ'X . V [Q'z = (PQ'X) ['Q'z . (PQ'X'z) P ( V'z) [*-4] h . Hp . D . a^S' . <S6\ . SP^V *-42. |-:.P, QeO.X-eOul.xC C'P'^' . p'(Tp^A Y\ : D:p = C'P'i nR[R\ a'PQ'X = PQ'X] .D.pC p'P'"'X . pcpQ"p c s'P''\ [*r3.r-H] r43. \-:¥AQ^'X.D.s'X'Q^'X^el.s'fp'X'Q^'Xel [*r0r02] r431. \-:Tp(lA.{Tp\^(lA Deni. |-.r43. Dh: fiTpX.D.ij,CX.^^X (1) |-.(1).*201-18. Dh.Prop *-432. h : Hp *203 . D . "I^i (P«)'X C /j'CTp*^ )'X [*-21 7] h :. Hp . P e /ci . yLt e (^V*.! )'X . /x {xl (T'p, \)| v . P ~ 6 z/ : D : g! z^ . D . a*S' . >SP^'P :. DI-:.Hp.Peyit.yL6e (rp*J.)'\ . P mill (P'O X : D : /x [^ (Tp, X)]v.'^\v.D.R€ V (1 ) [r431] t-:Hp.Pe/i.i;{^(Tp,X.)j/i.D.Pei; (2) h . (1) . (2) . D h :. Hp . D . P e i"^ P^'X : D : /^ e {Tp^Ayx . ^M.Pe/^ (3) h . (3) . D h . Prop *-433. h : Hp *-42 . p = C'P^^ nR(R[ a'PQ'X = PQ'X) . D.'^(P<0'/3 = tl(P'^)*X Dem. h :: Hpr42 . D :. SeC'P'^' : D : S[a'PQ'X = PQ'X . y.'^z.ze a'PQ'X . ^ ['Q'z = PQ'X [ Q'z . (S'z)P{PQ'X'z) . V . (PQ'X'z)P(S'z) :. 230 Miss Wrinch, On the exponentiation of well-ordered series [-r4] D I- :: Hp . D :. SeCP"^ .D . S e p :v: T e p .Dj ■ SP'iTiv.'g^U.UeX.UP^S: D h :: Hp . D :. T^eX . D,- . SP'^V : S eC'P'^ : DiSep.v.Tep.Dr-^'^P'^T: D I- :: Hp . D :. (T = ?7(FeX . D,- . UP'W) - max (P'iyp . SecTzD-.El max (P'^Yp . D . SP^ (max (P'^Yp) . max (P«) V epiv: i^'P« ^ = A . D . gT . .ST'-T . T e p : : D\-::lIl).D:.(T = U{Ve\. D,- . ?7P«F) - ^{P^p . D.aCP'^"p (1) h.(l).*205-193. DI-:.Hp.D:(7 = 0'(FeX.D,-- t^P'^F) -l'^(P«)V . D . ^(P«)'p u cr = ^(P'?)'p (2) h . *206-02 . D I- :. Hp . D : prec (P«)^X = ^' &( F 6 /,; . D . FP'^^ U) (3) h . (3) . :) h :. Hp . D : (7 = ^( Fe X . Dr ■ ^i"^'^^) - ^ (P«)'p . D . ^ (P«)'X = 1^ (P'-') Cp yj a) : |-.(2). Dh:.Hp.D:c7-f/(F6X.Dp.. f/^P^F) - rmJ (P«)> . D . prec (P<^)'X = nmJ (P''*)'p (4) I- . *-432 . D I- : Hp . D . i^ (P«)'X = A (5) h . (4) . (5) . *207-02 . D h : Hp . D . ^ (P^Yp = tl (P^)'^ r44. h : Hp r42 . p = C'P^ nR{R\- d'PQ'X = PQ'X) .pel. D . T'p = tl (P«)'\ [*205-lS . *-433] *-45. h : . Hp r42 . p - C"P« n P (P p d'PQ'X = PQ'X) : D : a'PQ'X =G'Q. = .p = I'PQ'X Dem. h :: Hp . D :. a'PQ'X = C'Q . D . PQ'X e G'P^ : Rep.D.a'R = a'PQ'X:: D I- :. Hp . D : a'PQ'X = C'Q . D . PQ'X = ^'p (1) h :. Hp . D : /^ = t'PQ'X . D . PQ'X e (7'P« :. Miss Wrinch, On the exponentiation of well-ordered series 231 [*176-19] D f- :. Hp . D : /? = i'PQ'X . D . G'PQ'X ■-= C'Q (2) l-.(l).(2).DI-.Prop *-451. t-:.P,QeO.\CC*P«.X-'eOul .p\Tp^Ay\ = A. D . a'PQ'X =C'Q:D.E ! tl (P'?)'\ [**45-44] r46. f- :, P, Q e a . \ C 0'P« . \ ~ e u 1 . p'(Tp*Ay\ = A . D^ . E! limin (P^Yfi [*rll-21 8-451] r5. l-.Hpr42.D.maJ<2^a'PQ'X = A Pe?u. [*-04] h. Hp.^ 6 a*PQ*\. D.'^fX,.fl€(Tp^AyX.2Q{Qm'f^) (1) [r04] h.Hp.yLie(rp*^yx. D.ai..^ = TpV.Q'QmV.ca'PQ'x (2) 1- . r21G . (1) . (2) . D I- . Hp . ^ e a'PQ'X . D.'^z'.zQz'.z'ea'PQ'X (3) I- . (3) . D h . Prop r51. l-:.Hpr42:D:a*PQ'\=C'Q. V . a^ . ^ e c"Q - a'Q, . a'PQ'X = Q'^ [*-5] -r52. \-:.P,Qen.C'QC Q'Q, . D : ya ~ e . /^ C G'P^ . D^ . E ! limin (P«)V [*r46-51] r53. h : P, Q e n . C'Q C Q'Q, . E ! P'Cnv'P« . D . P<^' e Ded [**-52-l] r531. \-:P,Qea.C'QC a'Q, .~B'Cnv'P'i = A . D . P« e semi-Ded [**-52-l] r5401. h :. P, Qe n : D : p = B'Cnv'P^ .^.Rep. ■D,.I)'R = i'B'P [*17ryl9] *'541. f- :. P, Q e O . D : E ! 5'Cnv'P« . = . E ! 5'P [r5401] *-55. h : . P, Q e n . (7'Q C a^Qi : D : P« e semi-Ded : P« e Ded . = . E ! B'P [**-541-53-531] *-56. h : . P, Q e n . Nr'Q ^ « . D : P« e semi-Ded : P«eDed.= .E!7i'P [r55] 232 Miss Wrinch, On the exponentiation of tuell-ordered series *-6. h : . Hp *-42 . p = C'P'^ r^ R[E\ a'PQ'X = PQ'X] . p ~ e 1 : D : E ! B'P . = . E ! max (P'^'Yp Dem. [*176-19] I- :. Hp . D :R\a'PQ'\ = PQ'X . Q'E = C'Q . B'{B\- a'PQ'X) = i'B'P . =^ . E - max (P^O^ (^) t- . (1) . D h :. Hp . D : E ! B'P . = . E ! max (P'O'p r61. h:.P,Qe-Q.ao. \ = P (as . sQ^a . P = //1pT"Q'^ e; (/2p t^Q*'^) • a € C'Q, - a'Qi : D : E ! limin (P'-')'^ . = . E ! B'P Dem. h:.Hp:D:-(aP,AS.P,,S'6X.P4=6'.P[^"4.'a h :. Hp . D : ya 6 {Tp^AYX . D . Q^iV Q « (1) h.(l).r04. Dh:.Hp.D.a'PQ^\c'$a . (2) I- . (2) . D h : Hp . D . p - e 1 (3) h . **-6-433 . D h :. Hp . D : E ! limin (P«)'\ . s . E ! P*P r62. V:.P,QeVL.r^\G'Q- Q'Qi . D - ^ C C"P« . X - e : D : E ! limin {P'^yx . = . E ! P'P [^Ol] -r63. h :. P, Q 6 O . Nr'Q > « : D : P« e Ded . = . E ! B'P . B'P = A. = . P«~esemi-Ded [*r61-l] r7. h :. P, Q 6 O . D :. P« 6 Ded . = . E ! P'P [**-56-63] r8. h :: P, Q 6 O . D :. P«e semi-Ded . = . E! P'P : = : Nr'Q > co Dem. h.*r63-7. Dh::P,Qefl.D:.Nr'Q>ft): D : P« e semi-Ded . = . E ! B'P (1) |-.r56. Dh::P,Q6f2.D:.Nr'Q^ft). D:P«esemi-Ded:E!P^P.v.p'^P = A (2) |-.(1).(2). DI-::P,Qeft:.Nr'Q>a): = : P« e semi-Ded . = . E ! B'P Miss Wrinch, On the exponentiation of luell-ordered series 233 The definitions and method used in the earlier part of this paper (**"01 — •341) are suggested in Principia Mathematica *27G. There it is stated tentatively that g! p'iT^Ayx . D . ^YiTp^Ayx = min (P«)'X <- g! p\Tp^Ayx . D . PQ'X = prec (P^YX The first of these propositions is established in ***1 — •218 : the second seems to be untrue. If in the field of Q there is a term a with no immediate predecessor (as for example the term co if Q were the series of ordinals less than <w + 4), there is a X, a subclass of the field of P***, for which PQ'X is a relation covering with P-terms only the Q-terms which precede a (cp. *"61). In such a case PQ'X is not a P*^ term and so is not prec (P^yx. If P has a last term z, the relation agreeing with PQ'X as far as a and covering a and all subsequent Q places with z will be prec {P'^yx, and therefore the lower limit of X with respect to P^. Thus, while agreeing with the proposition if P and Q are well- ordered series and P has a last term, P^ is Dedekindian, and ex- tending it to the proposition if P and Q are well-orde7-ed series, P'^ is Dedekindian tuhen and only when P has a last term, we dis- agree with the conclusion that if P and Q are well-ordered series, P'^ with the addition of a term at the end is Dedekindian even if P has no last term. Instead we would substitute the propositions when P and Q are luell-ordered series, and Nr'Q ^ w, P^ with the addition of a term at the end is DedekiJidian whetJier or not P has a last term, and if Nr'Q •> o), P^ with the addition of a term at tJie end is Dedekindian luhen and only when P has a last term. 234 Mr Neville, The Gauss-Bonnet Theorem ^ The Gfinss-Bonnet Tlieorem for Multiply -Connected Jler/ions of a Surface. By Eric H. Neville, M.A., Trinity College. [Received 1 Dec. 1918: read 8 Feb. 1919.] Among the most delightful passages of differential geometry is the use of Green's theorem to prove the relation discovered by Bonnet between the integral curvature of a bounded region on any bifacial surface and the integrated geodesic curvature of the boundary. The fundamental equation is ,ds+l\Kd^S= I'^^ds, .V 'as where the line integrals are taken round the whole boundary and the surface integral over the region contained, Kg is the geodesic curvature of the boundary, K the Gaussian curvature of the surface, and f an angle to the direction of the boundary from the direction of one of the curves of reference. Though there is no allusion to curves of reference on the left of this equation, not only do these curves appear explicitly on the right, but the use of Green's theorem implies that there does exist some system of curvilinear coordinates valid throughout the region and upon the boundary, an assumption of which it is difficult to gauge the exact force. The primary object of this note is to express Bonnet's theorem in a form purely intrinsic. In the case of a simply-connected region not extending to infinity, whose boundary has continuous curvature at every point, the value of J(d^/ds)ds is 27r*. If the region is simply-connected and does not extend to infinity, but the boundary is a curvilinear polygon, formed of a finite number of arcs of continuous curvature, the sum of the external angles must be added to the integral to make the total of 27r ; in other words, j{d^/ds) ds is then the amount by which the sum of the external angles falls short of 27r. In the particular case of a curvilinear triangle, the amount by which the sum of the three external angles fails short of 'Itt is the amount by which the sum of the three internal angles exceeds tt, and is called the angidar excess of the triangle. The name is adopted to serve a wider purpose : whether a connected region of a surface is bounded by a single closed curve or by a number of * See a paper by G. N. Watson, "A Problem of Analysis Situs"', Froc. Loud. Math. Soc, ser. 2, vol. 15, p. 227 (1916). for Multiply-Connected Regions of a Surface 235 curves, the amount by which the sum of all the external angles of the boundary falls short of 27r is called the angular excess of the boundary. Whatever the number of curves forming the boundary of a region, the addition to the boundary of a simple cut, joining a point of the boundary either to a point of the cut or to a point of the boundary and described once in each direction, increases the sum of the external angles by 27r. If the cut divides the region into two parts, the angular excess of each part is the amount by which the sum of the external angles of that part tails short of 27r, and therefore the sum of the two angular excesses is the amount by which the sum of the external angles of the composite boundary falls short of 47r; this, being as we have just seen the amount by which the sum of the external angles of the original boundary falls short of 27r, is the angular excess of the original boundary. If on the other hand the cut leaves the region undivided, there is an actual decrease of 27r in the excess. It follows that if by a succession of n simple cuts the region is divided into m distinct parts, the sum of the angular excesses of the boundaries of the parts is less than the angular excess of the original boundary by 2 (?i — in + 1) IT. Suppose now that each of these parts is simpl}^- connected and that there are no singular points of the surface ni the original region or upon its boundary. Then since Bonnet's theorem in its simplest form is applicable to each of the parts, addition of the sum of the integral curvatures of the parts to the sum of the integral geodesic curvatures of the boundaries of these parts gives the sum of the angular excesses of the individual boundaries. But the sum of the integral curvatures of the parts is the integral curvature of the original region, and the sum of the integral geodesic curvatures of the boundaries of the parts is the integral geodesic curvature of the original boundary, since an arc described once in each direction adds nothing to JKgds. Hence the sum of the integral geodesic curvature of the original boundary and the integral curvature of the bounded region is less than the angular excess of the original boundary by 2 {n — ni + 1) tt. This result affords a proof that if only the dissection has reached a stage at which every part is simply-connected, the difference n — m is independent alike of the form of the cuts and of their number. Since a simply-connected region is divided by one cut into two pieces, the integer used to measure connectivity is not n — m but n — in -h 2, and Bonnet's theorem in its most general form asserts that If a bounded bifacial region of any surface has finite con- nectivity k and neither extends to infinity nor includes tvithin it or upon its boundary any singularities of the surface, the sum of the integral geodesic curvature of the boundary and the integral curva- 236 Mr Neville, The Gauss-Bonnet Theorem I ture of the region bounded is less than tlie angular excess of the boundary by 2(^' — 1) ir. In other words, the sum of the two integrals and the external angles of the boundary is 2 (2 — k) -k. Gauss' famous theorem on the integral curvature of a geodesic triangle, which may be regarded either as the simplest case or as the ultimate basis of Bonnet's theorem, is in no less need of modi- fication if the region contemplated is multiply-connected. If a geodesic triangle on any surface has internal angles A, B,C and connectivity k, and if the surface is regidar throughout the triangle and on its perimeter, the integral curvature of the triangle is A + B-\-G-{'2.k-l)7r. The application to the whole of a surface which, like a sphere and an anchor- ring, does not extend to infinity, but has no boundary, is interesting. A simple closed curve can always be drawn to divide such a surface into two distinct parts, and since its direction as the boundary of one part is opposite to its direction as the boundary of the other part, the sum of the external angles of the two boundaries is zero, and so also is the sum of their integral geodesic curvatures. It follows from Bonnet's theorem that, if there are no singular points on the surface and the connectivities of the two parts are i, j, the integral curvature of the complete surface is 2 (4 — i —j) it. Hence i +j is constant ; in order that a surface which, like a sphere, is cut by any simple closed curve into two simply-connected parts may be described as of unit con- nectivity, the connectivity is measured by the integer i+j—1, and If the connectivity of a bifacial surface which has no boundary and no singular points and does not extend to infinity is k, the integral curvature of the surface is 2 (3 — k) ir. A striking deduction made by Darboux from Bonnet's theorem may be mentioned here. If on a complete surface there is any family of curves such that the surface can be divided into a finite number of parts throughout each of which this family provides one set of curves of reference, the angle ^ of our first paragraph can be measured from the curve belonging to this family, and J{d^/ds)ds taken once in each direction over every part of an imposed boundary is necessarily zero. Hence For there to exist on an unbounded bifacial surface, which does not extend to infinity and is everywhere regidar, afainily of curves which covers the surface and is wholly withoid singularities, the surface must have integral curvature zero and must therefore be triply-connected. In conclusion the subject may be presented in another form. Let the angular excess of the boundary of a region of connectivity k reduced by 2(A;- l)7r be called the effective angular excess. If for Multiply-Connected Regions of a Surface ^ 237 a simple cut which is added to the boundary does nut divide; the region, the anguh^r excess is reduced by 27r, and, since the con- nectivity is reduced by unity, the etfective angular excess is unaltered. If, on the other hand, the cut divides the region into parts of connectivities i, j, not only is the sum of the actual angular excesses of the boundaries of the parts the actual angular excess of the original boundary, but, since /; is i +j — 1, the sum of i — 1 and J — 1 is ^' — 1 : the effective angular excess of the boundary of the whole is the sum of the effective angular excesses of the boundaries of the parts. Effective angular excess is therefore additive in precisely the same way as the surface integral of a single-valued function. If then Bonnet's theorem for a simply- connected region is expressed in the form that the sum of the integral curvature and the integral geodesic curvature is the effective angular excess, the restriction on the connectivity is seen at once to be superfluous. But to take this course implies a previous acquaintance with the theory of connectivity, whereas it is arguable that if Bonnet's theorem is used to establish the theory of connectivity the extent to which there is an appeal to intuition is materially reduced. 238 Mr 8hah and Mr Wilson, On an empirical formula, connected On an empirical formula connected with GoldhacJis Theorem. By N. M. Shah, Trinity College, and B. M. Wilson, Trinity Col- lege. (Communicated by Mr G. H. Hardy.) [Received 20 January 1919 : read 3 February 1919.] § 1. The following calculations originated in a request recently made to us by Messrs G. H. Hardy and J. E. Littlewood, that we should check a suggested asymptotic formula for the number of ways V (n) of expressing a given even number n as the sum of two primes. The formula in question is vin)^\{n) = 2Aj^^/^l^^^ (1), ^ ^ ^ ^ (log ny p-2q — 2 ^ ^ where ??, = 2'^p«^^ ... (a^l) and A denotes the constant CO J- I p assuming, in this product, the odd prime values 3, 5. 7, 11, 13, .... The formula (1) was deduced from another conjectured asymp- totic formula, namely X A{m)A(m')^2An^^^ (2), where A (m) is the arithmetical function equal to log p when m is a prime p, or a power oi p, and to zero otherwise, and the summation on the left is extended to all pairs of positive integers m, m' such that m + m = n. Formula (1) arises from (2) by replacing in the latter A{m) and A {m) each by log n. It is natural, however, to expect a more accurate result if we replace A (m) and A {ni) not by log n but by \og^n, or, better still, if we replace the left-hand member of (2) by — ^ log X log {n — x)dx (3). 1^ . The exact value of the expression (3) is found to be V (n) {(log ny - 2 log n + 2- i-tt^} (4). The various formulae thus obtained from (2) are, of course, all asymptotically equivalent ; but the modified formulae are likely to give more accurate results than (1) for comparatively small values of n. We used the formula v(n)^p(n)=2A-. -^. ^^^~i (5), ^ ^ • ^ ^ (log ny-2\ogn p — 2 q-2 ^ ^ obtained by ignoring the constant 2 — ^tt'^ in (4). luith Goldhach's Theorem 239 § 2. For the numerical data used we are indebted to two different sources. The most complete numerical results are con- tained in the tables compiled and published* by R. Haussner, which give the values of v(n) for all values of n not exceeding 5000. Tables extending up to 1000 and 2000 had been calculated earlier by G. Cantor and V. Aubry. Further data, less systematic, indeed, than those of Haussner, but extending to considerably larger values of n, were given by L. Ripertf in a number of short papers in V Intermediaire des mathe'inaticiens. The values given for v()i) in the accompanying table differ, in several respects, from those given by Haussner or Ripert. In the first place, 7n + m and m + m are here counted as different decom- positions, whereas the above two writers regard them as identical ; secondly we do not (as do Haussner and Ripert) regard 1 as a prime ; and thirdly we increase the values of v (n) obtained from their tables by addition of the number of ways in which n may be expressed as the sum of two powers of primes, i.e. the number of ways in which n=p'^ + (f', where j) and q are primes, and either a or b is greater than unity. The last two modifications make, of course, no difference to the asymptotic formula, but it seems natural to make them when the genesis of the formula (1) or (5) is considered. As regards the choice and arrangement of the numbers n in the table, the smaller numbers — i.e. the numbers not exceeding 5000 — are intended to be " typical " ; that is, they are specially selected numbers, taken in groups so as best to test or illustrate the accuracy of formula (1). Thus, for example, if the formula in question is true, a multiple of 6 may be expected, in general, to allow of an unusually large number of decompositions :|:. On the other hand a power of 2 may be expected to allow of an unusually small number. The numbers below 5000 have therefore been selected in groups of four or five, all the numbers of each group being as nearly equal as possible ; and each group of numbers contains, in general, one highly composite number (i.e. 2.3.5.7,11....), one power of 2, and one number which is the product of 2 and a prime. For values of n exceeding 5000, such choice of" typical " numbers was, unfortunately, impossible without a large amount of fresh calculation. Ripert, indeed, selected his numbers according to a system, and they, too, occur, in general, in gToujJs of approximately equal magnitude; but he selected them with different objects, so that his numbers are, from our point of view, neither " typical " nor arbitrary. * Nova Acta tier Akad. der Natur/orscher (Halle), vol. 72 (1897), pp. 5-214. t See, for example, vol. 10 (1903), pp. 76-77, 16(3-167. X It was first pointed out by Cantor, on the evidence of his numerical results previously mentioned, that this is actually so. VOL. XIX. PART V. 17 240 Mr Shah and Mr Wilson, On an empirical formula connected The accompanying table gives the number of decompositions — actual and theoretical — for thirty-five numbers ; the value found for the constant A was 0"66016. In the second column the first number is the number of decompositions, using prime numbers only, and the second the number of decompositions involving powers of primes higher than the first. § 3. Table of decompositions. n V («) P («) v{n):p{n) 30 = 2.3.5 32 = 2" 34 = 2.17 36 = 22.32 6+ 4= 10 4+ 7= 11 7+ 6= 13 8+ 8= 16 22 8 9 17 •45... 1-38... 1-44... •94 210 = 2.3.5.7 214 = 2.107 216 = 23.33 256 = 28 42+ 0= 42 17+ 0= 17 28+ 0= 28 16+ 3= 19 49 16 32 17 •85 1-07 •88 1-10 2,048 = 211 2,250 = 2.32.53 2,304 = 28.32 2,306 = 2.1153 2,310 = 2.3.5.7.11 50 + 17= 67 174 + 26= 200 134+ 8= 142 67 + 20= 87 228 + 16= 244 63 179 136 69 244 1-06 1^11 1^04 r26 1-00 3,888 = 2*. 35 3,898 = 2.1949 3,990 = 2.3.5.7.19 4,096 = 212 186 + 24= 210 99+ 6= 105 328 + 20= 348 104+ 5= 109 197 99 342 102 1-06 ro6 1^02 1-06 4,996 = 22.1249 4,998 = 2.3.72.17 5,000 = 23.5* 124 + 16= 140 288 + 20= 308 150 + 26= 176 119 305 157 1-18 roi 1-12 8,190 = 2.32.5.7.13 8,192 = 213 8,194 = 2.17.241 578 + 26= 604 150 + 32= 182 192 + 10= 202 597 171 219 1^01 roe •92 10,008 = 23.32.139 10,010=2.5.7.11.13 10,014 = 2.3.1669 388 + 30= 418 384 + 36= 420 408+ 8= 416 396 384 396 1^06 1^09 1^05 30,030 = 2.3.6.7.11.13 36,960 = 25.3.5.7.11 39,270 = 2.3.5.7.11.17 41,580=22.33.5.7.11 1,800 + 54 = 1,854 1,956 + 38 = 1,994 2,152 + 36 = 2,188 2,140 + 44 = 2,184 1,795 1,937 2,213 2,125 1-03 1^03 •99 1-03 50,026 = 2.25013 50,144 = 25.1567 702+ 8= 710 674 + 32= 706 692 694 1^03 ro2 170,166 = 2.3.79.359 170,170 = 2.5.7.11.13.17 170,172 = 22.32.29.163 3,734 + 46 = 3,780 3,784+ 8 = 3,792 3,732 + 48 = 3,780 3,762 3,841 3,866 1^00 •99 •98 ivith Goldbach's Theorem 241 §4. Goldbach asserted that every even number is the sum of two pnmes, and this unproved proposition is usually called 'Gold- bach s Theorem'. It is evident that the truth of Hardy and Littlewood's formula would imply that of Goldbach's theorem, at any rate for all numbers from a certain point onwards. Previous writers, from Cantor onwards, had noted that the HTegularity m the variation of j/(«) depends on the structure of n as a product of pi-imes. In a short abstract in the Proceedings of the London Mathematical Society, Sylvester* suggested the formula , . 2n p-2 '^'^-logn^p^ (6), where, in the product on the right p assumes all prime values from 3 to Vw, except those which are factors of n. Sylvester gives but little indication as to how he arrived at the formula and indeed there is much m his paper which is not very clear. It is at once obvious that if 71, n' are two large, but approximately equal even numbers, the values furnished for the ratio v{n):v {n') by formulae (1) and (6) will be the same. For if n =2"-p'^ q^ ... and n = 2'^ p'^' q'^ both formulae will give, as an approximate expression for this ratio the quotient ' p-2 q-2'" I p -2 q -2"" The actual values of v {n) would however be different. For from formula (6) we should deduce \ognp-2q-2-j,'^lji-l- Now n ^-2= n Pil:z^ 'A U (1-- p<\'n \ PJ where A is the same constant as in formula (1). Also it is known f that IN 2e-y n 1- P<sJn\ p) log/i ^■,1 r ^1°"- f °"^«" ^^««''- ^oc-. vol. 4 (1871), pp. 4-6 (il/atft. Papers, vol. 2, pp 709- 711). See also Math. Paj^ers, vol. 4, pp. 734-737. > I'r « c; t Landau, Handbuch der Lehre vuii der Verteilung der Primzahlen, p. 140. 17—2 242 Mr Shah and Mr Wilson, On an empirical formula connected so that (6) is equivalent to ^ ^ (logw)^ j[9 — 2 g — 2 Hence the asymptotic values furnished for v{n) by (6) and by (1) are in the ratio 2e~'>' : 1, i.e. in the ratio 1123 : 1. A quite different formida was suggested by Stackel*, viz. (log?i)2 0(n) where ^ (n) denotes, as usual, the number of numbers less than n and prime to n. This is equivalent to V (n) ' P .(9). (log ny p — lq — 1 Since p/(p—l) is nearer to unity than (p — l)/(p — 2), the oscillations of v (n) would, if Stackel's formula were correct, be decidedly less pronounced than they would be if (1) were correct. As between the two formulae, the numerical evidence seems to be decisive. Thus the ratio z^(8190) : z/(8192) is 3-32, whereas ac- cording to (1) it should be 3"48, and according to Stackel's formula it should be 2*37. Stackel's i-esult is obtained by considerations of probability which ignore entirely the irregularity of the distribution of the primes in a given interval ii^N, and it is not surprising, therefore, that it should be seriously in error. On the other hand it should be observed that Sylvester's for- mula (7) gives, within the range of the table on p. 240, very good results, not much worse than those given by (5), and decidedly better than those given by (1). This is shown by the table which follows, in which decompositions into powers of primes higher than the first are neglected. n Formula (7) v{n):2e-y\{n) Formula (1) v{n) -.X {n) 2,048 = 2" 2,250 = 2.32.53 2,304=28.32 2,306 = 2.1153 2,310 = 2.3.5. 7. 11 •95 1-17 1-18 1-17 1-12 1-06 1-31 1-33 1-31 1-26 10,008 = 23.32.139 10,010 = 2.5.7.11.13 10,014 = 2.3.1669 1-11 1-12 1-17 1-25 1-27 1-32 170,166=2.3.79.359 170,170 = 2.5.7.11.13.17 170,172 = 22.32.29.163 1-06 1-05 1-04 1-19 1-18 1-16 Gottinger Nachrichten (1896), pp. 292-299. with Goldbach's Theorem 243 § 5. It has been shown by Landau* that Sv(/0~„-7r— ^ (10); 1 ^ ^ 2(logn)2 ^ ^' and that Stackel's formula (8) is inconsistent with (10), and ac- cordingly incorrect. The same test can be applied to the formula (1) and Sylvester's formula (7). In fact Messrs Hardy and Little wood have shown f that (10) is a consequence of (1) : from which it follows, of course, that the asymptotic formula of the type of (10), furnished by Sylvester's formula, would be in error to the extent of a factor 2e~'>'= ri23 ; that Sylvester's formula is therefore also incorrect ; and that if any formula of this type is correct, it must be (1). It may seem at first surprising that, in these circumstances, Sylvester's formula should give, for fairly large values of n, results actually better (as is shown by the results in the table on p. 242) than those given by (1). The explanation is to be found in the nature of the error term in (1). The modified formula (5), which we have already shown to be likely to give better results than (1), for moderately large values of n, differs from (1) by a factor of the type 9 1+r^ +.... logn This factor does not affect the asymptotic value of v (n), but it makes a great deal of difference within the limits throughout which verification is possible : thus when n= 170,170 it is equal to 1"166. When n= 10^", it is equal to 1"087, and its difference from unity is negligible only when n is quite outside the range of computation. It is only such values of n that would reveal the superiority of the unmodified formula (1) over Sylvester's formula. 1 6. Shortly after the writing of the preceding sections had been completed, Mr Hardy informed us of the existence of a third pro- posed asymptotic formula for v (n), given more recently by V. Brun;|:. The formula to which Brun's argument leads is v{n)^2Bn^'~l^^ (11), p—z q—2 where 5= { 1 - " ] ( 1 - " ) ( 1 -^ ) ... ( 1 -r =(-!) ('-?) (^ 2' 7, h<\'u f = n (1 -I)- 7i = 3 V " GiHtinger Nachrichten (1900), pp. 177-186. t See their note which follows this paper. :!: Archiv for Mathematik (Christiauia), vol. 34, 1917, no. 8. See also § 4 of Hardy and Littlewood's note. 244 Mr Shah and il/?- Wilson, On GoldbacJiS Theorem By an argument similar to that used in §4, in the reduction of Sylvester's formula, it may be shown that this is equivalent to the formula i.(n)~8^e-^y^ ,,^^...=4e-=yX(70 (12). (log nf p-2 q- 2 Thus this asymptotic value for v (n), and the Hardy-LittleAvood value, are in the ratio 4e~-T : 1 = 1'263... : 1. Sylvester's is their geometric mean. The formulae (11) and (12) would furnish a quite close ap- proximation for V (n) for those values of oi on which it could be, in practice, tested. Thus, for n = 170,170, we find that v{n)/4^e-'yX{n) = -9S.... But the ultimate incorrectness of the formula may be proved in the same way as that of Sylvester's formula, namely by use of Landau's asymptotic formula (10). Brun knew of the memoirs of Stackel and Landau, but appears to have been unacquainted with Sylvester's work. Mr Hardy and Mr Littlewood, Note on Messrs Shah, etc. 245 Note on Messrs Shah and Wilson's paper entitled: 'On an empirical formula connected with Ooldbach's Theorem '. By G. H. Hardy, M.A., Trinity College, and J. E. Littlewood, M.A., Trinity College. [Received 22 January 1919: read 3 February 1919.] 1. The formulae discussed by Messrs Shah and Wilson were obtained in the course of a series of researches which have occupied us at various times during the last two years. A full account of our method will appear in due course elsewhere*: but it seems worth while to give here some indication of the genesis of these particular formulae, and others of the same character. We have added a few words about various questions which are suggested by Shah and Wilson's discussion. The genesis of the formulae. 2. Let f{x) = %A (n) X'' = SA (n) e-"y = F{ij) and /, (x) = F, {y) = Ix^ (n) A (n) e"".", where A (n) is equal to logp when n is a prime p, or a power of ^j, and to zero otherwise, and x< (^0 i^ one of Dirichlet's ' characters to modulus (/'+. Also let X = xe'^'^''?, where p is positive, less than q, and prime to q ; and suppose that X tends to unity by positive values. It is known that n ^X'^{v)A{v) = o{n\ 1 unless Xk is the ' principal ' character Xi> ^^ which case n n 1 1 It follows that (2-1) A(K)'^-- ^ l-x and (2-2) f^^''^ = ^{l^ ^'^^l^- * An outline of one of its most important applications is contained in a paper entitled ' A new solution of Waring's Problem ', which will be ijublished shortly in the Quarterly Journal of Mathematics. t See Landau, Haudbuch, pp. 391 et seq. ■I 246 Mr Hardy and Mr Littleivood, Note on Now (2-3) /(^) = 2A(?2)x"e2n/.'^^'9= S e=-^^^'^'"? 2 A(n)x>\ i = l n=,i If J is prime to q, we have* (2-4) S A (n) X- = -^ 'S X. ( j) /. (X), where %« is the character conjugate to %«, and ^(5') is the number of numbers less than and prime to q. It follows from (2'1) and (2-2) that (2-5) S A (n) x» ^ 144 r^- = X?-M ^ • n=j </)(g) 1-X <l){q)l-K If on the other hand j is not prime to q, the formula (2'4) is untrue, as its right-hand side is zero. But in this case A (??) = unless n is a power of q, so that (2-6) S Ain)K^^=^o(~). From (2-3), (2-5), and (2-6) it follows that (2-7) •^^^•)~l~x' where ^ ^ ' <P{q)7 ^{q)7' ' the summation extending over all values of j less than and prime to q. The sum which appears in (2*71) has been evaluated by Jensen and Ramanujanf, and its value is /Li{q), the well-known arithmetical function of q which is equal to zero unless 5 is a product JJ1P2 ■•• Pp of different primes, and then equal to (— 1)p. Thus (^•«> f^^>-mTh.t- 3. The sum (3-1) co{n)= t A(m)A(??0, ■m + m'=n * Landau, I.e., p. 421. m t J.L.W.V. Jensen, 'EtnytUdtryk for den talteoretiske Funktion 2M(n) = M{my, 1 Saertryk af Beretning om den 3 Skandinaviske Matematiker-Kongres, Kristiania, 1915 ; S. Eamanu jan, ' On certain trigonometrical sums and their applications in the theory of numbers', Trans. Camb. Phil. Soc, vol. 22, 1918, pp. 259-276. J If ^ (q) is zero, this formula is to be interpreted as meaning /w=o(r^) Messrs Shah and Wilsons paper 247 which appears on the left-hand side of Shah and Wilson's equation (2), is the coefficient of a;'* in the expansion of [/(«))". And when a; ^ e2i?7ri7(? along a radius vector. Our general method ac- cordingly suggests to us to take n(n) = nt\^i(i^'e-'-^^P-il<i, where the summation extends over 5'= 1, 2, 3, ... and all values of jt) less than and prime to q, as an approximation to co (n). Using Ramanujan's notation, this sum may be written (3-2) n{n)=nl\^^l^%,(n). The series (3'2) can be summed in finite terms. We have (3'3) c,(7i) = SS/.(| the summation extending over all common divisors S of q and n*; and it is easily verified, either by means of this formula or by means of the definition of Cq{n) as a trigonometrical sum, that Cqq'{n) = Cq{n)Cg'(n) whenever q and q' are prime to one another. We may therefore write n(n)=n:ZAq = nUx-r!r, where the product extends over all primes -or, and since Aq contains the factor /j,(q) and A^^^., A ^-3, ... are accordingly zero. If n is not divisible by zj, we have c^ (n) = /u, (ot) = — 1 and A =_^ 1 = L__. while if n is divisible by zr we have Cw('0 = At(t3-) + t3-/A(l)= OT - 1, . _ 1 CT — 1 Hence * Ramanujan, I.e., p. 260. 248 Mr Hardy and Mr Littlewood, Note on where 11' applies to primes which divide n and II" to primes which do not. It is evident that O (n) is zero if n is odd. On the other hand, if n is even, we have "W = 2»n|i-(-'-3y,}n -^)/|-.^-:. = 2^nn-^^ln^P-^ where tn- now runs through all odd primes and p through odd prime divisors of n. The formula w (n) ^- il (n) is formula (2) of Shah and Wilson's paper*. The incorrectness of Sylvester's formula. 4. It is easy to prove that if any form tda of the type (4-1) fw (7^) ~ CO (n) be true, then G must be unity. In other words, our formula is the only formula of this type which can possibly be correct. This may be shown as follows. Let (4-2) f(^) = t'l^l where n runs through all even values; and let s — 1 = ^. The series is absolutely convergent if s > 2, ^ > 1. Replacing 12 (n) by its expression in terms of the prime divisors of n, and splitting up f{s) into factors in the ordinary manner, we obtain say, where A is the same constant as in Shah and Wilson's paper, and OT runs through all odd primes. Let + (*) = n (i + r^--.) = n [^}^_) = (1 - 2-0 f (0. and suppose that ^^1. Then x(0.nifi+^-4,*"4Vfi+i-^)} 'y\r{t) [\ OT-2 1- •37 V(^-2)| ^'1(37-1)^-11 A' When fi(n) = 0, the formula is to be interpreted as meaning w(H) = o(/t). Messrs Shah and Wilsons paper 249 and so (4-3) f{s) ~ 2^^ (0 -2(1- 2-0 ?(0 - ^1 = ,--2 • This is a consequence of our hypothesis : the corresponding consequence of the hj^pothesis (4"1) would be (4-31) /(*^>~^- On the other hand, it is easy to prove* that (4-4) &)(l) + ft)(2)+ ... +&)(7i)~i«'; and from this to deduce that <^(.) = 2 on (n) 1 w* s — 2 when s—>2. This equation is inconsistent with (4"1) and (4'31), unless (7 = 1. It follows that Sylvester's suggested formula is definitely erroneous. It is more difficult to make a definite statement about the formula given by Brun. The formula to which his argument naturally leads is Shah and Wilson's formula (12); and this formula, like Sylvester's, is erroneous. But in fact Brun never enunciates this formula explicitly. What he does is rather to advance reasons for supposing that some formula of the type (4"1) is true, and to determine G on the ground of empirical evidence^. The result to which be is led is equivalent to that obtained by taking C= 1-5985/1-3203 = 1-2107 %. The reason for so substantial a discrepancy is in effect that explained in the last section of Shah and Wilson's paper. Further results. 5. The method of § 2 leads to a whole series of results con- cerning the number of decompositions of n into 3, 4, or any number of primes. The results suggested by it are as follows. Suppose * Since SA (71) .r™-:; as a;-*-l, we have 2w(H)a;"= (SA («) a;"}-~ ,- - -', and the desired result follows from Theorem 8 of a paper published by us in 1912 (' Tauberian theorems concerning power series and Dirichiet's series whose coefficients are positive', Proc. London Math. Soc, ser. 2, vol. 13, pp. 174-192). This, though the shortest, is by no means the simplest proof. The formula (4-4) is substantially equivalent to Landau's formula (10) in Shah and Wilson's paper. t Evidence connected not with Goldbach's theorem itself but with a closely related problem concerning pairs of primes differing by 2. See g 7. 1 1-5985 is Brun's constant, while 1-3203 is 2A. 250 Mr Hardy and Mr Littlewood, Note on that Vr (n) is the number of expressions of n as the sum of r primes Then if r is odd we have (5-11) v,(n) = o()i>-') if V is even, and if n is odd, p being an odd prime divisor of ??, and (513) B=n[i^^^^^^^. where tn- runs through all odd primes. On the other hand, if r is even, we have (5-21) Vr(n) = o{n''-^) if n is odd, and where (5-23) c=n|i- (^-ir if 71 is even. The last formula reduces to (1) of Shah and Wilson's paper when r = 2. We have not been able to find a rigorous proof, independent of all unproved hypotheses, of any of these formulae. But we are able to connect them in a most interesting manner with the famous ' Riemann hypothesis ' concerning the zeros of Riemann's function f (5). The Riemann hypothesis may be stated as follows : ^(s) has no zeros whose real part is greater than ^. If this be so, it follows easily that all the zeros of ^(s), other than the trivial zeros s = — 2, s = — 4, ..., lie on the line <t = 'R{s) = ^. It is natural to extend this hypothesis as follows: no one of the functions defined, luhen a- > 1, hy the series n" possesses zeros luhose real part is greater than ^. We may call this the extended Riemann hypothesis. This being so, what we can prove is this, that if the extended Riemann hypothesis is true, then the formidae (5"11) — (5'23) a?'e true for all values of r greater than 4. The reasons for supposing the extended hypothesis true are of the same nature as those for supposing the hypothesis itself true. It should be observed, however, that it is necessar}", before we generalise the hypothesis, to modify the form in which it is usually stated; for it is not proved (as it is for ^{s) itself) that L{s) can have no real zero between ^ and 1. 1 Messrs Shall and Wilson's paper 251 6. A modification of our method enables us to attack a closely related problem, that of the existence of pairs of primes differing by a constant even number k. We have 2 A (n) A (n + k) r^''+'' = J- f '" \f{re^^) \ ' e'^'^ cW, where f(x) is the same function as in § 1, and r is positive and less than unity. We divide the range of integration into a number of small arcs, correlated in an appropriate manner with a certain number of the points e"P''^"J, and approximate to {/(j'e'")!^ on each arc by means of the formula (2-8). The result thus suggested is that ^A{n)A (n + k) r- ^ ^^~ U (^ £ ^) , where A has the same meaning as in § 2 and p is an odd prime divisor of k. From this it would follow that (6-1) S A (v) A (v + k) - 2AnU P^) ; and that, if A^^. (?2) is the number of prime pairs less than /?, whose difference is k, then T.r / N 2ylri „ /p — 1\ (6-2) ^V'«~(i^^,n(P-2). This formula is of exactly the same form as (1), except that p is now a factor of k and not of n. In particular we should have ,_ . , 2An (6-3) ^^(»)~(lo-g»r and (6-4) ^^"<»>~(Ttg-LV We should therefore conclude that there are about two pairs of primes differing by 6 to every pair differing by 2. This conclusion is easily verified. In fact the numbers of pairs differing by 2, below the limits* 100, 500, 1000, 2000, 3000, 4000, 5000, are 9, 24, 35, 61, 81, 103, 125; while the numbers of pairs differing by 6 are 16, 47, 73, 125, 168, 201, 241. * To be precise, the numbers of pairs ( j), p') such that p' =p + 2 and p' does not exceed the limit in question. 252 My^ Hardy and Mr Littlewood, Note on The numbers of pairs differing by 4, which should be roughly the same as those of pairs differing by 2, are 9, 26, 41, 63, 86, 107, 121. 7, Brun, ni his note ah'eady referred to, recognises the corre- spondence between the problem of §§ 2—4 and that of the prime- pairs differing by 2, and realises the identity of the constants in- volved m the formulae ; but does not allude to the more o-eneral problem of prime-pairs differing by k. He does not determme the fundamental constant A, attempting only to approximate to it empirically by means of a count of prime-pairs differing by 2 and less than 100000, made by Glaisher in 1878*. The value of the constant thus obtained is, as was pointed out in § 4, seriously in error. The truth is that when we pass from (6-1), which, when k = 2, takes the form 2 A{v)A{v + 2)r^^An, to (6-3), the formula which presents itself most naturally is not (6-3) but "^ (7-1) i\r3(n)o.2yir--^. J (log^)- This formula is of course, in the long run, equivalent to (6-3) But (log xf (log ny \ ^ log n "^ (log nf "^ " ' 7 ' and the second factor on the right-hand side is, for n = 100000 far from negligible. Thus (6-3) may be expected, for such values of n, to give results considerably too small. }^C^^ *^^® *^® ^^^^®^^ ^"^^^* ^^ integration in (7-1) to be 2 we find that the value of the right-hand side for n = 100000 is to' the nearest integer, 1249, whereas the actual value of i\^., (92) is, accord- ing to Glaisher, 1224^ The ratio is 1-02, and the agreement seems to be as good as can reasonably be expected. Tir "Sf ^^l^^^ation of prime-pairs has been carried further by Mrsfetreatteild, whose results are exhibited m the following table: *'ri '^^■- • ^^e number of pairs below 100000 is 1225 t iiie series is naturally divergent, and must be closed, after a finite number of terms with an error term of lower order than the last term retained ^ Glaisher reckons 1 as a prime and (1, 3) as a prime-pair, making 1225 in all. Messrs Shah and Wilsons paper 253 11 N,(n) 2 A f" ''"■ Ratio 1 1 100,000 1224 1249 ! 1-020 200,000 2159 2180 1-010 300,000 2992 3035 1-014 400,000 3801 3846 1-012 500,000 1562 4625 1-014 600,000 5328 5381 1-010 8. In a later paper* Brun gives a more general formula relating to prime-pairs {p, p) such that p = ap + 2. This formula also involves an undetermined constant k. It is worth pointing out that our method is equally applicable to this and to still more general problems. Suppose, in the first place, that v{n) is the number of expressions of n in the form n = ap + hp, where pi and p' are primesf. We may suppose without loss of generality that a and h have no common factor. The results suggested by our method are as follows. If n has any factor in common with a and h, then "<">='' {(log. o-^}' and this is true even when n is prime to both a and h, unless one of n, a, b is even|. But if n, a and b are coprime, and one of them even, then 2J. n P-i ab (log iif \p— 2 where A is the constant of § 2, and the product is now extended over all odd primes which divide n or a or b. * ' Sur les nombres premiers de la forme ap + h\ Archiv for Mathematik, vol. 24, 1917, no. 14. t We might naturally include powers of primes. + These results are trivial. If n and a have a common factor, it divides hp', and is therefore necessarily p' , which can thus assume but a finite number of values. If n, a, h are all odd, either ^^ ox p' must necessarily be 2. 254 Mr Hardy and Mr Littleiuood, Note etc. Similarly, suppose N(n) to be the number of pairs of solutions of the equation a})' — hp = h such that p' < n. It is supposed that a and h have no common factor. Then N(n) = o l(logw)- unless k is prime to both a and b, and one of the three is even. If these conditions are satisfied where p is now an odd prime factor of k, a, or h. Mr Harrison, The distribution of Electric Force, etc. 255 The distribution of Electric Force bettueen two Electrodes, one of which is covered with Radioactive Matter. By W. J. Harrison, M.A., Fellow of Clare College. [Read 17 February 1919.] It has been shown by Rutherford* that it is probable that the ionisation due to an a particle per unit length of its path is in- versely proportional to its velocity, provided the velocity exceeds a certain minimum necessary to effect ionisation. It follows that the ionisation per unit time is constant at all points of the path. Suppose radioactive matter distributed uniformly over the sur- face of a large plane electrode assumed to be infinite in order to obtain simplicity in calculation. Consider the a particles projected from a point P of the electrode. These particles are projected equally in all directions, hence the rate of ionisation per unit volume at a point Q will be proportional to l/PQ^ provided PQ< R, where R is the range of the particles. The total rate of ionisation at a point Q distance x (j: < R) from the electrode will be proportional to ■\/Ji---'e"27rrdr x- + r^' where r is the distance of a point P on the electrode from the foot of the perpendicular from Q. Now •sjitr—x^ 27' dr Jo x^ + r'' = log log (x- + r^) R sJlP-X' Hence rate of ionisation 1 ^ ^ = ^0 log -; . X The equations determining the distribution of electric force are given by Thomson, Conduction of Electricity through Gases, 1906, chap. III. The notation of this book is adopted as being sufficiently well known. The differential equation for the electric force X is of the form d'X^^ a (dX-'\' b , R „ _— =0, x> R. dx~ * Radioactive Substances and their Radiations, 1913, p. 158. VOL. XIX. PART v. 18 256 Mr Harrison, The distribution of Electric Force hetiueen ttuo The numerical solution may be obtained for any particular values of the constants a, b, c, q^, R by approximate methods. In the absence of any definite experimental results with which to compare the calculations, the labour involved in integration is not worth undertaking. The case, however, of the saturation current is the most impor- tant, and the integration is simple. It is assumed that recombi- nation of ions does not take place in this case, and therefore the equations reduce to = 0, x>R. Write Si7eq,{^ + y\=K. Then, for x< R, for X > R, 9 log 7. -\-ix- + Bx + G (vide Rutherford, Radioactive Substances, etc., p. 67), A, B, G are constants of integration. Now the conditions are (1) at ^ = 0, ni = 0, if ^ = be the positive plate, (2) at a; = J?, Wg = 0, (3) at a; = i^, n^ is continuous, (4) Sit x = R, X \B continuous. {vide Conduction of Electricity through Gases, chap. iii.). dX^ _ _ Sttj ' ' dx kz ' (2) and (3) lead to the same condition, which is the same as (1), if i = eRqQ. Now since there is no recombination . [^ R 1=1 eqolog - dx = eRqo. Jo ^ Electrodes, one of which is covered luith Radioactive Matter 257 Hence conditions (1), (2), (3) are identical and determine B. Condition (4) supplies a relation between G and A, A^K{G-IR^). Hence X' = K ^x~ log h f h Rx + BR' X' = K CO R. X ft-l ~f~ rCo 0<x<R, where BR- = C, The constant B can be determined when the potential differ- ence between the electrodes is given*. The general character of these results can be shown by numerical calculation for the cases k, = L, l-25k,=k^, A;i = 1-25 A-., (corresponding to the case in which the positive ion moves more slowly, as usual, than the negative ion, and the radioactive matter is spread on the negative plate), and for distances R, 2R, SR between the electrodes, and for B = O'l, 0-5, I'O. In order that the current may be the satu- ration current it is necessary in practice that B should exceed a certain limit. This limit is dependent on the particular conditions of any given experiment. The distribution of the electric force X is shown on the graph below. The curves marked (1), (2), (3) are for the cases kj = 1-25 k.2, ki = ^'2, A.-2 = 1-25 k^, respectively. The potential difference V between the electrodes is given in the following table, d being the distance between the plates. V RKK^' ki = l-25k.2 ^1 = ^2 k.2^1-25ki D=0-l d=R d=2R d=3R -343 1-056 2-034 -379 1-147 2-193 •412 1-232 2-343 D = Ob d=R d=2R d=3R -725 1-676 2-841 -743 1-740 2-963 -762 1-800 3-078 D = l-0 d=R d=2R d=3R 1-014 2-203 3-566 1-027 2-250 3-663 1-041 2-298 3-759 These forms of X are not strictly valid in the immediate neighbourhood of the electrodes, as the natural agitation of the ions has been neglected in this theory Vide Pidduck, Treatise on Electricity, 1916, p. 505. 18—2 258 Mr Harrison, The distribution of Electric Force, etc. I '0 R Z'O R DISTANCE BETWEEN ELECTRODE^. 3-0 R Mr Purvis, The conversion of saiv-dust into sugar 259 The conversion of saw-dust into sugar. By J. E. PuRViS, M.A. [Read 17 February 1919.] The production of sugar from wood is well known. In the Classen process, saw-dust is digested in closed retorts with a weak solution of sulphurous acid under a pressure of between six and seven atmospheres. The products contain about 25 °/^ of dextrose, and other substances are pentose, acetic acid, furfurol and formal- dehyde. Cellulose material can also be converted into sugar by other acids. The following results were obtained by digesting saw-dust from ordinary deal with different acids of varying concentrations ; estimating the amount of sugar in the liquid in the usual way from the amount of cuprous oxide precipitated from Fehling's solution, and converting this oxide of copper to cupric oxide. The numbers were then calculated in terms of dextrose. (1) 25 grams of saw-dust were digested with 300 c.c. distilled water and 50 c.c. strong H2SO4 (1 c.c. H2S04 = 1*78 grms. H2SO4) for 5^ hours in a sand bath at a temperature just below the boiling point and the mixture was constantly stirred. This was then filtered ; the residue well washed and the filtrate made up to a litre ; 10 c.c. of the filtrate were neutralised with sodium carbonate and the cuprous oxide from Fehling's solution was precipitated, filtered, dried and ignited to cupric oxide. This gave 0"215 grm. CuO which is equivalent to 39 °/^ of dextrose on the original amount of saw-dust. (2) 25 grams of saw-dust to which were added 500 c.c. of distilled water and 25 c.c. of strong H2SO4 of the same strength as in experiment (1) and digested for 5 hours under the same conditions. This gave 13 °/^ of dextrose. (3) 50 grams of saw-dust were digested with 500 c.c. of distilled water and 50 c.c. of the strong H2SO4 for 5f hours. The yield was 11 "5 % dextrose. (4) 25 grams of saw-dust were digested with 250 c.c. of tap water and 10 c.c. of strong H2SO4 for 2 hours. This yielded 10*5 7o dextrose. (5) 25 grams of saw-dust were digested with 720 c.c. of tap water and 10 c.c. strong H2SO4 for 2 hours. This produced 3*35 "/^ dextrose. (6) 50 grams of saw-dust were digested with 500 c.c. water and 50 c.c. N/1 HCl (= 1-825 grms. HCl) for 3 hours. This gave 3-35 % dextrose. 260 Mr Purvis, The conversion of saw-dust into sugar (7) 50 grams of saw-dust were digested with 500 c.c. water and 100 c.c. N/1 H2SO4 (= 2-45 grms. H.SO4) for 2 hours. This produced 1'82 °/^ dextrose. (8) 25 grams of saw-dust were digested with 700 c.c. water and 5 grams P0O5 for 12 hours at the temperature of the room (about 15° C), and then for 3 hours just below the boiling point. This gave 12'66 °/^ dextrose. The results show that the amount of sugar which can be obtained depends on the nature of the acid and its strength relative to the amount of saw-dust, and on the time of digestion. The greatest amount was obtained when the strongest sulphuric acid acted for a considerable time. In the other experiments not so much was obtained as by the Classen process. For the commercial production of sugar from such a cheap material as saw-dust the question to be decided would be the relative cost of the Classen process compared with the cost under the conditions of these experiments. That would include a comparison of the cost of the various acids and the recovery of these acids for further use. The conversion of sugar into alcohol and acetone presents no difficulty ; and it would be important to consider whether such useful chemical substances could not be produced from a waste product like saw-dust at a cheaper rate than by the present costly methods. Mr Purvis, Bracken as a source of potash 261 Bracken as a source of potash. By J. E. Purvis, M.A. [Read 17 February 1919.] The Master of Christ's College, Cambridge, in the autumn of 1917, had some correspondence with Mr J. A. A. Williams of Aberglaslyn Hall, Beddgelert, in regard to the use of bracken as a fertiliser. Mr Williams had burnt the bracken growing on a peaty soil on his estate at Beddgelert, ploughed in the ashes and obtained highly satisfactory crops of potatoes. It seemed to be of some importance to find out what amount of potash could be obtained from the ash; and in October 1917 a sample of bracken from the Botanic Gardens, Cambridge, was analysed. This grows on a poor sandy soil. It is known that bracken contains larger quantities of potash in the summer months than in the autumn and more complete investigations were deferred till the summer of 1918. Meanwhile in the April (1918) number of the Journal of Agriculture (vol. 25, no. 1, p. 1) Messrs Berry, Robinson and Russell published an article on " Bracken as a source of potash " which contained the results of the analyses of material collected from various districts in England, Scotland and Wales from May to October 1916, and from June to October 1917. The numbers show that the amount of potash is much higher in the summer months than in the autumn. For example, bracken gathered June 1st, 1917, from Harpenden Common, Rotharnsted, which is mainly gravel and clay, produced 4"1 ° I ^ of potash (KoO) on the dried material and only 1'8 7o when gathered September 1st, 1917. The authors also considered that their evidence indicates a more rapid falling off of the potash from bracken growing on sandy and peaty soils than on heavier soils rich in potash : and that, therefore, its chances of success as a fertiliser would be greater in these heavier soils. In view of these results the investigations were continued with the bracken growing in the Botanic Gardens, Cambridge, and also with that on Mr Williams's Welsh estate. The following tables summarise the results. Generally, the numbers are of the same order as those obtained by Messrs Berry, Robinson and Russell, and confirm the opinion that in the summer months there is more potash than in the later months. Also there is a clear indication that, on an average, the Welsh peaty soil yields more potash than the Cambridge poor sandy soil. 262 Mr Purvis, Bracken as a source of potash Cambridge Bracken. Date when sample was gathered Percentage of dry matter in fresh bracken Percentage of ash in dry matter Percentage of potash (K2O) in fresh bracken dry bracken 16 October, 1917 27-60 7-51 0-29 0-82 1 June, 1918 15-34 6-81 0-46 3-00 2 July, 1918 21-58 5-02 0-52 2-45 1 August, 1918 30-26 5-96 0-50 1-70 31 August, 1918 26-50 7-86 0-30 1-07 1 October, 1918 29-06 7-93 0-33 1-13 Welsh Bracken 3 June, 1918 24-4 6-55 0-77 3-19 4 July, 1918 25-8 5-78 0-83 3-22 31 July, 1918 40-7 3-84 0-42 1-45 1 September, 1918 30-97 7-02 0-53 1-71 3 October, 1918 34-54 4-82 0-45 1-32 To estimate the cost of collection is difficult as the conditions of transit and labour are variable and estimates for one locality would be useless for another. It is evident, however, that bracken is a valuable source of potash : but its economic application as a fertiliser will be controlled by the requirements and conditions of the neighbourhood where it grows. I have to thank Mr Williams for supplying the Welsh bracken, and Mr Lynch, of the Cambridge Botanic Gardens, for samples from the gardens. Dr Shearer, The action of electrolytes on the electrical, etc. 263 The action of electrolytes on the electrical conductivity of the bacterial cell and their effect on the rate of migration of these cells in an electric field. By C. Shearer, Sc.D., F.R.S., Clare College. (From the Pathological Laboratory, Cambridge.) {Read 17 February 1919.] If a thick creamy emulsion of the meningococcus or B. coli is made up in neutral Ringer's solution (that is, one in which the sodium bicarbonate is left out), and the conductivity measured by means of a Kohlrausch bridge and cell; it is found that its resistance is more than treble that of the same solution without the bacteria : that is the greater part of the resistance is due to the presence of the bacteria. This determination was made as follows: a 24 hour culture of the meningococcus or B. coli on trypagar (2-t plates) was washed off in a considerable quantity of Ringer's solution, centrifuged down and re washed several times in a similar manner to remove all traces of serum or any salts derived from the culture medium. The centri- fuged deposit was then made up to standard strength in neutral Ringer's solution, so that it was not too thick to be sucked up in a medium sized pipette and transferred to a Hamburger cell and its conductivity determined. It was found that the conductivity of such standard emulsions when measured under similar conditions of temperature was fairly uniform*. When sufficient care was taken to get the emulsions of the right thickness, resistances of 110 ohms could be pretty constantly obtained. The same quantity of Ringer's solution alone had about 26"7 ohms resistance under the same conditions. If, however, in place of the Ringer's solution we make up the bacterial emulsions in pure sodium chloride of the same conducti- vity as that of the Ringer's solution, i.e. one in which the resistance is 26'7 ohms (which corresponds to a NaCl solution of about 0'85 °/^), we obtain as in the case of the emulsion in Ringer's solution an initial resistance of 110 ohms. Within a few minutes, however, this gradually drops and at the end of 30 or 40 minutes the emulsion now has the same conductivity as that of the bare sodium chloride solution without the bacteria, i.e. 26'7 ohms resistance. Thus pure sodium chloride of about the concentration as that present in the blood gradually destroys the resistance of the bacterial cell. If the bacteria are allowed to lie in this solution for several hours it will be found that at the end of this time, on subculture, they are ••" All measurements were made at constant temperature 25° C. Resistance con- stant of conductivity cell = 29 8 x 10~^. 264 Dr Shearer, The action of electrolytes on the dead. If they are only allowed to remain in the NaCl for a short time and then transferred to neutral Ringer again they immediately return to their normal resistance and grow freely on subculture. If when the resistance of the bacterial emulsion has fallen in NaCl solution a little CaCL is added it again regains its normal conductivity and is uninjured. Thus we get the usual antagonistic action of CaCla to NaCl. It was found that KCl, LiCl, MgCl^ acted like NaCl in reducing the resistance offered by the bacteria, while BaCls, SrClg have no action on the resistance but act like CaClg. Thus it is clear that in the bacteria as with so many other plant and animal cells the entrance of the ions of NaCl, KCl, LiCl, MgCla is prevented by the presence of very small quantities of CaCL, BaCla or SrClg. Bacterial emulsions made up in BaClo, SrCla and CaClg , having the same conductivity as Ringer's solution, showed no change in resistance on being kept in these solutions for some time, invariably remaining normal. The interest of these experiments consists in that they agree completely with the results obtained by Loeb, Osterhout and a large number of other workers on animal and plant cells. In Laminaria, Osterhout finds with CaCL and presumably also with BaCla and SrCl.. there is invariably a brief temporary rise in resistance when placed in these solutions of the same conductivity as sea-water which is followed by a gradual fall. With the bacterial cell no such preliminary rise can be distinguished, w^hile the fall due to the toxic action of the solution is much delayed and slower. In view of the remarkable action of tri-valent ions on artificial membranes as shown by the work of Perrin, Girard and Mines, and the action on the permeability of cell wall as shown by the work of Mines, Osterhout and Gray, it is of great interest to consider their action on the bacterial cell. While the tri-valent positive ion of lanthanium nitrate brings about a rapid rise of resistance in Laminaria according to Osterhout and in the Echinoderm egg according to Gray, when this salt is used in such dilution as not to affect the conductivity of the solu- tion itself, no such action can be distinguished in the case of bacteria by means of the Kohlrausch method. The resistance remains unchanged until it begins to fall on account of the in- creasing strength of the salt added. In the same way the positive tri-valent ions of CeCL, neo-ytterbium chloride and the tri-valent negative ions of sodium citrate appear to have no action in in- creasing or decreasing the resistance of the bacterial cell as deter- mined by the conductivity method. It should be pointed out that these salts can only be used in very dilute solutions. In the case of lanthanium nitrate this salt readily flocculates living bacteria Avhen used in stronger solutions than ^ y\j^ M. It would seem remarkable in view of the sharp action of La on electrical conductivity of the bacterial cell, etc. 265 the Echinoderm egg when used in a strength of g^*^ M. that some similar action should not be found with bacteria, but repeated experiments with centrifuged solid bacterial deposits of both the meningococcus and B. coli using the same type of electrodes used by Gray for the Echinoderm egg and obtaining resistances as high as 150 ohms failed to show any initial rise of resistance. It was possible that in the case of bacteria, their enormous surface would render the preliminary rise of resistance so temporary that, before the electrodes could be placed in position and the bridge readings adjusted, it would be over and passed. To test this point a small quantity of La was added while the bridge telephone was kept to the ear, but in every instance no change could be detected. It would seem that the bacterial cell is normally in a state of maximum impermeability and that this can not be further increased by the presence of CaCL and the tri-valent salts. In distinction to the absence of effect of the tri-valent salts on bacteria as demonstrated by the conductivity method, is the marked action of these salts and especially lanthanium nitrate in changing the rate of migration of these cells in an electric field. This can be determined by the ultramicroscopic or still better the U tube method. If 10 c.c. of a thick growth of B. coli in spleen broth be run into a U tube under neutral Ringer's solution of the same conducti- vity as the broth, then on passing an electric current through the tube, the temperature being constant, an even rapid migration of the bacteria takes place towards the anode. That practically all bacteria carry a negative charge and migrate to the anode has been repeatedly confirmed by numerous workers, but what is of interest here is that this charge can be materially modified by various tri-valent salts, especially La. If to the 10 c.c. of B. coli emulsion in spleen broth run into the U tube in the above experiment 1 c.c. of a -^^ M. lanthanium nitrate solution is added, it will be found that the rate of migration of the bacilli under the same conditions of electric field and temperature is now halved. If 2 c.c. of the solution is added, little or no migra- tion takes place and the emulsion soon flocculates and is preci- pitated to the bottom of the tube. In terms of the Helmholtz-Lamb theory of the double electric layer the addition of the La has considerably altered the nature of the charge on the bacterial cell wall. The conductivity method however fails to show any change under this condition. This result is possibly of some interest in view of Mines' theory of the polarising action of certain ions on the cell membrane. It is of course possible that the resistances obtained in the conductivity experiments were too low to bring out the real changes taking place. 266 Miss Haviland, The bionomics of Aphis The bionomics of Aphis grossulariae Kalt., and Aphis viburui Schr. By Maud D. Haviland, Bathurst Student of Newnham College. (Communicated by Mr H. H. Brindley.) [Read 17 February 1919.] Aphis grossulariae Kalt. is a serious pest 'of currant and goose- berry bushes in this country. It attacks the young shoots in May, and when present in numbers, it distorts them to such an extent that growth ceases and a dense cluster of leaves is formed, under which the aphides swarm. The bionomics of this aphis are incompletely known. It appears on red currants in May, and remains there until the middle or end of July. The sexuales have never been found. In 1912 Theobald (Journ. Econ. Biol., vol. Vli. p. 100) first pointed out its resemblance to Aphis viburni Schr., a common species, which is found on the guelder rose ( Viburnum opulus) in spring and summer, while the sexual forms have been recorded from the same plant in the autumn. Aphis viburni has a very characteristic appearance, owing to the row of lateral tubercles on the abdomen. Such tubercles are not very common among the Aphidinae, but they are prominent like- wise in Aphis grossulariae. In fact there seems to be no structural difference between the two species; though in spirit specimens, the guelder rose aphis frequently stains the alcohol dark brown, while the currant form has no such property. In May 1918, I had under observation some red and black currant bushes, and two guelder rose shrubs, which all grew close together. Early in the month all were free from aphid attack, but on May 31st three colonies, each consisting of a single winged female with a few new-born young, appeared on the guelder roses, and the same evening four sprigs of currant were likewise each infected. During the following week, numerous other winged forms appeared both on the guelder roses and on the currants. The method of attack was the same in both cases. The migrant crept into the axil of a leaf, and from thence her progeny gradually spread up the stem and along the midrib. About the same time, I found a Viburnum tree swarming with winged females of Aphis viburni in a shrubbery a hundred yards away; and as these were in- distinguishable from the migrants on the Viburnum and currants, I have little doubt that this was the source of infection. Assuming that A. viburni and A. grossulariae are identical, I began experiments to test how far the host plants were interchange- able. Unfortunately, owing to heavy rains, the experiments with the original winged migrants were all inconclusive, and during grossulariae Kalt., and Aphis viburni Schr. 267 June and July I worked with alate and apterous individuals of later generations. The results are set out in the accompanying tables from which it will be seen that out of thirteen attempts to transfer A. viburni to Ribes rubrum, in only two cases did the resulting colonies survive more than ten days, while reproduction was very feeble and never occurred beyond the third generation. In one case (Table A, Number IX) an attempt was made to re-transfer the third generation back from the currant to the guelder rose, but the result was that the aphides all died within twenty-four hours. Similar attempts were made to transfer A. grossulariae from currant to guelder rose, but the colonies never survived more than six days, and reproduction was very feeble. Meanwhile the natural colonies on guelder rose and currant flourished from the end of May to the middle of August and end of July respectively. Aphis grossulariae has not been recorded from other food plants, but during June I observed three instances where winged migrants had established themselves on the flower heads of the Canterbury Bell {Campanula) and the resulting colonies persisted for two or three weeks. The conclusions suggested by the foregoing observations are that, as Theobald points out, A. grossulariae is probably identical with A. viburni. The first migrant from the birth plant ( Viburntmi) can form colonies either on Viburmim, which is the natural host, or else on Ribes. The descendants of the migrants to Viburnum may with some difficulty be established on currant although the resulting colonies are not so strong as those derived from an early migrant. On the other hand the descendants of the migrants to currant cannot be re-established on Viburnum. It seems as if in two or three generations some change takes place in the currant form which prevents it from flourishing on the guelder rose. One explanation is that there is some change in the constitution of the guelder rose plant — an increase of tannins for instance — and that the strain on guelder rose can gradually adapt itself to altered conditions which the newly transferred currant reared stock cannot tolerate. But this explanation is not wholly satisfactory because the dates show that unsuccessful transferences took place in the second and third generations while the plants were still young, while the most successful attempt was made in July when the shoots were mature. It is also worth noticing that while the more successful attempts were made with winged parents, yet in several of the Viburnum-io-cnvYajxt experiments, wingless females were found to feed and reproduce on the new host. Theobald {op. cit. p. 100) suggests that A. grossulariae maybe the alternating form of A. viburni, but says that he has twice failed to transfer the former to Vibui^num — a result confirming my own experiments in Table B. On the other hand, it is possible that Table A. Results of transference of Aphis viburni from Viburnum opulus to Ribes rubrum. Number Date of transference Forms transferred Death of last survivor Number of Generations born On new host I 12 . VI . 18 alate and apterous 21 . VI . 18 ?2 II 13 . VI . 18 alate 17. VI. 18 1 III 17 . VI . 18 alate and apterous 22 . VI . 18 ?2 IV 24. VI. 18 apterous 29 . VI . 18 1 V 29 . VI . 18 apterous 2 . VII . 18 VI 13. VI. 18 alate and apterous 26 . VI . 18 ?3 VII 5 . VII . 18 apterous 6. VII. 18 1 VIII 30 . VI . 18 — 9 . vii . 18 2 IX 9 . VII . 18 alate and apterous 25. VII. 18 3 X 6 . VII . 18 apterous 12. VII. 18 2 XI 5 . VII . 18 apterous 6 . VII . 18 XII 6 . VII . 18 — 7 . VII . 18 XIII 9. VII. 18 13. VII. 18 ?1 Table B. Table of transference of Aphis viburni, self-established on Ribes rubrum, to Viburnum. Number Date of transference Forms transferred Death of last survivor Number of Generations born on new host I 5 . VI . 18 apterous 12. VI. 18 1 II 2 . VI . 18 alate and apterous 8 . VI . 18 1 III 8 . VI . 18 — 10 . VI . 18 — IV 10 . VI . 18 — 14. VI. 18 1 V 22 . VI . 18 apterous 24 . VI . 18 — VI 30 . VI . 18 alate and apterous 1 . VII . 18 — VII 1 . VII . 18 apterous 2. VII. 18 — VIII 24 . VII . 18 alate and apterous 25. VII. 18 — Miss Haviland, The bionomics of Aphis grossulariae, etc. 269 A. grossulariae is not the natural summer form of A. viburni, but is merely a casual parasite of the currant. In those of the Aphidinae which have a regular migration between two plants, the change is usually from a woody stemmed primary, to a herbaceous secondary, host; and if in the case of ^. viburni, the currant should be found to be the normal second host, it would be a remarkable exception to this rule. Perhaps we have here a form that has not yet adapted itself to the conditions of modern fruit growing. In a natural state, the aphides are probably able to follow the whole life cycle on Viburnum, but the spread of the cultivated currant has presented them with an increasing supply of alternative food which induces a change that makes a return to Viburnum impossible. Whether sex-producing forms can arise from the currant stock, and thence return to the guelder rose, is not known. If not, and the early date of the disappearance from the currant is against this view, we must consider that the infestation of the currant is an unfortunate accident in the history of the species, which entails a waste of migrating individuals upon a cultivated plant that might otherwise have perpetuated themselves on the natural host. However this does not mitigate the danger of the pest from a fruit grower's point of view, and infected Viburnum ought not to be allowed in the neighbourhood of currant bushes. Note on an experiment dealing with mutation in bacteria. By L. DoNCASTER, Sc.D., King's College. [Read 17 February 1919.] (Abstract.) It was noticed that the recorded ratio of occurrence in cases of meningitis of the four agglutination-types of Meningococcus corre- sponded very closely with the ratio of occurrence of the four iso- agglutinin groups of blood in a normal human population. It seemed possible, therefore, that by growing Meningococcus of one type in media containing human blood of ditferent groups, mutation to other types might be induced. Experiment showed that con- siderable differences in type of agglutination resulted, but it was concluded that this was caused by the sorting out of races of different agglutinability from a mass culture, rather than by true mutation. CONTENTS. PAGE On Certain Trigonometrical Series lohich have a Necessary and Sufficient Condition for Uniform Convergence. By A. E. Jollifpe. (Com- municated by Mr G. H. Hardy) 191 Some Geometrical Interpretations of the Concomitants of Ttvo Qtiadrics. By H. ^Y. TuRNBULL, M.A. (Communicated by Mr G. H. Hardy) 196 Some properties ofp{^n), the number of partitions ofn. By S. Ramanujan, B.A., Trinity College 207 Proof of certain identities in combinatory analysis : (1) by Professor L. J. Rogers; (2) by S.- Ramanujan, B.A., Trinity College. (Com- municated, with a prefatory note, by Mr G. H. Hardy) . . .211 On Mr Ramanujan's congruence properties of p (n). By H. B. C. Darling. (Communicated by Mr G. H. Hardy) 217 On the exponentiation of well-order^ series. By Miss Dorothy Wrinch. (Communicated by Mr G. H. Hardy) . . . . . .219 The Gauss-Bonnet Theorem for Midtiply -Connected Regions of a Siorfaee. By Eric H. Neville, M.A., Trinity College 234 On an empirical formida connected with GoldhacK's Theorern. By N. M. Shah, Trinity College, and B. M. Wilson, Trinity College. (Com- municated by Mr G. H. Hardy) 238 Note on Messrs Shah and Wilson^s pamper entitled: '■ On cm empirical formida connected xoith GoldhacK s Theory \ By G. H. Hardy, M. A., Trinity College, and J. E. Littlewood, M.A., Trinity College . 245 The distribution of Electric Force betioeen tivo Electrodes, one of whixih is covered with Radioactive Matter. By "W. J. Harrison, M.A., Fellow of Clare College 255 The conversion of soAV-diist into sugar. By J. E. Purvis, M.A. . . 259 BracTcen as a sotorce of potash. By J. E. Purvis, M.A 261 The action of electrolytes on the electrical conductivity of the bacterial cell and their effect on the rate of migration of these cells in an electric field. By C. Shearer, Sc.D., F.R.S., Clare College . . .263 The bionomics of Aphis grossiilariae Kcdt., and Aphis viburni Schr. By Maud D. Haviland, Bathm'st Student of I^wnham College. (Com- municated by H. H. Brindley) 266 Note on an expe7'iment dealing tvith mutcUion in bacteria. By L. Don- caster, Sc.D., King's College. (Abstract) . . . . , 269 * ' 1 '^.^ PKOCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY VOL. XIX. PART VI. [Easter and Michaelmas Terms 1919.] (JDambttligt: AT THE UNIVERSITY PRESS AND SOLD BY DEIGHTON, BELL & CO., LIMITED, AND BOWES & BOWES, CAMBRIDGE. CAMBRIDGE UNIVERSITY PRESS, C. F. CLAY, MANAGER, FETTER LANE, LONDON, E.G. 4 1920 Price Three Shillings and Siccpence Net February 1920. NOTICES. 1. Applications for complete sets of the first Seventeen Yolumes (in Parts) of the Transactions should be made to the Secretaries of the Society. 2. Separate copies of certain parts of Volumes I. — XI. of the Transactions may be had on application to Messrs BowES & Bowes or Messrs Deighton, Bell & Co., Limited, Cambridge. 3. Other volumes of the Transactions may be obtained at the following prices: Vol. xii. £1. 10s. Qd.\ Vol. xiii. £1. 2s. %d. Vol. XIV. £1. 17s. U. ; Vol. xv. £1. 12s. U. ; Vol. xvi. £1. 10s. dd. Vol. XVII. £1. 2s. U. ; Vol. xviii. £1. Is. Qd. ; Vol. xix. £1. 5s. Od. Vol. XX. £1. 10s. Od.\ Vol. XXI. £1. 14s. Od; Vol. xxii.No. 1, Is. 6d. No. 2, 2s.; No. 3, Is. 6d; No. 4, Is. 6d; No. 5, 2s.; No. 6, Is. Qd. No. 7, 2s.; No. 8, 2s.; No. 9, 2s.; No. 10, Is.; No. 11,2s.; No. 12, 3s. 6d. No. 13, 2s.; No. 14, 3s. 6d; No. 15, 3s. 6a!.; No. 16, 2s. 6d; No. 17, 2s.; No. 18, 2s. 6d 4. Complete sets of the Proceedings, Volumes L — xix., may also be obtained on application to the Secretaries of the Society. 5. Letters and Communications for the Society should be addressed to one of the Secretaries, ! Prof. H. F. Baker, St John's College. [Mathematical.] ! Mr Alex. Wood, Emmanuel College. [Physical] Mr H. H. Brindley, St John's College. [Biological] 6. Presents for the Library of the Society should be ad- dressed to The Philosophical Library, New Museums, Cambridge. ^{jHl|| 7. Authors of papers are informed that the Illustrations and Diagrams are executed as far as possible by photographic "process" work, so drawings should be on a large scale and on smooth white Bristol board in Indian ink, 8. Members of the Society are requested to inform the Secretaries of any change of address. PROCEEDINGS OF THE Colourimeter Design. By H. Hartridge, M.D., Fellow of King's College, Cambridge. [Received 7 October 1919; read 10 November 1919.] In a previous paper (1) I have described certain factors which affect the efficiency of the spectrophotometer. The colourimeter has been found to be similarly affected, so that various modifica- tions in the usual designs are indicated. The comparison field is in most instruments divided at a diameter, so that one half receives light which has passed through one limb, and the other half light that has passed through the other limb of the instrument. In a few designs the bull's-eye and the central strip fields have been employed. All these fields have the disadvantage that local stimulation of the retina may occur that sets up after image phenomena greater in degree in one part than in another, thus preventing accurate determinations. And, further, they do not make the best use of the effects of simultaneous contrast. A better type of field is the one which I have previously described in connection with the spectrophotometer, namely, one which is subdivided into a number of strips, of which alternate numbers receive light from the two limbs of the instrument. With this field the eye does not select any one part for examination, but tends rather to judge of the field as a whole. When the adjustment of intensity has been correctly made the whole field should become uniform. The effects of retinal fatigue therefore tend to become uniformly distributed. The contour of this type of field is of considerable length compared with its total area ; the conditions are therefore beneficial for the development of contrast. The absence of visible lines of junction still further increases this effect. The prisms A and B by which the beams of light through the two limbs of the instrument are combined at the compound field VOL. XIX. PART VI. 19 272 Br HarLridge, Colourimeier Design Eyepiece Cup - Ramsden eyepiece • Comparison field Comparison field Plunger ■ Staa;e - Tail piece to carry lamp f*ush-on terminals Horse-shoe foot Opal glass disc Watt lamp Dr Hartridge, Colourimeter Design 273 described above, are similar in shape to those used in the spectro- photometer. They are shown in the diagram of the apparatus. It will be observed that the interface of the prisms is silvered, the metallic film being removed by means of a simple ruling machine, so that narrow strips of the silver alternate with strips from which the whole of the silver has been removed. Examination of the diagram will show that by this arrangement the field seen on looking down the eyepiece is formed of alternating narrow beams which have either been transmitted from one limb of the instrument through the spaces between the silver strips, or reflected from the other limb by the silver strips themselves. The lengths of the prisms A and JB should be such that the two entering beams have passed through equal lengths of glass. The troughs are adjustable on both limbs of the instrument, in colourimeters of usual design. This arrangement has the dis- advantage that if there should be any backlash in the micrometer mechanism which is used for adjusting the position of the movable troughs, or error in the setting of the scale, these will affect both the thickness of the pigment solution to be estimated, and also that of the standard. Such errors can be eliminated so far as the standard is concerned by the use of a special cell, the distances between the sides of which are determined by accurately ground distance pieces, which may be made of either glass or metal. Rustless steel would appear to be a suitable metal because it resists the corrosive action of ordinary solvents. I have shown that in the case of the spectrophotometer there are important reasons for the use of troughs with double compart- ments on both limbs of the instrument. In both ti'oughs the com- partment near the light source should contain the solvent only, the other being filled with the solution of the pigment. Double troughs should be used with the colourimeter for similar reasons, namely, (a) in order that absorption by the solvent may be com- pensated, since the thickness is the same on both sides of the instrument ; (b) that pigments accompanying the one under esti- mation may be compensated for; (c) that specific surface reflection at the sides of the troughs which contain pigment may be similar on both limbs of the instrument. With regard to the type of trough that should be employed I have previously considered the advantages of the double wedge trough in conjunction with the spectrophotometer. In the case of the colourimeter the plunger type usually employed has the advantage of not requiring calibra- tion with a micrometer microscope as wedge troughs do. The method of employing double compartment plunger troughs and standard troughs is shown in the diagram. In some colourimeters the troughs are bell mouthed, and are manufactured from black glass. These points are to be recommended. It should be noted, 19—2 274 Dr Hartridge, Colourimeter Design however, that reflection can still take place at the sides of the troughs, so that it is necessary carefully to restrict the light illu- minating the troughs to narrow vertical pencils of just sufficient diameter fully to illuminate the comparison fields. Since scattered or reflected light may increase the apparent brightness of one of the fields it is essential that this be reduced to a minimum. Special care should therefore be taken in designing the instrument to pre- vent the entrance of stray light, and to employ an illuminating system that will limit the entering beams to the narrow pencils above referred to. The illumination in the majority of colourimeters is obtained from the sky by means of a plane mirror. In some instruments this may be replaced at will by a finely matted white surface. The illumination therefore in either case consists of a large number of divergent pencils, which enter the lower ends of the troughs in all possible directions. Scattered light is therefore at a maximum. In the case of the microscope a similar practice used to be in vogue, but it has given way to the use of illuminating lens systems in which the corrections and alignment are well nigh as perfect as those used in the objective and eyepiece. Now, in the case of the spec- trophotometer I have shown that the beams illuminating the two limbs of the instrument should proceed from identical parts of the light source. This condition should be realised in the case of the colourimeter also. The arrangement of the illuminating apparatus is shown in the diagram. The light source is similar to that which I have applied to the microscope (2), consisting of a slab of white opal glass finely ground on both sides. This is lit from behind by means of a small half watt electric lamp, which obtains its current from a small accumulator or dry cell, or from the town supply through a suit- able resistance. The lamp is enclosed in a brass box, which is silver plated inside, and is finished dead-black outside so as to radiate heat. The life of the lamp is increased by connecting it with a press switch so that it is in circuit during observation only. The lamp box is attached to the tail-piece of the instrument so that it forms an integral part of the apparatus. The whole may thus be tilted or moved from place to place without requiring re- adjustment. Immediately above the opal glass is a metal dia- phragm, the aperture in which limits the surface exposed to a disc 4 mm. in diameter. Attached beneath the stage of the in- strument and 60 mm. above the diaphragm of the light source is a plano-convex achromatic lens of 26 mm. diameter and 60 mm. focal length. The divergent rays from each point of the source are rendered parallel by this lens, and at once pass through two achromatic plano-convex lenses of 18 cms. focal length and 14 mm. diameter. These lenses have a clear aperture of 12 mm. and form Dr Harfridge, Colourimeter Design 275 a focussed image of the diaphragm of the light source, which is magnified in the ratio of the focal lengths of the lenses; since the ratio is 3 to 1 this image has a diameter of 12 mm. The beams that emerge through the lenses Tl and T'2 do not therefore anywhere exceed 12 mm. and the light does not spread, for this reason, to the sides of the troughs during its passage and therefore stray light is reduced to a minimum. The beam from the lens T2 passes vertically upwards through a hole in the stage to the standard trough which rests upon it. Having passed through both the layer of solvent and also that of the solution of pigment, the beam enters prism B', and is totally inter- nally reflected at its inclined surface on to the silvered strips of the comparison field. The beam that has passed through Tl is deflected by internal reflection at the right angled prism C which is cemented to it, and falls on the silvered sui'face between the two halves of the prism D, so that the beam is directed vertically through a second hole in the stage on to the lower fixed cup of the adjustable trough, which is filled with solvent. It then passes through the movable cup which contains the pigment, and enters the prism A to fall on the silvered strips of the comparison field. The passage of this beam through the intervals between the strips, and the reflection of the beam from the other limb of the instrument at the strips themselves, has already been described. It will be noted that the reflection of the one beam by internal reflection within the prism C, and by ordinary reflection within the prism D, causes this beam to compensate for the internal reflection and reflection at a silvered surface which occurs within prism B in the case of the other beam. As it has been found that silvered surfaces vary in the intensity of rays of different wave-length which they reflect, it is advisable that both mirror D and prism B be silvered with the same solution at the same time. The lengths of the paths of the beams through the instrument are found to be in the case of the left-hand beam an actual dis- tance of 19'5 cms., that is an effective distance of 18 cms. since 2"2 cms. of glass is passed through ; in the case of the right-hand beam the total and the equivalent lengths are the same as those on the left. The comparison field therefore is illuminated by two super- posed images of the diaphragm of the light source, one of which has passed through the standard trough and the other through the adjustable trough. When the instrument is in correct adjustment these two images exactly coincide, so that if there should be any slight inequality between the intensity of illumination of different parts of the light source both images will be similarly affected, and therefore the match between their different parts will remain unchanged. Such a condition is not secured in the usual forms of 276 Br Hartridge, Colourimeter Design colourimeter, since it is due to the particular method of illumina- tion described above. The eyepiece used in the du Bosq type of colourimeter consists of a Eamsden lens system, at the upper focal plane of which has been placed a diaphragm pierced with a small aperture. This has the effect of limiting the rays reaching the eye to those which have passed as approximately parallel bundles up the limbs of the instrument. To be effective the aperture has to be small, and this has the disadvantage of making the intensity of illumination of the fields somewhat low. When this type of eyepiece is in use it is found that the eye has to be inconveniently close to the aperture in order that the whole field shall be seen at one and the same time. This is due to the fact that the diaphragm is a considerable distance below the effective pupil of the eye, even when the eye has been placed as close as possible, and as a result some of the rays which spread out from the diaphragm may not enter the pupil. The difficulty is in fact similar to that met with in high power microscopic eyepieces of the Huygenian type. To avoid this diffi- culty a more elaborate type of eyepiece has been devised, in which an erecting lens system has been placed above the Ramsden ocular and its diaphragm (8). This causes a sharp image to be seen on looking down the eyepiece, and at the same time the image of the small aperture is formed at a considerable distance above the top lens, so that the eye does not have to be placed inconveniently close to the eyepiece in order to obtain a full view of the field. These improvements are obtained, however, at a certain sacrifice of definition, which is unimportant in the usual types of colouri- meter in which the fields are of simple design, but is of relatively greater importance if the more detailed type of field be used which has been described above. It will have been observed that in the colourimeter which I have described above the illuminating beams are formed by the special method of illumination employed. Under which circumstances it is found that the Ramsden disc of the ocular contains the overlapping focussed images of the restricting aper- tures of the lenses Tl and T2, which when the instrument is in correct adjustment exactly overlay one another. It is therefore unnecessary that the eyepiece should contain any diaphragm to restrict the beams, and therefore the difficulties introduced by such a diaphragm are not met with. The eyepiece itself should be achromatic and should slide in a tight-fitting jacket so that the observer may set it at the best focus. It should magnify about 3 diameters. The angle at which the comparison field lies will be seen to be 45 degrees. But since it is enclosed between two pieces of glass, the apparent angle to the eye is reduced in the ratio of the refi'ac- tive indices of glass and air. The apparent angle would therefore Dr Hartridge, Colourimeter Design 277 be about 29 degrees. Now, the dimensions of the field seen by the eye are 8 mm. by 6 mm., the latter being in the direction of the slope. The apparent diiference of focus is therefore less than 4 mm., which would be equivalent to 12 mm. at a distance of 25 cms. Such a small change of focus would be at once met by a trifling change in the degree of accommodation of the eye, which would be effected subconsciously and involuntarily. No difficulty is to be met with therefore from this cause. The Mechanical System. The metal work of the colourimeter follows closely that of the microscope. The horse-shoe foot, stage and coarse adjustment all resemble those used in that instrument. The adjustment has a range of 40 mm. only, because, as will be shown later, the use of standard solutions of 20 mm. thickness makes a bigger movement than this unnecessary. An accuracy of one-quarter per cent, should be sufficient, and this is readily provided by a scale gi^aduated in half mm. and reading by a vernier to one-twentieths. The adjust- ment should have long, well-made V slides so as to eliminate lost motion. The scale should be attached to the moving member, the vernier being attached to the fixed. A simple lens and 45 degree mirror should make a magnified image of this visible to the ob- server. To the moving member is first screwed and afterwards sweated with soft solder a strong brass ring. To this is attached by means of a three-prong bayonet catch the ring fixed to the upper lip of the movable trough. The trough is cemented into a groove turned in this ring by means of plaster of Paris or Caemen- tium. Where plaster has been used the joint should be covered by a thin coat of Robiallac. The prisms and eyepiece are attached to a strong projection at the top of the pillar which forms the handle of the instrument. The removal of the troughs for filling and cleaning and their replacement is a simple process which should not take more than a few seconds. To remove the adjustable troughs, first swing the substage to one side ; this allows the lower trough to drop verti- cally through the hole in the stage until it can be removed. The upper trough is now gripped between the finger and thumb, and the trough rotated so as to free the bayonet catches ; this trough is then lowered through the hole in the stage and removed. The plunger and the troughs can now be cleaned, refilled and returned. The standard double trough simply rests on its side of the stage, so that its removal takes but a moment. 278 Dr Hartridge, Colourimeter Design The Colourimeter in Practice. Experiment has shown that if two solutions of the same colour contain different pigments in solution, then the thicknesses re- quired for a match vary not only with the observer and with the quality of the light, but also with the same observer from time to time. It is for this reason that the technique has been introduced of using the same pigment for the standard as that required to be estimated. Thus creatinin is no longer estimated by comparing the colour which develops when picric acid and soda are added with the colour of a solution of potassium dichromate ; but a standard solution of creatinin is used, picric acid being added to it at the same time as it is added to the solution to be standardised. If, then, the thick- ness of the standard is 20 mm. and that of the unknown 17 mm., it is assumed that the strengths of the solutions are in the inverse ratio of those numbers. Such is not the case however, because the sodium picrate itself absorbs rays from the same part of the spectrum as does the sodium picramate, and therefore, although the light may encounter the same number of coloured radicals in both limbs of the instrument, yet the sodium picrate absorption is greater on one side than the other, because the fluids are not of the same thickness. It is principally for this reason that I have adopted an instrument in which double troughs are used, on both sides of the instrument ; the lower pair on both sides being filled with sodium picrate solution in the case taken above as example, the upper pairs containing the picric acid plus creatinin. In this way the number of picrate radicals is kept approximately constant, since the total thickness of sodium picrate solution is the same on both sides of the instrument. The balance is not perfect however, because a certain amount of picric acid is used up in forming the sodium picramate, and this amount cannot be ascertained without assum- ing that the estimation to be done has already been accurately performed. The problem is, in fact, represented by a simultaneous equation involving two unknowns. I find that the matter can be solved in the following manner. Having diluted both the standard and the unknown solutions with equal amounts of standard picric acid and soda solutions, and having allowed the colour to develop in the ordinary manner, an estimate is made of the relative strengths of the solutions in the colourimeter. Having found that, say, a 20 mm. thickness of the standard has the same tint as 13"4 mm. of the unknown solution, a fresh sample of the unknown is taken and 13'4 c.c. of it diluted with water to bring the total to 20 c.c. The solution of the unknown has thus been brought to approximately the same concentration as the standard. (Where the approximate strength is known a preliminary dilution before making the initial estimation is beneficial.) The correctly diluted solution of the un- Dr Hartridge, Colourimeter Design 279 known is now treated, ab initio, with fresh picric acid solution and soda, and is then estimated against the standard in the colourimeter. It is now found that a 20 mm. thickness of the standard has the same tint as one of, say, 19"85 of the unknown after dilution. The strength of the unknown is thus ascertained, with considerable accuracy, because the conditions of equilibrium under which the sodium picramate develops and exists, and the quantities of picric acid used up in the determination are approximately constant. It should be pointed out that the above technique presents no difficulties, and takes little longer than the ordinary method. The p)rinciple may with advantage be applied to all estimations made Avith the colourimeter. The Accuracy of the Colourimeter. Since colour is due to absorption the colourimeter depends for its utility on the fact that a change in the number of coloured radicals encountered by light causes a change in the retinal stimu- lus when that light falls on the eye. We may, therefore, arbitrarily state that the accuracy of the determinations depends, firstly, on the rate of change in the quality of the light which is passed through the pigment, and, secondly, on the acuteness of the per- ception of the eye for the change in quality of the light. The greater the rate of change and the greater the acuteness of percep- tion of that change, the greater will be the accuracy. Many bodies which absorb light do so selectively, that is, they have a gref^ter effect in one part of the spectrum than in another ; they therefore show colour, that is, they are pigments. Under ordinary circum- stances the greater the absorption the stronger the colour and the less the intensity of the transmitted light. As the concentration of a pigment is altered, and therefore the degree of absorption, the strength of colour and the brightness of the transmitted light both vary. The colourinietric determination, therefore, depends on the simultaneous occurrence of both these changes. The important questions that arise are : (1) on what do the magnitudes of these changes depend ? (2) which is the more important ? and (3) how can the changes be increased for a given alteration in concentra- tion ? A study of absorption band formation gives a definite answer to each of these questions as follows: (1) The changes for a given alteration of concentration are greater the flatter and broader the absorption band. If, therefore, there were two pigments of the same concentration and the same colour, i>ne of which had a sharp well- defined band, while that of the other was broad and flat, the latter pigment would be found to give the more accurate readings in the colourimeter. (2) Of the two changes, that of colour is usually the more important, particularly with pigments showing single absorp- tion bands. In pigments with multiple bands the intensity change 280 Dr Hartridge, Colourimeter Design may be the more important : for example, a pigment absorbing to an equal extent in two complementary parts of the spectrum will cause the light to suffer no change in colour at all, while the in- tensity is altered. (3) The changes in the case of any one pigment can be increased by increasing the intensity of that part of the spectrum which is suffering change or by decreasing that of parts which do not show alteration. Of the two methods the latter is the easier to carry out and the more efficient. If colour filters are used they must be carefully adjusted according to the position in the spectrum of the absorption band of the pigment to be estimated. If a spectral illuminator is used the apparatus virtually becomes a spectrophotometer, and this elaboration is hardly necessary for ordinary work. The possibility should not be overlooked of the existence of alternative colour reactions to those at present in use in which pigments having less steep absorption bands are used and which therefore permit greater accuracy in their colourimetric estimation. The factors which influence the acuteness of perception of the eye remain for consideration. Firstly, it is clear since the accuracy of the determination depends on the correctness of the match ob- tained, that the eye should not be suffering from fatigue. The reading of small print and the exposure of the eyes to excessive light should, therefore, be avoided for a reasonable time before the determinations. The absence of refractional errors, eye strain, want of eye-muscle balance and the possession of good general health are all factors of importance. In my own case the period after tea is the best, provided that the morning's work has not been arduous. The presence of after images is most harmful for accurate estimations ; the best method of eliminating them is, I find, to look for a fcAV moments at a uniformly lit grey surface. All the above points may seem obvious ; it is however my experience to find that they are sometimes overlooked. The apparatus itself is best placed in a dark room, or at all events where the full light of a window cannot fall on the eye of the observer. In the latter case the eyepiece cup may be made deep with advantage, so as to protect the periphery of the retina from stimulation and thus bring about an increase in the diameter of the pupil. With regard to the use of colour filters, experiment shows that the theoretical conclusions arrived at above are amply justified, namely, that the accuracy of the determinations is increased if either the rays absorbed by the pigment are increased in intensity, or those not absorbed are decreased or removed altogether. The removal by means of colour filters is however usually attended by so great a diminution in the intensity of the light that a powerful source such as an arc lamp becomes necessary. It is a fortunate circumstance, therefore, that the retina should be even more sensi- Dr Hartridge, Colourimeter Design 281 tive to change in shade than it is to change in intensity. I have found, further, that the point of greatest sensitiveness is obtained when the fields are nearly neutral in colour. Such a condition is obtained by the use of a suitable colour filter which absorbs in that part of the spectrum which is occupied by the complementary colour to that absorbed by the pigment. Suppose, for example, a yellow pigment is to be estimated, then a blue solution of a dye is placed in the path of the light from the source of such a thickness and concentration that the comparison field seen in the instrument is of a neutral grey colour. Permanent colour films between glass should be used if much work is likely to be done with any given pigment. Such a technique is very simple, and I find that in my hands it increases the accuracy of the determinations by about three times (when estimating sodium picrate), the method of mean squares being used to calculate the average error of the experi- mental determinations both with and without the complementary filter. The probable error of the determinations was found to be 0"8 per cent., using home-made apparatus and the complementary screen. It should be possible to halve this amount if the precau- tions outlined above be taken and well-designed apparatus be used. Summary. (1) The comparison field seen on looking down the instrument should cause the greatest contrast and at the same time should not produce after images. (2) On both limbs of the instrument double troughs should be used, so that the thickness of pigment to be measured may be varied at will, while the absorption caused by other pigments remains constant. (3) An artificial light source should be used, and the lighting system be so designed that narrow beams are produced of just sufficient width as to completely illuminate the comparison field. The amount of reflected and scattered light may thus be reduced to a minimum. (4) If experiment shows that the change in colour produced by a given change in thickness or concentration of the pigment can be increased by modifying the relative intensity of different parts of the spectrum of the light source, then suitable colour filters should be prepared for use during the determinations. It was found in a test case that this modification alone increased the accuracy by three times. (5) The general design of the instrument should conform to microscopic practice, fixed troughs being supported by the stage and the movable trough actuated by the rack and pinion course 282 Dr Hartridge, Colourimeter Design adjustment screw. The illuminating system should be fitted beneath the stage so that the instrument may be tilted or moved from place to place without disturbing the alignment. For certain purposes it may be found beneficial to employ smaller quantities of liquid than those required in the ordinary colourimeter. I find that a modification in the design of the troughs should make 1 to 2 c.c. of liquid sufficient ; and further, by modifying the optical system as well, as little as "001 c.c. could be worked with. It should be pointed out however that such quantities could only be employed with solutions of considerably greater con- centration than those usually estimated ; e.g. about ten times the usual concentration for 1 c.c, and one hundred times for '001 c.c. References. (1) Hartridge, Journ. Physiol, l, p. 101 (1915). (2) Hartridge, Joum. Qitekett Micro. Soc. Nov. 1919. (3) Kober, Journ. Biol. Ghem. xxix, p. 155 (1917). Mr Snell, The Natural History of the Island of Rodrigues 283 The Natural History of the Island of Rodrigues. By H. J. Snell (Eastern Telegraph Company) and W. H. T, Tams^. (Communi- cated by Professor Stanley Gardiner.) [Read 10 November 1919.] Rodrigues lies some 350 miles east of Mauritius, and is a rugged mass of volcanic rock closely resembling Mauritius and Reunion. It is surrounded by a coral reef, the edge of which at the eastern end is within 100 yards of the beach, whilst on the north and south it extends outwards to a distance of three to four miles, and on the west to two miles. There is an irregular channel inside the reef close to the shore, extending round most of the island, sufficiently deep for boats at any state of the tide, and at the south-east end a small lagoon of three to ten fathoms, with a passage through the reef. The usual anchorage is Mathurin Bay, in the reef to the north. The reef is studied with islets, those nearer the shore being mostly of volcanic nature, and situated on the north and west, whilst the rest are of limestone, modern accumulations of debris, and situated on the south. The island itself is eleven miles long by five miles broad, and has an area of just over forty square miles. There is a central lofty ridge extending from east to west, with a break about one- third of its length from the west. The western bastion of the range is Mount Quatre- Vents, 1120 feet high, while at the eastern end is Grande Montaigne, 1140 feet. The highest point is Mount Limon (1300 feet), which lies with two other peaks a little out of the general line of mountains. The sides of these peaks are cut into numerous ravines, these being deeper and more frequent on the south side than on the north. At their upper ends these ravines are often bordered by perpendicular columnar basaltic cliffs, sometimes exceeding 200 feet in height, extensively cut into many coulees by small streams which often descend in a series of cascades. The volcanic ridge descends on the south-west gradually, and passes into a broad coralHne limestone plain, with occasional hills up to 500 feet high, indicating a comparatively recent elevation of at least a like amount. This tract of limestone is honeycombed with caves, in which stalactites and stalagmites are abundant. There are many holes and fissures, and often deep hollows occur, at the bottom of which lie large fragments of limestone in irregular heaps; these are apparently old caves, the roofs of which have fallen in. The floors of these hollows are covered with soil, often 1 The second author is solely responsible for the names of the insects herein recorded. 284 Mr Snell, The Natural History of the Island of Rodrigues witli lumps of volcanic rock on the surface. The limestone is not found along the northern or southern shores, except at their eastern extremity, where patches occur at the mouths of the valleys, occasionally at some distance from the shore. Some of the patches of limestone found in the volcanic region indicate an elevation of perhaps 500 feet, and the raised beaches on the south shore, some 20 feet in height, may point to a further subsequent change of level. The position of old volcanic craters has not been accurately determined, but the main ones appear to have been situated about the Grande Montaigne and Mount Malartic, The island is comparatively dry, and during the warm season many of the streams are dried up, though they assume in the rainy season torrential proportions. The climate is like that of Mauritius. The rainfall is very irregular; during the north-west monsoon from November to April the weather is wet and warm, and early in this season there are frequently severe hurricanes. From May to October the south-east monsoon prevails, and the weather is then cool and dry. Fogs are rare, and climatic conditions render the island healthy to live in. Rodrigues was discovered in 1510, by a Portuguese commander, whose name it bears. In 1691 the Dutch landed several fugitive French Huguenots there, among whom was M. Fran9ois Leguat, who wrote an account of the island in 1708. The island was later cultivated by the French East Indian Company, and maize and corn were grown ; these, with dried fish, turtles and land tortoises, were exported to Mauritius. It was occupied by the British in 1809, and made the base of operations against Mauritius. It is still cultivated as a garden for Mauritius, its main exports being beans, acacia seed, maize, salt fish, cattle, goats and pigs. The population is about 5000, mostly settled around Port Mathurin, the only town in the island. The people are mainly French Creoles, with a few Chinese and Indians, and are subject to the Government of Mauritius, which suppHes a Resident Magistrate. The island is a station of the Eastern Telegraph Company, connecting to Cocos- Keeling. Each family usually cultivates an acre or several acres of land, whereon they grow maize, sweet potatoes, haricot beans, pumpkins, various herbs, onions, etc. They depend, in fact, largely on their own plantations for food. At one time a species of mountain-rice, which does not require an abundance of moisture, was grown in large quantities, but its cultivation was abandoned owing to the depredations of small birds. Tobacco grows well. Haricot beans are still exported. There have lately been, however, only five ships per year, and these small sailing ships of 500 tons down to 100 tons register; this makes it very difiicult to market the produce of the island. The maize grown is barely enough for local consumption. Mr Snell, The Natural History of the Island of Rodrigues 285 One of the most profitable products of this island is acacia seed, which is exported to Mauritius for cattle feeding. The acacia {Lucaena glauca), which was introduced about seventy years ago, now grows wild and flourishes everywhere, covering the ground for acres, and forming a dense almost impenetrable scrub, beneath which nothing will grow. The cattle and goats are exceedingly fond of the leaves and pods, and this is probably the reason for its spreading so extensively, the original plantation having been in a valley near Port Mathurin. Amongst other things which have been successfully grown may be mentioned coffee, vanilla, sugar-cane, oranges and lemons. Bananas and plantains, custard apples, strawberries and raspberries are found wild. Many other com- modities such as ginger, safran (turmeric) and arrowroot have also been grown. There is very little real pasturage in Rodrigues, the largest area being in Malgache Valley. Besides this there are barren tracts round the coast covered with coarse grass, which provides in- sufficient subsistence for the stock. Most of the inhabitants own goats and pigs, on which they rely for their milk and meat supply, and which are also exported. They were allowed to run wild, but measures have now been introduced by the Government to control them. Poultry, ducks and geese also thrive in the island. Rodrigues was originally covered with dense forests of lofty trees, with corresponding undergrowth. Indeed, according to early descriptions its vegetation partook of the nature of a regular tropical moist woodland. Here were to be found flightless birds, the Solitaires, and giant land tortoises. When Leguat saw this island first, the scenery was such as to call forth from him such designations as "a lovely isle," "an earthly paradise." To-day its grandeur and beauty have vanished. There remains a bare parched pile, on which it is difficult if not impossible to discover any corner in its original condition. Many agencies are responsible for this destruction and denudation. It has been swept by fire many times, accidentally and intentionally. The goats devour the young shoots and leaves of any vegetation within their reach. Pigs have done their share, especially with regard to the Latanier Palm (Pandanus), of the nuts of which they are very fond. Then there are the introduced plants, which have in many cases crowded out the native vegetation. A notable example is seen in the acacia, previously mentioned, which has spread into almost every valley in the island. A certain amount of destruction has been done by the inhabitants, who have cut timber over large tracts without discrimination. Though a check has been placed on this by the government, there still remains a source of destruction, in that the inhabitants are in the habit of acquiring year by year fresh tracts of woodland, the undergrowth of which they cut down and burn, 286 Mr Snell, The Natural History of the Island ofRodrigues and here they plant their haricot beans. They utiHse a tract of land for one season, and abandon it the next. Thus the work of destruction continues. Many of the older inhabitants, at present living on the island, say that they remember large tracts, which are now almost bare except for a few Vacoas (Screw-pines), being originally covered with almost impenetrable forest, but nobody remembers the large expanse of coralline limestone at the south- western end of the island in any other than its present state, though there are unmistakeable traces, in roots and stumps em- bedded in the ground and charred by fire, shoAving that this region was also at one time completely afforested. The large rifts are often thirty feet or more deep, and fifteen to twenty yards wide, and contain many fine old indigenous trees which have escaped destruc- tion. The Valley of St Frangois, at the north-east end of the island, is perhaps the only other tract which has escaped destruction. The commonest trees in the island are the Vacoas or Screw-pines (Pandanus), of which there are two species, both endemic. Three other species have been recorded by various authorities, one being a native of Asia, and the other two Madagascar species. None of them occurs in Mauritius or Reunion, and the evidence of their occurrence in Rodrigues is faulty. There are three species of endemic palms, belonging to three genera, which are all Mascarene. Probably half the plants have been destroyed, but from what is left — 297 species of Phanerogams, and 175 species of Cryptogams (excluding Marine Algae) — it is clear that the endemic flora was large and of Mascarene aSinities. There are only about twenty species of ferns, the scarcity of this group being accounted for by the present dryness of the island, in confirmation of which it may be remarked that the tree-ferns of the other Mascarene islands are not represented. The present day fauna is not large. The extinct fauna has proved to be of very great interest, particularly in the case of the Solitaire (Pezophaps solitaria, Gmel.), the extinct Didine bird related to the i)odo of Mauritius. Considerable collections of the remains of this bird have been made from the limestone caves, where also the remains of other extinct birds and of the giant Land Tortoise have been found. Our main knowledge of the recent fauna is due to the labours of the naturalists attached to the Transit of Venus Ex- peditions carried out in 1874-5. The marine fauna is in general of the Indo-Pacific type. The only indigenous mammal found in the island is a fruit-bat, Pteropus rodericensis, Dobson, which is peculiar to Rodrigues. The introduced mammals, other than those already mentioned, are deer, rabbits, rats, mice and cats, the latter being left by the Dutch to destroy the rats. Sir Edward Newton, K.C.M.G., published a list of Rodrigues Mr Snell, The Natural History of the Island of Rodrigues 287 birds in his "List of the Birds of the Mascarene Islands" {Trans. Norfolk and Nonvich Naturalists' Society, vol. iv, President's Address). The Fresh Water Fishes, as far as known, belong to species which inhabit the fresh waters of the Mascarene Islands generally, with the exception of two Grey Mullets, which were collected by the Transit of Venus Expedition, and were described as new. Further collections in certain groups have recently been made by Mr H. P. Thomasset and Mr H. J. Snell, who visited the island during the period August to November, 1918, with a view to im- proving our knowledge of the insect fauna. Mr Snell visited practically every part of the island, with the exception of the valley of St Francois, and a small district round the Riviere Coco. The best collecting ground he found to be un- doubtedly the Grande Riviere Valley, which he worked right up to Mount Limon. The islands on the reef were also visited, but con- tained very little of interest, as they have been burnt over in recent years, and are now covered with rough coarse grass and short scrub {Tournefortia, Pemphis, etc.). These islands, particularly Gombranil and Flat, were formerly nesting places for sea-birds, which seem to have disappeared, only a few white terns and boobies being found on Sandy and Coco Islands, which were some years ago planted with firs. In the deepest ravines were commonly seen the fruit-bats or flying-foxes, feeding on the flower of a kind of aloe, of which they seem very fond, and also on wild figs, mangoes, etc. Geckos were abundant in warm and sheltered spots, particularly in all habita- tions. Their eggs were frequently found in nests (usually composed of dry Sow-thistle bloom) under rocks and in crevices. Two species only have been recorded: Gehyra mutilata, Gray, and Phelsuma cepedianum, Gray; the latter is common in Madagascar, Mauritius and Reunion, but is rare in Rodrigues. Freshwater fishes were found in many of the streams, in which also eels were quite common. There are in the island a Land Planarian, Geojplana whartoni, Gull., and a Land Nemertean, Tetrastemma rodericanum, Gull. Both are peculiar to Rodrigues, but the former has not been ade- quately described. (Mr Thomasset subsequently obtained a Land Planarian from Mauritius, a new locality for these.) They were found under decaying logs, sometimes on the bark, under the bark, or in the wood; the Nemertean appeared to exist in far greater quantities than the Land Planarians, but they often live together in the same situation. Earthworms were not abundant. Amongst the Crustacea collected, large numbers of an Amphipod were found under stones, dead leaves, etc., wherever the ground was moist. In all the streams were to be found freshwater shrimps VOL. XIX. PART VI. 20 288 Mr Snell, The Natural History of the Island of Rodrigues and a crayfish. Woodlice were abundant in deca}dng vegetable matter, the largest specimens being obtained from rotting banana stems. Myriapoda were common throughout the island. Large centi- pedes live on the corals on the west side of the island, attaining sometimes a length of twelve inches. Hardly a lump of debris can be turned over without disclosing one or more of these creatures. The Transit of Venus Expedition obtained twelve species of Myriapods, of which eleven were new. There is a single species of scorpion, Tityus marmoreus, Koch, and in addition the Transit of Venus Expedition obtained twenty-seven species of Arachnida, eleven being new ; unfortunately Mr Snell could not obtain a supply of alcohol adequate to preserve these. In the Insect collections among the Orthoptera, the Forficulidae are represented by eleven specimens, probably Anisolabis varicornis, Smith. Of the Blattidae, Periplaneta americana, Linn, and Leu- cophaea surinamensis, Fab. are among the five species previously recorded, whilst there are two other species in Mr Snell's collection at present undetermined. One species of Mantidae occurs in the island, viz. Polyspilota aeruginosa, Goeze, of wide distribution. Of the Gryllidae there are three species in the present collection: Acheta bimaculata, de Geer, found also in Africa and S. Europe; Curtilla africana, Beauv., found also in Africa, Asia, Australia, and New Zealand (introd.?); and a species of Ornebius near syrticus, Bolivar, but larger and more brightly coloured than the Seychelles specimens of this species. Besides the first of these, the Transit of Venus Expedition obtained three other species. Among the Phasgonuridae we have Conocephaloides differens, Serv. and Anisoptera iris, Serv., both previously recorded by the Transit of Venus Expedition. In addition the present collection contains a specimen of apparently another species of Anisoptera, resembhng A. conocephala, Linn., which occurs in Spain, Africa, and the Seychelles. There are two species of Locustidae: Locusta danica, Linn., a cosmopolitan species, and Chortoicetes rodericensis, Butl., described from Rodrigues, and not found elsewhere. The Neuroptera comprise a few specimens of a Termite, and specimens of one species of Hemerobiidae and of one species of Chrysopidae. It may here be mentioned that Dr H. Scott found a species of Termite working in the wood at the bottom of a fighter in Victoria harbour, Mahe, Seychelles. This indicates a possible explanation of the existence of Termites in such a locafity as Rodrigues, where any indigenous Termites would probably be exterminated by the fires which have repeatedly devastated the island. Until the Termites in Mr Snell's collection have been identified, no statement of course can be ventured regarding the distribution of this species. Mr Gulfiver, on the Transit of Venus Mr Snell, The Natural History of the Island of Rodrigues 289 Expedition, secured one specimen of Myrmeleon obscurus, Rambur. This species was described from Mauritius, and is widely distributed in Africa. The Odonata consist of six species, as follows : Pantala flavescens. Fab., occurs in all the warmer parts of the world, but not in Europe. Tramea limhata, Desj., a very variable species of wide dis- tribution, described from Mauritius. Orthetrum hrachiale, P. de Beauv. Found elsewhere in Zanzibar, Congo, etc. Anax imperator mauricianus, Rambur. Agrees with a specimen in the Museum of Zoology, Cambridge, named by Campion. The species was also taken by Gulliver, on the Transit of Venus Ex- pedition. Ischnura senegalensis, Rambur. Widely distributed in tropical Asia and Africa. Agrion ferrugineum, Rambur. One specimen was taken by GulHver. The present collection contains several specimens. The collection of Hymenoptera, exclusive of Ants, contains two species of Tubulifera, eleven species of Aculeata, and approxi- mately 170 specimens (of about twenty species) of Parasitica. The two species of TubuUfera, for the identification of which I am indebted to Mr F. D. Morice of the British Museum of Natural History, are Chrysis {Pentachrysis) lusca, Fab., found also in India, Ceylon and Mauritius, and Philoctetes coriaceus, Dahlb., known also from East and South Africa. Of the Aculeata, the Formicidae are not yet determined, and a species of Halictus is at present unidentified. The remainder of the Aculeates are as follows : Megachile disjuncta, Fab. Common in India; recorded also from Mauritius. (M. lanata, Fab., is recorded by Smith as having been taken by Gulhver on the Transit of Venus Expedition.) Megachile rufiventris, Guer. Found elsewhere in East and South Africa, Mauritius and Seychelles; previously taken in Rodrigues by GulUver. Apis unicolor, Latr. Previously taken in Rodrigues by Gulhver. Found in the Seychelles, Amirantes, Chagos (Diego Garcia, Peros Banhos). Commoner in Madagascar. Odynerus trilobns, Fab. This species has not been previously recorded from Rodrigues. It is common and widely distributed, being known from Madagascar, Mauritius, Reunion and South Africa. Polistes macaensis, Fab. Previously taken by Gulhver and listed as P. hebraeus, Linn, There seems to have been considerable confusion over these names, as Cameron {Trans. Linn. Soc. (2), vol. XII, p. 71) hsts this species as P. hebraeus, Fab., stating that it is known from Rodrigues. Dr R, C. L. Perkins has, however, 20—2 290 Mr Snell, The Natural History of the Island of Rodrigues demonstrated the differences between the male P. macaensis and male P. hebraeus. (See Ent. Mo. Mag. (2), vol. xii, 1901, p. 264.) P. macaensis is known also from Seychelles, Amirantes, Chagos (Salomon Islands, Diego Garcia), and Mauritius. Scolia (Dielis) grandidieri, Sauss. I am indebted to Mr Rowland E. Turner of the British Museum of Natural History for the identification of this species. He states that the specimens under review are of "a form of D. grandidieri, Sauss. from Madagascar, with a few more punctures on the abdomen than in that species." Ampulex compressa, Fab., not previously recorded from Rodrigues. Common from Eastern Europe to China, and also in Africa. Passaloecus (Polemistus) macilentus, Sauss. Mr R. E. Turner has kindly identified this species for me. He states (in litt.) that "Mr Morice considers that Philoctetes coriaceus, Dahlb. is probably parasitic on this, as species of Passaloecus are often attacked by small Chrysids." The species was described from Madagascar. Sceliphron hengalense, Dahlb. ( = Peolpaeus convexus, Sm.). Mr Turner has confirmed my identification of this species. He adds: "This is probably an imported species, as species of the genus build mud nests on ships and are carried in that way from place to place." Trypoxylon errans, Sauss. Not previously recorded from Rodrigues. Found also in Mauritius and the Seychelles. There are approximately 750 specimens of Coleoptera, of pos- sibly 100 species; 640 specimens of Diptera, of at least seventy species; and 360 specimens of Hemiptera, of some forty-five species. These have not yet been critically examined. In the Lepidoptera, seven species of Butterflies were collected by Mr Grulliver on the Transit of Venus Expedition. Of these one species is not represented in Mr Snell's collection, viz. Hesperia forestan, Cr. The list of Butterflies is as follows : '^Melanitis leda, Linn. '\*Zizera lysimon, Hiibn, *Danais chrysippus, Linn. "f^Polyommatus boeticus, Linn. Precis rhadama, Boisd. *Tarucus telicanus, Lang. *Hypolimnas misippus, Linn. Parnara borbonica, Boisd. "f^Atella phalantha, Drury Among the Moths (Heterocera), exclusive of the Pyralidae, Tortricidae, and Tineidae, though Gulliver's collection contained only twelve species, five of these were species not represented in Mr Snell's collection, Mr Snell obtained three species of Sphingidae, * Of wide distribution. ■]• Not previously recorded from Rodrigues. Mr Snell, The Natural History of the Island of Rodrigues 291 one species of Arctiidae, twenty-five species of Noctuidae, and two species of Geometridae, as follows : ^Acherontia atropos, Linn. "f^Erias insulana, Boisd. 'f*Herse convolvuli, Linn. *An,ua tirhaca, Cr. "I" Hippotion aurora, Roth. & Jord. Achaea trapezoides, Guen. i*Utetheisa pulchelloides, Hamps. Achaea finita, Guen. 'f*Chloridea obsoleta, Fab. *Parallelia algira, Linn. "f^Agrotis ypsilon, Linn. *Chalciope hyppasia, Cr. ■f*Cirphis loreyi, Dup. "f^Mocis undata', Fab, ■j" Cirphis leucosticha, Hamps. *Phytometra chalcytes, Esp. ( = insulicola, Saalm.) *Cosmophila erosa, Hiibn. "f^Perigea capensis, Guen. "f^Dragana pansalis, Walk. ^^Eriopus maillardi, Guen. ^*Magulaha imparata, Walk. *Prodenia litura, Fab. ^^Hydrillodes lentalis, Guen. *Spodoptera abyssinia, Guen. "f^Hypena masurialis, Guen. Athetis expolita, Butl. ^*Hyblaea puera, Cr. "fEublemma apicimacula, Mab. f*Craspedia minorata, Boisd. *Amyna octo, Guen. ^*Thalassodes quadraria, Guen. The five species collected by Mr Gulliver and not represented in the present collection are as follows: *Argina cribraria, Clerck. (Hypsidae). *Nodaria externalis. Walk, (redescribed as Diomea bryophiloides, Butl.) (Noctuidae). Pericyma turbida, Butl. (Noctuidae). Peculiar to Rodrigues. *Achaea catella, Guen. (Noctuidae). *Mocis repanda, Fabr. (Noctuidae). Butler listed a species as Laphygma cycloides, Guen., apparently in error, as Sir George Hampson has in his Catalogue placed the record under Spodoptera abyssinia, Guen. There are about 180 specimens, of some thirty species, of Micro-lepidoptera. These have not yet been worked out. The collections made by Mr Snell are of importance as showing more definitely the relations of Rodrigues with the other islands in the vicinity. Undoubtedly the fauna has, with the flora, suffered considerably from the devastating effects of the fires which have so frequently swept the island, but investigation of the collections of the groups not yet worked out, will undoubtedly show that con- siderable traces of the indigenous fauna still exist, and will serve to indicate with greater accuracy the affinities of Rodrigues with the neighbouring islands. *. Of wide distribution. f Not previously recorded from Rodrigues. 292 Mr Snell, The Natural History of the Island of Rodrigues Bibliography. Legtjat, Franqois. Voyages et aventures en deux lies Desertes des Indes Orientales (1650-1698). Grant, C. History of Mauritius and the neighbouring Islands (1801). (Pridham, C?) An Account of the Island of Mauritius and its Dependencies (1842). Strickland and Melville. The Dodo and its Kindred (1848). HiGGiN, E. "Remarks on the Country, Products and Appearance of the Island of Rodrigues, with opinions as to its future Colonization." Joum. Boy. Geogr. Soc. xix, Pt I, 1849, p. 17. Newton, E. "Notes of a Visit to the Island of Rodrigues." Ibis, vol. i, (new series), 1865. Balfour, I. B. and others. "An Account of the Petrological, Botanical and Zoological Collections made in Rodrigues during the Transit of Venus Expeditions in 1874-5." Phil. Trans. Boy. Soc. vol. CLXvm (extra volume), 1878. Oliver, S. P. (Edit.) The Voyage of Frangois Leguat of Bresse (2 vols. 1891). (Transcribed from the Original English Edition for the Hakluyt Society.) Gardiner, J. S. "Islands in the Indian Ocean, Mauritius, Seychelles, and Dependencies, 1914." In the volume on Africa, Oxford Survey of the British Empire. Newton, Sir E., K.C.M.G. "The Birds of the Mascarene Islands." Presi- dential Address. Trans. Norfolk and Norwich Naturalists' Society, vol. iv. Encyclopaedia Britannica. Miss Haviland, Note on the Life History of Lygocerus 293 Preliminary Note on the Life History of Lygocerus {Procto- trypidae), hyperparasite of Aphidius. By Maud D. Haviland, Fellow of Newnham College. (Communicated by Mr H. H. Brindley.) [Read 10 November 1919.] Plant lice are frequently parasitized by certain Braconidae of the family Aphidiidae. The parasite oviposits in the haemocoele of the aphis, and the larva, during development, consumes the viscera of the host. At metamorphosis nothing remains but the dry skin, within which the Aphidius spins a cocoon for pupation. At this stage, the Aphidius itself is liable to be parasitized in turn by certain Cynipidae, Chalcidae, and Proctotrypidae. The two former are known to be hyperparasites, but the Proctotry- pidae have hitherto been considered doubtful, although some writers have suspected that they are hyperparasites of the Aphidius, and not parasites of the aphis. Gatenby in his paper: " Notes on the Bionomics, Embryology, and Anatomy of certain Hymenoptera Parasitica" (Journ. Linn. Soc. 1919, vol. xxx, pp. 387-416) says: ". . .1 am inclined to support the view that the Proctotrypid is a parasite, and not a hyperparasite." The following is a summary of some observations made in the summer of 1919, on two Proctotrypids of the genus Lygocerus. I am much indebted to Professor Kieffer, who has kindly identified them for me as L. testaceimanus, Kieff., hyperparasite of Aphidius salicis, Hal., parasite of Aphis saliceti, Kalt., from the willow; and L. cameroni, Kieff., hyperparasite of Aphidius ervi, Hal., parasite of Macrosiphum. urticae from the nettle. The following notes probably apply to both species, but the observations were made more especially upon the latter. It was found also that in cap- tivity L. testaceimanus would oviposit on Aphidius ervi. The Proctotrypids do not confine their attacks to the Aphidiidae, but their larvae may also be found feeding on the larvae of other Chalcid or Cynipid hyperparasites of that family ; and indeed once or twice were observed upon dead pupae of their own species. One remarkable instance of hyperparasitism came under notice. An aphis {Macrosiphum urticae) was parasitized by an Aphidius (A. ervi). The latter had been hyperparasitized by a Chalcid, of species unknown, which immediately after pupation had been attacked by another hyperparasite, either Chalcid or Cynipid, whose identity is not yet determined. This second hyperparasite in turn had been attacked by Lygocerus cameroni, and the larva was in the second instar when the cocoon was opened. We may ask, where are the limits to this hyperparasitism? 294 Miss Haviland, Note on the Life History of Lygocerus Lygocerus cameroni was fairly common round Cambridge in 1919, from mid-July to the end of August. The female selects an aphis-cocoon containing a full-grown larva or newly transformed pupa of Aphidius, and runs round it with much excitement, tapping it with her antennae. Oviposition takes from 30-60 seconds, the insect meanwhile standing either on the top of the cocoon facing the anterior end, or on the leaf behind, with her back to it. Either way, the ovipositor is brought into the angle of the host's body, as it lies curled inside. Sometimes two or three eggs, the result of successive ovipositions by different females, are found on the same host. The egg, which is laid on the upper surface of the abdomen of the Aphidius, measures -25 x -10 mm. It is translucent, white, and elliptical, with marked longitudinal striae of the chorion, and a minute stalk at one end. Treatment of the egg with lacto-phenol and cotton-blue showed the presence of bodies resembling the symbiotes from the pseudovitellus of Aphides. The egg hatches in about twenty hours. The larva of the first instar is a maggot shaped form, with thirteen body segments and a head furnished with two minute papillae. The mouth, which is circular and very small, contains two simple chitinous mandibles set well behind the hood-hke labrum and the labium. The mid-gut, which at this stage does not communicate with the rectum, is large and globose, and its con- tents tinge the transparent body pale yellow. Later on, when the host dies, they become brown. The tracheal system consists of two lateral longitudinal trunks, united by an anterior and posterior commissure. When newly hatched, there are two open spiracles between the first and second and on the fourth segments, but soon afterwards the spiracles of the third and fifth segments become functional. The larva is active and crawls over the host's body. This instar lasts from twenty to twenty-four hours, and the dimensions are about -45 x -22 mm. The larva of the second instar differs from that of the first chiefly in the size, which is -70 x -35 mm., and in the tracheal system. The ramifications of the latter are more numerous, the dorso- ventral branches of the second segment become visible, and the spiracular trunks of segments six, seven, and eight appear, though their spiracles are not open. The duration of this instar is about thirty-six hours, and at this time the host usually dies, and its body becomes blackened and shrunken. In the third instar, the papillae on the head disappear, the body becomes more globose, and the greater proportionate development of the three first segments causes the head to be bent round to the ventral side. The dimensions are about 1-00 x -75 mm. The spiracles of the sixth, seventh and eighth segments open, and the spiracular {Prodotrypidae), hyperparasite of Aphidius 295 trunk of the second segment becomes visible. In addition, two short spiracular trunks can be made out on the ninth and tenth segments; but these never become functional, and they disappear in the later stages of development. This instar lasts from about thirty-five to forty hours. In the fourth instar, which lasts about two days, the Procto- trypid grows rapidly, and when mature measures 1-67 x -83 mm. The remainder of the host is quickly consumed, and, just before metamorphosis, the mid-gut opens into the rectum, and its con- tents are voided into the cocoon. The larva is active and wriggles about freely inside the aphis skin, aided possibly by a curious caudal appendage; and by these movements the faeces, together with the host's skin, are kneaded into a moist compact pellet on the ventral side of the body. The full grown larva is yellowish white, and each segment has a double row of short chitinous spines. The thorax is large and broad, while the abdominal segments taper away somewhat to the eleventh, which bears a short stout appendage furnished with spines. The head is turned completely under the thorax, and the tracheal system does not differ essentially from that of the pre- ceding instar. No larval antennae nor maxillary nor labial palpi seem to exist at this stage. Lygocerus does not produce silk, but pupates in the cocoon made previously by the Aphidius inside the skin of the aphis. The period of pupation is fourteen to sixteen days. When ready to emerge, the imago gnaws a hole somewhere on the upper side of the cocoon, and creeps out. So far, no parthenogenetic ovipositions have been observed, and two broods, certainly, and possibly more, may occur in the season. The life of the imagoes is generally five or six days, but they may live as many as ten. Examples in captivity were observed to feed on the sap oozing from cut leaves, and on honey- dew dropped by the aphides, but they seemed to live as long and to remain as vigorous when no food was supplied. 296 Mr Warburton, Note on the solitary wasj), Note on the solitary wasp, Crabro cephalotes. By Cecil Warbueton, M.A., Christ's College. [Read 10 November 1919.] Last summer a small colony of C. cephalotes took possession in my garden of a log of elmwood which was kept as an example of a woodpecker's nest. The entrance hole of the woodpecker was there, and just below it the log had been sawn through so that the internal cavity could be examined. The first advent of the wasps was not noticed, but in the first week of August a wasp was observed entering the hole, and this led to an investigation of the log, which presented signs of boring in the half-decayed heart-wood. One of the wasps had attacked the log from the top and its operations could be noted with more or less exactness, but the others passed in and out by the wood- pecker's hole, and it was impossible to recognise individuals or to follow their work without constantly disturbing it by opening up the log, with the risk of inaccurately replacing the two halves. The log was nevertheless opened several times during the first half of August, but it was then thought better to let the wasps finish their work without further disturbance. That the wasps are not easily diverted from their labours the following facts sufficiently demonstrate. The log was moved several yards, to a spot more convenient for observation. The wasp working on the top (hereafter referred to as wasp No. 1) was captured in a glass tube and examined for identification, but on being liberated continued working as before. Close observation, with a hand lens, did not deter this wasp from entering its burrow without hesitation in the course of its operations, nor were the other wasps disconcerted by the removal of the lid on several occasions at an early stage of their work. As a rule no attention was paid to anyone sitting silently near the log, but it must be recorded that on one occasion a wasp returning with a fly appar- ently objected to the dress — light with dark spots — of a lady sitting near at hand, and after a close investigation from many points of view, retired instead of entering the log. To ascertain if wasp No. 1 were at home or not I was in the habit of placing a stout straw in its burrow — protruding an inch or more. One would have thought that on returning home and finding such an object impeding its entrance the insect would manifest some perturbation and either refuse to enter or take some measures to remove the obstacle. It did nothing of the kind, but absolutely disregarded the straw, pushing past it even when laden with a fly. It was several times Crabro cephalotes 297 ejected together with the frass from new tunnelling operations, but never otherwise. Continuous observation of work that went on for many hours a day for about three weeks was, of course, impossible, but on several days, especially during the week Aug. 18 — 25, operations were watched for spells of an hour or two at a time, and the exact times of ingress and egress carefully noted. The notes which immediately follow especially concern wasp No. 1. The hole was sometimes clear, sometimes choked with "saw- dust." After watching for a time the "sawdust" would be seen to heave up and form a mound over the hole. Then the wasp would emerge and proceed to remove the frass, butting it away from the neighbourhood of the hole with its head. Sometimes in the course of its excavations the wasp would emerge, fly away for a time, and return empty handed to resume its digging. On Aug. 19 it was seen to be carrying home flies, and the per- formance was watched for an hour, and the following times were noted: Returned with fly, 9.37, 9.48, 10.18, 10.31. Emerged, 9.40, 9.55, 10.25, 10.39. Thus four flies were caught in the hour, and the times spent in capturing three of them were 8', 23' and 6' respectively, while 3', 7', 7' and 8' were occupied in packing the four flies into the burrows. To find, capture, paralyse and bring home the right kind of fly in six minutes strikes one as a remarkable feat. From further observations it appeared that the operation usually occupied about a quarter of an hour. None but "hover flies" (Syrphidae) were taken by any of the wasps, and the prey was generally Syrphus halteatus, a species almost as large as the wasp itself. It was, nevertheless, carried with perfect ease, arranged longitudinally, head foremost beneath its captor, and, I believe, venter to venter. No preliminary examination of the hole was ever made before carrying the fly in, such as Fabre has recorded in the case of some wasps. About noon on Aug. 21 this wasp apparently ceased workino;. There were no signs of activitv that afternoon nor the following morning. On Aug. 22 about 3 p.m. a wasp (wasp No. 2) was seen to come out of the woodpecker's hole and alight on the top of the log, which it proceeded to explore. It found No. I's burrow and entered it for a short distance, after which it flew away. Nothing further was noted till the evening of Aug. 23, when on returning home at 5.30 I noticed a heap of frass on the top of the hole. At 6.20 a wasp arrived and after pointing at the main entrance, seemed to change its mind and alighting on the top, entered No. I's hole. Its behaviour convinced me that it was not No. 1, but it 298 Mr Warburton, Note on the solitary wasp, might very well be wasp No. 2. Anyhow it entered the burrow, and by 7.50 it had turned out more "sawdust" containing several of the flies so carefully stored up by wasp No. 1 ! The explanation that first occurred to one was that the wasp wanted to dig, and naturally found it easier to work w^here someone had been before. Such a defective instinct would, however, militate against the preservation of the race. Moreover there were no further develop- ments, and No. 2 remained satisfied with undoing some of No. I's work. A wild suggestion did occur to me, which I will give for what it is worth. Is it possible that one of those working from the interior became aware of operations from the outside which might imperil the results of its own labours, and proceeded to put a stop to them? With regard to the remaining wasps, which entered by the woodpecker's hole and worked from the inside, the following notes may be given. The earlier hasty inspections of the interior showed that the cavity of the woodpecker's nest was being gradually filled with the "sawdust" of their workings, and conspicuous on the "sawdust" were a number of Syrphid flies, apparently dead. At the final investigation at the beginning of October about a hundred and twenty of these derelict flies were found in the central cavity, and as there were certainly not more than six wasps at work at any time, and as two were early captured and retained for identification, it is probably safe to estimate the average numbers of the wasps responsible for discarding them at five. This allows twenty-four discarded flies to each wasp — about six hours strenuous labour by each insect entirely wasted! As wasp No. 1 was never seen to discard a captured fly this phenomenon was apparently attributable to the conditions prevailing inside. There all the burrows com- menced with a horizontal boring at the junction of the two sections of the log, at some little distance from the main opening. After alighting at the main entrance they had, therefore, either to fly across or to crawl round the central cavity, and it seems as though a number of flies had been accidentally dropped. It would be quite in keeping with what has been observed in the case of allied insects that a wasp which had accidentally dropped a fly should make no attempt to retrieve it, but should simply go away and catch another. These discarded flies were in any case very useful as evidence of the particular prey selected by Crabro cephalotes. At the beginning of October some of these flies had been reduced to fragments by other predaceous creatures, but of 113 recognisable specimens 60 were S. halteatus. My friend Mr N. D, F. Pearce very kindly undertook to identify the remainder for me and he finds among them five species of Syrphus, three of Platychirus, two of Melanostoma, and one of Crabro cephalotes 299 Rhingia, Catabomba and Helophilus respectively. No family of flies except the Syrphidae was represented. The complete list is as follows : Syrphus balteatus 60 S. luniger 5 S. vitripennis 4 S. corollae 4 S. auricoUis 3 S. albistr ictus 1 Platychirus albwianus 9 9 P. scutatus ? 2 P. peltatus 1 Melanostoma mellinum 7 M. scalar e ? 2 Rhingia campestris 13 Catabomba pyrastri 1 Helophilus pendulus 1 113 Early in October the log was thoroughly explored, and an attempt was made to follow out the windings of the galleries, but the extreme friability of the decaying heart-wood made this very difficult. The first thing that struck one was the absence of any attempt to seal or mask the tunnels which were entirely open to any chance intruder. Indeed a family of wood-lice was found three inches down the tunnel of wasp No. 1. There was nothing to prevent any enemy from entering. While at work the wasps had never manifested any interest in other insects in the neighbourhood of their burrows, nor did they finally make any provision for keeping them out. While watching the operations of wasp No. 1 a few insects had been seen to enter the tunnel, including Phoridae, one of which was secured, and a Muscid fly ( ? Tachina) and an Ichneu- monid which unfortunately evaded capture. The main tunnels were clear, and penetrated the wood for several inches, with abrupt turnings on no definite plan. From these proceeded side galleries in which were found " sawdust," the debris of flies, and the brown cocoons containing the fully-fed wasp larvae. Sections of the log showed that these were dotted here and there throughout the soft heart- wood precisely hke the raisins in a Christmas pudding. k 300 Mr Aston, Neon Lamps for Stroboscopic Work Neon Lamps for Stroboscopic Work. By F. W, Aston, M.A., Trinity College (D.Sc, Birmingham), Clerk-Maxwell Student of the University of Cambridge. [Read 19 May 1919.] For the accurate graduation and testing of revolution indicators and similar technical purposes the stroboscopic method is probably the most reliable. This depends on the fact that if a rotating disc is illuminated N times per second by very short flashes, a regular figure drawn symmetrically on the disc will appear at rest when the number of revolutions of the disc per second is some exact multiple or submultiple of N depending on the number of sides of the regular figure. The value of N — in practice 50 — can be s^t and easily kept extremely constant by the use of an electrically driven tuning-fork so that the success of the method rests principally upon the illuminating flashes ; its accuracy will depend upon their shortness of duration and brightness; its convenience as a practical method upon their brightness and quality as affecting the eye of the observer. The first experiments were tried with naked Ley den jar sparks obtained from the secondary of an ordinary ignition coil, the tuning-fork being introduced into the primary circuit as an interrupter. These showed the principle of the method to be excellent but spark illumination left much to be desired; it was noisy, feeble in intensity, and being mostly of short wave-length, caused rapid and excessive eye-strain even when used in a dark room. The remarkable properties of Neon seemed to offer an almost ideal solution of the illumination problem. A form of lamp to replace the spark was therefore devised which appeared likely to give good results and several of these were filled from the author's stock of Neon at the Cavendish Laboratory. The success of these lamps was immediate, eye-strain disappearing completely. The present paper is a description of the lamps and their behaviour during continuous use. The Form of Lamp. The original form of the lamp, which it has not been found necessary to alter materially, is shown in the sketch. As, in the discharge in Neon, nearly all the light is in the "Positive Column" and its brightness increases with the current density, the lamp was designed to give a positive column as long and narrow as Mr Aston, Neon Lamps for Stroboscopic Work 301 possible consistent with the potential available in the spark, and consists essentially of two relatively large spaces containing the electrodes connected by a very long capillary tube which is the counterpart of the filament in an ordinary glow lamp. In the lamps flNOoe: -h C<^THOOE Neon vacuum lamp for Stroboscopic work. Two-thirds actual size. in use the filament is about 60 cm. long by 1 mm. diameter and is coiled up inside the space containing the anode. This was done for convenience and strength, but it has another and important advantage, for this type of construction is strongly unsymmetrical to the discharge, allowing it to pass much more easily in the direc- I 302 Mr Aston, Neon Lamps for Strohoscojnc Work tion indicated in the figure than in the opposite, hence it effectually stops the "reverse" current from the secondary of the coil. Other important results depending on the length of the fila- ment will be discussed later, it should be roughly one hundred times the length of the spark the coil is capable of giving in air when running on the tuning-fork break. It is hardly necessary to state that the shape into which the filament is wound is not in the least essential and could be varied to any extent in lamps for special purposes. The electrodes are of aluminium and may be of any form so long as they are not too small. Method of Filling Lamps. As Neon, like the other gases of the Helium group, has the remarkable property of liberating gas from aluminium electrodes which have been completely run in for other gases, the operation of filling necessitates the contamination of a comparatively large volume of Neon, so that this can only be done economically and conveniently where liquid air is available for re-purifying. So far all the lamps have been filled on the author's Neon fractionation apparatus at the Cavendish Laboratory^, The gas for filling is contained in charcoal cooled in liquid air. A quantity is admitted to the exhausted lamp which is then sparked at a pressure of 1 to 3 mm. with a small coil for a time. The dirty gas is then pumped off with a Toepler mercury pump, a fresh supply of pure gas admitted and the tube run again. These operations are repeated until spectroscopic and other observations show the desired conditions of purity have been reached and are not altered seriously by prolonged running. The full charge of 5 to 10 mm. of gas is now let in and the lamp sealed off. The whole operation takes about 3 hours, three lamps being filled at once. The pressure, purity and time of running in are all matters of some nicety as will be seen from consideration of the life of the lamp. Life of the Lamps. Apart from accident the lamps are serviceable until the pressure of gas within them becomes too low for the spark to light them adequately. Their life appears to consist of two distinct periods, the first during which chemically active impurities derived from the electrodes and walls of the tube are being slowly and completely eliminated (at least as far as a spectroscopic observation goes) and the second during which sputtering of the cathode takes place and the inactive Neon itself slowly disappears until the pressure gets too low for use. During the first period the luminosity steadily 1 V. Lindemann and Aston, Phil. Mag. sxxvii, May 1919, p. 527. Mr Aston, Neon Lamps for Stroboscopic Work 303 improves, remaining almost constant afterwards till near the end of the second period when it rapidly decreases. The first set of lamps were filled with very carefully purified Neon at 1-2 mm. pressure and run till sputtering had commenced before being used; they may therefore be considered to have had no first period at all. These lamps had a life of 500-1000 hours. Experiments soon showed that the less preliminary running and the higher the pressure of filling the longer the life would be, but on the other hand, if the preliminary running is not sufficient the impurities derived from the electrodes turn the light' of the lamp a dull grey and render it absolutely useless and pressures above 10 mm. are not advisable as these increase the spark potential of the lamp too much. One lamp was actually so nicely balanced in these respects that though it became grey and useless after about 1 hour's use it completely recovered its original brightness after a day's rest. This is clearly a case of carbon compounds being given off by the elec- trodes while running, which are reabsorbed on standing and there is little doubt that were it worth while very prolonged running would render this lamp quite satisfactory. Very slow production of gases from the electrodes is advantageous, as prolonging the first period of the life, so that these should be of a fairly solid pattern. So far, the best results have been obtained from a batch of lamps filled at about 10 mm. pressure, some with pure Neon, some with a mixture of Neon and about 10 per cent. Helium. One of the latter had a working life of well over 3000 working hours. Helium disappearing from its spectrum after the first few hundred. As there is every reason to assume that for any given lamp the life is determined by the total number of coulombs passed through it, the light obtained per coulomb should be arranged to be a maximum. This will be the case when the filament is made as long as possible, consistent with the potential available from the coil. Cause of Disappearance of Gas from the Lamps. The exhaustion of gas by continuous running has long been observed in the case of spectrum discharge tubes. It is doubtless allied to the phenomenon of "Hardening" in X-ray bulbs, but difi'ers from the latter in that under the relatively high pressures in spectrum tubes, and the Neon lamps under consideration, the mean free-path of a charged molecule is so small that it can only fall freely through a potential of a few hundred volts and so never attain the very high velocities reached in the X-ray bulbs which are supposed to cause the gas molecules to become permanently embedded in the glass walls. VOL. XIX. PART VI. 21 304 Mr Aston, Neon Lamfs for Strohoscopic Work The disappearance of gases of the HeHum group in spectrum tubes is invariably associated with sputtering of the electrodes which, at high pressures, only takes place when the gas is spectro- scopically free from chemically active gases. It is generally sup- posed that the gas so disappearing remains embedded or adsorbed in the layer of sputtered aluminium on the sides of the tube near the cathode, the idea of true chemical combination not being acceptable without very rigorous proof. In order to obtain information on this point, a completely run out specimen of the first batch of lamps, which was of course very heavily sputtered, was taken for test. First the sputtered cathode end was gradually heated to near the softening point of the glass (when it cracked) without any substantial or apparent increase in the internal pressure of Neon. The end was then cut ofi, broken into small pieces and heated in a quartz tube in a high vacuum apparatus provided with a spectrum tube. At a temperature about the softening point of the glass a good deal of gas was released which showed the hydrocarbon spectrum (but may nevertheless have contained some Neon as this is easily masked) ; this gas was pumped off and on heating further to a red heat, as the glass started to melt, Neon was given off, the spectrum showing quite clearly. Apparatus for measurement and analysis of the gas so released was not available, but it is hoped to repeat this interesting experi- ment, which shows definitely that the Neon is contained either in the sputtered aluminium or very near the surface of the glass so that it is released by heat. Use of other Gases instead of Neon. Ordinary chemically active gases give very feeble illumination, CO being about the best. Helium gives a bright discharge but not nearly so valuable in quality for visual work as Neon ; its presence as an impurity in the latter gas renders the discharge more rosy red but up to 10 per cent, does not affect its brightness seriously. Mercury vapour as used by C. T. R. Wilson in his photography of ionisation tracks would probably give very bright flashes, but the fact that the lamp has to be kept very hot is a serious objection. Reason for Superiority of Neon. The brilliant orange-red glow of the discharge in Neon is com- posed almost entirely of lines in the region 5700-6700 a.u. and is in such striking contrast to sunlight that strohoscopic observations can even be done in broad daylight if necessary, the ordinary appearance of the rotating disc having merely a grey background added, looking bluish by contrast. The actual amount of light radiated per unit of energy, i.e. Mr Aston, Neon Lamias for Stroboscopic Work 305 the real efficiency of the discharge in Neon, is not markedly greater than that in e.g. mercury vapour, but the apparent efficiency is enormously enhanced by the fact that it consists so largely of red light. Victor Henri and J. L.desBancels have shown (" Photochemie de la Retine," Jl. Phys. Path, xiii, 1911) that the Fovea Centralis of the eye is immensely more sensitive to red light than the outlying portions of the retina^, thus a Neon lamp as a source of general illumination is very disappointing, but when viewed directly appears surprisingly bright. As the spinning disc of the stroboscope subtends a comparatively small angle the Fovea is the only part of the observer's eye used in testing, which is probably the reason for the eye strain with the spark. Nature and Duration of the " Working Flash.'" If one analyses the flash of a short spectrum type Neon tube in a rotating mirror it is seen to consist of two separate parts, an extremely short flash followed by a flame or "arc." The first is probably due to the simultaneous ionisation of the gas throughout the whole length of the tube, the second to the further carriage of current by the ions formed during the first. The structure of the latter, which appears to consist of bright striations travelling from anode to cathode at velocities of the order of that of sound in the gas, is of great theoretical interest and is at present under investi- gation. Discussion of its nature is needless in the present paper for its duration being of the order of thousandths of a second it is useless for stroboscopic work and, by the employment of a suffici- ently long filament tube, it can be eliminated altogether. In a lamp properly proportioned to the power of the coil in use the v.'hole energy of the discharge is absorbed in the first flash. In order to get some idea of the duration of this "working flash" the following experiment was performed. A plain mirror, silvered outside to avoid double images, was mounted vertically on the axis of a large centrifuge and the image in this of the Neon lamp at a distance of 3 metres was observed by means of a telescope with a micrometer eye-piece. Each division in the micrometer subtended 4-2 x 10^^ radians and when the centrifuge was running at 3500 revolutions per minute corre- sponded to 5-75 X 10~' seconds. The lamps were run with the tuning-fork attachment used in actual testing and were viewed directly and also through ground glass with a V-shaped slit to be certain of getting the effect of the ' The difference of retinal effect between red and green light can be easily ob- served by looking at an ordinary luminous wrist watch in the faint red light of a photographic dark room. On shaking the watch so sluggish is the green light in recording its position on the retina compared with the red that the figures seem to be shaken completely off the dial, giving a most curious and striking effect. 21 — 2 306 Mr Aston, Neon Lamj)s for Sirohoscopic Work total duration of the flash. In neither case was the fuzziness of the image of a measurable order. After careful observation under good conditions the conclusion of three observers was, that it was probably less than one-tenth of a division and certainly less than one-fifth. This gives the maximum duration of the working flash as one-ten-millionth of a second, so that it can be taken as perfectly instantaneous for the purpose employed. Other Technical Afflications . Of the many uses besides measuring velocity of rotation to which Neon lamps may be put with advantage in engineering and other problems it is sufficient to mention two in which they have been very successful. Any rapidly rotating mechanism such as an airscrew, if illuminated by a lamp the break of which is operated mechanically at each revolution, will appear at rest, flicker being small at speeds well over 1000 r.p.m., so that strains or movement of parts can be examined with great accuracy under actual working conditions. A still more striking effect can be obtained by illuminating a high speed internal combustion engine by a lamp whose break is operated mechanically at e.g. 99 breaks per 100 revolutions of the engine shaft by the use of a creeping gear. The engine then appears to be rotating quite smoothly at one-hundredth its normal speed so that such instructive details as the movements of the valves and springs, the bouncing of the former on their seats, etc., can be studied with ease. It is of course necessary for the speed of rotation to be fairly rapid to give appearance of continuity to the eye and in conse- quence one cannot apply this method to the analysis of such a thing as the movement of a chronometer escapement. As the technical importance of Neon lamps is rapidly on the increase it is very desirable that liquid air engineers in this country should consider the erection of a fractionating plant for recovering the gas from the air (which contains -00123 per cent, by volume) such as has been used with such success by Mons. Georges Claude of Paris, to whom the author is indebted for the Neon with which these experiments were performed. Mr Harrison, The pressure in a viscous liquid etc. 307 The ^pressure in a viscous liquid moving through a channel unth diverging boundaries. By W. J. Harrison, M.A., Fellow of Clare College, Cambridge. [Read 24 November 1919.] If non- viscous liquid is flowing along a tube having a cross- section which is increasing in area in the direction of flow, the pressure will also increase, in general, in the same direction. On the basis of this remark an explanation has been given of the secretory action of the kidneys. The author's attention was drawn to this explanation by Dr Ffrangcon Roberts. The physiological aspect of the question and a more detailed numerical consideration will be dealt with by Dr Roberts and the author in a separate paper. In the present paper two problems are considered, viz. the flow of liquid in two and three dimensions when the stream lines are straight lines diverging from a point. Two-dhnensional 'problem. Let the boundaries of the channel be ^ = ± a, where (r, d) are two-dimensional polar coordinates. The motion in which the stream lines are straight lines passing through the origin has been ob- tained by G. B. Jeffery^. With a slight change of notation the results of his solution are as follows. Let the velocity at any point be ujr, where m is a function of d only. Then 2 A ^^** , U^ = — 4:VU — V -77i5- + a, dO^ where v is the kinematic coefiicient of viscosity, and a is a constant of integration. Whence u = — 2i^ (l — m^ — m^k^) — Qvkhn^ sn^ {md, k), where k and m are constants, which may be determined from the conditions that u must vanish at ^ == ± a, and that the total rate of flux may have a given value. Instead of the latter condition it is simpler to assume that the velocity is given for ^ = 0, i.e. u = Uq for 6 = 0. Thus the conditions are — 2v {1 — m^ — m^k^) = Uq, (1 — ni^ — m^k^) + 3kh)i^ sn^ {ma, k) = 0. 1 Phil. Hag. (6), vol. xxix, p. 459. 308 Mr Harrison, The pressure in a viscous liquid These may be written m2 = (1 + uJ2v)l{l + B), 'I + uJ2v\^ J 1 + F sn' (^-^1?)-^ 3F (1 + 2i//wo) ' If the values of Uq and a be given, the last equation serves for the determination of h. Writing h-y = l/k, the equation has the same form in k-^ as in Jc. Hence, if k is a solution, l/k is also a solution. Therefore, of real values of k, it is only necessary to consider such that satisfy ^ ^ ^ 1 . Treat a as small, and assume that ( — = — %-^ — ) a is also small. V l + k^ J (1 + 7(^2)2 We have a^ = ^-.^ .^ \, , — tk— . • 3P (2 + 2v/uq + uJ2v) The least value of a for a given value of Uq/2i', if k is real, is given hj k = 1. In this case, if Uq/2p = 1, a^ = ^, a = -58. This value of a is not small enough for the approximation to hold good. Put ^ = 1 and 2v/uq = 1 in the original equation, and we find a = "65, approximately. For smaller values of a, k will be a complex imaginary quantity. As uJ2v is either increased or decreased, a real value for k can be obtained for smaller values of a. It will be found sufiicient for the purposes of the present paper to restrict the consideration of the solution to the ranges of values of a and Uq/2v for which k has a real value. We proceed to discuss the pressure variation in the case for which k is real ; the variation in the case for which k is complex can be inferred by considerations of continuity. Let J) be the mean pressure at the point (r, 9) in the liquid, and p its density. We obtain from, the two-dimensional polar equations of motion u^ _ Idp V d'^u ~^^ ~pdr^r^W' _ \ dy 2vdu ^'^^^ ^-~2^-2^aP + ^(^) substituting for ^^ from the differential equation satisfied by u. Also ^=J^+/(y) p "'^ moving through a channel with diverging boundaries 309 Hence - = — - — --^ + C, where C is a constant. Now the lateral p r^ zr^ stress in the liquid is pee , where p p r Hence pgg is independent of d, and is the normal stress (of the nature of a tension) exerted by the liquid on the boundary. If a is negative the normal pressure on the boundary decreases as the channel widens, and if a is positive the normal pressure increases. Now by substitution of the solution for u given above in the differential equation satisfied by ii, we find a = 4i;2 [_ 1 + m^ (1 - F + J^)] = iv^ [-! + (! + uJ2vf (1 + k^)/{l + k^fl (!) Writing a = 0, we can immediately discriminate between those cases for which the pressure on the boundary decreases and those for which it increases. If a = 0, we have 1 + uJ2p - (1 + k^r/{i + k^)^, and 1 + 2i//mo = (1 + k^f/{{l + k'^f - (1 + k^)^ Hence s„^ \( A + ^Y., 4 = (1 + P)* - (1 +^e)t iV(l + k^y/ ) sk^ (1 + F)* The following diagram shows how the value of uJ2v for which Pffg is independent of r varies with a, for those cases in which k is 310 Mr Harrison, The pressure in. a viscous liquid real. It clearly indicates that when a is small the critical value of Uq/2v may be somewhat large. If a > 7r/4, the lateral pressure increases for all values of Uq. (2) It is a simple matter to discuss the variation of the pressure when Uq/2i> is large. We have, approximately uJ2v sn' 1 + k^ 3F = m" = a = 1 + P' (1 + F)3 ' k will be real provided F > |, and, corresponding to real values of k, a will be small. In the absence of viscosity, so that u = Uq for all values of d, 1^69 = 2r2 C. 1 + k^ Thus the lateral pressure increases at a rate which is ■ ,^3 of the rate for a non- viscous liquid. The following table will indicate the character of the results when k is real. uJ2. a {l+k^)/{l+k^r 100 10° 30' 9° 30' •30 •27 1000 1° 3' 0°57' •30 •27 10,000 0° 6' •28 m For larger values of a than those given above, and for the corre- sponding values of uJ2v, k is unreal. When a is small there is apparently an approximation which JefEery gives, viz. u= — 2v {1 — m^ — m^k^) — Qvk^m'^d^, 1 moving through a channel with diverging boundaries 311 where = - (1 - m2 - m^F) - - Skhn^a^, leading to Qvm^ (1 — m2),„„ = «„ (1 - ff'ja?). -«2) This gives a = — 2vuja'^, and Pee ^Uq , ^, _ — ..9...9. + ^ • /3 a^r" Thus the lateral pressure apparently decreases for all values of Uq. But if Uq= — 2v {I — m^ — m^k^), and = - (1 - w2 - yn^) - Skhn^a^, we have Uq/2u = 3khn^a^, and therefore ma is not necessarily small. Hence the approxima- tion is only valid for values of Uq/2v below some limiting value. If this condition be satisfied the expression for pgg given above is an approximation to its value for small values of a. Three-dimensional problem. Let the boundary of the channel be ^ = a, where {r, 9, (f)) are polar coordinates. This problem has been considered by Prof. A. H. Gibson^. In his solution Cartesian and Polar Coordinates are confused, and he assumes that the stream lines are straight lines diverging from the origin, a state of motion which is impossible if the inertia terms are retained in the equations of motion, as he retains them. One result of these errors is that in his solution p is a function of 6 although the preliminary assumption is virtually made that p is independent of 6. His expression for the pressure appears to be quite wrong. Assume, in the first place, that the stream lines are straight lines diverging from the origin, so that u = f{d)jr^, v = 0, w = 0. The polar equations of motion reduce to du 1 dp or p or dht 2 du cot 6 du 1 c^u 2u dr^ r dr r^ dd r^ dd^ r^ ^ _ 1 dp 2v du pr^^^dd' = - -^ ^ p d(f) 1 Phil. Mag. (6), vol. xviii, p. 36, 1909. 312 Mr Harrison, The pressure in a viscous liquid etc. We have i| = |-V^ [cot «./'+/"], 1 dp _2v ., Hence eliminating f, ^ + ,-4 [/'" +/" cot d -/' cosec2 e + 6/'] = 0. Therefore ff = 0, and /'" +/" cot 9-f' cosec2 d+6f' = 0. Hence/' (6) = 0, and the boundary conditions cannot be satisfied^ since u becomes independent of 6. For slow motion, or any motion in which the inertia terms can be neglected, we have /'" +/" cot e -f cosec2 d + 6/' = (1). A first integral is /"+/'cot0+6/+C = O (2). The solution of (2) suitable for the present purpose is /(6') = Z)(2-3sin2^)-iC. Let f{e)-^u„ 6=0, f{e) = 0, d^a. We have D = uJ3 sin^ a, (7 = 2 (2 — 3 sin^ a) ^o/sin^ a. Hence u = Uq (sin^ a — sin^ 6)/r^ sin^ a. Integrating the equations of motion, we have f=-|^(/'cot^+r) + i^x(^) ^"d P^=p+F,ir). Hence ,^ ^ '^V(^) + ^^^ + 5, and Vm^_19^_b. p 3 r* The lateral pressure will continually increase as the channel widens if C be negative, that is, if sin a > (f)^, or a > 54° 45'. If a < 54° 45', for sufficiently small values of Uq the pressure will continually diminish. Mr Gray, The Effect of Ions on Ciliary Motion 313 The Effect of Ions on Ciliary Motion. By J. Gray, M.A., Fellow of King's College, Cambridge. [Read 10 November 1919.] The ciliary mechanism of the gills of Mytilus edidis has been described by Orton^. There are at least four distinct sets of cilia. . whose movements form a complex but highly coordinated system by which food particles are filtered from the sea-water and passed up to the mouth. This coordinated system is entirely free from any nervous control and continues for many days in detached portions of the gill. These gill fragments therefore form an admirable material for the physiological study of ciliary motion. The effect of the hydrogen ion on ciliary action is very easily studied. Normal sea- water has a Ph of about 7-8; when the con- centration of hydrogen ions is increased to about 6-5 rapid cessation of movement occurs. In sea-water of Ph 6-7 the rate of ciliary movement is checked at first, but within f-l| hours complete recovery takes place. If gill fragments whose cilia have been stopped by the more acid solution are returned to normal sea- water, complete recovery takes place in less than 20 minutes although the cilia may have been motionless for several hours. A large number of experiments have been performed from which it is clear that if the concentration of hydrogen ions is only slightly greater than normal, the cells can react to the environment and recovery take place in the acid solution. In stronger acid, however, recovery only takes place on removing the gills to a more alkaline solution. In still stronger acid the cells become opaque and are killed. Gills which are exposed to an abnormally high concentration of hydroxyl ions behave in a remarkable manner. In such solu- tions ciliary action is either not affected at all or proceeds at an abnormally rapid rate, but the individual cells of the ciliated epithelia break away from each other and move about in the solution owing to the movement of their cilia. Since such cells are no longer in their normal environment, it is impossible to determine any upper limit of hydroxyl ions which will permit normal ciliary action to go on. Since the hydrogen ion has a most marked effect on ciliary activity,. it is necessary to adjust the hydrogen ion concentration of all artificial solutions during a study of the effects of various salts on ciliary action. In the case of the salts of the alkali metals this is satisfactorily performed by the addition of an appropriate 1 Journ. Marine Biol. Assoc, vol. ix, p. 444 (1912). 314 Mr Gray, The Effect of Ions on Ciliary Motion buffer such as sodium bicarbonate. In the case of the salts of the alkaline earths it is impossible to obtain pure isotonic solution of the same hydrogen ion concentration as sea-water, and it is there- fore necessary to compare the effects of the pure solutions with that of sea-water whose hydrogen ion concentration is abnormally high. A number of experiments have been performed which prove that sodium, potassium, calcium and magnesium are all necessary to maintain gill fragments in a normal state of ciliary activity for a protracted period, viz. four days. If one or more metals are omitted, the individual cells of the ciliated epithelia show the same disruptive phenomenon as in sea-water of abnormally high con- centration of hydroxyl ions. Solutions containing only one metal show this phenomenon to a very marked degree although they may be more acid than normal sea-water; the effect of solutions containing two metals is less marked than that of solutions contain- ing only one metal, but more marked than that of solutions con- taining three metals. No evidence was obtained of specific ion action or of antagonistic action between monovalent and divalent ions. These experiments afford another example of the intense action of the hydrogen ion upon physiological activity and of its reversible nature if the acid treatment is not too severe. The same action of acids is found in the activity of the heart and in the movement of spermatozoa. Mr Saunders, Photosynthesis and Hydrogen Ion Concentration 315 A Note on Photosynthesis and Hydrogen Ion Concentration. By J. T. Saunders, M.A., Christ's College. [Read 10 November 1919.] Last April (1919) I was testing the hydrogen ion concentration of the water of Upton Broad, a small broad in Norfolk. I had determined the hydrogen ion concentration of the water of the broad itself to be 8-3 and I found this varied very Httle whether the water was taken from the surface or the bottom, from near the edge or the centre of the broad. The determination of the hydrogen ion concentration was made by the use of standard solutions and indicators as recommended by Clark and Lubs. When however the water in the shallow lodes and ditches surrounding the broad was tested, great variations in the hydrogen ion concentration occurred. The water became more acid as soon as the broad was left and the ditches entered. At one end of the broad where the water was shallow, not more than 18 inches deep, and when there was no wind to mix it with the open waters of the broad which was 6 feet deep, the hydrogen ion concentration would fall to 8' 15. In the lode itself the hydrogen ion concentration was 7-65. After boiling and rapidly cooling, water from the middle of the broad and from the shallows both showed a hydrogen ion concentration of 8'4, while that from the lode after the same treat- ment was 8"15. At one point in the lode, however, I found surprising varia- tions. Dippings of water from the same place gave readings of the hydrogen ion concentration varying from 7-7 to 8-6. At this point there was a certain amount of Spirogyra growing and I found that if I took water from the centre of a mass of Spirogyra I could get a reading as high as 9-0. I took some of the Spirogyra back with me and placed it in test-tubes in tap-water which I coloured with indicator solutions. The hydrogen ion concentration was 7-2 at the commencement of the experiment. After standing the test-tube in a window in sun- light the hydrogen ion concentration rose after an hour to 8-6 and in two hours the phenolphthalein indicator had turned bright pink, indicating a hydrogen ion concentration of more than 9-0. I had no standard solutions with me which I could use to test higher values than 9-0 so that I was unable to determine accurately the ultimate result. I left the test-tubes until the next morning, when I found the hydrogen ion concentration had fallen to 7-6. After again placing the test-tubes in sunlight the hydrogen ion concentration rose above 9-0. On my return to Cambridge I repeated these rough experi- ments. It is easy to prove that the rise in alkalinity is not due to alkali dissolved out of the glass, nor is it due alone to the abstrac- 316 Mr Saunders, Photosynthesis and Hydrogen Ion Concentration tion of the dissolved carbon dioxide out of the water. The hydrogen ion concentration of the Cambridge tap-water which I used for these experiments was 7-15 when the water was tested immediately after being drawn from the tap. On standing at a temperature of 13° C. the hydrogen ion concentration rises to l-i. After boihng and rapidly cooling the hydrogen ion concentration was 7-9 and bubbling through air free from carbon dioxide produced the same result. By incubating tap-water for 36 hours at a temperature of 40° C. and then cooling the hydrogen ion concentration could be made to rise to 8-15, but in no case did the value of the control tap-water approach near that of the tap-water containing Spiro- gyra filaments. The following is a record of a typical experiment. The Spirogyra was placed in 25 c.c. of tap- water in a boiling tube and exposed to light at a window. Control boiling tubes containing tap-water only were used. All these tubes were half immersed in a glass bowl of running water so that the temperature was maintained fairly constant. Hydrogen Ion concentration Date Time (G.M.T.) Temp. Remarks 1 Control Spirogyra 1. V. 19 11-10 a.m. 14-0° C. 7-15 7-15 DuU day. 12-10 p.m. 13-0° C. 7-4 8-3 1.10 p.m. 12-5° C. 7-4 8-6 2.10 p.m. 12-5° C. 7-4 8-6 3.10 p.m. 12-5° C. 7-4 8-8 5.30 p.m. 13-0° C. 7-4 8-5 I have tried using Elodea instead of Spirogyra and it gives much the same result. Both in darkness and in daylight the contents of the living cell of Spirogyra show an acid reaction when stained with neutral red. When Spirogyra is killed by heating to 40° C. and then placed in tap-water the hydrogen ion concentration falls considerably since the cell membranes are broken or dead and the contents of the cell are now free to pass out into the water. In a large pond the mass of the plants in proportion to the water is not sufficiently great to affect the hydrogen ion concen- tration very much. I have however found slight variations. On one occasion I noticed a fall in the hydrogen ion concentration of 0-1 after several dull days and a subsequent rise of 0-2 after sunny days. This variation may possibly be due in some degree to the photosynthetic activity of the plants present. Mr Aston, Distribution of intensity 317 The distribution of intensity along the positive ray parabolas of atoms and molecules of hydrogen and its possible explanation. By F. W. Aston, M.A., Trinity College (D.Sc, Birmingham). Clerk- Maxwell Student of the University of Cambridge. [Read 19 May 1919.] No one working with positive rays analysed by Sir J. J. Thomson's method can fail to notice the very remarkable intensity variation along the molecular and atomic parabolas described by him under the term ' beading.' It will be sufficient for the reader to refer to Plate III of his monograph on the subject {Rays of positive electric, p. 52) to realise how striking these can be. Beadings at points corresponding to energy greater than the normal have been quite satisfactorily accounted for by multiple charges {I.e., p. 46), but the ones with which this paper is concerned have a smaller energy than the normal, actually half, and fractional charges are presumably impossible. Nevertheless they seem capable of a simple explanation and an opportunity of putting this to the test occurred recently while making some experiments to determine the best form and position of the cathode pre- liminary to the design of an apparatus to carry the analysis to higher degrees of precision. The observations were made with an apparatus essentially of the form now well known {I.e., p. 20) the discharge tube being arranged to be removable with the minimum trouble to change or move the cathode. As no camera suitable for photographic recording was immediately available or necessary a willemite screen and visual observation was employed. This form has many obvious disadvantages and in addition, owing to the enormous difEerence in sensitivity between the parabolas of hydrogen and those due to heavier elements the latter can only be seen with difficulty. It has however one notable advantage, namely that sudden and even momentary changes in intensity can be observed and correlated in time with changes in the discharge or in the intensity of other lines. As no accurate measurements were intended a large canal ray tube was employed so that the H^ and H^ parabolas could be easily seen even with the less effective types of cathode. It was soon realised that the appearance on the screen was in general the sum of two superposed effects which could be only unravelled like the writings on a palimpsest by eliminating one of them. This by good fortune it was found possible to do under certain conditions. For the sake of clearness it is proposed to I 318 Mr Aston, Distribution of intensity consider these two extreme types and their explanation before going on to describe the conditions under which they may be attained or approached. In the diagrams the fields of electric and magnetic forces are horizontal and such that positive ions will be deflected to the right and up, negative ones to the left and down. Brightness is roughly indicated by the width of the parabolic patch drawn. O Fig. 1. Atomic Type. Atomic type of discharge. Fig. 1 illustrates the first or 'Atomic' type in which apparently the whole of the discharge is carried up to the face of the cathode by ions of atomic mass. Those which pass through the fields without collision produce the true primary streak on parabola m=l, the head of which corresponds in energy to that obtained by the charge e falling through the full potential of the discharge. Now the pressure in the canal ray tube is never negligible being on the average at least half that in the discharge tube, and the ionisation along its length very intense so that in passing through it a large number will collide with electrons, atoms or molecules. The collision and capture of a single negative electron will result along positive ray parabolas of atoms and molecules of hydrogen 319 in a neutral atom striking the screen at the central undeflected spot while the capture of two will cause the faint negative parabolic streak a^ as has already been described (I.e., p. 39). But besides these forms of collision by which the velocity of the atom is practically unaffected there is distinct evidence that it may collide with and capture another hydrogen atom. If the atom struck is negatively charged the resulting molecule will strike the central spot but if it is neutral and the collision is inelastic the resulting positive ray will have the same momentum (the^ atom struck being relatively at rest) but double the mass so that it will strike the molecular parabola at a point the same height above the JT-axis as would the atom which generated it. Molecular rays formed in this manner will therefore form the streak b^ which, allowing for the geometrical difference in the curves will show a similar distribution of intensity to a^. Collision with a positively charged atom wiU obviously be unlikely to result in capture and those with heavier atoms will be referred to later. It is to be noted in connection with the brightness of these secondary streaks a^ and 63' which may conveniently be called 'satellites' to distinguish them from the 'secondary lines' already fully described {I.e., p. 32), that a^ is always very much fainter than its primary but b^ can be equally bright. This atomic type of discharge with its pendant bright arc on the molecular parabola corresponding to similar momentum and half normal energy is most beautifully illustrated in Fig. 29 of Plate III already referred to. It was this photograph which suggested the above theory of its explanation. Molecular type of discharge. The extreme form in which the whole discharge is carried up to the cathode by ions of molecular mass is unattainable so far in practice and is probably impossible but its share in the illumina- tion of the screen can be deduced by eliminating the superimposed atomic type and is indicated in Fig. 2, The principal feature is a short and very bright spot of light b^ on the molecular parabola at the point corresponding in energy to a fall through the full potential of the discharge. It will be shown that all the ions causing this are probably generated in the negative glow. Besides this there are two symmetrical and equally bright positive and negative satellite patches ag ^^^ (^2 on the atomic parabola but of half the normal energy. The proposed explanation of these is somewhat similar to that considered by Sir J. J. Thomson {I.e., p. 94) and is as follows. The collision with and capture of a single negative electron by a positively charged molecule will not necessarily merely neutralise it and cause it to VOL. XIX. PART VI. 22 320 Mr Aston, Distribution of intensity hit the central spot but may result in it splitting into two atoms one with a positive one with a negative charge. The energy of impact may be itself capable of causing this, if not some other cause, e.g. radiation, may effect the dissociation. In any case it would give exactly the observed result, i.e. two bright patches lying symmetrically on the extension of the line joining the primary spot to the origin at twice its distance from the latter, corresponding to half the mass but the same velocity. •^ __i ^ /c. Fig. 2. Molecular Tyi^e. The general appearance on the screen when both types of discharge are present is indicated in Fig. 3. Effect of different forms of cathode. Experiments were performed with plane, concave and convex cathodes. Convex cathodes are the least efficient in producing bright effects but give the molecular type with the least atomic blurring. Concave ones are most efficient and throw the maximum energy into the atomic type which can be obtained practically pure with them under a moderate range of conditions. The original along positive ray parabolas of atoms and molecules of hydrogen 321 shape of cathode {I.e., p. 20) may be said in a sense to combine both forms and was designed to give long and bright parabolas at the same time allowing the discharge to pass easily at very low pressures. The present results however lead one to recommend a concave cathode similar to those used in X-ray focus tubes but pushed further forward into the neck of the bulb, for though this form requires a rather higher pressure this objection is more than counterbalanced by the great increase in efficiency. Plane cathodes, as was expected, give effects midway between the other forms. O / Fig. 3. General Type. Under very exact conditions of pressure, etc. it is possible to obtain the pure atomic type with plane cathodes but no conditions have yet been found under which convex ones will give it. These results seem to indicate that atomic ions are formed by the passage of the stream of cathode rays through the Crookes dark space molecular ones tending rather to be formed in the negative glow. The axial intensity of the cathode stream is enormously increased by the concavity of the cathode while that of the negative glow does not appear to be affected to anything like the same extent. 22 2 322 Mr Aston, Distribution of intensity Behaviour during change of pressure. The pressure in a freshly set up bulb always increases with running owing to the liberation of gas by heat etc. so that the changes due to gradual alteration of pressure can be observed most conveniently by exhausting highly, starting the coil and watching the events on the screen. Thus using a concave cathode of about 8 cms. radius of curvature set just in the neck of the discharge bulb the following sequence of events was observed. At very low pressures with a potential of about 50,000 volts the parabolas are very faint but correspond to the general type, the primary streak a^ and spot h-^ being much brighter than their satellites (doubtless due to few collisions). As the pressure rises the discharge becomes curiously unsteady the spots on the screen become much fainter and change with flickering into the pure atomic type (Fig, 1), 6i having practically disappeared. This form of discharge which is evidently abnormal lasts for a certain time depending on the rate of increase of pressure. Then with absolute suddenness h-^ flashes out intensely bright and with it appear at the same instant its satellites a^ and a^. At the same time the current through the bulb increases, the discharge settles down and the negative glow makes its appearance. As far as it was possible to judge the satellites a^, and a^ are of equal brightness and generally much brighter than the negative atomic satellite «!. The appearance of the discharge bulb while the pure atomic type is shown on the screen is difficult to describe but quite characteristic and different from the general. Near its critical upper limit of pressure it was found possible to effect the change to the general type by bringing a magnet near the cathode and so disturbing the discharge. On removing the magnet the discharge at once reverted to the atomic type. This form of controlled change from the one to the other gave an excellent opportunity of testing the invariable association between the primary spots and their appropriate satellites. Possible cause of disappearance of primary molecular rays. It is unlikely that change of pressure is itself the determining factor in the disappearance of the molecular type. This seems to be due to some disturbance in the discharge by the cathode stream (not caused by the diffuse one given by a convex cathode) which makes the formation of the negative glow impossible. The facts so far may be brought into line fairly well by the somewhat speculative assumption that molecular rays can only originate freely in parts of the discharge where the electric force is very small, e.g. the negative glow, ionisation by more violent alo'ng positive ray parabolas of atoms and tnolecules of hydrogen 323 means in strong fields tending to cause simultaneous disruption of the molecule into its atomic constituents. This agrees with the observed fact that in general molecular arcs, or at least true primary molecular arcs, are shorter than atomic ones. It would also mean that a very short arc infers as origin a molecule capable of disruption. If this is so it offers interesting confirmatory evidence, if such were needed, that the substance X^ is molecular as this body often makes its appearance on the photographic plate as a short arc. Effects with heavier elements. The inelastic collision of a hydrogen atomic positive ray with the atom of a heavy element would clearly result in the formation of a molecular ray of such low velocity that it might not be detected by a screen or plate and would in any case be deflected completely off the ordinary photograph. The visual evidence on the screen although faint leaves little doubt that the formation of satellite arcs also takes place by atoms of heavier elements colliding to form molecules. There is also some evidence of this in many of the photographs, thus in Fig. 26 (I.e., p. 46) taken with oxygen all four maxima are suggested. In Fig. 17 (p. 26) the satellite on the molecular parabola caused by the capture of oxygen atoms by carbon atomic rays (or vice versa, but this is less likely) is unmistakable, in fact attention is called in the text to this remarkable increase in brightness. Should the above theory of collision with capture prove correct the formation of compound molecules by this means opens an extremely interesting field of chemical research. Another important question raised is in what form the energy of the collision is radiated off by the rapidly rotating doublet formed. In conclusion the author wishes to express his indebtedness to the Government Grant Committee for defraying the cost of some of the apparatus used in these experiments. 324 Sir Joseph Larmor Gravitation and Light. By Sir Joseph Larmor, St Johirs College, Lucasian Professor. [Read 26 January 1920.] 1. Newton's provisional thoughts on the deep questions of physical science were printed at the end of the second edition of the Opticks in 1717. As he explains in the Preface " . . .at the end of the Third Book I have added some questions. And to shew that I do not take Gravity for an Essential Property of Bodies, I have added one Question concerning its Cause, chusing rather to preface it by way of a Question, because I am not yet satisfied about it for want of Experiments." In the first and next following Queries he gives formal expression to the idea that "Bodies Act upon Light at a distance and by their action bend its Rays. ..." What was thus propounded in general terms as an explanation of the diffraction of light in passing close to the edge of an obstacle, assumed a more definite but different form in the hands of the physically-minded John Michell*; in Phil. Trans. 1767 he insisted that the Newtonian corpuscles of light must be subject to gravita- tion like other bodies, therefore that the velocities of the corpuscles shot out from one of the more massive stars vrould be sensibly diminished by the backward pull of its gravitation, and thus that they would be deviated more than usual by a glass prism, a supposi- tion which he proposed to test by experiment. He also speculated that the scintillation of the stars might be due to the small number of corpuscles which reach the eye from a star, amounting perhaps to only a few per second. The forces, of molecular range, that would have to be con- cerned, on the lines of Newton's Query, in the diffraction of light would be of course enormously more intense than gravitation : but the other Newton-Michell theory of the gravitation of light rays is paralleled in both its aspects with curious closeness in certain modern physical speculations. It will be observed that this notion of light being subject to gravitation makes its velocity exceed the limiting velocity c, which on electrodynamic theory could not be attained by any material body. But there need not be a discrepancy there: for the limit arises because a material body is supposed to acquire more and more inertia, belonging to energy of its motion, without limit as its velocity increases, whereas the quantum of energy in the hypo- thetical light-bundle presumably would remain sensibly the same — - at any rate we would be free to make hypotheses in absence of any knowledge. * See Memoir of John Michell (of Queens' College), by Sir A. Geikie, Cambridge Press, 1918. Gravitation and Light 325 Forty years a.oo there was a phase of strong remonstrance in this country against the famihar uncritical use of the phrase centrifugal force. The implication was that the term force should be restricted to intrinsic unchanging forces of nature, which are determined physically by the mutual configuration of the system of bodies between which they act: these forces are then held responsible for the accelerative effects specified by the Newtonian second law of motion. In this sense, centrifugal force so-called would not be a force of nature, but would be the reaction postulated in the scheme of the Newtonian third law to balance an imposed centripetal acceleration. This formative principle, the Newtonian third law, of balance everywhere between appHed forces and reactions against palpable changes of motion, as amplified in the Scholium an]iexed to it — which so widely reached forward towards modern theory as Thomson and Tait especially have remarked — would then assert that the forces of nature that act on the framework of a material body and the forces of reaction that are thereby induced in it, form together a system of forces that preserve statical equilibrium in relation to the constraints of that framework, as tested by the principle, also Newtonian in its origin, of virtual work. This became in time the Principle of d'Alembert (1742), who did not invent it, but exhibited its power and developed its method by applying it to a great dynamical problem of unrestricted form, that of the precession of the equinoxes. As a preliminary to its solution he had to develop in general terms the equations of static equilibrium of a system of forces considered as applied to a single rigid body such as the Earth, that is, to create a formal science of Statics: and it may be said to be the mode of development rather than the principle itself that constitutes his essential contribution to general dynamical theory. Cf. the historical introductions in Lagrange's Mecanique Analytique. 2. The principle of the relativity of force has recently become prominent again, and pushes along further on the same lines; it now even puts the question — Are there intrinsic forces of nature at all? May not all force, including universal gravitation, be ex- pressible as reaction against acceleration of motion, just after the manner of the obviously unreal centrifugal? On such a view, wherever there is a force of gravitation in evidence, its presence must be replaced by an acceleration common to all of the material bodies at each place and relative to our frame of measurement, of amount equal and opposite to the intensity of the force. That would be the end of the matter, if any frame of reference could be found to satisfy this condition. There being then no forces left, the Principle of Least Action would make orbits simply the shortest paths in the frame. Newtonian uniform space and time certainly 326 Sir Joseph Larmor could not permit this transformation: nor could the fourfold uniform continuum of interlaced space and time of the earlier relativity theory be adapted to it. Will such a fourfold, deformed into a non-uniform and therefore non-flat heterogeneous space, permit it? This is the problem raised by Einstein's idea of the relativity of gravitational force. Perhaps it goes even further, and asks whether if this will not do, there can be some other corpus of abstract differential relations invented, that will transcend the notion of spacial continuity altogether but will in compen- sation for that formidable complexity succeed in effecting this object. In any case we may recognise that this merging of all the forces of nature into spacial relations satisfies one requirement which is not quite the claim that is explicitly made for it. The question is immediately insistent; why should intrinsic forces be measurable with Newton in terms of second gradients of type (Ps/dt^ and not by a more complex formula involving others as well? The answer supplied by the theory would be that the idea of the curvature of a deranged space is expressed by a measure which does not involve higher gradients. It is interesting to reflect nowadays that in referring to the doctrines of action at a distance in the preface to the Electricity and Magnetism, in 1873 Maxwell classifies them as "the method which I have called the German one," and that notwithstanding Helmholtz's very powerful critical work on Maxwell's theory, be- ginning in 1870, that description remained substantially true until after Maxwell's death in 1879. Though he lived for nine years longer he seems to have taken no part in these discussions with exception of a reference to Helmholtz in connexion with Weber's theory {Treatise, § 254), but worked chiefly at the development of the theory of stresses in gases regarded as molecular media, and so in some respects parallel to his theory of an electric medium. He seems to have been content to leave his electric scheme to germinate and expand in the fulness of time. In connexion with the recent efliorts to transcend both action at a distance and an aethereal medium, his explanations, in an Appendix to the Memoir on the determination of the ratio of the electric units, Phil. Trans. 1868 and the critical chapter on ' Theories of Action at a Distance' in the Treatise, §§ 846 — 866, are far from being obsolete. This hypothesis as to gravitation, which asserts that it is essentially of the same nature as the apparent increase of weight which is experienced by an observer going up in a lift with ac- celerated motion, naturally involves many consequences, and raises questions regarding the relation of gravitation to physical agencies such as light, the answer to which may be ambiguous until yet further postulates intervene. Gravitation and Light 327 Thus in the preliminary stage it occurred to Einstein that the period of a train of light waves would be no longer uniform throughout its course. Let us consider a mass of hydrogen gas at P, say in the Sun, sending light- waves to an observer Q, both being situated in a region in which there is a field of gravitation of intensity represented by </, directed from Q to P. In terms of the postulate of the relativity of that force this statement would mean that the spacial frame to which the underlying events are referred is rushing as a whole from P toward Q with acceleration g. Let V be the velocity of the frame at the instant when a specified light- wave passes any intermediate point Q' : by the time this wave has reached Q the velocity of the frame as a whole has risen to V -\- g.Q'Qjc approximately, where g is mean intensity along the range from Q' to Q. Thus to the accelerated observers the waves emitted become longer with distance traversed, in the ratio ^ + 9 -Q Q/c^, owing to this velocity of recession from the source : that is, the apparent wave-length undergoes change so that during the progress from Q' to Q it is altered in the ratio 1 — SF/c^, where 87 is the rise of potential (or fall of gravitational potential energy) along that path. The period of the light will thus appear to be increased to different observers on the line PQ, all of them travelHng along with the same acceleration g, in different degrees according to their positions. This is what will happen if the observers and their space and optical instruments form a world of their own rushing past, or through, an underlying actual world, with this acceleration g, instead of the actual world rushing past them with the opposite acceleration produced by a force of gravitation. For these alter- natives are not now the same: the finite velocity of propagation c is constant with respect to the actual underlying world, not the observers' moving space. If the radiating hydrogen belongs to the actual underlying world, and the spectroscopes of the observers belong to their own spacial scheme that is imposed on that world, this description is complete: the period of each wave as apparent to observers along its path will increase as the wave travels away to places of lower gravitational potential. The spectral lines of solar hydrogen as observed on the earth ought to be displaced towards the red, by the amount corresponding to the total fall of potential between Sun and Earth. But the postulate of two worlds seems to be here necessarily involved. Which of them would a mass of radiating hydrogen situated half-way to the Sun belong to?* The larger Doppler-Fizeau effect due to the motion of the source itself relative to the observers' frame has not here been * All the bodies in the space, being subject to the same gravitation, would move along with it: the waves of light alone would seem to be regarded as inde- pendent: yet they have energy and so inertia. 328 Sir Joseph Larmor mentioned: that is included satisfactorily in tlie earlier uniform relativity formulation. This relation of light to gravitation is thus one of the questions raised by the postulate of the relativity of that universal force. Einstein answered in 1911* in one way, that the spectrum of solar hydrogen, when compared with terrestrial hvdrogen which is con- nected with the observer, should be displaced slightly towards the red: but it is a question whether the consistent development of that train of ideas would not rather require that it be not displaced at all. In connexion with his later formal theory of gravitation the same effect is described as due to varying local scales of time, which seem to be carried without change, by the pulsations of the rays, from the place of their origin to all the other parts of the universe: whereas in the above the apparent period f changes as the ray advances. The observers along the ray are supposed to be in communication with one another. In so far as their space moves forward as a whole it is not stretched or shrunk: in that case it can be only their scales of apparent duration of time that are lengthened localh'^ by a factor, the inverse of 1 — V jc^. This involves that the scale of apparent velocity in the unchanged space will be altered in the direct ratio: and rays of light in a field of varying potential, if they were paths of stationary time, might be thought to be deflected. But fundamentally the path of the ray is determined by the number of wave-lengths in its course being made stationary, as compared with neighbouring courses: and this is, in the present case, not the same as minimum time of transit, for apparent time has lost its uniform scale while space has not. Thus the path of a ray would be determined by the condition that SSs/A summed along it shall be stationary: but if there is correspondence between the two systems of reference which changes all lengths around each point in the same ratio then hsjX will be everywhere the same in both systems. The circumstances of the path would thus not be altered by this change of view regarding gravitation, and there ought to be no special deviation of the rays involved in it. But if g is not uniform along the path r of the ray, is a shrinkage of the accelerated apparent space involved? The answer * His exposition which has here been paraphrased is in Ann. der Physik, 35, 1911, §3, p. 904. The argument of this and the next two paragraphs is based on the implication that in a theory of transmission by contact, radiation like other things, the so- called clocks included, must conform to local measure: the alternative, described at the end of the paper,- that racUation is extraneous in so far as it imposes an absolute scale of space-time of its own on the whole cosmos, was here taken to be excluded in advance from this type of theory. •j- Measured on a fundamental scale. Gravitation and Light 329 is given that, passing to the general problem, the demands of the universal gravitational correspondence (to be evolved immediately, infra) require that the apparent space of the observers must be constructed so that S/^ — c"^ht^ where c is a function of r shall be invariant. This requires slight warping of the fourfold space, so that the section in the plane r, t is curved away from its tangent plane. But is the warped element of extension ^r' .c'ht thereby altered only to the second order from its corresponding previous normal value Sr.cS^? If that be so, the scale of t must be altered in the inverse ratio to the scale of velocity c' or (what is the same in another aspect) of time t : and in fact it is partly this secondary change of scale of r that modifies the astronomical gravitation, as will presently appear. The answer to this question might at first be imagined to be as follows : any change in the element of surface may be made in two stages, a stretching on the original plane and a displacement along the direction normal to that tangent plane: it is only the former that can produce a first-order effect: but this is only an apparent change, a mere alteration of coordinates, because in it the curvature of the plane is conserved, so it cannot affect the concatenation of relations or events which alone counts : the latter does affect them, e.g. disturb the law of gravitation, but only to the second order. But as will appear presently this relation of conservation of extent is between coordinate systems that most closely correspond, so is a real imposed condition which cannot be adjusted by change to another set in the fiat. It is the expression of, or at any rate is involved in, a restriction that in the containing fivefold the distance between corresponding points on the two systems is everywhere small, so that approximate methods can apply con- sistently throughout, of which otherv/ise, in making continuations in an uncharted extension, there would be no guarantee. 3. Now let us survey this problem of transcending gravitation from the other side, on which it originated. With Minkowski the very incomplete relativity of electrodynamics, referring only to uniform translatory convection, crystallised into the complete pro- position that events occur in a uniform fourfold of mixed space and time, determined by the consstitutive spacial equation Here c has nothing to do with the velocity of radiation : it is simply the dimensional factor, prescribing a scale of measurement, that is needed to make time homogeneous with length and may be taken as unity. Gravitation remains outside this electrodynamic scheme, being formulated in the different Newtonian reckonings of space and time. Can it be forced in, either exactly or approxi- mately? dt = 330 Sir Joseph Larmor The complete circumstances of the orbits in a field of force of potential energy —V per unit mass (in a gravitational field V is TiSm/r) are condensed into the single variational Least Action equation of Lagrange-Hamilton, with integration between limits of time fixed and unvaried. This suggests comparison with the equation for the shortest or most direct path in a modified fourfold involving Euclidean space com- bined with a measure of time varying from place to place: for that equation is Sjda=0 where Sa^ ^ Sx'- + Sy^ + Sz^ - c'^Bt^ in which c' is a function of x, y, z. Let us write C'2 = C2 (1 -f K), where K _is very small on account of the greatness of c. The equation is now or approximately up to the fourth order ■&--i---im-m-m. dt^ 0. The time-limits being unvaried the first term — c^ can be omitted : thus this variational equation of most direct path coincides with the previous orbital equation if - |Zc2 = F. Thus the forces are absorbed into a varying scale of time; and the motion being now free under no force, the orbit is, as was antici- pated, a geodesic or straightest path. The orbits have become however straightest paths, not in their original Newtonian separ- ated space and time, but in the uniform space-time fourfold of relativity as slightly deranged by the not quite constant scale of time. Thus the orbits in any field of attraction have actually been fitted into the mixed space- time frame of electrodynamic relativity, at the expense of doing slight violence to that frame, by making the measure of time vary from place to place while the positional specification remains uniform. But this transformation does more than is needed. It ought somehow to be restricted to the one universal force of nature, that of gravitation with its inverse-square law. It is here that the special feature of the Einstein theory seems to come in. For Gravitation and Light 331 velocities beyond actual astronomical experience, not small com- pared with that of light, mass comes to depend on speed; thus it is not any longer available as a definite dynamical constant. On the earlier uniform relativity it emerged however definitely in another way as a feature of every permanent collocation of energy and proportional to its amount E, equal in fact to Ejc^. This follows immediately if Least Action is fundamental. Thus it is grouped energy that possesses located momentum: and it is this energy that has to gravitate, mass confined to matter alone having proved inadequate to a Least Action formulation in the mixed space-time of universal limited relativity. Dynamical principles had therefore to take the form of a theory of conservation of energy and of abstract momentum as they travel through a medium, at the same time receiving additions by the operation of an internal stress to which the medium is to be subject. In other words, general dynamics cannot be more detailed than a mere description of the migration of energy and of momentum in a medium under the influence of some internal system of stress adjusted to fit the equations as simply as possible. This stress is what has to stand for or represent the agencies of nature. The theory is borrowed and generalised from the Maxwellian theory of stress in the aether, which was an isolated, apparently rather accidental, feature that did not fit well into the substance of Maxwell's scheme, because in fact it could not be connected with a strain expressive of its origin. Now however, inertia of bodies having failed as the standard measure of force, energy and momentum, and a postulated ad- justing stress entirely at our choice, are promoted to occupy the vacant place. Only it is not called a stress: the idea of a physical medium is avoided, so it is named an algebraic tensor. There is no law of elasticity involved, or relation of stress to strain, such as makes elastic problems determinate. Thus the scheme may have accidental features, is perhaps far from being unique. Another parallel to it is Maxwell's theory of stresses in a gas due to varying temperature: but that continuous theory could never have been constructed in definite form without the foundation of the be- haviour of the individual molecules. When however the fourfold frame is very nearly flat, the rela- tions of energy-momentum-stress appear to fall in with the law of gravitation, with energy as the source of its potential instead of matter. When the deranged spacial frame nowhere differs much from the flat, it may be expected that the extent of its fourfold element will be altered from the value for coordinates of the corresponding type on the flat only to the second order, for the same kind of reason as applies in comparing a slightly deranged plane sheet with the original plane. In fact, if the displacement is everywhere small, 332 Sir Joseph Larmor this extent taken over a small region would have a stationary value for the flat, changing in the same direction on both sides of it. Cf. supra, p. 329. Thus for a spherically symmetrical field the constitution of the fourfold must be determined in polar coordinates by the equation Sct2 - {c/c'f 8/2 + {rSdf + (r sin O^f - c'^Si, showing that the positional part of the extension is very slightly non-uniform and so not quite Euclidean. It appears to be this secondary feature, not the energy-momentum-stress tensor con- ditions, that modifies gravitation from the Newtonian law. The expositions of relativity do not mention an extended fourfold, which would be foreign to the cardinal idea that space is constructed from physical origins, only in so far as it is needed — even though it has to be implied that it is reproduced unerringly each time. But the instrument of such construction or continua- tion of a metric space is an infinitesimal linear measuring rod supposed to have complete free mobility without change of in- trinsic length : and it would seem to be a tenable view that such a mobile apparatus must determine an underlying flat space of higher dimensions* in which the physical system may be supposed imbedded. It is to be noted here that a surface defined intrinsically in the Gaussian manner by the distance relation on it Ss^ =fSp^ + 2gSpSq + hSq^, remains the same surface when the coordinate quantities p, q are changed to others p , q' which are any assigned functions of them both, so that Ss^=.f'8p'-^ + 2g'8p'8q' + h'8q'^, provided 8s is measured by the same infinitesimal unchanging measuring rod extraneous to the surface in both cases. These two equations represent the same surface, only the generalised co- ordinates of the same point on it are changed from {p, q) to {p, q'). The intrinsic curvatures are the same from whichever form they be calculated: if one form represents a flat, so does the other. On this definition by an intrinsic differential relation surfaces are indistinguishable, if one can be bent to fit the other without stretching. So in the Riemann theory of spaces of more than two dimensions it is the functional forms of the coefficients in the quadratic function of differentials and the mobile absolute mea- suring rod that determine the nature of the space; any transforma- tion of coordinates changes the coefficients (or potentials in the gravitational formulation) but so that the space remains un- * For a radial field it need be of onlj' one more dimension. Gravitation and Light 333 changed, being only referred as regards the same points to the other generalised coordinates. But the apparent extent ^pdq does alter when the coordinates are changed, and it would be a limita- tion to keep it constant. See Appendix infra. The feature that remains unfathomed as yet is the fact that the velocity of transfer of energy of radiation in undisturbed regions of space is equal to the merely dimensional constant that renders time comparable with space on the fourfold frame of reference : it at any rate suggests a dynamical origin for that mixture of the effective relations of time with those of space *. The locus in the fourfold in which a never changes and so ha vanishes has some claim to be called the 'absolute,' in a sense parallel to the ' absolute ' of Cayleyan geometry which for Euclidean space is represented by the equation x^ + y^ + z^ = 0. Everywhere on this locus S.s = c'ht ; thus velocity of displacement is everywhere c', and the rays in it are the paths of shortest time with this velocity. It separates the disparate regions in which ha measures real distance when time is unvaried and in which iSct measures real time when position is unvaried. 4. It would appear (as infra, p. 335) that if we are prepared to replace a field of potential energy of gravitation or any other type of universal force by a field of varying time-scale without change of the uniform scale of space, on the lines sketched above, this formal change ought not sensibly to affect radiation either as regards its path or its period. To each element of extent there would be a cor- responding element, and all events and measures in one pass over to the other according to rule. But we now pass from kinematic discussion of frames of refer- ence to physical considerations. If we are to assert, in agreement with the doctrine of relativity plus Least Action, that inertia is a property of organised energy and proportional to it, therefore not solely of matter, and if we are to admit with Einstein, in the same and other connexions, that light is made up of small discrete bundles or quanta of energy, it would appear to follow that each bundle is subject to gravitation. Therefore if a bundle comes on from infinite distance with velocity c, when it has reached a place of potential V near the Sun its velocity c must be given by ic'2 - F - ic2, in other words, is increased in the ratio 1 + V jc^. It will swing round the Sun in a concave hyperbolic orbit, and as the result, the direction of its motion will suffer deflection away from the Sun by half the amount that has been astronomically observed. This reasoning would not be estopped by the principle that c is the upper limit of possible material velocities: for that is because * See final paragraphs. 334 Sir Joseph Larmor a moving body acquires energy and therefore inertia without hmit as its speed approaches c, whereas the energy of a Hght quantum is not supposed so to increase. This is all on the older notions: the velocity c is far too great for the new approximate gravitation analysis to be applicable. But the idea of wavefronts and phases must also be introduced somehow. If we imagine a row of these corpuscles of energy coming on abreast, the more distant ones would fall behind in swinging round the Sun and their common front would become oblique to their direction of motion, the exactly transverse directions being now the loci of equal Action not of equal time. If we superposed the Huygenian principle of propagation normal to the front, the orbital deflection would thereby be just cancelled by the swinging back of the front which would retain its direction : and there would be no deflection of direction of propagation. But such ideas are plainly incoherent. The earlier development of Einstein sketched above* was driven on other grounds to conclude that light must gain energy in a field of gravitation, but the gain was named potential energy. In the finally developed theory there seems to be no longer energy of motion or other types : energy becomes a single analytic scalar in what is left of the field of interplay of momentum, energy and stress. These earlier considerations have doubtless crystallized into the formal theory of which also the result has been illustrated above, in a way which transforms the variational equation of free orbits in ordinary space and time into the variational equation of straightest lines in a non-uniform space-time fourfold given differen- tially. The coordinates are carried over unchanged in values, into this fourfold, but their differentials no longer express in it direct measurements of length and time; these are now imported in the Riemann manner as regards any element of arc or interval of time by the value of the absolute element 8a. As compared with the underlying absolute time determined by Scr, the element of apparent time St of a gravitational world, which is taken over into its expression is variable, proportional to c'"^, with locality. The quantities x, y, z, t which are the measures of space and time as apparent in the world of gravitation are now mere co- ordinate quantities in the new differentially given world in which there are elements of absolute length and time both measured by Sct. The final expression for Scr^ with radial symmetry ,f Sr2+ ... + ... - (^y^^S^^ shows that the element of apparent time in the gravitational world * Ann. der Physik, 35, 1911, §2, p. 902. Gravitation and Light 335 is the unchanging element of absolute time divided by c' Ic, or that the scale of apparent time is variable with locality in the ratio cjc' : also that the scale of apparent radial length is variable in the ratio c jc: and therefore the scale of radial velocity is variable as their quotient c^jc"^. How then with respect to the velocity of rays of light whose absolute value is the same as the dimensional constant c? Referred to these variable scales its apparent value along any element of arc ought to be changed at the same rate as any other velocity along that element of arc would be changed, if rays are not to remain outside the correspondence between hx, 8y, Sz, St representing time-space in the apparent gravitational world and the same quantities, now elements of mere coordinates in a difEerentially given world in a curved space-time which has absorbed gravitation. This maintenance of correspondence is secured if we determine the ray- velocity along any element of arc by making Scr = : and the modified theory of radiation for the apparent space of gravitation must be such as can accept this value of the velocity of propagation *. The correspondence takes over the same values of the coordinate differential elements. In the apparent gravitational world they represent its space and time, in the new world differentially specified, they belong to mere coordinates: absolute elements of space and of time are there ex- pressed by 8o-, but a relation of scales can be established from the formula which expresses Sct^. The transformation which changes orbits into geodesies in the difEerentially given space-time does not turn rays into rays: their velocity is too great and moreover their minimum property is relative to their locus 8a = 0. But if the ray is supposed to have a constant underlying absolute period of pulsation and a constant absolute wave-length (and therefore to be a straight line in an auxiliary uniform fivefold) its apparent period in the gra^dtational world must vary with locality as (c'/c)"^, also its apparent element of length inversely as the scale of length pertaining to its direction on that locality, and its apparent velocity as before specified. Its apparent path in the gravitational world will correspond to the true absolute path Sfda/X^ == 0, therefore will be given by Sjds/X = 0, complications being avoided as fortunately t is not involved ex- plicitly in these equations. But at the same place the scales of apparent 8s and apparent A would alter on the same ratio owing to the presence of gravitation : therefore its influence is eliminated in the quotient, and the path is not affected by the gravitation, is the same whatever be its intensity. A ray passing near the Sun ought not to be deflected on this view: an observed deflection, * On this and the following paragraphs, cf. however the end of the paper. VOL. XIX. PART VI. 23 336 Sir Joseph Lannor whicli a priori was well worth looking for, would seem to await explanation on other lines. Again would there be an observable change of periods of spectral lines according as the vibrating source was at the Sun or at the Earth? The underlying absolute periods of radiating hydrogen molecules would be always and everywhere the same: thus the apparent period in the gravitational world would vary inversely as the local scale of time, and be longer at the Sun. But this is a local apparent period. The waves sent out from the solar molecule are observed at the earth: we have seen that their length changes as they progress, being inversely as the local scale of length, and their speed changes also, so that their period changes inversely as the local scale of time. Thus when they have reached the Earth their period conforms to the local scale and would agree with that of the radiation of a similar terrestrial molecule. In fact if complete correspondence is established*, element for element, as above, all periods or intervals of time measured at any element are changed in the same ratio depending on the locality alone. Any other conclusion would make the pulsating rays into signals establishing absolute time throughout the apparent universe, which could hardly be a result of a theory of relativity. The condition 8ct = prescribes a definite ray- velocity for each element of arc, the same forwards as backwards, only when Scr^ involves St'^ but no products of St with other differentials: in other cases it gives two velocities, not equal and opposite, and this spacial scheme of rays seems to fail. If rays are to be pro- perties of the space a very severe restriction is thus imposed on the form of 8cr^, but one which seems to be satisfied for the slight modifications that would be involved in the actual gravitation of experience. In the modifications of the expression for 8cr^ which absorb gravitation the coefficients do not involve the time explicitly: therefore the ray-paths are fixed in the space, and it almost looks as if they were guides imposed by the nature of the space alone, as thus modified, for the alternating energies of radiation to run along. Any inference that because a ray is fixed in space, as many waves must run in at one end as run out at another, would be at variance with the very notion of relativity, by providing a scale of absolute time throughout the universe. Such an argument seems to amount in more general form essentially to this: when the expression for 8a^ does not contain t explicitly it will make no * As has been estabKshed for the more general case in a beautiful analysis by Prof. Th. de Bonder, of Brussels, Comjptes Rendus, July 6, 1914, Archives du Musee Teyler, Haarlem, vol. iii, 1917, pp. 80-180. [It is merely continuity with non-gravitational fields, and not correspondence, that is established.] Gravitation and Light 337 difference to the cosmos if t is everywhere increased by the same constant: therefore the scale of time must be everywhere the same — which excludes any possibility of local scales of time, A change of origin of measurement for time is not the same as progress of events in time, unless the scale of time is everywhere the same. The matter may be put from a different angle as follows. To obtain the time of transit of a ray from P to Q it is not possible to add elements of heterogeneous local times such as 8^*. What can be done is to find the true underlying time of transit. If this homogeneous true time is delayed at the start, at one end of the path at P, it is delayed by an equal amount at arrival at the other end, as the equations of transit do not involve this time explicitly: hence apparent times at the two ends are delayed not by equal amounts, but by amounts inversely as their local scales, so that a ray cannot (as has been impKed) transmit apparent time along its path. The alternative development is, as above, that 8ct^ being the underlying unchanging standard there are local scales of time, and local scales of length which may involve direction, and therefore also of velocity (including that of the rays) which is their quotient. The path of a ray from point to point is determined by making the number of wave-lengths from the one to the other minimum, that is by Sjds/X = : but Ss and A are both altered to the same scale; thus there is no alteration due to gravitation in the varia- tional equation determining the ray-path, so that it would suffer no deflection. The essential feature in the argument is that, whether rays may be regarded as the limiting case of free orbits or not, their specification has been postulated so that the ray- velocities correspond in the same way as all other velocities in the two frames. Appendix. — On Space and Time. Let us try for a closer realization of these abstract positions. The Gauss-E,iemann theory for an ordinary curved surface will be wide enough to serve as an illustration. The theory involves coordinates p, q: they must represent something. The very least we can do for them is to regard the surface as twofold extension dotted over with points, so that the coordinates express their order of arrangement according to some plan of counting them with respect to this extension in which they lie. There is no metric idea at all in this numeration, and nothing to distinguish one surface from another. Now bring in an infinitesimal unchanging * Yet it is just such elements of quasi-time d.v^ that are added together, ■infra p. 343. It is the so-called shifting clock-time and absolute time running parallel that are the source of all this confusion. 23—2 338 Sir Joseph Larmor measuring rod, which can make play in each element of extension represented by SpSq and also be transferred from place to place: and we can thereby impart or rather superpose metric quality on the twofold which hitherto was purely positional or rather tactical. The simplest plan is to follow Euclid, on the basis of the Pytha- gorean theorem, and expressing absolute length according to measuring rod by a symbol Ss, to impose a scale-relation of form 8s^ = Sp^ + Sq^. But this metric cannot be applied consistently over a curved surface, unless it is of the very special type that can be rolled out flat: for other surfaces it is necessary to have the more general type of relation S52 =fSp^ + ^g^pBq + hSq^ in which/, g, h are functions of the coordinates p, q. This specification of an imported metric thus determines the surface: starting from a given small region of it, the form of the surface in an outer threefold space can be gradually evolved by prolongation so as to fit in with consistent application of this metric. It is this idea of prolongation of a non-uniform manifold, equivalent to its geometrical continuation within a flat one of higher dimensions, that was Riemann's contribution to the ideas of geometry. But the manifold itself is supposed to be given only tactically or descriptively; and it is the metric that is imposed on it that, by its demand for consistency in measurements, deter- mines for it a form, as located in a higher flat manifold. This form is expressed in detail analytically by the ' curvature ' at each place, as specified by a set of functions (one in the case of a surface) of the successive gradients of the set /, g,h, If we keep the system self-contained by avoiding the immersion of it in a uniform auxiliary manifold of higher dimensions, our resource is to deter- mine the curvature as the simplest set of functions that are invariant for local changes of coordinates. But, in order of evolution at any rate, this invariance may be held to be only a derived idea. In any case the nature of the non-uniform manifold, as thus determined by a metric imposed on formless space, has nothing to do essentially with the coordinates p, q, ... to which it may happen to be referred: it is settled by the algebraic form of the functions/, g, h, ... expressed in terms of jj, q, ..., or in geometric terms by the 'curvature' as so expressed. As a consequence, if we transform a surface from internal or intrinsic coordinates p, q, to others p', q', which are assigned functions of the former, so that we obtain §§2 = f'8p'^ + 2g'8p'8q' + h'8q'^ and construct the surface implied in this new equation by the 4 Gravitation and Light 339 process of continuation, it will prove to be just the same surface as before. Whether it is expressed in terms of p' , q' or of jp, q is intrinsically of no consequence : the coordinates are of no account, it is only the functional forms of/, g, h that are essential. This last statement, developed in terms of the criterion of invariance in order to avoid a representation by immersion in a uniform geometrical manifold of dimensions higher than the given four of space and time, appears to cover the general relativity of Einstein. The/, g, h, ... can be named the potentials which deter- mine the space. In the special relativity, before gravitation was absorbed into the metric of extension, all spaces were flat, so /, g, h, ... were constants ; which is all that is left, for that particular case, of these relations of invariance. In this flat fourfold, relativity implied merely that a physical system is determined by its own internal relations, so that the position that may be assigned to it in the fourfold is of no account, any more than is the position of a surface or a system of bodies in space. In the later general relativity the manifold must be supposed given descriptively by coordinates, which represent numerical counts arranged to suit the number of dimensions that are involved : it only gains internal form when a metric is imposed upon it. If the Euchdean metric §s^ = Sp^ + Sq^+ ... is imposed it becomes a Euclidean space everywhere uniform and also flat, in which bodies are mobile without change of form. If a metric varying with position is imposed, the expressions in this manifold of the metric relations of nature will become complicated, and the relations so changed be described as a modified set of laws. The original non-metric continuum might be marked for instance by gradations of colour: the colour-scheme of Newton as developed by Young, Helmholtz, and Maxwell, is the standard example of a non-metric threefold extension. May we not here have refined down to the unresolvable essence of space, as the mere possibihty of descriptive continuity of three- fold type which is an essential feature in our mental world ? Within this a priori datum of threefold uncharted pure continuity we may construct types of charted spaces almost without limit, by imposing metrics of various types. Any particular space is not however determined by the system of coordinates of reference p, q, ... but by the variable coeSicients f, g, h, ... of the imposed metric expressed as functions of them. But yet it is only under special conditions when it is uniform and flat that finite difl'erences of these co- ordinates can be involved, this being part of the expression of the mobility of solid bodies in the space. It is in this narrower sense, that "the system of coordinates is accidental, that relativity has 340 . Sir Joseph Larmor now expelled general metric ideas of position. Would it be entirely wrong to assert that local or sectional relativity has been retained for nature, so far as this order of ideas extends, by transferring the laws of nature into a space-time frame which itself no longer possesses that quality? The distinction has thus been made between an ultimate idea of space as mere threefold continuity, marked but uncharted, and the metric that may be imposed on it by which it becomes a frame fit for the purposes of description of nature. There is only one space: but its practical aspect, whether Euclidean or elliptic or merely heterogeneous, depends on the metric that we choose to assign to it. The metric would thus appear to pertain more closely to the order of nature for which it is to form the most convenient frame for description, than to space itself. For space is primarily bare threefold continuity; though a set of descriptive coordinates jp, q, ... is unavoidable as a foundation of thought, any set is as valid as any other. For ultimately, the count or census of the points or marks that pervade the continuity and render it descriptively given to us, is the same count however it be made. May we say that the insistent, originally uncritical, notion of relativity reduces itself ultimately into this postulate, that as nature is presented to us, it is such that in mental operations we need attend only to one portion of the spacial continuity at a time? This makes the onefold time, or rather mere temporal succession as representable by the 8a of Minkowski, the fundamental feature*, which however diverges spacially into a manifold: according to Hamilton long ago, algebra was the science of pure time. In the above, space is given by a manifold array of points, of which the coordinates p, q, ... express one of the varieties of numerical census. Is then space-time absolute, or is it continually being constructed by physical science as it ranges over the void, for its own purposes, just to the extent that it may be required? May we say that the formless manifold is the fundamental feature, that the array of points and their census do not need to be definite in any respect a priori, and that the metric which is imposed on it and makes it into a definite working type of space is related to the physical world and so is to be regarded as evolved in connexion with our organic description or mapping of nature, and to be just as permanent? What remains of the original notion of relativity after this sifting of ideas would then coincide with the principle of Newton, Faraday and Maxwell, originated by Descartes, that the operations of nature are elaborated in fourfold extension according to a scheme purely differential, that is by transmission from element to element * The spacial sign here attached to 8(r^ is an accident of the order of exposition. J Gravitation and Light 341 of the cosmos, in no case leaping across intermediate elements as action at a distance would imply. The early stage of formulation of the confused notion of relativity is the postulate that position and change of position are purely relative: the final solution is to abolish the idea of immediate ^m/e change of position altogether. But that does not imply that a portion of the cosmos can evolve itself without constant interference from all the rest. To a question as to what is gained by absorbing gravitation in space an answer would be that it need make no difference as regards gravitation ; but if other relations of an assumed space- time fourfold (e.g. stress-tensor theory) have to go in also in a simple way, it may be convenient or even necessary to assist them by choosing a space which requires some alterations of the recognised laws of gravitation and, if these suggested discrepancies are verified, that may presumably have a claim to be the real type of space. The aim is not primarily to reduce gravitation to a quality of space, — perhaps is not even relativity, which has evaporated, — but is to get it out of Newtonian space and time into the mixed space-time fourfold which was strongly suggested by the form of the Max- wellian electrodynamic relations of free space, and would make that scheme valid for great velocities of convection beyond ex- perience, even up to the speed of light. An expansion of the Einstein ideas on general relativity has been worked out by H. Weyl {Ann. der Physik, 59, 1919) in which a further metric scale of vector character appears to be imposed on a non-uniform space-time, which has here been itself ascribed to the imposition of a Gauss- Riemann metric on the formless spacial threefold that is inherent in the mind. There would seem to be no formal obstacles to such piling up of metric upon metric, in an unlimited play of thought. The physical analysis perhaps not very remote to this new elaboration of metric is, as I think Prof. Schouten remarks, a theory of an elastic aether in which at each point p, q, ... a vector displacement ^, 17, ... of the element of the medium is supposed, involving a strain and an elastic stress determined in terms of the strain by assigned laws. Only it is to be remembered that time is now in a fourth dimension, in which the historical world- process is all spread out once for all; so that the feature of elastic wave propagation becomes a static relation. The idea that the single fundamental electric vector is represented by a superposed metric is thus correlative with the usual dynamical hypothesis that electric force is a stress in an aether. It thus affords another illustration of this kind of speculation: the interlacing of space and time for purposes of electrodynamics having upset the his- torical development of dynamical principles on a Newtonian basis of separate space and time, order has to be re-constituted by 342 Sir Joseph Larmor piecing together a cognate analytical scheme on a symmetrical fourfold basis which tries to make no difference between them. It is not improbable that these remarks merely turn over ground that has already been explored by cultivators of hyper- geometry. But it may be claimed that the interest of this range of ideas extends far beyond the analytical technique, and that their naive expression in a form of language outside its conventions may prove to be helpful in other regions of speculation. The argument above has been based on the supposition that the mathematical analysis must establish a complete correspondence, element for element, between the activities in the new space-time and in the Newtonian space and time. That however is not the case. There is a gravitational correspondence into which radiation and its rays do not enter. As regards the latter no conclusions could be drawn at all, except in the special circumstances in which the coordinate X/^ that stands nearest to time * does not enter explicitly into the quadratic expression determining the space. If that is postulated the equations of propagation of radiation have their solutions periodic as regards x^^^ treated as a quasi-tim.Q, therefore every beam of radiation carries with it a scale of X/^^ throughout its course |. Moreover, if the spacial quadratic contained hx^ in a product term, the velocities of the waves of radiation in forward and backward directions would not be the same : their half difference would thus be the local velocity of the frame of reference in that direction. Where hx/^^ does not occur in the first power, the frame of reference is thus fixed locally with respect to the waves of light and their assumed underlying uniform fourfold extension with regard to which they are propagated. Thus, under these postulated circumstances of x^ not occurring explicitly in So-^, the mere fact that isotropic vibratory radiation exists with its absolute velocity c is sufficient, not merely to de- termine absolute measurements both in space and time, at every locality in the extension, but also to determine the rate of motional change of the coordinates as referred to the uniform space-time of the radiation. It is gravitational correspondence, subject to this general control of the whole range of space-time by observations of light, with its isotropic and uniform qualities, that has led to verifiable conclusions. Cf. letter in Nature, Jan. 22, 1920: also Monthly Notices R. Astron. Soc. * That is the one coordinate the square of whose differential is affected in dcr- with a negative sign, which marks it off from the others. t It is the alleged measurement of this abstract coordinate x^ by a travelling clock, which connotes a physical system, that is a main source of confusion. Gravitation and Light 343 We have absorbed gravitation into space and time by distorting the latter from its essential Newtonian uniformity: but there can be no illusion about the matter either way, for the theoretical measuring bar of the differential spacial theory is not our only instrument; in the practical world rays of light provide the essential isotropic measures, and the spectroscope is always available to reveal to us what spacial adjustments have been made, in relation to the underlying frame with regard to which the propagation of light is isotropic and has its standard absolute velocity. Light, instead of conforming to local relativity, imposes its own absolute space-time*. The argument may be directed tow^ards yet another type of conclusion, as follows. When change is made from Newtonian space and pure time to the uniform space-time fourfold, the equation of a straight path is altered from h^ds = to SJ(Zct = 0. The free orbits in any field of force of potential energy function — F can readily be altered so as to preserve continuity with this change, as above, that is, so that where F becomes negligible they tend to straight lines: they are then given by h\{d<j^ + 2Vdt^Y = ^. The interpretation is at hand, to regard them as the analogues of straightest paths in a modified space-time, referred to a set of coordinates represented now by colourless symbols x^, x^, Xz, x^ and given in terms of them by 8a2 = Sa;i2 + Sx^^ + Sx^^ - c^ (1 - 2c-2F) Sx^^ As Sct^ does not here involve x^^ explicitly, the differential equations of propagation of free radiation, as expressed in this space-time in terms of these coordinates, have solutions involving the quasi- time, x^ only in the form e'^^*: therefore the radiation from any source, however far it has travelled, retains the same period in regard to x^ as it had at the start. Around a radiating molecule the extension can be taken as practically uniform: therefore the interval of absolute time is equal to (1 — c'W) hx^. It follows thus from the periodicity as regards x^ that the periodic time of a ray alters as it travels so as to be proportional to 1 — c~^V . If the ray belongs to a definite molecular period at the Sun, it has changed when it reaches the Earth so as to agree no longer with that period as reproduced by a local vibrator. All this is true only to the first order, but it applies to any law of potential, and is irrespective of any special energy-tensor theory. The point to be brought out is that if influence of gravitation on * Prof. Eddington in a recent article, Quarterly Review, Jan. 1920, seems not to disagree with this conclusion: at any rate he contemplates the possibility of an aether. 344 Sir Joseph LanTwr, Gravitation and Light spectral periods were definitely disproved, then it would appear that any hope of bringing orbits into direct relation with the electrodynamic space-time fourfold must be abandoned altogether*, on the threshold. This drastic conclusion is perhaps an argument in fa^^our of the existence of the effect. The other two verifiable effects, the influence on the planetary perihelia and the deviation of light passing near the Sun, arise in part from first order and in part from second order causes. Unlike the previous one, their exact verification is thus a test of the special theory of Einstein, or the equivalent Least Action formulation. Its original recommendation was that it restricts the universal forces of nature to the one type of gravitation: possibly it would be difficult to imagine ways in which there could be room for any different result. * A formulation of the original Nordstiom type, starling from d^Vda = 0, is to some degree an exception. C. T. R. Wilson, On a Micro-voltameter 345 On a Micro-voltameter. By C. T. R. Wilson, M.A., Sidney Sussex College. [Read 19 May 1919.] Experiments were described with a mercury voltameter, in which one elctrode consists of a sphere of mercury deposited on the end of a fine platinum wire and measured by means of a microscope. Quantities of electricity varying from a few hundred electrostatic units to about one coulomb may be measured by it. The almost instantaneous change of size of the drop when a capacity of one tenth of a microfarad, charged to 1 volt, is dis- charged through the instrument is easily observed. A magnet inserted in or removed from a coil connected to the terminals of the voltameter produces an easily measured effect. Experiments were also mentioned which suggest the possibility of its application in measurements of much smaller electrical quantities. 346 R. Whiddington, The self-oscillations of a Thermionic Valve The self -oscillations of a Thermionic Valve. By R. Whid- dington, M.A., St John's College. [Read 19 May 1919.] (Abstract.) It lias been found possible to produce oscillations of almost any frequency from a three electrode vacuum valve, without employing the usual capacity-induction circuits. Thus a. valve with two suitable batteries, one in the anode circuit, another in the grid circuit, will produce quite powerful oscillations, whose frequency will be determined by the value of the grid potential. The phenomenon can be explained by supposing that the oscil- lations are due to surges of mercury ions closing in on the filament from the grid with a frequency given by the approximate formula 2 2^ T/ n = . V md^ ' e • where — is the usual charge to mass ratio, d is the radial distance m o ^ filament to grid and V is the positive grid voltage. Experiments conducted so far indicate that the monatomic Hg ion with one live charge is mainly responsible. PROCEEDINGS AT THE MEETINGS HELD DURING THE SESSION 1918—1919. ANNUAL GENERAL MEETING. October 28, 1918. In the Comparative Anatomy Lectui'e Room. Prof. Marr, President, in the Chair. The following were elected Officers for the ensuing year : President : Mr C. T. R. Wilson. Vice-Presidents: Dr Doncaster. Mr W. H. Mills. Prof. Marr. Treasurer: Prof. Hobson. Secretaries : Mr Alex. Wood. Mr G. H. Hardy. Mr H. H. Brindley. Other Members of Council: Dr Shipley. Prof. Biffen. Mr L. A. Borradaile. Mr F. F. Blackman. Prof. Sir J. Larmor. Prof. Eddington. Dr Marshall. Prof. Baker. Prof. Newall. Dr Fenton. The following was elected an Associate of the Society : G. A. Newgass, Trinity College. The following Communications were made to the Society: 1. Proof of certain identities in combinatory analysis. By Prof. L. J. Rogers and S. Ramanujan, B.A., Trinity College. 2. Some properties of p (n), the number of partitions of n. By S. Ramanujan, B.A., Trinity College. 348 Proceedings at the Meetings 3. On the exponentiation of well-ordered series. By Miss D. Wrinch. (Communicated by Mr Gr. H. Hardy.) 4. On certain trigonometrical series which have a necessary and sufficient condition for uniform convergence. By A. E. Jolliffe. (Communicated by Mr Gr. H. Hardy.) 5. Some geometrical interpretations of the concomitants of two quadrics. By H. W. Turnbull, M.A. (Communicated by Mr G. H. Hardy.) 6. On Mr Ramanujan's congruence properties of p (n). By H. B. C. Darling, B.A. (Communicated by Mr G. H. Hardy.) 7. On the correct Generic Position of Dacrydium Bidwillii Hook f. By B. Sahni, M.A., Emmanuel College. (Communicated by Professor Seward.) February 3, 1919. In the Balfour Library. Mr C. T. R. Wilson, President, in the Chair. The following were elected Fellows of the Society : S. R. U. Savoor, B.A., Trinity College. S. C. Tripathi, B.A., Emmanuel College. The following was elected an Associate : P. W. Burbidge, Trinity College. The following Communications were made to the Society : 1. The Gauss-Bonnet Theorem for multiply-connected regions of a surface. By E. H. Neville, M.A., Trinity College. 2. On the representations of a number as a sum of an odd number of squares. By L. J. Mordell. (Communicated by Mr G. H. Hardy.) 3. On certain empirical formulae connected with Goldbach's Theorem. By N. M. Shah and B. M. Wilson. (Communicated by Mr G. H. Hardy.) 4. Note on Messrs Shah and Wilson's paper entitled : On certain empirical formulae connected with Goldbach's Theorem. By G. H. Hardy, M.A., Trinity College and J. E. Littlewood, M.A., Trinity College. Proceedings at the Meetings 349 February 17, 1919. In the Comparative Anatomy Lecture Room. Mr C. T. E. Wilson, President, in the Chair. The following Communications were made to the Society : 1. Note on an experiment dealing with mutation in bacteria. By Dr DONCASTER. 2. Electrical conductivity of bacterial emulsions. By Dr Shearer. 3. The bionomics of Aphis grossulariae, Kalt., and Aphis viburni. Shrank. By Miss M. D. Haviland. (Communicated by Mr H. H. Brindley.) 4. (1) The conversion of saw-dust into sugar. (2) Bracken as a source of potash. By J. E. Purvis, M.A., Corpus Christi College. 5. Terrestrial magnetic variations and their connection with solar emissions which are absorbed in the earth's outer atmosphere. By S. Chapman, M.A., Trinity College. 6. The distribution of Electric Force between two electrodes, one of which is covered with radioactive matter. By W. J. Harrison, M.A., Clare College. May 19, 1919. In the Cavendish Laboratory. Mr C. T. R. Wilson, President, in the Chatr. The following were elected Fellows of the Society : E. V. Appleton, M.A., St John's College. W. G. Palmer, M.A., St John's College. S. P. Prasad, B.A., Trinity College. The following was elected an Associate : Mrs Agnes Arber. The following Communications were made to the Society: 1. (1) Use of Neon Lamps in Technical stroboscopic work. (2) The distribution of intensity along the positive ray parabolas of atoms and molecules of Hydrogen and its possible explanation. By F. W. Aston, M.A., Trinity College. 2. On a Micro-voltameter. By C. T. R. Wilson, M.A., Sidney Sussex College. 3. The self-oscillations of a Thermionic Valve. By R. Whiddington, M.A., St John's College. INDEX TO THE PROCEEDINGS with references to the Transactions. M Abel's Theorem and its converses (Kienast), 129. Amos, A., Experimental work on clover sickness, 127. Aphidius, Life History of Lygocerus (Proctotrypidae), hyperparasite of (Haviland), 293. Aphis grossulariae Kalt., Bionomics of (Haviland), 266. Aphis viburni Schr., Bionomics of (Havtland), 266. Appleton, E. v.. Elected Fellow 1919, May 19, 349. Arbek, a., Elected Associate 1919, May 19, 349. Aston, F. W., Neon Lamps for Stroboscopic Work, 300. The distribution of intensity along the positive ray parabolas of atoms and molecules of hydrogen and its possible explanation, 317. Axiom in Symbohc Logic (Van Horn), 22. Bacteria, Mutation in (Doncaster), 269. Bailey, P. G., see Punnett, R. C. Bessel functidns of equal order and argument (Watson), 42. Bessel functions of large order (Watson), 96. Bionomics of Aphis grossulariae Kalt., and Aphis viburni Schr. (Havtland), 266. BORRADAILE, L. A., On the Functions of the Mouth-Parts of the Common Prawn, 56. Bracken as a source of potash (Purvis), 261. Brindley, H. H., Notes on certain parasites, food, and capture by birds of the Common Earwig (Forficula auricularia), 167. Buchanan, D., Asymptotic Satellites in the problem of three bodies. See Transactions, xxii. Burbidge, p. W., Elected Associate 1919, February 3, 348. Cambridgeshire Pleistocene Deposits (Marr), 64. Caporn, a. St Clair, The Inheritance of Tight and Loose Paleae in Avena niida crosses, 188. Cells, Action of electrolytes on the electrical conductivity of (Shearer), 263. Chapman, S., Terrestrial magnetic variations and their cormection with solar emissions which are absorbed in the earth's outer atmosphere. See Transactions, xxii. Colourimeter Design (Hartridge), 271. Convergence, Uniform (Jolliffe), 191. Convergence, Uniform, concept of (Hardy), 148. Index 351 Convergence of certain multiple series (Hardy), 86. Corals, Reactions to Stimuli in (Matthai), 164. Crabro cephalotes. Solitary wasp (Warburton), 296. Cubic Binomial Congruences with Prime Moduli (Pocklington), 57. Darling, H. B. C, On Mr Ramanujan's congruence properties of ^ {n), 217. See MacMahon, P. A. Dirichlet, Theorem of (Todd and Norton), 111. DoNCASTER, L., Note on an experiment dealing with mutation in bacteria, 269. DuTT, C. P., On some anatomical characters of coniferous wood and their value in classification, 128. Electric Force between two Electrodes (Harrison), 255. Electrometer, A self-recording, for Atmospheric Electricity (Rudge), 1. Empii'ical formula comiected with Goldbach's Theorem (Shah and Wilson), 238. Exponentiation of well-ordered series (Wkinch), 219. Eish-freezrng (Gardiner and Nuttall), 185. Forficula auriciiluria, Common Earwig, parasites, food, and captm-e by birds of the (Brindley), 167. Gardiner, J. Stanley, and Nuttall, G. H. F., Fish-freezing, 185. Gauss-Bonnet Theorem (Neville), 234. Gibson, C. Stanley, Elected FeUow 1918, May 20, 189. Goldbach's Theorem (Shah and Wilson), 238. Gravitation and Light (Larmor), 324. Gray, J., The Effect of Ions on Cihary Motion, 313. Green, F. W., Elected Fellow 1916, November 13, 126. Hardy, G. H., On a theorem of Mr G. Polya, 60. On the convergence of certain multiple series, 86. Sir George Stokes and the concept of uniform convergence, 148. See Rogers, L. J., and Ramanujan, S. Hardy, G. H., and Littlewood, J. E., Note on Messrs Shah and Wilson's paper entitled: On an empirical formula connected with Goldbach's Theorem, 245. Hargreaves, R., The Character of the Kinetic Potential in Electromagnetics. See Transactio7is, xxii. Harrison, W. J., The distribution of Electric Force between two Electrodes, one of which is covered with Radioactive Matter, 255. The pressure in a viscous liquid moving through a channel with diverging boundaries, 307. . Hartridge, H., Colom-imeter Design, 271. Haviland, M. D., The bionomics of Aphis grossulariae Kalt., and Aphis viburni Schr., 266. vol. XIX. part VI. 24 352 Index Haviland, M. D., Preliminary Note on the Life History of Lygocerus (Procto- trypidae), hyperparasite of Aphidius, 293, Hill, M. J. M., On the Fifth Book of Euclid's Elements (Foui'th Paper). See Transactions, xxn. Horn, see Van Horn. Hydrodynamics of Relativity (Weatherburn), 72. Hydrogen Ion Concentration (Saunders), 315. Identities in combinatory analysis (Rogers and Ramanitjan), 211. Ince, E. Lindsay, Elected Fellow 1918, February 18, 189. Intensity along the positive ray parabolas of atoms and molecules of hydi'ogen (Aston), 317. Ion, Hydrogen, Concentration (Saunders), 315. Ions, Effect of, on Ciliary Motion (Gray), 313. JoLLiFFE, A. E., On certain Trigonometrical Series which have a Necessary and Sufficient Condition for Uniform Convergence, 191. Jones, W. Morris, Elected Associate 1916, October 30, 126. 1 KiENAST, A., Extensions of Abel's Theorem and its converses, 129. Lake, P., Glacial Phenomena near Bangor, North Wales, 127. Shell-deposits formed by the flood of January, 1918, 157. Larmor, J., Gravitation and Light, 324. Liquid, Viscous, pressiire in a (Harrison), 307. LiTTLEWOOD, J. E., see Hardy, G. H. Logic, Primitive Propositions of (Nicod), 32. Logic, Symbolic, an axiom in (Van Horn), 22. Lygocerus (Proctotrypidae), hyperparasite of Aphidius, Life History of (Haviland), 293. Lynch, R. I., Elected Fellow 1916, November 13, 126. Exhibition of the Fruit of Chocho Sechium edule, 127. Mackenzie, K. J. J., see Marshall, F. H. A. MacMahon, p. a., On certain integral equations, 188. MacMahon, p. a., and Darling, H. B. C, Reciprocal Relations in the Theory of Integral Equations, 178. Madreporarian Skeleton (Matthai), 160. Marr, J. E., Submergence and glacial cUmates during the accumulation of the Cambridgeshire Pleistocene Deposits, 64. Marshall, F. H. A., and Mackenzie, K. J. J., On extra mammary glands and the reabsorption of milk sugar, 127. Matthai, G., Is the Madreporarian Skeleton an Extraprotoplasmic Secretion of the Polyps?, 160. On Reactions to Stimuli in Corals, 164. Micro-voltameter (Wilson), 345. Index 353 Modular Functions (Mordell), 117. Moduli, Prime, Quadratic and Cubic Binomial Congruences with (Pockling- ton), 57. Mordell, L. J., On Mr Ramanujan's Empirical Expansions of Modular Functions, 117. Multiple series. Convergence of certain (Hardy), 86. Mutation in bacteria (Doncaster), 269. Neon Lamps for Stroboscopic Work (Aston), 300. Neville, Eric H., The Gauss-Bonnet Theorem for Multiply-connected Regions of a Surface, 234. Newgass, G. a.. Elected Associate 1918, October 28, 347. NicoD, J. G. P., A Reduction in the number of the Primitive Propositions of Logic, 32. Norton, H, T. J., see Todd, H. NuTTALL, G. H. F., see Gardiner, J. Stanley. Oldham, F. W. H., Elected Fellow 1917, February 5, 127. Palmer, W. G., Elected Fellow 1919, May 19, 349. Partitions of n, Some properties of p {n), the number of (Ramanujan), 207, Phase, Limits of applicability of the Principle of Stationary (Watson), 49. PoCKLiNGTON, H. C, The Du-ect Solution of the Quadratic and Cubic Binomial Congruences with Prime ModuU, 57. P6lya, G., see Hardy, G. H. Prasad, S. P., Elected Fellow 1919, May 19, 349. Prawn, Common, Functions of the Mouth-Parts (Borradaile), 56. Primitive Propositions of Logic (Nicod), 32. Proceedings at the Meetings held during the Session 1916-1917, 125. 1917-1918, 187. 1918-1919, 347. PuNNETT, R. C, and Bailey, P. G., Inheritance of henny plumage in cocks, 126. Some experiments on the Inheritance of weight in Rabbits, 188. Purvis, J. E., The conversion of saw-dust into sugar, 259. Bracken as a source of potash, 261. Quadratic and Cubic Binomial Congruences with Prime Moduli (Pockling- TON), 57. Quadncs, Geometrical Interpretations of the Concomitants of Two (Turn- bull), 196. Ramanujan, S., Elected Fellow 1918, February 18, 189. On the expression of a number in the form ax'^ + hy'^ + cz^ + du^, 11. Empu'ical Expansions of Modular Functions (Mordell), 117. On certain Trigonometrical sums and their applications in the theory of numbers. See Transactions, xxii. 354 Index Ramanujan, S., On some definite integrals, 188. Some properties of ^ (n), the number of partitions of n, 207. 8ee Rogers, L. J. Reciprocal Relations in the Theory of Integral Equations (MacMahon and Daeling), 178. Relativity, Hydrodynamics of (Weatherburn), 72. Rodrigues, Natural History of (Snell and Tams), 283. Rogers, L. J., and Ramanujan, S., Proof of certain identities in combinatory analysis, 211. RuDGE, W. A. D., A seK-recording electrometer for Atmospheric Electricity, 1. the I Sahni, B., On an Australian specimen of Clepsy drop sis, 128. Observations on the Evolution of Branching in the Ferns, 128. ■ ■ On the branching of the Zygopteridean Leaf, and its relation to probable Pinna-nature of Oyropteris sinuosa, Goeppert, 186. • The Structure of Tmesipteris Vieillardi Dang, 186. — — • On Acmopyle, a Monotypic New Caledonian Podocarp, 186. Saunders, J. T., On the growth of Daphne, 126. A Note on Photosynthesis and Hydrogen Ion Concentration, 315. Savoor, S. R. U., Elected Fellow 1919, February 3, 348. Saw-dust, Conversion of, into sugar (Purvis), 259. Shah, N. M., and Wilson, B. M., On an empirical formula connected with Goldbach's Theorem, 238. Shearer, C, The action of electrolytes on the electrical conductivity of the bacterial cell and their effect on the rate of migration of these cells in an electric field, 263. Shell-deposits formed by the flood of January, 1918 (Lake), 157. Snell, H. J., and Tams, W. H. T., The Natiu-al History of the Island of Rodrigues, 283. Stokes, Sir George, and the concept of uniform convergence (Hardy), 148. Symbolic Logic, An Axiom in (Van Horn), 22. Tams, W. H. T., see Snell, H. J. Theorem of Dirichlet (Todd and Norton),. 111. Theorem of Mr G. Polya (Hardy), 60. Thermionic Valve, Self-oscillations of a (Whiddington), 346. Todd, H., and Norton, H. T. J., A particular case of a theorem of Dirichlet, 111. Trigonometrical Series which have a Necessary and Sufficient Condition for Uniform Convergence (Jolliffe), 191. Tripathi, S. C, Elected Fellow 1919, February 3, 348. TuRNBULL, H. W., Some Geometrical Interpretations of the Concomitants oi Two Quadrics, 196. Van Horn, C. E., An Axiom in Symbohc Logic, 22. i hlet, I I Index 355 Waeburton, C, Note on the solitary wasp, Crabro cephaloies, 296. Wasp, Crabro cephalotes (Warburton), 296. Watson, G. N., Bessel functions of equal order and argument, 42. The hmits of applicability of the Principle of Stationary Phase, 49. Bessel functions of large order, 96. Asymptotic expansions of hypergeometric functions. See Transactions, xxn. Weatherburn, C. E., On the Hydrodynamics of Relativity, 72. WmDDiNGTON, R., The self-oscillations of a Thermionic Valve, 346. Wilson, B. M., see Shah, N. M. Wilson, C. T. R., Methods of investigation in atmospheric electricity, 126. On a Micro-voltameter, 345. Wood, T. B., The siu-face law of heat loss in animals, 126. Woods, H., The Cretaceous Faunas of New Zealand, 127. Wrinch, D., On the exponentiation of well-ordered series, 219. Yamaga, N, Elected Associate 1916, November 13, 126. CAMBRIDGE: PRINTED BY .1. B. PEACE, M.A., AT THE UNIVERSITY PRESS I CONTENTS. PAGE Colourimeter Design. By H. Hartridge, M.D., Fellow of King's College, Cambridge. (One Fig. in text) 271 The Natural History of the Island of Rodrigues. By H. J. Snell (Eastern Telegraph Company) and W. H. T. Tams. (Communi- cated by Professor Stanley Gardiner) 283 Preliminary Note on the Life History of Lygocerus {Proctotrypidae), hyperparasite of Aphidius. By Maud D. Haviland, Fellow of Newnbam College. (Communicated by Mr H. H. Brindley) . 293 Note on the solitary wasp, Crabro cephalotes. By Cecil Warburton, M.A., Christ's College 296 Neon Lamps for Stroboscopic Work. By F. W. Aston, M.A., Trinity College (D.Sc, Birmingham), Clerk-Maxwell Student of the Uni- versity of Cambridge. (One Fig. in text) 300 The pressure in a viscous liquid moving through a channel loith diverging boundaries. By W. J. Harrison, M.A., Fellow of Clare College, Cambridge. (One Fig. in text) 307 The Efect of Ions on Ciliary Motion. By J. Gray, M.A., Fellow of King's College, Cambridge 313 A Note on Photosynthesis and Hydrogen Ion Concentration. By J. T, ■ Saunders, M.A., Christ's College 315 The distribution of intensity along the positive ray parabolas of atoms and molecules of hydrogen and its possible explanation. By F. W. Aston, M.A., Trinity College (D.Sc, Birmingham), Clerk-Maxwell Student of the University of Cambridge. (Three Figs, in text) . 317 Gravitation and Light. By Sir Joseph Larmor, St John's College, Lucasian Professor 324 On a Micro-voltameter. By C. T. R. Wilson, M.A., Sidney Sussex College 345 The self-oseillations of a Thermionic Valve. By R. Whiddington, M, A., St John's College 346 Proceedings at the Meetings held duiing the Session 1918 — 1919 . 347 Index to the Proceedings with references to the Transactions . . 350 PEOCEEDINGS OF THE CAMBKIDGE PHILOSOPHICAL SOCIETY VOLUME XX PRINTED IN ENGLAND AT THE CAMBRIDGE UNIVERSITY PRESS BY J. B. PEACE, M.A. PKOCEEDINGS OF THE CAMBEIDGE PHILOSOPHICAL SOCIETY VOLUME XX 26 January 1920—16 May 1921 CAMBRIDGE AT THE UNIVERSITY PRESS and sold by deighton, bell & co., ltd. and bowes & bowes, cambeidge cambridge university press c. f. clay, manager, fetter lane, london, e.c. 4 1921 I CONTENTS. VOL. XX. PAGE On the term by term integration of an infinite series over an infinite range and the inversion of the order of integration in repeated infinite integrals. By S. Pollard, M.A., Trinity College, Cambridge. (Communicated by Prof. 6. H. Hardy) 1 Note on Mr Hardy^s extension of a theorem of Mr Polya. By Edmuxd Landau. (Communicated by Prof. G. H. Hardy) .... 14 Studies on Cellulose Acetate. By H. J. H. Fenton and A. J. Berry . 16 An examination of Searle's method for determining the viscosity of very viscous liquids. By Kurt Molin, Filosofie Licentiat, Physical Institute, Technical College, Trondhjem. (Communicated by Dr G F. C. Searle.) (Four figs in Text) 23 Preliminary Note on Antennal Variation in an Aphis (Myzus ribis, Linn.). By Maud D. Haviland, Fellow of Newnham College. (Communicated by Mr H. H. Brindley) ..... 35 The effect of a magnetic field on the Intensity of spectrum lines. By H. P. Waran, M.A., Government Scholar of the University of Madras. (Communicated by Professor Sir Ernest Rutherford.) (Plates I and II and one fig. in Text) ........ 45 Further Notes on the Food Plants of tite Common Earwig (Forficula auricularia). By H. H. Brindley, M.A., St John's College . . 50 Lagrangian Methods for High Speed Motion. By C. G. Darwin . . 56 A hifilar method of measuring the rigidity of wires. By G. F. C. Searle, Se.D., F.R.S., University Lecturer in Experimental Physics. (Five figs, in Text) 61 The Rotation of the Non-Spinning Gyrostat. By G. T. Bennett, M.A., F.R.S., Emmanuel College, Cambridge ...... 70 Proof of the equivalence of different mean vahies. By Alfred Kienast. (Communicated by Professor G. H. Hardy) ..... 74 Notes on the Theory of Vibrations. (1) Vibrations of Finite Amplitude. (2) A Theorem due to Routh. By W. J. Harrison, M.A., Fellow of Clare College ■ 83 Experiments with a plane diffraction grating. By G. F. C. Searle, Sc.D., F.R.S., University Lecturer in Experimental Physics. (Ten figs, in Text) 88 The Shadow Electroscope. By R. Whiddington, M.A., St John's College. (One fig. in Text) 109 Mathematical Notes. By Professor H. F. Baker and C. V. Hanumanta : On the Hart circle of a spherical triangle . . . . . .116 On a property of focal conies and of bi circular quartics . . . 122 On the construction of the ninth point of intersection of two plane cubic curves of which eight points are given ...... 131 vi Contents Mathematical Notes (continued); On a proof of the theorem of a double six of lines by projection from four dimensions. (Three figs, in Text) . . . . . .133 145 147 On transformations urith an absolute quadric .... On a set of transformations of rectangular axes. (One fig. in Text) On the generation of sets of four tetrahedra of which any tivo are mutually inscribed .......... 155 On the reduction of homography to movement in three dimensions. (One fig. in Text) 158 On the transformation of the equations of electrodynamics in the Maxwell and in the Einstein for ms . . . . . . .166 On the stability of periodic motions in general dynamics . . . 181 On the stability of 7'otating liquid ellipsoids ...... 190 On the general theory of the stability of rotating masses of liquid . . 198 Sur le principe de Phragmen-Lindelijf Par Marcel Riesz. With Note by G. H. Hardy 205 A note on the nature of the carriers of the Anode Rays. By G. P. Thomson, M.A., Fellow of Corpus Christi College .... 210 Proceedings at the Meetings held during the Session 1919 — 1920 . . 212 The Problem of Soaring Flight. By E. H. Hankin, M.A., Sc.D., late Fellow of St John's College, Cambridge, Chemical Examiner to Government, Agra, India. (Communicated by Mr H. H. Brindley.) With an Introduction by F. Handlby Page, C.B.E., F.R.Aer.S. . 219 Preliminary Note on the Superior Vena Ca,va of the Cat. By W. F. Lanchester, M. a., King's College, and A. G. Thacker . . . 228 A Note on Vital Staining. By F. A. Potts, M.A., Trinity Hall. (One fig. in Text) 231 Preliminary Note on a Cynipid hyperparasite of Aphides. By Maud D. Haviland, Fellow of Newnham College. (Communicated by Mr H. H. Brindley) 235 A method of testing Triode Vacuum Tubes. By E. V. Appleton, M.A., St John's College. (Two figs, in Text) 239 The Rotation of the Non-Spinning Gyrostat. By Sir George Greenhill and Dr G. T. Bennett 243 On the representation of the simple group of order 660 as a groiop of linear substitutions on 5 symbols. By Dr W. Burnside, Honorary Fellow of Pembroke College ......... 247 On the representation of algebraic numbers as a sum of four squares. By L. J. Mordell. (Communicated by Professor H. F. Baker) . . 250 On a Gaussian Series of Six Elements. By L. J. Rogers. (Communicated by Professor G. H. Hardy) 257 Note on Ramanujan^s trigonometrical function Cg(n), and certain series of arithmetical functions. By Professor G. H. Hardy .... 263 On the distribution of primes. By H. Cramer, Stockholm. (Communicated by Professor G. H. Hardy) 272 Note on the parity of the number which emimerates the partitions of a number. By Major P. A. MacMahon ' . 281 Note on constant volume explosion experiments. By S. Lees, M.A., St John's College. (Two figs, in Text) 285 Contents vii PAGE On the Latent Heats of Vaporisation. By Eric Keightley Rideal, M.A., Trinity Hall .... 291 Oil the fimction [x]. By ViGGO Brun (Drobak, Norway). (Communicated by Professor G. H. Hardy) 299 A theorem concerning summahle series. By Professor G. H. Hardy . . 304 Standing Waves parallel to a Plane Beach. By H. C. Pocklington, M.A., St John's College 308 The Origin of the Disturbances in the Initial Motion of a Shell. By R. H. Fowler and C. N. H. Lock. (One fig. in Text) . . . .311 Tides in the Bristol Channel. By G. I. Taylor, F.R.S. (Four figs, in Text) 320 Expenments ivith Rotating Fluids. By G. I. Taylor, F.R.S. . . . 326 Experiments on focal lines formed hy a zone plate. By G. F. C. Searle, So.D., F.R.S., University Lecturer in Experimental Physics. (Five figs, in Text) 330 The Tensor Form of the Equations of Viscous Motion. By E. A. Milne, B.A., Trinity College 344 Insect Oases. By C. G. Lamb, M.A 347 A Note on the Hydrogen Ion Concentration of some Natural Waters. By J. T. Saunders, M.A., Christ's College 350 The Mechanism of Ciliary Movement. By J. Gray, M.A., Balfour Student, and Fellow of King's College, Cambridge. (Three figs, in Text) . 352 A Note on the Biology of the '■ Crown-GaW Fungus of Lucerne. By J. Line, M.A., Emmanuel College. (Seven figs, in Text) . . . 360 On some Alcyonaria in the Cambridge Museum. By Sydney J. Hickson, M.A., F.R.S., Professor of Zoology in the University of Manchester. (One fig. in Text) 366 The Influence of Function on the Conformation of Bones. By A. B. Appleton, M.A., Downing College. (Five figs, in Text) . . . 374 Animal Oecology in Deserts. By P. A. Buxton, M.A., Fellow of Trinity College, Cambridge 388 Venational Abnormalities in the Diptera. By C. G. Lamb, M.A. (Four- teen figs, in Text) 393 The Cooling of a Solid Sphere with a Concentric Core of a Different Material. By Professor H. S. Carslaw. (Three figs, in Text) . 399 Symbolical Methods in the theory of Conduction of Heat. By Dr T. J. I'a. Bromwich, F.R.S. (Two figs." in Text) 411 0?!. the effect of a magnetic field on the intensity of spectrum lines. By H. P. Waran, M.A., Government of India Scholar of the University of Madras. (Communicated by Professor Sir Ernest Rutherford, F.R.S.) (Three figs, in Text and Plate III) 428 On a property of focal conies and of bicircular quartics. Qj C. V. Hand- MANTA Rao, LTniversity Professor, Lahore. (Communicated by Professor H. F. Baker) 434 Convex Solids in Higher Space. By Dr W. Burnside, Honorary Fellow of Pembroke College ......... 437 Note on the Velocity of X-ray Electrons. By R. Whiddington, M.A. (One fig. in Text) 442 A Laboratory Valve method for determining the Specific Indxictive Capacities of Liquids. By R. Whiddington, M.A. (One fig. in Text) 445 viii Contents The Theoretical Value of Stitherland' s Constant in the Ki-netic Theory of^ Gases. By C. G. F. James, Trinity College, Cambridge. (Communi- cated by Mr R. H. Fowler.) (One fig. in Text) . . . .447 On the Stability of the Steady Motion of viscous liquid contained between two rotating coaxal circidar cylinders. By W. J. Harrison, M.A., Fellow of Clare College, Cambridge 455 The soaring flight of dragon-flies. By E. H. Hankin, M.A., Sc.D., Agra, India. (Three figs, in Text) 460 The Gluteal Region of Tarsius Spectrum. By A. B. Appleton. 11! (Plate IV) 4^^ 11 An unusual type of mcde secondary characters in the Diptera. By C. G. Lamb, M.A. (Four figs, in Text) 475 A Note on the Mouth-parts of certain Decapod Crustaceans. By L. A. BoRRADAiLE, M.A., Fellow and Tutor of Selwyn College, Cambridge, and Lecturer in Zoology in the University ..... 478 An Apparatus for Projecting Spectra. By H. Hartridge . . . 480 Note on true and apparent hermaphroditism in sea-urchins. B}' J. Gray, M.A., Balfour Student, Cambridge University 481 On Certain Simply -Transitive Permutation-Groups. By Dr W. Burnside, Honorary Fellow of Pembroke College 482 Proceedings at the Meetings held during the Session 1920 — 1921 . . 485 Index to the Proceedings with references to the Transactions . . 492 PLATES. Plates I— III. To illustrate Mr Waran's papers .... 48, 433 Plate IV. To illustrate Mr Appleton's paper 474 CORRECTION. MoRDELL, p. 250, line 5, after conjugate numbers insert in the reed conjugate fields. PEOCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCDETY VOL. XX. PART I. 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Authors may receive, on request, fifty ofiprints of a paper in the Proceedings, or twenty-five offprints of a paper in the Transactions, without charge. Additional copies may be had by payment to the University Press. 6. Members of the Society are particularly requested to inform the Secretaries of any change of postal address. PROCEEDINGS OF THE Cambriirgc ISljtlasapIjkal Bomi^. On the term by term integration of an infinite series over an infinite range and the inversion of the order of integration in repeated infinite integrals. By S. Pollard, M.A., Trinity College, Cambridge. (Communicated by Prof. G. H. Hardy.) [Received 1 January, 1920. Read 8 March, 1920.] The problem for infinite series. 1. The problem to be solved is that of determining conditions, under which the equation 00 /•=» /"» 00 2 Un(oc)dx==l S Un(oi;)dx, (1) n=l J a J an=l is true. It is discussed in detail in Bromwich's Infinite Series, pp. 452-455, where various conditions are given. All these con- ditions will be found to involve uniform convergence, the fact being that the infinite integrals there considered are obtained as limits of Riemann integrals and, in the theory of the latter, con- siderations as to the validity of the equation rb m rb oo lim I 2 M„ (x) dx = j S Uji (w) dx, (2) almost always involve uniform convergence. Thus conditions for term by term integration over an infinite range, being built up from the conditions for term by term integration over a finite range, involve uniform convergence. Now the condition of uniform convergence is by no means a necessary one : it occurs because of the lack of power in the methods of the Riemann theory. Much wider conditions can be obtained by the use of the Lebesgue theory. It is the object of this paper to give these. VOL. XX. PART L 1 2 Mr Pollard, On the term by term integration Conditions for passage to the limit under the sign of integration, the range of integration being finite. 2. We give, for the sake of reference, the two principal elementary conditions. (C 1) If u^ (sc) is positive for a^x ^b; n = l, 2, 3 . . ., then if either side of (2) is finite the equation holds, and if either side is infinite both are. V (C 2) Ifl'^Un (oc) I < i|r (a;) for a-^a:^b,v=l,2,S,..., tuhere n=l '>^ is summable in (a, b), then both sides of (2) eccist and are finite and equal*. Resume of theorems of double limits. 3. As the use of double limits is fundamental in the theory about to be developed, we give a short summary of the results required. (a) If the double limit lim S^, y exists, and lim 8x, y exists for all sufiiciently large y ; then lim (lim S^, y) exists and is equal to the double limit. Similarly for the limit lim (lim S^^y). (^) If Sx, y is increasing in x and y, and any one of lim Sx,y, lim (lim *S^a;, j/X lim (lim>Sa;, j,) exist; then all three exist and are equal. (y) If 8x, y can be expressed as the difference of two functions S'x, y, S"x, y ^^ch of which is increasing in x and y and lim {S'xy + S"xy) X-»-oo , y^"Xi exists and is finite ; then lim Sx,y, lim (lim Sx,y\ lim (lim Sa;,y) all exist and are finite and equal. The condition (7) is especially convenient when S.,y=n''fil,v)d^dv. J aJb * De la Valine Poussin, Cours d^analyse infinites imale, t. i. , 3rd Ed., p. 264, theorems iii and 11. of an infinite series over an infinite range 3 For if lim f" ^ \f{l v)\d^dr) exists and is finite, then S^^y satisfies the condition of (7). We have in fact ^x, y — ^ x,y ^ x,yj fx ry where S'x,y= I \f{^,v)\d^dr}, J aJb S\y=r ("[\f{^,V)\-f{lv)]d^dv, J aJ b and both S'cc,y, 8"x,y are increasing in x and y and have a finite double limit — the former by hypothesis and the latter because Note. The above results still hold when either or both of the variables x, y take only positive integral values. Definition of infinite integrals. 4. Let /(a;) be any function which is summable in (a, X) for all X greater than a. If lim f{x) dx, X-*-ix J a where the integral is taken in the sense of Lebesgue, exists and is finite, we say that J f (x) dx converges and attribute to it the value of the limit. This definition is evidently consistent with and more general than that usually given, where /(a:;) is assumed to be integrable in Riemann's sense in (a, X). It has the special advantage of not being restricted to functions which are bounded in every (a, X). And we lose nothing by adopting it, as the two theorems on which the theory of infinite integrals rests, the first and second mean value theorems, are still true when we abandon the restriction that f{x) is to have a Riemann integral and make only the assumption that /(a;) is summable*. General theorems. rX m rX m 5. I. If the double limit lim I 2 m„ (x) dx exists and m-^-oo, X-*-<x>J an=l (a) I Un (x) dx converges for all n, J a Ibid. t. II. 2nd Ed., p. 53. 4 Mr Pollard, On the term by term integration 00 (b) 2 u^ (x) converges for X'^a, n=\ rX m rX CO (c) lim S Un (oc) dx, 2 ?/„ {x) dx, m-*-ao J a n=l J an=l exist and are equal for all X ; then both sides of {!) exist and are equal. rX m Proof. Write S u^ {cc) dx = Sm, x, J a n=l and let lim Sm,x = S. m-*-'x>,X^'OD Since I w„ («) dx converges for all n J a rX m lim % u^{x)dx, i.e. lim Sm,. X exists for all m. Hence lim (lim >S^,x) ='S^ (3)» Wl^-oo X-^cc by(«)- . In virtue of (b) and (c) rX m ^ lim 1 2 Un (^) dx m^-as J a n=l rX 00 exists and is equal to 2 u^ (x) dx. Jan=l Thus lira Sm, x exists for X ^ a. Hence lim (lim Sm,x) exists and is equal to S. X-^-CC 77l-*-00 rX 00 Taking lim S^, x in the form 2 u^ {^) dx, we see that m-*-oo J an = l /-■» 00 S Un (^) dx = S. (4) J an=l And (3) and (4) give us our theorem. 00 II. J/ 2 I w„ (x) I converges for x^a and the double limit rXm rXm lim 2 I w„ (a;) I dx ■00, X->-<x) J an=\ exists and is finite; then without further condition both sides of (1) exist and are finite and equal. o/ an infinite series over an infinite range 5 rx m Proof. By (7), if lim I S | w„ {x) \ dx exists, so does «-»■», X-*-ao J a n=X rX m lim S w„ {x) dx. m-^-oo , X-*-<x> J an=l CX m Also Sm, x=\ ^ \Un{x)\dx^ 8, J a n = l for all X and m. But Sm,x increases with X for each m. Hence rx m % \u„(x)\ J an=l rX m lim I 2 I w„ (ic) I dx X^oo J an=l exists for each m and is less than 8. And therefore rX f rx m rX m-1 "j lim 1 I u„ (x) I dx = lim j I S 1 1*„ (x) \dx — I % \Un{x)\dx> , X^-co J a X^i-co [J an=l J a n=l J exists for each m, i.e. 1 | ^^^ (/») | dx and therefore I ?/.„ (x) dx con- ^ a J a verges for each m. This is (a) of (I). Again, 8m, x increases with m for each X. Hence lim S.m,,x exists and is finite for each X. So from (C 1) [X 00 X \Un (x) I dx Jan=l 00 is finite. Thus 2 | m„ (x) I is summable in (a, X). But mm 00 71=1 n = l 71 = 1 and so by (C 2) rx m rx CO lim I S Un (x) dx, I 2 w„ (a;) c?^, m-*-oo .' a n. = l ./a7i.= l exist and are finite and equal. This is (c) of (I). Now (b) of (I) is satisfied by hypothesis. Thus all the conditions of (I) are satisfied and so both sides of (1) exist and are equal. Deductions from the general theorems. 6. A. // u^{x) = (f>(x)f„{x), where 2 /„ (x) converges for x'^a, n = l V I ^ fn{x)\< G, for x'^a and all v, n = l 6 Mr Pollard, On the term by term integration and I ^ {x) I dx converges; then both sides 0/ (1) exist and are J a finite and equal. B. If either of % i \un(cc)\dx, 2 I M„ (x) I dx, w = l J a J a n = l eadst and are finite; then both sides of (1) exist and are finite and equal. rX m rx <x> C If lim S Un (x) dx, 'S. Un (^) dx, m-»-oo .' a«,= l ■!an=l exist and are finite and equal, and 2 Un (x) dx J a converges uniformly for a^^ x, and each u^ (x) dx J a converges; then both sides of(l) exist and are finite and equal. V D. 7/ I 2 M„ («) I < -^x (^) for a^ x^ X and all v, where -^x n=l is summable in {a, X), and rx S Un (x) dx J a converges uniformly for a^ X, X being arbitrary, and each Un {x) dx J a converges; then both sides of {1) exist and are finite and equal. D is a special case of C obtained by making use of (C 2). A, B, D may be regarded as generalisations of theorems A — C, pp. 452-455 of Bromwich's Infinite Series. Proofs. A. If m' > m, m' m' m we have 2 /„ {x) = X fn{x)-X fn {x), and therefore m' Til' m I 2 fn{x)\^\ Xfn{x)\ + \ ^ f,{x)\^2G. of an infinite series over an infinite range 7 Hence rX' m' rX' m' I S Un {x)dx\^\ I S </> {x)f^ (x) I dx JX n=m+l J X n=m+l rX' m' < \<^{x)\t \fn {OC) I dx J X n = m ^2g[ \d> {x) I dx. J X Now, given any positive number e, we can, since \<f) {x)\dx J a converges, find Xo such that \(f){x)\dx< € J X forX, X'>Xo. Hence rX' to' I 1 S ii„ (x) dx\< € Jx n = m+l for X, X' > Xf) and all rn, m . Thus the double limit exists. Further rx m lim / S i<„ ix) dx •■»,X-*"X J a n = \ n=l x=l and G^ 1 (ic) I is summable in C«, X) for all X greater than a. Thus by (C 2) CX m rX 00 lim X Un (x) dx, i u„ (x) dx J a n=l J a n=l exist and are equal and finite. All the conditions of (I) are now satisfied, and our theorem follows. B. If we write r^ m rX m Sm,X= I 2 \Un{x)\dx, •I a n=l then lim 8m,x exists and is either finite or positive infinity. In the first case our theorem follows at once by (II). In the second case, both the repeated limits lim (lim Sm,x), lim (lim 8m, x), m-^cD X^'x> X^oD m-*oo are infinite. Suppose now that S I I M„ (x) I dx »=1 J a 8 Mr Pollard, On the term by term integration exists and is finite. Then lim (lim ^«t,x) exists and is finite, and we get a contradiction. And if ^ \Un{x)\dx J an=l exists and is finite, then so does rx m \ 2 I ti„ {x) I dx J a 7i=l for all X greater than a. Hence as in theorem II rX oa rX TO I "E \ Un {w) \ dx = lim X I Wn (^) I ^^> J a 71=1 Wi-»-oo J a 71 = 1 and it follows that r"" 00 E \un\x)\dx= lim (lim S^^x), Ja n = l X-*-oo 7j,-*-oo and we again get a contradiction. Thus the first case alone is possible, and this is the case in which our theorem is true. C. Write r u,{w)dx = g,{X), - « m tgn{X)^Sm,X. 71=1 00 Since S g^ (X) converges uniformly for a ^ X, given e > we 71=1 can find No such that I i gAX)\<^, {X>a,N^N-,). n=N+l Thus \Sm,x- i 9n (X) I < e, {X^a, m ^ N^). 71 = 1 00 Hence if lim S gn{^) exists and is finite, so does lim aS'^^^ x , X-*-^ n=l 7?l-»Q0 , JC-».Xi and the two are equal. Now I i: g^ (ZO -i g, (Z") I ^ i f g, (Z) - % g, (Z") j tt=l 71=1 71=1 n=l + 1 i gn{X')\ + \ i 5'n(X")I n=N+l n=N+l ^1 f 5r,(Z')-i5'«(X")l + 26. 71=1 71=1 But since lim g^ (Z) (= u^ (w) dx) exists and is finite for X-*oo J a o/ an infinite series over an infinite range 9 N each n, so does lim S gn{^) ^-nd we can find X^ such that 1 f 5r„ (Z') -I g, {X") 1 < 6. {X', X" ^ Zo). 71=1 »=1 Hence 1 I ^„ (X) - 2 ^„ (Z") | < 3e, (Z', Z" ^ Z„) and therefore, by the general principle of convergence 00 lim 2 gr, {X) X-»-t» 71=1 exists. Thus lim *S^„i,x exists. The other conditions of (I) are satisfied by hypothesis and our theorem follows. The problem for infinite integrals. 7. We have to determine conditions under which the equation dx\ f{x,y)dy==\ dy \ f{x,y)dx (5) J a J b J b J a is true. The methods adopted above apply almost without change and we get conditions almost identical with those already given. We quote them without "proof, as the proofs can be made up immediately on the lines of those already given. As regards the nature oif{x, y), we assume throughout that f {x, y) is summable in the region {a^x^X, b^y^Y), for all Z ^ a, Y ^b ; so that, by Fubini's theorem*, the repeated rX rY fY rX integrals dx \ f{x,y)dy, I dy \ f{x,y)dx exist and are J a ■ b J b J a equal to the double integral. General theorems. rX rY and 8. I'. If the double limit lim I j f{x, y) dxdy exists (a) I f{x,y)dx, converges for y^b, * (b) I f{x,y)dy, converges for x '^ a, rX rY rX r^ (c) lim dx f{x, y) dy, dx f{x, y) dy, F-w-oo J a J b J a J b * De la Valine Poussin, Integrales de Lebesgue etc., p. 53. 10 Mr Pollard, On the term by term integration exist and are finite and equal for X ^ a ; then both sides of (5) exist and are equal. IT. If the double limit lim \ f{x, y)\dxdy exists X-*-ao , Y-^x J a J b and is finite; then without further condition both sides of (5) exist and are finite and equal. Deductions from the general theorems. 9. A'. // f{x,y) = <^{x)d{x,y), where j 6 {x, y) dy \ < G for x^a, y ^b, Jh f{x, y) dy converges for x^a,' 'b and \(f>{x)\dx converges; then both sides of (5) exist and are J a finite and equal. B'. If either of 1 dx \f{x, y) I dy, dy \f{x, y) \ dx, J a J b J b J a exist and are finite; then both sides of {5) exist and are finite and equal*. rX rY rx r=° ■ C. If lim dxl f{x,y)dy, dx \ f{x,y)dy, Y-*-oo J a J b J a J b exist and are finite and equal, and dy fix, y) dx Jb Ja converges uniformly for a^ X, and o f{x, y) dx / J a converges for y ^ b ; then both sides of (o) exist and are finite and equal. * This is de la Valine Poussin's theorem. See Bromwieh, Infinite Series, p. 457. The hypothesis given by Bromwieh to the effect that both the integrals /"OO /"OO are convergent is unnecessary, the existence of one (the one necessary to the existence of the repeated integral) is sufficient. That of the other is implied by the existence of the double hmit, see Note 2. o/ an infinite series over an infinite range 1 1 D'. If \r fix, y)dy\^^x{x) for a^ x^ X, 6 ^ F, J b ivhere yjrx is summahle in (a, X), and \ dy \ f {x, y) dx Jb J a converges uniforndy for a ^ X, X being arbitrary, and f{x, y) dx converges for y^b; then both sides of {b) exist and are finite and equal. 10. Note 1. Results B are especially valuable, as they are easy to remember and convenient to apply. The power of the Lebesgue theory is shewn very clearly here in that by using it we are enabled to make the hypothesis which ensures the exist- ence of the double limit* ensure also the passage to the limit under the sign. Note 2. It is well to be precise as to the meaning of the word "exists" as used in connection with repeated Lebesgue integrals. Suppose /(^, y) is measurable in x, y in the rectangle ^a^x^X\ Kb^y^Yj' We know that the function y (a:;, y) considered as a function of X, is measurable in (a, X) for each y in (6, Y) a set of zero measure being excepted. It may not, however, be summable in (a, X), i.e. f{x,y)dx may not exist, for all ?/ concerned. But, if/(^, y) is summable over the rectangle, i.e., if the double integral rx rY fix, y)dxdy ■ b exists ; then it can be shewn that rx fix, y)dx f J a exists for all values of y in (b, Y), save possibly those of a set of measure zero. CO ,-co * The existence of 2 I j u^^ (x) \ dx implies the existence of the double limit n=lj a by (7) of §3 ; and in addition, by the use of (C 1) on | u„(x) | , it will be found to imply the validity of the passage to the limit under the sign. 12 Mr Pollard, On the term by term integration Now in the Lebesgue theory the integral of any summable function over a set of zero measure is zero, and consequently we may neglect a set of measure zero without affecting the value of _ the integral. Hence when we are faced with the problem of find- l{ ing the value of a function which is indefinite or infinite at the points of a set of measure zero, we simply neglect these points and find the value of the integral over the residue. This is taken to be the value of the integral over the original set. With the above convention it is true that, if I /(^, y) dxdy Ja Jb exists, so does dy f{x, y) dxdy, . h J a although there may be points in (6, Y) at which the single integral rx f{x,y)dx f J a does not exist. It is always to be understood in dealing with repeated Lebesgue integrals (finite or infinite) that the inner integrals need only exist at all the points of the range of integration of the outer integral save those of a set of measure zero. Let us apply the foregoing remarks to theorem B'. Suppose rY rx dy \ \f{^,y)\dx ■lb J a exists. Then we know that lim 11 fix, y) I dxdy X-*-x , F-9.20 J a J b exists. It follows that I dx\ \f{x,y)\dy, J a Jb on of Y, is bounded £ lim / dx \f{x,y)\dy T-^x J a J b considered as a function of Y, is bounded as F tends to infinity. Thus rx rY Y- exists and is finite. It follows that \f{x, y) \ dy b converges at all the points of (a, X) save possibly those of a set of of an infinite series over an infinite range 13 measure zero, because if it did not the above limit would be infinite ; and so, for our purposes J a J b exists. Our convention has enabled us to infer the existence of the inner integrals from the existence of the double limit. Note 3. A thorough treatment on different lines of the subject of this paper will be found in two papers by Prof W. H. Young : (1) " On the change of order of integration in an improper repeated integral," Trans. Gamh. Phil. Soc, xxi. p. 361. (2) "The application of expansions to definite integrals," Proc. Lond. Math. Soc, ix. (1910), p. 463. In this paper we content ourselves with giving simple generalisations of well-known results with proofs depending on comparatively elementary theorems. There is no attempt to obtain comprehensive results. 14 Mr Landau, Note on Mr Hardy's extension of a theorem, etc. Note on Mr Hardy's extension of a theorem of Mr Poly a. By Edmund Landau. (Communicated by Prof. G. H. Hardy.) [Received 10 December 1919. Read 26 January 1920.] In a recent note in these Proceedings* Mr Hardy has estab- lished an improved form of a theorem of Mr Pdlya, viz. : Suppose that g {x) is an integral function, and M (r) the maxi- mum of I ^ (^) I for \x\^r. Suppose further that g {x) is an integer for x = 0, 1, 2, 3, ..., and that M{r)=o{2'). Then g (x) is a polynomial. As Mr Hardy remarks at the beginning of his note, it is sufficient (after the analysis given already by Mr Pdlya), to prove the one formula ^,2-r _ — ^ — =0(1). — TT n (2n - s cos 6) Mr Hardy's proof of this formula may be replaced by the following shorter proof Since nl2^n^ = nl2^n(^^-^,2^nn^l=OWn), n(2n-s) ^ ^' s=l it is enough to prove '[ir{e,n)de==o{^), where ylr(e,n)=u( ^ ^^ ~ ' . ^ ^ ' s=i\2n — scos6 Now l-cos6/_2-24 + ---^2 24^2 24 ~ 12 ' for — TT < ^ ^ TT, and 1 + 2/ * Vol. XIX. (1919), pp. 60-63. Mr Landau, Note on Mr Hardy's extension of a theorem, etc. 15 for ^ y ^ 1. Hence 1-7? 1 1 1-^cos^ l+_^(l-cos^) ^+va-cos0) 1 -7] ^ g-i^ (1-cosfl) < g-Tll*^ for ^ t; ^ |, — TT ^ 6 ^tt; and so 1 — — - 02 n -^{d, n) = n — — ^ e~487tif = e-TyV(«+i)9^ ^ e-?Vne^^ *='l-^cos6' In for — TT ^ ^ ^ TT and ?? = 1, 2, 3, .... Therefore [" A/r (^, n)de^j e- ^^^''dd = ('-^) . GoTTlNGEJf, 4 December 1919. 16 Dr Fenton and Mr Berry, Studies on Cellulose Acetate Studies on Cellulose Acetate. By H. J. H. Fenton aii( A. J. Berry. [Read 8 March 1920.] The enormous demand for cellulose acetate and the serious, shortage of acetone and certain other materials used in the manu- facture of aeroplane dopes during the war originated a systematic research on cellulose acetate, especially as regards the behaviour of this material towards solvents and its chemical properties generally. The research has been pursued in a number of directions, the most important of which have been (a) substitutes for acetone as solvents, (b) the preparation of cellulose acetate and a study of the influence of the mode of preparation on the properties of the resulting product, and (c) the analytical chemistry of cellulose ace- tate. Most of our experiments, especially those relating to aeroplane dopes were necessarily of a technical character, but as a few results- of general chemical interest have been obtained in the course of the work, we have thought it desirable to give a brief account of them in the present communication. Solvents. At the time of the difficulty caused by the serious shortage of acetone we were urged to discover efficient substitutes for this, solvent for use in aeroplane dopes. It should, in passing, be observed that the properties of acetone make it an ideal solvent: its conveniently low boiling point, rapid solvent action on cellulose acetate, non-poisonous character, and, in normal times, cheap and abundant supply. All other liquids which have so far been suggested show a deficiency in. some one or other of these particulars. In August, 1917, we suggested that in case of emergency the three following solvents might be employed, viz. aeetaldehyde^ acetonitrile, and nitrobenzene with certain additions. Quite early in the investigation (October, 1916) we suggested acetic acid and ethyl formate as solvents. We also suggested the use of cyclo- hexanone and of beechwood creosote as substitutes for tetrachloro- ethane or benzyl alcohol as high boiling solvents. We were never informed whether these solvents were actually employed. It is- remarkable that at considerably later dates, patents have been taken out for the use of both acetaldehyde and cyclohexanone as- dope constituents. (British Patent 131647, July 4th, 1918 (acet- aldehyde) and Ibid. 130402, February 15th, 1918 (Cyclohexanone).) Dr Fento7i and Mr Berry, Studies on Cellulose Acetate 17 Our experiments have demonstrated that the destructive effect of acids upon fabrics is dependent on the strength of the acid in the physico-chemical sense. Hitherto it had been supposed that esters were objectionable as dope constituents on account of the possibilities of free acids resulting from hydrolysis. This, however, we found not to be the case. As far as weak acids only are concerned, tensile strength determinations gave excellent results; and fabrics doped with acetic acid as the principal solvent compared most favourably with others. In our experiments a large number of liquids have been examined, not only from the purely practical point of view, but also from a desire to obtain if possible some information with regard to possible relationships between the nature of the liquid and its solvent action. It is of course impossible to define strictly the solubility of cellulose acetate in any given solvent owing to the colloidal nature of the products. The term "positive" is used in the following lists to imply that the liquid named has the property of gelatinizing cellulose acetate and subsequently converting it into a clear homogeneous "sol" without the aid of heat. All the results were obtained with a sample of the material which yields 54 per cent, of acetic acid on cold alkaline saponification. Positive. Liquid ammonia, liquid sulphur dioxide, liquid hydrogen cyanide, acetaldehyde, benzaldehyde, salicylaldehyde, acetone, methyl ethyl ketone, suberone, acetonitrile, propionitrile, formic acid, acetic acid, butyric acid, formamide, ethyl formate, ethyl oxalate, ethyl malonate, etbyl acetoacetate, aniline, phenyl- hydrazine, ortho-toluidine, piperidine, pyridine, tetrachloroethane, nitrobenzene*, nitromethane, cyclohexanone, guaiacol, chloro- form*. Although cellulose acetate is insoluble in water and in absolute ethyl alcohol, a mixture of these two liquids dissolves it freely on boiling. On cooling, however, precipitation takes place almost completely. Negative. Liquid air, liquid ethylene, liquid nitrous oxide, liquid hydrogen sulphide, benzene, toluene, turpentine, carbon disulphide, carbon tetrachloride, alcohol, ether, ethyl chloride, acetal, dimethyl acetal, nickel carbonyl, and many other liquids. No general conclusion can be drawn as regards the chemical nature of a liquid and its solvent action on cellulose acetate. It is, * Nitrobenzene requires certain additions. Chloroform had only a partial solvent action on this specimen of the material. VOL. XX. PART I. 2 18 Dr Fenton and Mr Berry, Studies on Cellulose Acetate however, worthy of note that there appears to be some relation (with undoubted exceptions) between the dielectric constant and solvent action. Influence of methods of preparation upon the properties of cellulose acetate. The materials obtained by acetylating cellulose with acetic anhydride diluted with acetic acid in presence of various catalysts such as concentrated sulphuric acid, ferric sulphate, ortho tolui- dine bisulphate, may show considerable variations in properties depending upon the temperature, length of time of acetylation, and numerous other factors. When cellulose is acetylated and the product at once precipitated by water, it is nearly insoluble in acetone. Various methods have been adopted in order to convert the product so obtained into an acetone-soluble modification. The most widely used of these methods is that of Miles. This consists in heating the acetic acid solution of the cellulose acetate with water in rather greater quantity than that required to combine with the residual acetic anhydride. Sodium acetate may also be added to react with the catalyst if still present. The results are usually supposed to be due to chemical hydration. In our experiments, cellulose was acetylated under the influence of various catalysts, and the effect of treatment, by the Miles process was subjected to a critical examination. The most marked effects of this process are the changes in solubility in acetone and chloroform, most cellulose acetates being soluble in chloroform and insoluble in acetone before the treatment. This change in physico-chemical properties was found to be accompanied by a fall in the acetyl number. In one case the untreated cellulose acetate with an acetyl number of 60-9, yielded a product after the Miles process carried out at 100° for 48 hours with an acetyl number of 46-7. In another case when the treatment was carried out at the same temperature for 23 hours, the acetyl number fell from 60-5 to 50-4. The specific gravity of the cellulose acetate is also greatly reduced after the treatment. The influence on the heat test is not well marked but the decomposition point appears to be lowered somewhat. In our view these results are to be ascribed to partial hydrolysis of the cellulose esters, not to hydration as is commonly supposed*. Apart from the diminution of the acetyl number already mentioned, we have carried out a series of experiments which have demon- strated that cellulose acetate does not form a hydrate. These * Oux view that the effect of the Miles process is essentially hydrolytic and not due to chemical hydration has been expressed subsequently by Ost {Zeifsch. angeic. Chem. 1919, xxxii, 66, 76, and 82). Dr Fenton and Mr Berry, Studies on Cellulose Acetate 19 experiments originated in connexion with our determinations of tlie water contained in commercial samples of cellulose acetate. As is well known, the water is readily expelled by exposure of the material over concentrated sulphuric acid in a desiccator or by heating to 100°. It has frequently been supposed that the approxi- mately constant proportion of 5 or 6 per cent, of water usually met with indicates a definite hydrate. In order to obtain positive information on this point, we determined the pressure-concentra- tion relationship in the manner originally adopted by van Bemmelen in his well known researches on silicic acid {Zeitsch. anorg. Chem. 1896, XIII. 233). Weighed quantities of the material were exposed in a series of exhausted desiccators over sulphuric acid of various determined concentrations, and the corresponding vapour pressures were found by reference to Landolt and Bornstein's tables. The weights were found to be constant after 24-48 hours, and the pressure concentration relationship showed that no chemical hydration occurs. The phenomenon is to be regarded as one of adsorption, probably with subsequent difiusion, and is precisely similar to the absorption of water by cellulose itself. (Compare Masson and Richards (Proc. Roy. Soc. 1906, lxxviii. 421), Trouton and Pool {Ibid. 1906, lxxvii. 292) and Travers {Ibid. 1906, lxxviii. 21, and 1907, lxxix. 204).) Characterization and Analysis of cellulose acetate. In the technical analysis of cellulose acetate, it is usual to examine the product by the heat test, solubility, acidity, and viscosity of the solutions, in addition to the determinations of acetyl (as acetic acid), copper reducing power, water, ash, and impurities. We have made an exhaustive investigation of various methods of carrying out these determinations, especially of the acetyl number, and have also carried out many ultimate analyses for carbon and hydrogen in some commercial specimens of the material. The methods of determining the acetyl group may be classified under the two heads of alkaline saponification and acid hydrolysis. In the former the substance is saponified by excess of standard alkali, either at the ordinary temperature or at some higher tem- perature, and the excess of alkali determined by titration. In the latter, the substance is hydrolysed by strong acid, usually sulphuric or phosphoric, and the resulting acetic acid separated by steam distillation (Ost), or alcohol is added and the resulting ethyl acetate distilled off and collected in excess of standard alkali (Green and Perkin). The following is a summary of the principal results obtained in our experiments. (1) Cold alkaline saponification (Ost and Katayama, Zeitsch. angew. Chem. 1912 (25), 1467). A known weight of the substance 2 2 20 Dr Fenton and Mr Berry, Studies on Cellulose Acetate is soaked with alcohol, then a measured volume of normal alkali is added and allowed to stand for 24 hours. The excess of alkali is then determined by standard acid. The mean result was 54 per cent, of acetic acid calculated for the dry substance. (2) Cold alkaline saponification (Boeseken, van der Berg and Kerstjens, Rec. Trav. Chim. 1916, xxxv. 320). The substance is treated with strong aqueous potash for one or two days. A measured excess of normal hydrochloric acid is then added, the liquid then boiled for a moment to expel carbon dioxide and the resulting solution titrated with baryta water. The mean result calculated as above was 53-5 per cent, of acetic acid. (3) Hot alkaline saponification (Barthelemy, Moniteur Scienti- fique, 1913 (3), ii. 549). In this method the saponification is effected by heating the substance with normal soda for about 16 hours at 85°. The excess of alkali is then determined by titration with standard acid. Several experiments were made in which the condi- tions were subjected to considerable variations as regards length of heating and amount of excess of alkali. The extreme variations in the acetyl number calculated as above were 60-0 and 62-1 per cent. (4) Hot alkaline saponification (Green and Perkin, Trans. Chem. Soc. 1906, 812). The saponification is carried out at the boiling point with semi-normal alcoholic soda and the excess of alkali titrated by standard acid. Our experiments yielded results of 60 per cent, of acetic acid, the extreme variations being 58-2 and 61-9 per cent. These numbers are in agreement with those of Green and Perkin (loc. cit.). It is evident that the methods of hot alkaline saponification invariably yield results which are considerably higher than those obtained by cold saponification. There can be little doubt that the higher results are due to the action of alkali on the regenerated cellulose. Support to this contention was obtained by digesting two equal weights of filter paper with 50 c.c. of normal soda for two days, one at the ordinary temperature, the other at 85°. In the former case no alkali was consumed, while the heated product showed a loss of nearly 2 c.c. of normal alkali on titration. (5) Acid hydrolysis (Ost, loc. cit.). The substance is first digested with 50 per cent, (by volume) sulphuric acid. After 24 hours the liquid is diluted considerably and the acetic acid separated by steam distillation, and titrated with baryta water. In our experiments phosphoric acid was substituted for sulphuric acid in order to avoid error due to possible formation of sulphur dioxide. The results varied from 51-5 to 55-0 per cent, of acetic acid. (6) Acid hydrolysis (A. G. Perkin, Trans. Chem. Soc. 1905, 107). In this method the cellulose acetate is treated with ethyl alcohol and sulphuric acid, and the resulting ethyl acetate distilled into Dr Fenton and Mr Berry, Studies on Cellulose Acetate 21 excess of standard alkali. The ester is then saponified and the excess of alkali determined by titration. In our experiments phosphoric acid was used instead of sulphuric acid for the reason already mentioned. The results varied from 52-2 to 54-4 per cent, of acetic acid. In our opinion, preference should be given to the method of cold alkaline saponification of Ost. Not only are the results more uniform, but they agree well with those obtained by acid hydro- lysis. The latter methods are exceedingly tedious to carry out. We have also carried out some experiments with the use of hot baryta water as a saponifying agent and subsequent gravimetric determination of the barium, the results averaging 57-58 per cent, of acetic acid. The materials met with in commerce known as cellulose acetate are most probably mixtures or solid solutions of various acetates, not definite chemical individuals. If, however, it were desired to represent cellulose acetate as a chemical individual, the results of our analyses of a number of specimens do not correspond with the formula of the triacetate C6H7O2 (0C0CH3)3 which is commonly supposed. They agree better with the formula of a pentacetyl derivative of C12H20O10 and still better with that of a heptacetyl compound of CjgHgQOig. Thus Carbon Hydrogen Acetic acid CcH-Oa (OCOCH3)3 requires 50-0 5-5 62-1 per cent. C12H15O5 (OCOCH3)5 „ 49-4 5-6 560 C.sHaA (OCOCHs)^ „ 49-2 5-64 53-8 Our most reliable results average carbon 49-2, hydrogen 5-5, and acetic acid 54 per cent. Certain authors have stated that sodium ethylate may be used for the determination of acetyl in cellulose acetates. In investi- gating this reaction, we were surprised to find that ethyl acetate was always produced along with a yellow sodium derivative of cellulose. Quantitative experiments were performed in which the ethyl acetate was distilled into an excess of standard sodium hydroxide, and after saponification determined with standard acid. The residue was washed with alcohol to remove the unaltered sodium ethylate and this solution was titrated with standard acid. The residue was then treated with water to decompose the sodium compound and titrated also. It was found that the quantity of acetic acid converted into ethyl acetate to that becoming sodium acetate appears to depend to some extent on the proportion of sodium ethylate employed. The results can be explained, if the average commercial cellulose acetates are represented by the formula C12H15O5 (OCOCH3)5, by the equation: 22 Dr Fenton and Mr Berry, Studies on Cellulose Acetate C12H15O5 (0C0CH3)5 + CgHsONa + 4C2H5OH = CiaHigOgONa + 5CH3COOC2H5 which may be taken to represent the main reaction. In support of this, the yellow sodium compound from a similar experiment, after thorough washing with alcohol, was digested for several hours in a reflux apparatus with excess of methyl iodide, and the methoxy group in the resulting product determined by Zeisel's method. The result obtained was 9-2 per cent, of methoxyl , in agreement with that calculated for the formula C12H19O9OCH3. The adsorption of basic dyestuffs by cellulose acetate. Certain dyestuffs, such as gentian violet are adsorbed in con- siderable quantities from aqueous solution by cellulose acetate, the solid being coloured blue. Cellulose, it is true, also adsorbs the dye, but to a much smaller extent, and the solid becomes violet. This property may be utilized to identify unaltered cellulose in commercial preparations of cellulose acetate. Methyl orange gave negative results, but methyl red was adsorbed in considerable quantity, the solid becoming red. Free dimethylaminoazo benzene gave negative results, but the hydrochloride of this base was strongly adsorbed, the solid cellulose acetate assuming a pinkish yellow colour and the colour of the aqueous solution being almost completely discharged. The authors desire to express their grateful thanks to Mr J. W. H. Oldham, M.A., of Trinity College, for much valuable assistance in connexion with this investigation. Mr Oldham has also carried out a large number of experiments on the influence of the mode of preparation upon the resulting properties of cellulose acetate, and it is hoped that his results when completed may form the subject of a future communication. Mr Molin, An examination of SearWs method, etc. 23 An examination of Searle's method for determining the viscosity of very viscous liquids. By Kurt Molin, Filosofie Licentiat, Physical Institute, Technical College, Trondhjem. (Communicated by Dr G. F. C. Searle.) {Read 9 February 1920.] § 1. The determination of the coefficient of internal friction in very viscous liquids has been the object of measurements by many different methods. A review of these will be found in Reiger*. A number of more recent methods are given by Kohlrauschf, and among them is a method of Searle'sJ. An examination of this method is the object of the present paper. In his paper, "A simple viscometer for very viscous liquids," Dr SearleJ gives an account of a viscometer he has constructed. The method consists in causing a vertical cylinder to rotate within a coaxal cylinder containing liquid, and in determining the angular velocity of the inner cylinder for a known value of the driving couple. The couple is produced by the weights of two loads acting on a drum by two threads. The time, T seconds, of one revolution of the cylinder is found, and the length, I cm., of the inner cylinder immersed in the liquid is observed. Newton's statement is that f--^Tn' <1) where/ is the force per unit area which acts against the direction of motion and at right angles to the normal, n, to the surface, dV/dn is the velocity gradient, and rj is the coefficient of viscosity. In this statement the motion of the liquid is supposed to take place parallel to a fixed plane. Treating the liquid as incompressible, and modifying (1), by substituting the rate of shearing for dV/dn, so as to suit the case of rotation, we obtain the following formula: gD (a2 _ 62) fMT\ ^ (MT\ Here D is the effective diameter of the drum, a and h are the radii of the cylinders, and M is the mass of each of the two loads, which are required to move the inner cylinder with the constant angular velocity Q, such that 2ttJQ. = T. * R. Reiger, Ann. d. Phys., 19, p. 985, 1906. t F. Kohlrausch, Lehrbuch d. praktischen Physik, xii. Aufl., p. 268. % G. F. C. Searle, Proc. Cambridge Phil. Soc, 16, p. 600, 1912. 24 Mr Molin, An examination of Searle's method The angular velocity of the liquid about the axis of the cylinders, at a distance r from the axis, is given by _ 277 62 /^ ^ ^ T ' a^-bAr^~ When r = b, the radius of the inner rotating cylinder, oj = 1^ = 27r/r, and when r = a, the internal radius of the outer fixed cylinder, CO =0. This problem was first treated, not quite accurately, by Newton. The above results were given substantially by Stokes *, and are also given by Lambf and by SearleJ. The rate of shearing, rdco/dr, varies somewhat as r increases from b to a, as is shown by the formula doi 27r 2a%^ r dr T ' (a2 _ §2) ^2 • We have only taken into account the friction between the coaxal cylindrical layers of the liquid and not the friction between the horizontal layers in proximity to the bottom surface of the movable cylinder, and have not considered the conditions that arise near that surface. In practice, only the lower end of the rotating cylinder is exposed to viscous action ; Dr Searle makes an allowance for this end by writing ^^^•r+i' (2) where I is the length by which the height, /, of the liquid, in the simple theory, must be increased, in order that the increase of couple shall correspond to the viscous action in proximity to the end surface and the edge of the rotating cylinder. Dr Searle gives a graphical method of determining k. The values of MT are plotted against I, and he says, "It will be found that the points lie on a straight line, which cuts the axis of I at a distance k from the origin." Dr Searle adds "If the corresponding total load hung from each thread be M grammes, it will be found, on repeating the observation with various loads, that MT is constant for a given level of liquid. This result confirms the fundamental assumption that the viscous stress at each point is proportional to the rate of shearing of the liquid." * G. G. Stokes, Brit. Ass. Report, p. 539, 1898. t H. Lamb, Hydrodynamics, Third Ed., p. 546, 1906. loal ^' ^' ^' ®^*^^®' ^°^- ^^f-' P- 602. Compare C. Brodman, Wied. Ann., 45, p. 163, for determining the viscosity of very viscous liquids 25 § 2. In my experiments I used Dr Searle's viscometer, as supplied by Messrs W. G. Pye and Co., Cambridge*. I determined the viscosity of treacle, as Dr Searle refers to a determination of 77 for that liquid. I found 26 = 3-74 cm., 2a = 5-01 cm., and D = 1-95 cm. Since g = 982 cm. sec. -2 at Trondhjem, the con- stant C has the value C = 3-070 ± 0-035. From the data given by Dr Searle, I find for the constant of the instrument used by him, Cg = 3-153. In my instrument the rate of shearing for radius r is given by ^_ _27r 15-80 '^ dr~ T ' r^ ' § 3. To examine how MT depends upon M, when I is kept constant, six series of observations were taken with six values of I varying from 10-0 to 2-15 cm., and in each series M was made to vary from 5 to 205 grammes. Since the viscosity of highly viscous substances diminishes very rapidly as the temperature increases, as was shown by Reigerf and by Glaser J for values of rj of the magnitudes 4-8 x 10^ to 67-2 x 10®, and by Ladenburg§ for '>7 = 1-3 x 10^, great care must be taken to keep the temperature constant. The apparatus was, therefore, placed in a thermostat with electric temperature regulation, and a very constant temperature of 19-8° C. was /naintained. The apparatus was left in the thermostat for 24 hours before the measurements were begun, and, during the short time a rotation trial was in progress, only the outer wooden door of the thermostat was opened, since one could see into the thermostat through the inner glass door. The final measurements were all carried out in the course of a day; the observations were made at intervals of about 10 minutes, so that the unavoidable disturbances of tempera- ture, due to the manipulations, might have time to disappear. In other respects the measurements were carried out in ac- cordance with Dr Searle's II instructions. The revolutions were timed by aid of a stop-watch and the times were taken for different numbers of revolutions with odd numbers up to 9, as well as the average time for one revolution. As no decrease in the time of a single revolution could be noticed as the rotation continued, the divergences from the mean lying within the limits of the errors * Catalogue of Scientific Apparatus manufactured by W. G. Pye and Co., List No. 120, p. 39, 1914. t R. Reiger, loc. cit., p. 998. X H. Glaser, Ann. d. Phys., 22, p. 719, 1907. § R. Ladenburg, Ann. d. Phys., 22, p. 309, 1907. II G. F. C. Searle, loc cit., p. 603. 26 Mr Molin, An examination of Searle's method of observation, there was no observable acceleration. AVe may conclude that, even for the greatest values of M, the viscosity of the liquid remained sensibly constant, in spite of the fact that some potential energy was converted into heat. The values of T found in these experiments are given in Table 1. Table 1. Time, in seconds, of one revolution of cylinder. M Z = 10-0 Z = 8-45 Z=:7-65 Z = 5-50 Z = 3-30 l = 2-\5 grm. cm. cm. cm. cm. cm. cm. 5 129-0 7 114-6 50-7 10 120-4 108-7 100-2 71-3 44-7 12 93-3 85-0 77-0 54-3 15 71-5 65-3 59-0 42-3 26-6 19-5 20 52-8 47-0 41-7 29-7 25 4M 35-9 32-2 23-6 14-5 11-1 30 33-3 29-6 26-4 19-0 35 28-8 24-8 22-2 16-3 10-0 40 251 21-7 19-2 14-0 45 22-0 19-0 171 12-3 55 17-9 15-2 13-6 101 6-3 4-6 65 14-9 12-6 11-4 8-3 75 12-8 10-9 9-8 7-2 4-5 3-4 105 9-1 7-7 7-0 5-1 155 61 5-2 4-S 3-5 205 4-6 3-9 3-6 The results have been plotted in the form of six curves each for one value of I, as in Diagram 1. The curves are represented in the form T {MT, M )^.eonst. = 0. From the diagram it is clear that the function T {MT, T)i = does not represent a family of straight lines parallel to the ilf-axis, and that each of the six curves has a hyperbolic appearance. When M approaches a certain lower limit Mq, MT tends to infinity. The area covered by the group of curves can be divided by a parabolic boundary curve into two departments, in one of which MT is sensibly constant for a given value of I. § 4. I have, further, examined how MT depends upon I, when M is kept constant, and have found that the function F {MT, Z)^,=eonst. = for determining the viscosity of very viscous liquids 27 represents, not a single straight line*, but a family of approxi- mately straight lines. Each line can be represented by the equation MT =al + p. For this group of curves d {MT)ldl tends to a definite value as M increases, i.e. the curves approach a certain border line MT IhagTa; Til- I Wn;i i)-o. * t ^ l^ '~~~~ \\M l=1QO 1 \ \ -^ / / (1-5,3) T=a^5 \ --- / / 1=765 / * V J (1-H.3) ■^ r' il-3.'^] 1=5,50 - «K>0 V il [' (i-:i,35) \ \ •v.,^^^/' ]=S.SO \ / \ ■"--^ / fl-hl 1=Z.15 / / / N y Calculat e^forl- 3. ^ o lO 10 30 it-o so €0 70 60 90 100 110 1Z0 r3o i4o ;5o 160 170 180 190 200 aw which is comparable with Searle's straight line. The coefficients a and ^ have been calculated for each line by the method of least squaresf , using the formulae 1.1 . HMT - 6111 . IslMT ^ 1.1 . i:iMT - IIMT . ZV- (SZf - 6S/2 (S^)2 - 6SZ2 * G. F. C. Searle, loc. cit., p. 604. t F. Kohlrausch, Lehrbuch d. jn-aktischen Physik, p. 13, 1914. 28 Mr Molin, An examination of SearWs method the various observations being regarded as having equal weights. The values of a and ^ have been thus calculated for seven different lines, and the results are given in Table 2. Table 2. Values of a, § and k. M grm. , a ^ k cm. 12 105-86 78-40 0-740 15 102-64 65-61 0-639 20 99-29 53-54 0-539 35 108-80 47-79 0-439 65 90-78 43-83 0-482 75 90-27 43-63 0-483 105 89-79 45-47 0-506 1 When I = 0, then MT = ^, and Table 2 shows how ^ varies with M. The curve thus extrapolated for ^ = is marked "Calculated for Z = 0" in Diagram 1. When MT = 0, we have ^ = | ^ | = | /3/a | , where k is the correction for the lower end of the rotating cylinder. 0,7 0,6 qs Q3 \ Diagram % ^ \ 2(k, n]=q 'x o • — o— — «— 10 ZO 30 ^O 50 60 70 80 90 100 110 ISO ISO 1^^o I^ gr. Diagram 2 shows how k depends upon M. The facts here recorded show that equation (2) should be replaced by M,T -n^c i + k,' where k^ is the value of k corresponding to the load M^. .(3) for determining the viscosity of very viscous liquids 29 If the value of h^ corresponding to M^ is read off from the curve of Diagram 2, the viscosity 17 can be calculated by equa- tion (3). The values of k found from Diagram 2 have been used in forming Table 3. Table 3. Values of M^T/il + k^). M Z = lO-0 i = 8-45 1 = 1 -m ? = 5-50 Z = 3-30 grm. cm. cm. cm. cm. cm. 10 111-7 118-0 118-5 ] 13-4 109-5 12 104-2 111-0 110-0 104-5 15 101-2 108-0 106-8 103-2 1010 20 99-3 104-5 102-0 98-6 25 97-9 100-1 99-1 98-5 95-1 30 96-5 99-0 97-2 95-4 35 95-6 97-0 95-6 94-7 92-5 40 950 970 94-5 93-7 45 94-5 96-0 94-3 92-5 55 93-3 94-0 91-5 92-4 91-6 65 92-4 92-0 90-7 89-7 75 92-0 91-7 90-4 89-7 105 91-1 90-2 90-4 901 155 90-3 89-2 90-6 89-6 205 90-0 89-3 89-3 From Table 3 it appears that the area in Diagram 1 in which equation (3) holds good is restricted to that part of the diagram to which the parabolic boundary curve is convex. From the values of MT derived from Table 1 and plotted in Diagram 1, the equation of the parabola is found to be M'^ = 11-26 {MT). I have not been able to give the parabola any definite physical interpretation, and it ought to be regarded as representing a diffuse limit region. But it is only when we pay regard to this, that we obtain values of 7) differing from each other by amounts lying within the limits of experimental error*. To make a comparison with the values of M and I which Dr Searle has used, I have, in Diagram 1, plotted (the broken hne) his values of i/Tf (strictly speaking, MTjC, which are comparable in magnitude with my values of MT) against M. Dr Searle has pointed out to me that the effect shown in Diagram 1 might conceivably be due to pivot friction. I have carefully considered this possibility. Before the liquid was put into the apparatus, I adjusted the pivots so that the rotation due * Compare G. F. C. Searle, loc. cit., Table II, p. 606. I Calculated from Table 1, G. F. C. Searle, loc. cit., p. 605. 30 Mr Molin, An examination of Searle's method to the weights of the two empty pans (5 grm. each) was so rapid that I was hardly able to measure, for instance, 3T by using a stop watch. I have, therefore, not been able to take account of any pivot friction. This cause of error would, at any rate, produce effects much smaller than those actually found. § 5. From the results for M = 205 grm. given in Table 1 we find the mean value -q = TIA:-1 dyne sec. cm.-^, for the temperature of + 19-8° C. To show how t] depends upon 3^0 ( 3« ^j 1 Diagrams. ja 3« r 4 — 1— 1 - 1 1 1 S>«nt 1. ViS i * Vtt \ 320 — r^ \ 3. \ ■Q 315 310 305 io • — i: nngular veiociL^ oi the "^-q Rotating C^inder. qi GX \ '\ « L Tf(i3- 90'r ° \ J 290 285 280 •o\ A • \ « \ ^j ' * .> ^^ ^ e e 270 ■ , A 0,1 Qi 03 0^ q5 Q« Cr7 <;» <^9 ^O -p V^ t^ y^ \5 1/5 ^7 1^ the angular velocity O = 27t/T, the values of 17 and Q, obtained from the first three series, have been plotted in Diagram 3. The curve drawn among the plotted points suggests that the relation between -q and Q. can be expressed in the form 7] = 274-7 + <f) exp (- AO^). To find the constants cf), A and x, I considered the equation log, [rj - 274-7) - log,^ - A^^' (4) /or determining the viscosity of very viscous liquids 31 When the values of loge (-7 — 274-7) were plotted against O, the curve was roughly a straight line. Hence x may be taken as unity, and thus the number of constants to be found is reduced to two. By the method of least squares, I obtained logf<^ = 4-375 and A = 5-694, and thus r^ig.g = 274-7 + 79-44 e-^-''^'*". .(5) Equation (5) expresses the results of the observations when Q. exceeds 0-1, but not for smaller values of Q. § 6. Experiments carried out at different temperatures showed that the curves representing the function T [MT, M\ are of the same character as those given in Diagram 1. Table 4 gives the values of 7] found for various temperatures. In these experiments I was 10-0 cm. ; and, at each temperature, six different loads were used, in order that I might be able to decide with certainty that the values of M, used in calculating the value of 17 for each temperature, lay in the area to the right of the parabolic boundary line of Diagram 1. The same value of k, viz. the limiting value 0-48 cm. shown in Diagram 2, was used in calculating the Table 4. Values of 7] at various temperatures. Temp. V Temp. ■n t°C. Dyne sec. cm."^ t°C. Dyne sec, cm. ^ 19-8 274-7 8-75 1950 18-0 415 6-2 2700 13-0 860 60 2750 11-8 1140 2-8 4970 11-6 1200 various values of rj. These values are not claimed to be exact. In these experiments it was very difficult to keep the temperature constant during each series of observations, and thus a deter- mination of k for each temperature was out of the question. From the curve of the function -q =f{t), shown in Diagram 4, it follows that I drj/dt I rises rapidly as -q increases; this tallies with what was said above. "32 Mr Molin, An examination of Searle's metJiod 4000 9 10 Tl 1Z 13 IV IS IS 17 18 19 20 jTemptX- § 7. I thought it would be interesting to compare the results given by Searle's method with those obtained by Poiseuille's method. The utility of the latter method for very viscous liquids* is proved by the investigations of Kahlbaum and Eaberf for values of rj in the neighbourhood of 40, and by LadenburgJ for r] = 1-3 X 10^. Fausten§ has found that the length of the dis- charge tube must exceed 45 cm., if the simple Poiseuille formula is to represent actual facts. In the formula h = Height of liquid corresponding to difierence of pressure between ends of tube. R = Internal radius of tube. L = Length of tube. p = Density of liquid (= 1-4103 ± 0-0003 grm. cm.-^at 19-8° C). m = Mass of liquid discharged. t = time of discharge. For shorter tubes, Hagenbach's* correction must be employed; otherwise the value obtained for 77 will be too high. As the liquid flows out into the air in an even jet, it carries kinetic energy with it; in order to allow for this, the value of t] given by Poiseuille's. * H. Glaser, Eriangen Diss., 1906. t G. W. A. Kahlbaum and S. Raber, Acta Ac. Leap., 84, p. 204, 1905. X R. Ladenburg, Ann. d. Phys., 22, p. 298, 1907. § A. Fausten, Bonn. Diss., 1906. for determining the viscosity of very viscous liquids 33 formula must be multiplied, according to Hagenbach*, by a cor- recting factor slightly less than unity. As the thermostat could only accommodate tubes shorter than 45 cm., Hagenbach's correc- tion was calculated, but was found to be negligible. Ladenburgf points out that both Hagenbach's and Couette's corrections to Poiseuille's formula can be entirely ignored for liquids such that t] is of the magnitude 1-3 x 10^. The discharge vessel consisted of a wide glass cylinder; through the bottom of this was bored a hole through which the discharge tube was connected with the interior of the cylinder. The whole apparatus was placed in the thermostat and the same temperature, 19-8° C, was" maintained as was used in the earlier experiments, AVhen a tube whose internal radius was about 0-26 cm. was used, the liquid did not issue in a continuous jet but in drops. The values obtained for -q are given in Table 5. The mean value is r] = 271-1. The value obtained by Searle's method, viz. 274-7, differs from that obtained by Poiseuille's method by 1-3 per cent.; the agreement may be regarded as good. Table 5. Values of 7] by Poiseuille^s method. Rem.. Zcm. h cm. m grm. t sec. ■n 0-3168 46-48 49-36 49-93 54-421 53-568 1790 1757 269-9 272-3 § 8. The influence of the base of the rotating cylinder can be eliminated, without determining k, by using the relation J 77 = C . ^ -, = Cy, n ~ h provided that the points corresponding to M^T^ and M^T^ He to the right of the parabolic boundary line in Diagram 1. If we put Zj = 10-0 cm., we obtain the results given in Table 6. * F. Kohlrausch, Lehrbuch d. praktischen Physik, pp. 264 — 269, 1914. t R. Ladenburg, loc. cit., p. 298. i Compare C. Brodman, loc. cit., p. 163. VOL. XX. PART L 34 Mr Molin, An examination of Searle's method, etc. Table 6. Values of y. MT 944 804 734 537 342 251 215 I 10-0 8-45 7-65 5-50 3-30 J 90-2 89-3 90-5 90-0 88-4 Wlien the various values are given the same weight, the mean value of y is 89-7, and then 17 = 2754-. § 9. Diagram 3 and formula (5) show that 77 cannot be re- garded as independent of Q. unless Q. exceed a certain value, in this case 0-9. Since Q. is related to the rate of shearing rdw/dr, according to the formula it follows that 7y is a function of the rate of shearing. Hence, the assumption on which formula (1) is based, viz. that rj is independent of the rate of shearing, seems to be unjustifiable for small values of the rate of shearing, at least in the case of the highly viscous liquid used in these experiments. Miss Haviland, Note on Antennal Variation in an Aphis 35 Preliminary Note on Antennal Variation in an Aphis (Myzus ribis, Linn.). By Maud D. Haviland, Fellow of Newnham College. (Communicated by Mr H. H. Brindley.) [Read 8 March 1920.J In 1918, during an investigation of the life-history of the Red Currant Aphis, Myzus ribis, Linn., it was observed that consider- able variation occurred in the antennae of the winged partheno- genetic females; and the evidence pointed to the conclusion that this variation was induced by the food^. Antennal variation in certain Aphididae has been studied by Warren ^ Kelly^ Ewing- and Agar^. Warren's experiments on Hyalopterus trirhodus showed some diminution of the correlation co-efficient in passing back from parent to grandparent. Kelly, for Aphis rumicis, con- sidered that somatic variations of the parents were not inherited by the offspring. Ewing, who bred eighty-seven generations of Aphis avenae, concluded that the variations were not transmitted to the offspring. Agar found some evidence of a partial inheritance of individual variations in Macrosiphum antherini, but he showed that this might be due to causes other than true inheritance. Myzus ribis is a common pest of red currant bushes. The sucking of the aphides upon the leaves tends to cause red galls or blisters, within which the plant lice continue to feed and reproduce. The fifth and sixth antennal segments of the winged partheno- genetic females normally bear two sense organs of unknown function — one on the distal third of Seg. v., the other on the proximal third of Seg. vi. It was observed in 1918 that, in indivi- duals reared on red blistered leaves, these sensoria were placed comparatively close to the articulation of Segs. v. and vi. On the other hand, if the aphides were fed upon green unblistered leaves, the sensoria were placed further away from the articulation. For the sake of brevity, the first type of antenna will be referred to hereafter as the Red (or R) type, and the second as the Green (or G) type; but every degree of transition may exist between the two extreme types. The experiments of 1918 were incomplete, and were conducted with a polyclonal population. They were repeated in 1919 with a monoclonal population, but the results are still far from being- conclusive owing to the small numbers available in some genera- tions. Only the winged forms show the required character. The production of these forms is probably governed by environmental factors which at present are imperfectly understood, and, for some 3—2 36 Miss Haviland, Preliminary Note on Antennal reason, in the population used in 1919, it was unusually low. It is hoped to repeat and extend the range of the experiments in 1920. The character chosen is the distance between the sensoria of antennal segments v. and vi. and the articulation of these two segments, expressed as the percentage of the width of the head between the eyes. The ratios are shown separately for each segment, with a dividing line to represent the articulation. ^, ,„ , , ,, , Seg. VI. = 19% of the head- width Thus J# denotes that -^ ■^, — j— j — ^ — . ,^, . ^ Seg. V. = 8% of the head- width Each generation is designated by combinations of two letters: E, (= red leaves) and G (= green leaves) and numerals, which express its complete ancestry. Thus ^^^^ denotes the fourth generation from the fundatrix of the population, and the F^. generation after transference to Green leaves after two consecutive generations on Red blistered leaves. In the transferred generations, the aphides were removed to the new environment when less than twelve hours old. The individuals for transference were selected wholly at haphazard. Thus, if a brood mother Eg gave birth to four young in the day, two were transferred to red blistered leaves, and two to green leaves, and so on in equal numbers from day to day. The pure Red (RRR, etc.) lines, and pure Green (GGG) lines were used as controls. The latter unfortunately became extinct in the third (Gg) generation. Hence for later generations the next longest unbroken line on green leaves (R2G0, etc.) had perforce to be taken as the control, though as it had been fed for the first two generations upon red leaves, it cannot be regarded as wholly * satisfactory. In Table 1, the curves of error of the ratios of genera- tions R2, R4 and R2G4 are shown. Rg is the common ancestral generation. The mode of the curve of R4 tends to shift to the left, i.e. the ratios of the antennal segments to the head-width are smaller. For the sake of clearness, in the graph only the curve of R4 is shown, but those of Rg, R5 and Rg, though with a smaller number of individuals, are almost identical with it. The curves of the ratios of R2G1 and R2G2 are very similar to their red controls. The R2G3 generation produced very few winged individuals, but these indicate a somewhat greater range of variation in Seg. vi. The curve of R2G4, as shown in the graph, has a marked tendency to shift to the right, indicating that the ratio of the antennal joints to head-width has increased, and this tendency is maintained in the succeeding generations, R2G5 and R2Gg. The position in the generation series does not account for the change in the antennal structure, for the modes for the six Red generations are nearly identical. So far we have considered only the modes. The mean ratios of the different generations are dealt with in the succeeding tables. I Variation in an Aphis (Myzus ribis, Linn.) 37 Table 2 shows the mean ratios of the successive generations in four lines of descent, including the red and green controls. The extinction of the green control line was unfortunate, and in future experiments it will be very desirable to obtain a pure green line. At present the explanation that suggests itself of the variation of the RgGrg. . . line is that the influence of red feeding persists for at least two, and probably three generations after removal to different food, and this is somewhat confirmed by the R4G1. . . etc. line. Tables 3, 3a, 4, 4a and 5, 5a, give the effect of transference upon the mean ratios of the first, second, and third generations respectively, and below each is an analysis of the ratio of each segment, indicating its increase or decrease over previous genera- tions and the controls. Examination of the figures seems to show that the ratios of the first generation after transference vary irrespectively of the parental ratio. In transference to Red, the ratio of Seg. v. increases over that of the parental ratio, but in Seg. vi. it decreases (Table 3). In transference to Green, the results for both segments are quite inconclusive as regards the parental ratio (Table 3a). In the second generation after transference to Red, the results are like- wise inconclusive for both segments (Table 4). After transference to Green, the ratio of Seg. v. shows a tendency to rise above, and Seg. vi. a tendency to fall below, the parental and grand- parental ratios (Table 4a). In the third generation after transference to Red, the ratio of Seg. v. rises above the ancestral ratios, and that of Seg. vi. falls (Table 5). After transference to Green, the ratio of Seg. v. rises above those of the ancestral generations, and that of Seg. vi. rises in one case and falls in the other (Table 5a). These results are inconclusive, but examination of the control ratios shows that, with occasional exceptions, the ratio of a genera- tion with a mixed ancestry tends to rise above that of the Red control, but remains below that of the Green. Many more experi- ments in transference are required, and a much larger number of individuals must be examined before any conclusion can be reached; but at present the evidence suggests that the antennae of Myzus ribis are modified according to the food supplied, and that the effect induced by feeding in one generation is discernible in the succeeding three or four generations. It is difficult otherwise to explain the difference between the ratios of Rg and R2G4, and between Rg and R4G3, which, translated into the terms of human relationship, would be third cousins, and first cousins once removed, respectively, for all were produced by parthenogenesis, and, except for the food, reared side by side under identical environ- mental conditions. 38 Miss Haviland, Preliminary Note on Antennal Table 1. Curves showing the ratio of the distance of the sensoria from the articulation of antennal Segments V and VI to the width of head. The lower curves refer to the fifth, and the upper to the sixth segment. — B2 generation = E^Gi Table 2. Mean ratios of the successive generations of the lines^ Cto i?3 ..., JR/jG^ ..., Hcfii ..., and R2GQR1 II III IV V VI VII VIII (^2¥ GsU R2 ^^0- R3¥- R4¥ R5 ¥- Re -¥- — — )! i) » 5> )) }■> R4G1 ""7" R4G2 V ^iGsB — 1) H RA¥- E2G2 1§ R2G3 n R2G4 t"5 R2G5 M RgGe ft )> )) >5 ?) )) )) 55 55 R2G3R1 n R2G3R2 V — J Variation in an Aphis (Myzus ribis, Linn.) 39 Table 3. Mean ratios of the first generation transferred from Green leaves to Red blisters, ivith an analysis below. + = increase over ancestral ratio - = decrease from „ ,, = identical with ,, ,, Generation Parental Generation Red Control Green Control Ratio Ratio Ratio Ratio GiRi -V- G no winged forms R24? G2¥ G2R1 f* G, ^^ R3¥ Gsfl R2GjRi \f R2G1 ^^- R4¥ R2G2 i§ GlR2¥ GjRi -gi- R3¥ G3ff R4GXR1 ¥ R4G1 ■\-- R6¥ R2G4 ^ R2G3R1 f§ R2G3 \i R6¥ R2G4 f f R2G4R1 i\ ^i^i 15 Rc¥ R2G5 f 1 Segment V Generation Variation from Parental Ratio Variation from Red Control Ratio Variation from Green Control Ratio GiRi no winged forms — G2R1 + + - R2GiRj + + - G1R2 - R4GiRj + - R2G3R1 + - R2G4Rj^ - + — Segment VI Generation Variation from Parental Ratio Variation from Red Control Ratio Variation froln Green Control Ratio GiR, no winged forms + - G2R1 - + - R2GxRi - - G1R2 - + - R4GjRi - + - R2G3R1 + - R2G4R1 — + — 40 Miss Haviland, Preliminary Note on Antennal Table 3a. Mean ratios of the first generation transferred from Red blisters to Green leaves, with analysis as in Table 3. Generation Eatio Parental Generation Eatio Green Control Eatio Eed Control Eatio ExGi \«- R no winged forms G2¥ R,^^ R2G1 ¥ R2 ¥ G-3 ft R3¥ GiRiGi -V- GfiRi "V" Gaff R3¥ RsGi^ R3 -¥ - G3 f f E4 ¥ GiRA¥ G1R2 ¥ R2G2 1§ E4¥ R4G1 ¥ R4 ¥ K2G3 15 1 Segment V Generation Variation from Parental Eatio Variation from Green Control Eatio Variation from Eed Control Eatio RiGi no winged forms — R2G1 - - - GiRiGi + - + R3G1 - GjR2Gi + - + R4G1 - - - Segment VI Generation Variation from Parental Eatio Variation from Green Control Eatio Variation from Eed Control Eatio RjGi no winged forms _ _ R2G1 - - + GjRiGx - - + R3G1 - + G^RgGj + + R4(]ri + + + Variation in an Aphis (Myzus ribis, Linn.) 41 Table 4. Mean ratios of the second generation after transference from Green leaves to Red blisters, with analysis as in Table 3. Generation Ratio Parental Generation Ratio Grand-parental Generation Ratio Red Control Ratio Green Control Ratio G,R2 V- G2R2 if R2G3R2 V- GiRi -^ G2Ri U R2G3R1 f g Gj no winged forois G2 ¥ R2G3 H R3¥ Rg¥ G3tt G3 If K2G5 f § Segment V Generation Variation from Parental Ratio Variation from Grand-parental Ratio Variation from Red Control Ratio Variation from Green Control Ratio G,R2 G2R2 R2G3R2 + no winged forms + + — Segment VI Generation Variation from Parental Ratio Variation from Grand-parental Ratio Variation from Red Control Ratio Variation from Green Control Ratio (hR2 R2G3R2 — no winged forms + + — 42 Miss Haviland, Preliminary Note on Antennal Table ia. Mean ratios of the second generation after transference from Red blisters to Green leaves, with analysis as in Table 3. Geueratiop Ratio Parental Generation Ratio Grand-parental Generation Ratio Green Control Ratio Red Control Ratio R2G2 i§ R4G2 \^ R,Gi -V- R4G1 '^-f- R2 4P R4 V- G3 ti R2G4 ft Segment V Generation Variation from Parental Ratio Variation from Grand-parental Ratio Variation from Green Control Ratio Variation from Red Control Ratio R2G2 R4G2 + + + + - + + Segment VI Generation Variation from Parental Ratio Variation from Grand-parental Ratio Variation from Green Control Ratio Variation from Red Control Ratio R2G2 R4G2 - + - + + Variation in an Aphis (Myziis ribis, Linn.) 43 Table 5. Mean ratios of the third generation after transference from Green leaves to Red blisters, with analysis as in Table 3. Generation Ratio Parental Generation Ratio Grand-parental Generation Ratio Great-grand - parental Generation Ratio Red Control Ratio Green Control Ratio G1R3 ¥ G1R2 ^' G,R, i^i 110 winged forms K4V- GsM Segment V Generation Variation from Parental Ratio Variation from Grand-parental Ratio Variation from Great-grand- parental Ratio Variation from Red Control Ratio Variation from Green Control Ratio G1R3 + + no winged forms + - Segment VI Generation Variation from Parental Ratio Variation from Grand-parental Ratio 1 Variation from ! Variation Great-grand- ' from Red parental Control Ratio Ratio Variation from Green Control Ratio G1R3 - - 1 no winged + forms 1 - 44 Miss Haviland, Note on Antennal Variation in an Aphis Table 5a. Mean ratios of the third generation after transference from Red blisters to Green leaves, with analysis as in Table 3. Generation Ratio Parental Generation Ratio Grand- parental Generation Ratio Great-grand- parental Generation Ratio Green Control Ratio Red Control Ratio E2G3 u K4G3 !l R2G2 \% R4G2 -V- R2G1 -v- R4G1 5^ R4 V Gsff RsGsfl R5 ¥ Rg¥ Segment V Generation Variation from Parental Ratio Variation from Grand-parental Ratio Variation from Great-grand- parental Ratio Variation from Green Control Ratio Variation from Red Control Ratio R2G3 R4G3 + + + + + - + + Segment VI Generation Variation from Parental Ratio Variation from Grand-parental Ratio Variation from Great-grand- parental Ratio Variation from Green Control Ratio Variation from Red Control Ratio R2G3 — — — — + R4G3 + + + + + LITERATURE REFERRED TO IN THE TEXT. (1) Agar, W. E. (1914). "Experiments on Inheritance in Parthenogenesis," Phil. Trans. Roy. Soc, Series B, vol. ccv, pp. 421-487. (2) EwiNG, H. E. (1916). " Eighty-seven generations in a parthenogenetic pure line of plant lice," Biol. Bull., vol. xxxi, No. 2, pp. 53-112. (3) Haviland, Maud D. (1919). " On the Life History and Bionomics of Myzus rihis, Linn.," Proc. Roy. Soc. Edinburgh, vol. xxxix, pt. 1 (No. 8), pp. 78-112. (4) Kelly, J. P. (1913). "Heredity in a Parthenogenetic Insect," Ainer. Nat., vol. XLVii, pp. 227-234. (5) Warren, E. (1901). "Variation and Inheritance in the Parthenogenetic Generations of an Aphis, Hyaloptenis trirhodus, Walk.," Biometrika, vol. I. Mr Waran, Ejfect of magnetic field on Intensity of spectrum lines 45 The effect of a magnetic field on the Intensity of spectrum lines. By H. P. Waran, M.A., Government Scholar of the University of Madras. (Communicated by Professor Sir Ernest Rutherford.) [Read 8 March 1920.] [Plates I and II.] Since the discovery of the Zeeman effect the main attention has been directed to the detailed study of the phenomenon of the small change of wave length suffered by a monochromatic radiation in a magnetic field. The question whether a magnetic field affects the spectrum as a whole has not received much attention. While wiorking on the Zeeman effect with a mercury discharge tube run by an induction coil as the source, a small portion of the capillary tube being subjected to a magnetic field of about 5000 c.G.s. units as shown in Fig. 1, the light was observed to suffer a change in intensity and also in colour opposite the pole pieces when the field was thrown on. A spectroscopic examination revealed the existence of some selective changes in the spectrum in addition to the increased brilliancy of the general spectrum. It was also noticed that the changes taking place varied considerably with the pressure, at a low pressure the tube showing little change visually but greater changes in the general spectrum. Attention was concentrated on the latter. In the case of mercury which was the first spectrum investigated, the tube, containing a trace of residual air at very low pressure, gave the principal mercury lines, viz. : 5790-66, 5769-6, 5460-7, 4916-0, 4358-34 and the principal hydrogen lines 6563, 4861-5 and 4340-7. On applying the magnetic field, however, marked changes were observed, including a new set of lines at 5426, 5679, 5872 and 5889, and a very strong red line at 6152, brought out prominently by the field. Mercury lines have been recorded at these wave lengths and these lines brought out are probably due to mercury. The behaviour of the line 6152 was very remarkable. It was invisible under ordinary conditions but showed up brilliantly in the magnetic field, the effect being practically instantaneous. Exhausting the tube still further and increasing the current through the tube to about 5 m.a. Four faint lines appeared at wave lengths 6234, 6152, 6123 and 6072, 46 Mr Waran, The effect of a magnetic field and corresponding to these wave lengths mercury lines are recorded by* Stiles, Eder, Valenta, Arons and Hermann. But Arons and Fig. 1. Hermann have not recorded the line 6152, while Stiles records it as of equal intensity with the line 6234. Eder and Valenta have not observed the latter lines at all, but record the line 6152 as * Kayser, Handbuch der Spectroscopie, Band v. p. 538. on the Intensity of spectrum lines 47 one of veiy great intensity. Examining the efEect of the magnetic fields on these four lines, it is very interesting to note that the line 6152 alone increases about five times in brilliancy while the others if they suffer any change at all, decrease in intensity. It is also interesting to note that this line 6152 seems to be the same line that becomes so greatly enhanced when the tube contains a trace of helium as observed by* Collie. It seems very difl&cult to excite this line unless at least a trace of helium is present in the ajjparatus and at this stage it is not possible to suggest any explanation of its abnormal behaviour. In addition to these very prominent changes there are also many minor changes, among which is the disappearance of a faint trace of continuous spectrum, as well as of some of the nebulous bands and lines, the remaining lines being quite sharp on a dark background. The abnormal behaviour of the mercury spectrum in the visible region (the ultra violet spectrum has not yet been investigated) 'suggested the study of other spectra and the spectrum of helium was next examined. The discharge tube contained hydrogen and a slight trace of mercury vapour as impurity and the hydrogen lines and the prominent mercury lines were also visible. The effect of the magnetic field in this case was to enhance the helium lines very considerably, leaving the hydrogen lines practically unaffected or even slightly reduced in intensity. In this spectrum there were also a few faint lines not yet identified definitely which remain quite unaffected by the magnetic field. In the further study of the helium spectrum, the gas was contained in a separate tube from which any small quantity of it could be introduced into the discharge tube. At a pressure of 1 mm. of mercury the addition of a small trace of helium produced no perceptible effect on the spectrum of residual air which showed the prominent hydrogen lines and the nitrogen bands, but no trace of any of the helium lines. But on switching on the magnetic field, the helium lines flashed out prominently and disappeared again as soon as the field was turned off. The effect is shown in the accompanying photo- graphs (Plates I and II). In a plate taken with a greater percentage of helium the lines are visible without the magnetic field, but a great enhancement of these lines with the field is evident, and a dense new fine at 49334 a.u. is also noticed which has not yet been definitely identified. The spectrum of neon was also studied, and in a tube kindly lent to me by Dr Aston, there was a trace of hydrogen also present, showing the three principal hydrogen lines. Here also the effect of the field was to enhance very considerably the neon lines, * Proc. Roy. Soc. 71, 25, 1902. 48 Mr Waran, The effect of a magnetic field leaving the hydrogen lines comparatively unaffected, so that by a casual examination of the spectrum the hydrogen and the neon lines can be distinguished from one another. The oxygen spectrum is rather difficult to excite when mixed with other gases. Yet a mixture of hydrogen, oxygen and a trace of helium was tried with success and here again the monatomic helium lines were brouglt out by the magnetic field, leaving the diatomic oxygen and hydrogen lines comparatively unaffected as shown in the photographs. From these experiments the natural inference follows that in a mixture of the monatomic and diatomic gases, the monatomic gases alone seem to be selectively affected in a peculiar way resulting in their spectrum lines alone being very considerably enhanced or brought out prominently even when not visible at all previously. By this method minute traces of the monatomic gases when mixed with other diatomic gases can be detected. On this view we might also explain the abnormal mercury line 6152 and others as due to the radiation from the monatomic atom while the other lines may be classified as belonging to the molecule. Examining the spectrum of the atmospheric air at low pressure in this way the effect of the magnetic field is to bring out new lines which are not present without the magnetic field, as shown in the photographs. As far as their wave lengths have been determined, though one or two of them fit in fairly well with lines catalogued as belonging to oxygen and nitrogen, yet there are others which are difficult to identify while the absence of other stronger lines of oxygen and nitrogen make even these two or three fits inconclusive. Another interesting point noted in these experiments is the varying degrees of enhancement under the influence of the field for lines belonging to the same element helium. Preston has shown that the Zeeman effect is of the same magnitude for lines belonging to the same series, but differs in different series. Simi- larly we might expect the degree of enhancement of the lines in the magnetic field to depend on the series to which the line belongs. The exact nature of this phenomena and the mechanism of the reaction that brings about these novel changes in the general spectrum is not yet definitely known and it is not desirable to attempt an explanation until the study of the spectrum has been extended to the ultra violet. The current in the tube was usually about 3 m.a. and the effect of the field was to decrease the current by about 20 to 30 per cent. The changes of intensity observed cannot be attributed to this since the reduction of the current by a spark gap in series only brings about a proportionate decrease in brilliancy of the general spectrum. Phil. Soc. Proc. Vol. xx. Pt. i (a) Hydrogen and Mercury (6) Hydrogen and Helium (c) Hydrogen and Helium, low percentage (d) Hydrogen and Helium in larger per- centage (e) Neon On Oft (/) Air Ullltl I t K II On Off (gr) Air On Off Fig. 2. Photographs showing the enhancing effect of the field. The small lateral shift is due to the camera slider, and in («) the mercury line 6152 is incUcated by the dot, while in the other cases the lines that newly turn up are indicated by the arrows. I Phil. Soc. Proc. Vol. xx. Pt. i. Plate II O H, (1) Air, Oxygen, Hydrogen and trace of Helium (2) Oxygen, Hydrogen and trace of Helium (3) Excess of Helium and trace of Air and Oxy- gen On OfE (4) Oxygen, Hydrogen and Helium and trace of Air On Off Fig. 3. Photographs showing the effect in mixtures of gases studied. on the Intensity of spectrum lines 49 It may be of interest to note that in solar spectroscopy the spectrum of the sunspots is found to difTer in many respects from that of the photosphere, considerable numbers of enhanced lines occurring in the sunspot spectrum. The existence of a powerful magnetic field in sunspots has been demonstrated by the Zeeman effect and possibly the differences in the spectrum of the sunspot and the photosphere may be attributed to this new effect of the magnetic field on the spectrum. The further study of this effect and the examination of other spectra are in progress. Cavendish Laboratory, Cambridge. VOL. XX. PART I. 50 Mr Brindley, Further Notes on the Food Plants Further Notes on the Food Plants of the Common Earwig (For- ficula aiiricularia). By H. H. Brindley, M.A., St John's College. [Read 8 March 1920.] In a paper pubHshed in the Proceedings of the Cambridge Philo- sophical Society, xix, Part 4, July 1918, p. 170, I recorded certain observations in August and September, 1917, on the food plants of the Common Earwig, with the view of obtaining more exact infor- mation than was then available as to the damage likely to be done by this species in a flower or kitchen garden. The paper also epitomised recent literature on the subject, a consideration of which had revealed a considerable amount of diversity and want of exact information as to the favourite food plants of earmgs in the British Isles. The observations made by myself were on earwigs kept in captivity in connection with a statistical enquiry as to the variation of the forcipes which is still in progress. The observations in 1917 were on earwigs from St Mary's, Isles of Scilly, and those recorded in the present paper were made in the second half of the year 1918 on a collection from the Bass Rock, which swarms with earwigs. The animals were all adults and were kept in large glass dishes bedded with sand slightly damped occasionally. Earwigs re- main healthy in a soaked substratum if the ventilation is good, but in captivity in a warm room without circulation of air they suffer heavy mortality from fungoid attack, as I have already recorded {Proc. Camb. Phil. Soc, xvii, Part 4, Feb. 1914, pp. 335-338). The fungus appears to be usually Entomofhthora forflculae (Picard, BuU. Soc. Etude Vidg. Zool. Agric. Bordeaux, Jan.-Apr., 1914, pp. 25, 37, 62). The importance of ventilation and of normal tem- perature is well illustrated by the far fewer fungoid attacks and the low mortality when the new Insect House belonging to the Cambridge Zoological Laboratory became available in 1919. It is at present too early to say how far an improvement is obtainable in the survival of eggs and young which it is hoped to rear in the spring in normal outside temperatures in the Insect House. Earwigs offer a great contrast to cockroaches as regards desire for water; the latter thrive in captivity for months in ^ warm room on food which is entirely dry, while earwigs certainly visit water to drink, as I have seen in both the captive and wild conditions. I have previously recorded {Proc. Zool. Soc. Lond., Nov. 1897, p. 913) how Stylofyga orientalis in captivity seems to pay no attention to a damp sponge when that is the only source of moisture. We have however to bear in mind that the Common Cockroach is probably an of the Common Earwig (Forficula auricularia) 51 immigrant from warmer countries of the East. The earwigs under observation during the past three summers had no animal food save that afforded by those which died. In order to obtain information as to preference for one kind of plant above another they were usually given three different species, taken haphazard, at a time, for a period of two or more days. In the following summary the observations of 1917 and 1918, with a few made in 1919, are combined. The dates when the different foods were given are noted, as in the latter part of September, when the animals tend to become lethargic, and in the succeeding two months the desire for food is much lessened, even in the artificial temperature of a laboratory. The capital letters after the names of the plants indicate those which were given at the same time, and the numbers appended indicate the preference exhibited by the earwigs: e.g. in food group M, M^ was attacked more than M^, M^ more than M^; in group F, F^ after two plants indicates that they seemed to be attacked equally, and more readily than F^: while in group Q, Q*' indicates that the plant offered was not attacked at all. Similarly for the other groups. 24-26 Aug. '17. Alkanet, Blue {AncJmsa sp.) C-: leaves not attacked; petals gnawed considerably. 27-29 Aug. '17. Anemone, White Japanese {Anemone japonica) D^: leaves not attacked ; petals eaten moderately. 1-23 Sept. '17. Apple {Pyrus Mains) F^: rather unripe fruit with skin whole was not attacked; but when cut across was gnawed moderately: 24-28 Sept. '18, leaves holed. 24-28 Sept. '18. Ai-tichoke, Jerusalem {Helianthus tuberosus) W: leaves holed and edges gnawed down to midrib; tuber, cross sUce attacked vigorously and its buds also devoured. 20 Sept.-5 Oct., 3-17 Nov. '18. Asparagus {Asparagus officmalis) OS T": leaves gnawed a little ; fruit not attacked. 26-31 Aug. '18. Aster, Mauve China {Callistephics chinensis) K-: leaves not attacked ; petals and flower buds much eaten. 6-11 Sept. '18. Aster, Pink China {Callistephus chinensis): leaves shghtly nibbled; petals much eaten ; ^OM;ers used as a refuge. 15-20 Sept. '18. Balm, Pale Mauve {Melissa officinalis) J^ : leaves not attacked : petals of buds devoured. 22-23 Aug. '17. Bean, Dwarf {Phaseolus vulgaris) B^: leaves nibbled ver;^ shghtly. 30-31 Aug. '17. Bean, Scarlet Runner {Phaseolus multiflorus) E^: leaves, floivers and pods apparently neglected: 16-18 Oct. '18, leaves holed a good deal and edges gnawed down to veins. 20-28 Oct. '18. Beard Tongue, Scarlet {Pentstemon sp.) R": leaves and flowers not attacked. 22-23 Aug. '17. Beet {Beta vulgaris) B^: leaves much attacked, especially the petioles, which were opened out and their pith devoured. 20-24 Sept. '18. Bell Flower, White {Campanula sp.) K^: leaves not touched; petals completely devoured. 31 Aug.-6 Sept. '18. Bindweed, Common {Convolvulus sp.): leaves much holed. 11-13 Sept. '18. Blackberry {Rubus fruticosus): vipe fruit well gnawed. 4—2 52 Mr Brindley, Further Notes on the Food Plants 30-31 Aug. '17. Cabbage, Garden {Brassica oleracea capitata) W : leaves gnawed down to midrib and veins and ends of veins eaten off. 2-5 Oct. '18. Canterbury Bell, Blue {Campanula medium) N^: leaves and petals well devoured. 6-7 Sept. '18. Carrot {Daucus Carota): root not attacked where covered by skin, but cut end was much gnawed. 6-11 Sept. '18. Celery {Apium graveolens) H^: leaves holed and their edges gnawed. 29 Sept.-3 Oct. '18. Cherry {Prunus [Cerasus] sp.) M^: leaves not attacked. 20-23 Oct. '18. Chickweed {Stellaria media) R^: edges of leaves gnawed sUghtly. 31 Aug.-6 Sept. '18. Chrysanthemum, Garden {Chrysanthemum indicum): flower buds used as refuge, tips of petals apparently somewhat nibbled: 31 Aug.-6 Sept. '18, purple variety: edges of leaves much nibbled; flower buds used as refuge, tips of petals apparently somewhat nibbled: 31 Aug.- 6 Sept. '18, white variety: leaves not attacked; petals much eaten. 20-24 Sept. '18. Clematis, White {Clematis sp.) K^: leaves, a few eaten off at ends and edges gnawed here and there ; flowers entirely devoured. 23-27 Oct. '18. Cluvia miniata (Natal): leaves not attacked; petals gnawed a little along edges. 15-20 Sept. '18. Cornflower {Centaurea Cyanus) .P: leaves well eaten, only midrib ]eit; flowers entirely devoured. 29 Sept.-3 Oct. '18. Cups and Saucers {Cobaea scandens) M^: petals nibbled a httle. 27 Oct.-3 Nov. '18. Dandelion {Taraxacum oflicinale): petals of ray florets entirely devoured. 26-31 Aug. '18. Elephant's Ear, Pink {Begonia sp.): leaves much gnawed along edges and also holed ; flowers thoroughly devoured. 2-5. Oct. '18. Fern, Male {Lastraea fllis-mas) 0°: leaves not attacked. 15-20 Sept. '18. Feverfew {Pyrethrum sp.) J^: leaves gnawed down to midrib; flowers apparently not attacked. 21-28 Sept. '18. Fig {Ficus Carica): leaves not attacked; fruit neglected when whole, but cross section was well gnawed. 7-15 Oct. '18. Fox-glove {Digitalis purpurea) P^: leaves holed. 6-11 Sept. '18. Fuchsia, Crimson Garden {Fuchsia sp.) H^: neither leaves or flowers were attacked. 28 Sept.-2 Oct. '18. Geranium, Scarlet {Oeranium sp.) L^: petals eaten a httle. 20-24 Sept. '18. Gesnera, Orange and Pink {Gesnerasp. )K}: leaves not attacked; petals entirely devoured. 24-26 Aug. '17. Golden Rod {Solidago sp.) C^: leaves gnawed at edges here and there ; flowers apparently not attacked. 2-5 Oct. '18. Gooseberry {Ribes grossularia) O^: leaves not attacked. 11-15 Sept. '18. Hawthorn {Crataegus oxyeantha) P: neither leaves ov flowers were attacked. 24-31 Aug. '18. Hollyhock, Dark Crimson {Althaea rosea): leaves not attacked; flower buds used as refuge, petals apparently eaten to some extent. 10-20 Aug. '18. Honeysuckle {Lonicera sp.) G^: leaves not attacked; fruit gnawed considerably. 7-20 Oct. '18. Hydrangea, Pink {Hydrangea sp.) Q": neither leaves ov flowers were attacked. 7-15 Oct. '18. Larkspur, Garden variety {Delphinium sp.) Q^: leaves gnawed thoroughly down to midrib. 3-6 Nov. '18. Leek {Allium porrum) T^: leaves gnawed deeply towards base. 6-15 Sept. '18. Lettuce, Cabbage {Lactuca sativa): stem aljundantly gnawed and bored; leaves of "heart" entirely devoured. 7-27 Oct. '18. Lupin {Lupinus polyphyllus) S^: leaves gnawed to some extent. of the Common Earwig (Forficula auric ularia) 53 3-17 Nov. '18. Mallow [Malvus ? sylvestris): leaves holed and edges gnawed down to veins. 23 Oct.-17 Nov. '18. Marguerite, White-rayed [Chrysanthemum leucanthemum) S^, U^: petals of ray florets well gnawed. 20-21 Aug. '17. Marrow, Vegetable (Cucurbita ovifera) A^: leaves thoroughly devoured. 20-21 Aug. '17. Michaelmas Daisy (Aster sp.) A^, 'i>i^: leaves hardly touched, if at Sill; floivers also neglected. 11-15 Sept. '18. Mignonette (Reseda odorata): leaves gnawed down to midrib; flowers attacked but slightly or not at all. 16-18 Sept. '18. Mint (Mentha sp.): leaves, edges and ends nibbled; flowers entirely devoured. 20-23 Oct. '18. Xavew (Brassica campestris) R^ : leaves holed and edges gnawed a little; petals moderately attacked. 3-17 Nov. '18. Nettle (Urtica dioica) U^: leaves well gnawed down to veins. 31 Aug. -6 Sept. '18. Onion (Allium Cepa) L": inflorescence used as refuge, but apparently not eaten. 7-15 Oct. '18. Pansy (Viola tricolor) P^: leaves nibbled slightly. 10-20 Aug. '18. Parsley, Garden (Carum Petroselinum) G^: inflorescence nibbled moderately. 29 Sept. -3 Oct. '18. Peach (Prunus [Amygdahis] sp.) N^: leaves gnawed moderately. 28 Sept.-2 Oct. '18. Periwinkle, Blue (Vinca sp.) L^: leaves and petals gnawed moderatelv. 22-23 Aug. '17. Phlox, White (Phlox Drummondi) B^: leaves apparently not attacked; petals much gnawed and pollen found in gut of earwigs. 1-3 Sept. '17. Plum (Prunus communis) F^: fruit well eaten. 23-31 Aug. '18. Poppy, Garden (Palaver sp.): dried fruits very popular as refuges; some were holed to obtain entrance. 1-18 Sept. '17. Potato (Solanum tuberosum) F^: tuber in skin was neglected, but slices were thoroughly gnawed. 28-29 Aug. '17, 20-23 Oct. '18. Primrose, Evening, yellow variety (Oenothera sp.) D^; leaves not attacked; petals eaten thoroughly; pods neglected. 7-15 Oct. '18. Privet (Ligustrum vulgare) Q'^: leaves holed and edges gnawed; fruits not attacked. 20-21 Aug. '17. Radish, Horse (Raphanus sativus) A^: leaves nibbled slightly. 27-29 Aug. '17. Raspberry (Rubus idaeus) D": leaves not attacked, but earwigs assembled in crowds on their hairy undersides. 22-28 Sept. '18. Red hot poker (Kniphofla sp.) : cut end of stem gnawed; lea ves and petals not attacked. 11-15 Sept. '18. Rest-harrow (Ononis sp.) P: apparently neither leaves or floivers were attacked. 30-31 Aug. '17. Rhubarb (Rheum officinale) W: leaves well gnawed. 24-26 Aug. '17. Rose, White garden variety (Rosa sp. ) C^ : haves not attacked ;' petals devoured. 7-10 Oct. '18. St John's Wort (Hypericum sp.) P^: leaves holed and their edges gnawed ; floioer buds not attacked. 31 Aug. -6 Sept. '18. Scabious, Crimson Garden (Scabiosa atro-purpurea): leaves much holed ; floivers apparently not attacked. 23-27 Oct. '18. Scotch Kale (Brassica oleracea acephala) S^: leaves holed a very little ; curled margins a favourite refuge. 10-24 Aug. '18. Sea Kale (Brassica oleracea acephala) G^: leaves holed and gnawed away from edges to between veins. 6-11 Sept. '18. Snapdragon, Scarlet (Antirrhinum sp.): leaves gnawed moder- ately; petals apparently holed to some extent, also used as refuge. 54 Mr BrincUey, Further Notes on the Food Plants 23-30 Oct. '18. Sow thistle [Sonchus oleaceus): leaves holed slightly; flower huds not attacked. 3-17 Nov. '18. Strawberry (Fragaria vesca) W: leaves holed a little. 31 Aug.-6 Sept. '18. Tomato (Lycopersicum esculentum): leaves and vipe fruit gnawed thoroughly. 14-15 Sept. '18. Valerian, Red Garden ( Valeriana sp. ) : edges of leaves gnawed moderately ; petals entirely devoured. 21-24 Aug. '18. Vervain, Blue (Verbena sp.): leaves nibbled slightly, haiiy undersides used for assembhng; petals entirely devoured. 24-31 Aug. '18. Vetch, Mauve and White garden varieties (Vicia sp.): leaves attacked very slightly, if at all; petals entirely devoured. 23 Oct.-3 Nov. '18. Violet, Single and Double garden varieties {Viola sp.): leaves holed and edges gnawed moderately. 3-17 Nov. '19. Wartweed (Euphorbia helioscopia) T^: edges of leaves gnawed very shghtly. 15-18 Sept. '18. Wormwood (Artemisia sp ): leaves not attacked. These observations are of course subject to tbe drawback that in captivity animals which normally feed daily may take unusual food with apparent eagerness because no other is available; but the above record probably indicates normal preferences over a certain range of common plants, and also that some are disliked by earwigs; thus Wartweed was left entirely untouched for many days in the absence of any other food, the animals attacking potato tuber ravenously as soon as this was substituted. It seems natural that such stiff and dry foliage leaves as those of Raspberry, Haw- thorn, and Cherry, should escape attack, and there is no doubt that the more succulent leaves are preferred. The list of plants affords some information which may facilitate the destruction of earwigs when they become a pest by the indications obtained as to plants which are popular as refuges, and also by the mode in which the attack on leaves is made; thus, some leaves seem to be attacked by holing as well as by gnawing along the edges, and others only by the latter method. There is no doubt that earwigs have pre- ferences among the common plants of a flower or vegetable garden, and that if numerous they are likely to become a pest. In certain cases, as for instance, chrysanthemums, the actual damage done seems to be exaggerated by common report. Since the epitome of recent literature on the subject in my previous paper {Proc. Camb. Phil. Sac, xix, Part 4, 1918, p. 170) was written. The Review of Applied Entomology has recorded attacks on beets and sugar-beets in Denmark sufficiently serious to obtain mention by Lind and others in their Report on Agri- cultural Pests in 1915 {Beretning fra Statens Forsogsvirksomhed i Plantekultur, Copenhagen, 1916, pp. 397-423). As regards the carnivorous habit of F. auricularia, lean roast mutton without other food was given for several days to the ear- wigs under observation in 1918 and was gnawed sparingly, while of the Common Earwig (Forficula auricularia) 55 mutton suet substituted for it was eaten readily and extensively. In the Journal of the Bombay Natural History Society, xxvi, No. 2, May, 1919, p. 688, F. P. Connor records an unnamed earwig at Amara catching moths in its forcipes and in one case nibbUng its prey. F. Maxwell Lefroy {Indian Insect Life, p. 52) remarks: "The function of the forcipes is a mystery that will be cleared up only when their food habits and general hfe are better under- stood." They are very possibly "frightening" as well as defensive organs. Pemberton {Hawaiian Planters' Record, Honolulu, xxi, No. 4, Oct. 1919, pp. 194-221) mentions the benefit to cane fields arising from the destruction of the leaf-hopper parasite Perkin- siella optabilis by the black earwig Chelisoches morio. The importance of nocturnal observations on the feeding habits of Forficula auricularia to a satisfactory understanding of the economic effects of this insect in gardens, urged in my previous paper, may be referred to again. 56 Mr Darwin, Lagrangian Methods for High Speed Motion Lagrangian Methods for High Speed Motion. By C. G. Darwin. [Read 8 March 1920.] 1. In the later developments of Bohr's* spectrum theory, it is necessary to calculate the orbits of electrons moving \vath such high velocities that there is a sensible increase of mass. The selection of the orbits permitted by the quantum theory almost necessitates the treatment of such problems by Hamiltonian methods. Working on these lines Sommerfeldf and others have calculated with a very high degree of success those spectra which involve the motion of a single electron. But the application of the Hamiltonian function involves a knowledge of the momentum corresponding to any generalized coordinate, and in the formulation of most problems the momenta are not known a priori but must be calculated from the corresponding velocities. In other words the formation of the Hamiltonian function must in general be preceded by that of the Lagrangian. An exception occurs in precisely the problems referred to above; for, the electromagnetic theory furnishes directly values for the momentum and kinetic energy of a moving electron in terms of its velocity, and the velocity can be eliminated between them so as to obtain the Hamiltonian function. But in even slightly more complicated cases this simple relation is destroyed — thus the problem of a single electron in a constant magnetic field can only be solved by introducing the artificial conception of rotating axes • — and in general it will be necessary to follow the direct course of finding the Lagrangian function in terms of the generalized velocities, and then deducing from it the momenta and the Hamiltonian function in the usual way. If more than one particle is in motion another difficulty enters. For the interaction of two moving particles depends on a set of retarded potentials and the effect of the retardation is readily seen to be of the same order as the increase of mass with velocity. The calculation of the retardation can only be carried out by expansion and so the results are only approximate. This is not surprising since the methods of conservative dynamics cannot apply to such effects as the dissipation of energy by radiation, effects inevitably required • by the electromagnetic theory, though they do not occur in actuality. We can also see from the fact that these radiation terms are of the order of the inverse cube of the velocity of light, that it will be useless to expand beyond the inverse square. * N. Bolir, Kgl. Dan. Wet. SelsL, 1918. t A. Sommerfeld, Ann. Phys., vol. 51, p. 1, 1916. Mr Darwin, Lagrangian Methods for High Speed Motion 57 2. We first consider the motion of a single electron in an arbitrary electric and magnetic field varying in any manner with the time and position. If m is the mass for low velocities, the momentum is known to be mv/^, where ^ = V 1 — v^jc^. Starting from this we have quasi-Newtonian equations of motion of the type lir*}-^^ '^■^'- The force F^. is given from the field E, H as the vector eE + ^ [v, H], where v is the velocity vector of the particle's motion. E and H can be expressed in terms of the scalar and vector potentials in 1 3A the form E = — grad (t> ~ ^.^^ and H = curl A. C ct Then if r^ is the vector x, y, z we have as the vector equation of motion It \t '4 ^~'' ^''^ '^ ~ c W + c ^'1' '""'^ ^^ •••(^■^^' where ^^ = V 1 - V/C^. Let q be any one of three generalized coordinates representing the position of the particle. Take the scalar product of (2-2) by i=r^. Then since ^ = -^, we have oq cq cq dii d (nil . )\ d (m dq'dtX^^'^^U dt\^. dt dq dq '9r _ . ,\_ d4> where "Wq = j;^ — ^ ^^^ Lagrangian operator. Again - gj [J-, grad .^ j = - e^ ^ = e^Bc/,. The remainder can be reduced to CV''~dq)~c[dq'lt) ^-'^^' dA dA dA . dA . 8A . where -j7 = ^7 + ^^+^-y+'^^ dt dt ox Cy '^ oz 58 Mr Darwin, Lagrangian Methods for High Speed Motion and so is the total change of A at the moving particle. (2-3) can be reduced to — ^ IBq (fi, A), Thus the whole equation of motion can be derived from a Lagrangian function L=- m,C^^, - e[<j> + g, (ii, A) (24). This is valid for any fields of force including explicit dependence of ^ and A on the time. The first term in L, which reduces to the kinetic energy for low velocities, differs from it in general. It is very closely connected with the "world line" of the particle. 3. To treat of the case where several moving particles interact we shall start by supposing that there is a second particle present undergoing a constrained motion so that its coordinates are imagined to be known functions of the time. The same will then be true of the potentials it generates. The motion of e^ will then be governed by (2-4) if ^ and A are expressed in terms of the motion of e^. These potentials are given by / _ ^2 a _ ^ ^2 /o.-i \ In these expressions r^ = (ig — ij)^ and the values are to be retarded values. If the time of retardation be calculated and the result substituted in (3-1) we obtain /_e2, 62 \ i^^+{i^,r^-Ti) (f^, r^ - i,)^ \ ._e.,i^ where now ij, la refer to the same instant of time, cf) is an approxi- mation valid to C"^, but the value of A has only been found to the degree C~^ on account of the further factor C~^ in (2-4) which is to multiply it. Then substituting in (2-4) we obtain r _ ,^ P2/P ^1^2 6162 ( r2^+ (r2,r2-ri) - 2 (fi, f^) L-- m,C Id, --y-^, I The equations of motion are unaffected by adding to L the expres- sion - mgC^^a + ^ ^2 ^'^"^'~^'^ - The first is a pure function of the time and so contributes no terms to the equations of motion. The second contributes nothing because for any function / we have Mr Danvin, Lagrangian Methods for High Speed Motion 59 The new form of L then reduces to L = - m,C-^^, - m,C^^, - f + |gi j^^^ I (ri,ra -ri)(fa,r2-ri) | ^^.^^^ From the complete symmetry of this form the roles of e^ and eg ^^^-y be interchanged. Further from the covariance of the operator IB for point transformations, both may be included in the dynamical system, so that if q is any generalized coordinate involving both Tj and ig, the equations of motion will be of the form "313 gL = 0. For the sake of consistency, as the last term in (3-4) is only an approximation valid to C~^, the first two should be expanded only to this power. The first term will give - m^C^ + lm,i^^ + g^ mj^\ Generalizing our result to the case of any number of particles in any external field we have L = ^lm,i,^ + 2 g^, m,i,^ - He.cf. + 2 ^, (r,A) - SS '^^ + si; ^^ I^AiA^ + (ri,r2-ri) (^2>r2-ri) ) ^ /3.5)_ The double summations are taken counting each pair once only. 4. The transition to the Hamiltonian now follows the ordinary r) T rules. We find momenta f = -^ and solve for the g-'s in terms of the 2^'s. This can be done in spite of the cubic form of the equations in the g's by use of the approximation in powers of C. The Hamil- tonian function will then he H = Hpq — L and the equations of motion will be the canonical equations q = ^—, p = — -o— • K Pi be the momentum corresponding to Tj , the Hamiltonian in these coordinates will be «- ^ 2lJ - ^^ sit' + ^^'^ - ^ CS^ <""*» + ^^ t _ ss ^1^2 [ (Pi>P2) ^ (Pi,r2-ri) (P2,r.,-ri) All the applications of general dynamics, such as the Hamilton Jacobi partial differential equation, follow from this. As in ordinary dynamics, many problems can be conveniently solved in the La- 60 Mr Darwin, Lagrangian Methods for High Speed Motion grangian form. The solution will usually depend on finding integrals corresponding to coordinates which do not occur explicitly in L and if cj) and A do not involve the time explicitly there is also the energy integral. This has the form ^^^r/+2.,<^+SS^?^^ Simfi^ + Sf '-^l r/ + 2ei</. + SS -^-^ This completes the development of the method. Its direct applica- tions are naturally somewhat limited, since, even with the large order terms only, there are comparatively few problems that are soluble. A problem of some interest that can be solved completely is the motion of two attracting particles, where their masses have a finite ratio*. * A discussion of this problem by the present writer will be found in Phil. Mag. , Vol. 39, p. 537 (1920), together with a somewhat fuller account of the general theory. Dr Searle, A bifilar method of measuring the rigidity of wires 61 A bifilar method of measuring the rigidity of wires. By G. F. C. Searle, Sc.D., F.R.S., University Lecturer in Experimental Physics. [Read 3 May 1920.] § 1 . Introduction. In this method the couple due to the torsion of two similar wires is balanced against the couple due to the load carried by the wires and arising from bifilar action. The method is hardly suitable for accurate measurements of rigidity, but, as an exercise in the use of a bifilar suspension, it has proved useful at the Cavendish Laboratory. § 2. Bifilar cowple. We first consider two light flexible strings. Let the strings AB, CD, each I cm. in length, hang from two fixed points A, C, which are at a distance 2a^ cm. apart in a horizontal plane. The lower ends B, D of the strings are attached to a rigid body of mass M grm., the points B, D being 2^2 cm. apart. The centre of gravity of the body is symmetrical with regard to B and D and thus the tensions of the strings are equal. The line BD will then be horizontal. If, now, a couple, whose axis is vertical, is applied to the body, the body will be in equilibrium when the couple due to the obliquity of the strings balances the applied couple*. In Fig. 1, A', B', C, D' are the projections of A, B, C, D on a horizontal plane. In our symmetrical case, A'C, B'D' bisect each other in 0. When the body has turned through 6 radians from the zero position, in which the strings are in the vertical plane through A'C, qt then B'D' will make an angle 6 with A'C nT Let ON be the perpendicular from on A'B'. Let the tension in each string be T dynes. If the vertical distance of BD below AC is h cm., the vertical component of the tension is Th/l, and the horizontal component is T .A'B' jl. Since the weight of the body equals the sum of the vertical components, Mg = 2Thll. The horizontal component of the tension at B acts along a line whose projection is A'B', and hence its moment about the vertical * For the general theory of the bifilar suspension, see Maxwell, El. and Mag.^ Vol. n, § 459; A. Gray, Absolute Measurements in El. and Mag., Vol. i, p. 242;. Kohlrausch, Physical Measurements (1894), p. 226. 62 Dr 8earU, A bifilar method of measuring the rigidity of ivires axis through is T .A'B' .ON/l. Since the moment due to the two tensions equals that of the appHed couple, G dyne-cm., G^2T. A'B'. ON/l. But A'B' .ON is twice the area OA'B' and thus is fz/Zg sin 6. We thus obtain _^ _ A'B'. ON _ a^a2 sin d Mg- h h Fig. 2. Since h = {P - A'B'^}^, we see that, when A'B' is small compared with I, we may put h = I, and so obtain G=''-^^^.Mg. •(i; In the examples of § 7, ^ never differed from I by as much as 1 in 4000. Dr Searle, A bifilar method of measuring the rigidity of wires 63 II § 3. Ap'paratus. This is shown diagrammatically in Fig. 2. The wires are soldered into torsion heads S, T, which pass through a board XY held in a firm support. The lower ends of the wires are soldered into screws which pass through "clearing" holes in the bar EF, and are secured with nuts. The heads of the screws are made with "flats" to fit a spanner. Before the screws are secured to EF, the torsion heads are set to zero; the screws are then secured to EF so that, when the bar is only subject to the action of the wires and of gravity, the flats on both screws have the same directions as when the wires hung freely. The distance BD is, as near as may be, equal to AC. The load is carried by a knife-edge forming part of the hnk iV, Figs. 2, 3. The knife-edge rests in a V-groove in a plate, P, fixed to EF by screws passing through slots. By adjusting P, the tensions can be equalised; the notes emitted by the wires when plucked have the same pitch when the ten- sions are equal. A weight W (a few kilogrammes) is suspended by the rod Q from the link N. A slot in the lower cross-piece of N allows Q to be put into place; the nut drops into a recess. The weight should be so attached to Q that it cannot turn about a vertical axis relative to Q with any freedom; otherwise it will be difficult to reduce the system to rest. The bar may be fitted with two pointers K, L, and the readings of their tifs are taken on two horizontal scales. These scales are adjusted to be perpendicular to KL when the torsion heads read zero. If KqLq is the straight line through the zero positions of the tips and K, L are the tips when the bar has turned through d, Fig. 4 shows that Fig. 3. Lo — .^^^^^^^^"^ K L — ^;S===^ — H ^ ^ Fig. 4. sm U = KH _ yi+j/a LK V •(2) where y-^ = KKq, y^ = LLq and p = LK, the whole length of the pointer system. The deflexion of the bar is best observed optically. A metal strip R is screwed to EF, packing pieces being interposed to allow the link N free movement, and a plane mirror is fixed to R. The deflexion can be observed by aid of a telescope and scale, or of a lamp and scale. It is, however, simpler to employ a goniometer such as those which have been in constant use at the Cavendish Laboratory for several years. A description of the instrument and 64 Dr Searle, A bifilar method of measuring the rigidity of wires the method of using it for experiments of this type will be found in Proc. Catnb. Phil. Soc, xviii, p. 31, or in the author's Experimental Harmonic Motion, p. 35. The goniometer measures the tangents of angles. The motion of the suspended system, as so far described, being only slightly damped, it is consequently not easy to reduce the system to rest, and the vibrations of the building add to the diffi- culty. A simple damping device is therefore used. An annulus of thin sheet metal is carried by the bar GH, which is clamped to the rod Q. The annulus is immersed in motor lubricating oil or other highly viscous liquid contained in the annular trough U, which rests on the table. The rod Q passes through a hole in the table. By adjusting the height of GH, the annulus can be brought close to the bottom of the trough, and then the motion is so highly damped that tte system is practically immune to vibrations of the floor or the table. If the wires are overstrained by turning the heads through toO' large angles, the wires will no longer be vertical when the heads read zero, and it will be necessary to readjust the screws in the bar EF, To prevent overstrain, and at the same time to allow the heads to be turned through tt in either direction from their zeros, a movable safety device is used. A metal disk, about 1 cm. in diameter, can turn freely about its centre on a screw by which it is attached to the board XY (Fig. 2). A vertical pin is fixed excentrically in the disk, the greatest distance from the pin to the axis of the head being small enough to prevent the steel wire, which forms the index of the head, from passing the pin. The torsion head can then be turned only a little more than tt in either direction from zero. Care must be taken not to bend the wires near the soldered joints. A bend at 5 or Z) will alter the effective value of a^. If the wire AB is bent near A, the effect, when the torsion head is turned, will be the same as if the point A describes a small horizontal circle. This causes changes in a-^ as the head is turned, and, what is more serious, causes the bar EF to turn through angles which are by no means neghgible, in addition to the angles directly due to the torsion of the wires. For this reason, annealed wires are more suitable for the experiment than hard drawn wires, as they are more easily straightened. The torsion heads are read on circles divided at intervals of 45°, the dividing lines being scribed on the board XY. § 4. Theory of the method. If each torsion head is turned from its zero through </> radians in either direction, the bar EF will turn in the same direction until the bifilar and torsional couples are equal. If EF turns through 6, the whole twist of each mre is (f> — 6. Let the radius and the length of each wire be r cm. and I cm.. Dr Searle, A bifilar method of measuring the rigidity of wires 65 and the rigidity of the metal n dyne cm.~^. Since the wires are nearly vertical, the couple, due to torsion, exerted by the fair upon the bar is Trnr^ {(f) — 6) /I*, to a close approximation. The small couple due to the bending of the wires assists the bifilar couple; Kohlrauschf takes account of this small couple by writing in place of (1), G = ^^^^}^.Mg, (3) where V = I - r^ {iTrEjMgY, (4) and E is Young's modulus. Equating the torsional to the (corrected) bifilar couple, we have ^md=C{4>-d), (5) ^^^^^ ^ = ]^r ^^) Then n = ^^r.MC. (7) § 5. Experimental details. The distances AC = 2^1, BD = 'la^ are measured. The diameters of the wires are taken at a number of points and the mean radius is found. The total mass, M grm., of the system carried by the wires is found. The masses of the screws are found before they are soldered to the wires. The torsion heads are first set to zero, and the scales on which the pointers K, L are read are adjusted to be perpendicular to KL. If a goniometer is used, it is set so that its arm is in the central position when the goniometer wire coincides with its own image. To eliminate errors due to slight bends in the wires, the readings must be taken over the range — tt to tt for ^ ; the theory assumes absence of hysteresis. But in experimental work in elasticity we must realise that hysteresis effects are unavoidable, when the strains are more than infinitesimal. To ensure that the effects of hysteresis shall be orderly and not irregular, the torsion heads are taken through a complete cycle from tt to — tt and back to it. To make the two readings for <^ = tt agree as closely as possible, a preliminary half cycle from — tt to tt is done. To make the conditions uniform throughout the cycle and a half, the readings for the pre- liminary settings are taken and recorded ; this will secure approxi- mately constant time intervals between successive readings. Thus the heads are set in succession at the following multiples of 7r/4 : -4,-3,-2,-1, 0, 1, 2, 3, 4, 3, 2, 1, 0, - 1, - 2, - 3, - 4, - 3, - 2, - 1, 0, 1, 2, 3, 4. * G. F. C. Searle, Experimental Elasticity, § 39. •f- Kohlrausch, W jet?. ^rt,«., xvu, p. 737, 1882. VOL. XX. PART I. 5 66 Dr Searle, A bifilar 7nethod of measuring the rigidity of wires The first 8 are the preliminary readings, and only the last 17 are used. If cf) goes through a complete cycle, and 6 is plotted against (j), a narrow hysteresis loop will be obtained. When readings are taken as above, there will be two values of 6 for each value of </> except (j) = — 7T. With careful work, the two values of d for <f) = tt will be exactly or very nearly identical; for the wires used in § 7, I have seldom found a difference between these two values as great as one minute. As a rough method of eliminating the effects of hysteresis, the mean of the two values of 6 for each value of (f> is taken as the value of 9 for that value of cf) . The effect of bends in the wires near their upper ends. A, C (Fig. 2), will be the same as if these points described small horizontal circles about the centres Aq, Cq, as in Fig. 5. Let </> be measured Fig. 5. from CqAqX, and let AAqX = (p + a, CCqX = cf) + y, while AqA = r, CqC = s, AqCq = 2ai. Then, if e is the small angle between AC and AqCq, r sin ((/) + a) — s sin (^ + y) 2ai + r cos (^ + a) — s cos {<p + y)' When r and s are small compared with 2ai, tane may be replaced by 6 and the variable terms in the denominator may be neglected. Then, putting [r sin a — s sin 7)/2ai = P, {r cos a — s cos y)l'2ax = Q, we have 6 = P cos(/> + ^ sin ^. (8) Here P and Q are the values of e when cf) = and ^ = Jtt. If the line BD makes an angle 9 with AqCq when the heads read </>, the angle between BD and ^C is ^ — 6. The wires will not be quite free from torsion when the heads read zero; let rj be the mean twist of the wires when </> = 0. We must thus write sin {9 — e) for sin 9 and cf) +rj — 9 for ^ — ^ in the equilibrium equation (5), which thus becomes sm{9-€) = C{cf)+r)-9). (9) To evade difficulties, 9 is kept small. Then, since e is also small, we may replace the sine by the angle in (9), and thus obtain I J Dr Searle, A bifilar method of measuring the rigidity of wires 67 e = YTC i<f>+V)+^ = D{c/>+r))+e (10) If Oq, Cq correspond to ^ = 0, we have, since €q = P, e, = Drj+e,^Drj + P. (11) Thus 6 - 6^ = Dcf>+e- P = D<f) + P cos (l> + Qsincf) - P (12) Since this equation is linear in 6, we may take 6^ as corresponding to any initial position of the bar which is yiear its ideal zero position. Thus, if /S is the angle at any time between the bar and some nearly ideal zero position, ^ = Q-K (13) Since ^, though small — say less than 0*2 radian — is not infinitesimal, some correction should be made. An exact solution cannot be given, but accuracy is gained by writing sin ^ for ^, and then the final formula becomes sin^ = Z)(/. + P cos </. + Q?,\\\(j> - P (14) To eliminate P and Q, we combine the observations. Let /3^ correspond to (f> = imr/i. Then, putting cf) = tt and (f) = — tt, so that m = i and m = — 4, we have ttD = i (sin ^4 - sin ^_4). (15) A second value for ttD is found by giving m the values 3, — 3, 1, - 1. Then 7tD = sin ^3 — sin/3_3 — (sin^j — sin /3_i) (16) The two values of D are usually in good agreement, although, when P is plotted against <f), the curve differs considerably from a straight line. The mean value of ttD is used to find C. Thus ttD C' = -^. (17) Then n is found by (7). The actual values of P and Q are easily found. Thus P = - 1 (sin ^4 + sin ^_4), (18) Q = l (sin ^2 - sin ^.^ - ttD) (19) § 6. Conversion table. A goniometer, such as those used at the Cavendish Laboratory, gives the tangent of the angle iJj through which the arm is turned from its zero. To find sin ip we subtract from tan j/» the small quantity s given in the table. 5—2 Q8 Dr Semie, A hifilar method of measuring the rigidity of wires tan\/^ s tan\//- s tan\//' s •01 •00000 •10 ■00050 •19 •00334 •02 •00000 •11 •00066 •20 •0C388 •03 •00001 •12 ■00085 •21 •00448 •04 •00003 •13 ■00108 •22 •00514 •05 •00006 •14 ■00135 •23 •00585 •06 •00011 •15 •00166 •24 ■00663 •07 •00017 •16 •00201 •25 ■00746 ■08 •00025 ■17 •00240 •09 •00036 •18 •00285 Simple interpolation, by "proportional parts," will give s with an error not exceeding unity in the fifth place of decimals. Thus, if tan xfs = •124, we find s = "00095, and then sin i/f = tan i// - 5 - •12305. § 7. Practical example. The following results were obtained for a pair of soft brass wires. The distances AO, BD were each 6^00 cm. Hence aj = ag = 3 cm. Mean radius of wires = r = 0^0352 cm. Length of each wire = I = 47^30 cm. Mass of suspended system, excluding the weight W (Fig. 2) = 417^6 gm. The small correction for the buoyancy of the damper was neglected. The deflexions were observed by a goniometer. The distance from the centre of the pivot to the scale was 40^00 cm. The central, or zero, reading is 10-00 cm. The following goniometer readings were obtained for the last 17 of the values of specified in § 5. <f> \ Reading Reading Mean X tan/3 sin (^ sin /3 radians cm. cm. reading cm. = tI40 obsd. oalcd. IT 14-10 14-10 14-100 4-115 -1029 -1023 -1033 Itt 12-96 1 13-04 13-000 3-015 -0754 -0752 -0749 i^ > , 11-85 , 11-98 11-915 1-930 •0482 •0482 •0471 iT 10-80 10-96 10-880 0-895 -0224 -0224 •0220 9-90 10-07 9-985 0-000 -0000 •0000 •0000 -iTT . 9-16 t 9-33 9-245 -0-740 --0185 --0185 - -0204 — r/TT 8-30 8-43 8-365 -1-620 - -0405 - -0405 --0415 -|7r 7-37 1-U 7-405 - 2-580 - -0645 - -0644 - -0653 — TT 6-33 6-330 -3-655 --0914 - -0910 - -0921 The value of x was found by subtracting from the mean reading, as given in column 4, the mean zero reading 9-985 cm. corresponding to (^ = 0. The differences between the readings in columns 2 and 3 are due to hysteresis. The seventh column shows that sin i3 is not proportional to <p. Dr Searle, A bifilar method of measuring the rigidity of wires 69 m By (15), T^D = l (-1023 + -0910) = -09665, and by (16), nD - -0752 + -0644 - (-0224 + -0185) = -09870. Mean value of irD = 0-0977. Then, by (17) C= ""^ ^ = 0-03210. By (18), P= - i (-1023 - -0910) = - 0-0028, and by (19). Q=l (-0482 + -0405 - -0977) = - 0-0045. In the table, the column "sin/3 calcd." gives sin ^3 as calculated by (14), using the values of ttD, P and Q just found; there is fan agreement between the calculated and observed values of sin ,3. The total load M was 417-6 + 4999 = 5416-6 grm. Taking E = 10^^ dyne cm.--, we have r- {2TrE/Mg)^ = 1-35 cm., and hence, by (4), I' = 47-30 - 1-35 = 45-95 cm. Then, by (7), n = ?^!.MC = - ^'^J^~^l^, X 5416-6 x 0-03210 TvrH' IT X 0-0352* x 45-95 = 3-277 X 10" dyne cm -2 A similar set of observations, m which ilf was 3417-lgrm., gave the following values of sin /3: •1510, -1123, -0743, -0361, -0000, - -0345, - -0735, - -1122, - -1540. Mean value of ivD = 0-1532. Hence C = 0-05127. Also I' = 47-30 - 1-70 = 45-60 cm. Then 981 X 32 X 47-30 TT X 003.52* X 45-60 X 3417-1 X 005127 = 3-326 x 10" dyne cm." An independent determination of n was made by attaching a bar, of moment of inertia K = 4-766 x 10* grm. cm.^, to each of the two wires in turn; the mean periodic time of the torsional vibrations was T = 10-55 sec. Hence 87r X 4-766 x 10* X 47-30 „ „. _ ,_., , .3 10-55^ X 0-0352* = ^'^^^ ^ ^^ ^^^^ ^"^- " 70 Mr Bennett, The Rotation of the Non-Spinning Gyrostat The Rotation of the Non-Sf inning Gyrostat. By G. T. Bennett, M.A., F.K.S., Emmanuel College, Cambridge. [Read 8 March 1920.] § 1. The following extract is taken from an old examination paper* : "A symmetrical wheel free to rotate about its axle is moved from rest in any position by means of the axle and is finally restored to a position in which the axle again points in the same direction as formerly. Shew that the wheel, again at rest, will have rotated through a plane angle equal to the solid angle of the cone described by the varying directions of the axle." The proof of this result may be put briefly in a geometrical form. Translational and rotational movements being independent, the centroid of the wheel may be treated as stationary. As the gyrostat has no component rotation about its axis, the axis of rotation is at any moment some diameter of the wheel. This line has the central plane of the wheel as locus for the body-axode, and has a closed cone of arbitrary form as locus for the space-axode. The angular movement is therefore representable by the rolling of the plane on the cone. The angle of ultimate rotation of the wheel is thus (for cones of ordinary type) the excess of the four right angles of the plane surface above the total surface-angle of the cone. This difference is equal to the solid angle of the reciprocal cone described by the axis of the wheel. And hence follows the result quoted; namely, that the solid angle described by the axis of the wheel is equal to the circular measure of the plane angle of the resultant displacement of the wheel about its axis. Further, the sense of the displacement accords with the sense of circulation associated with the solid angle. § 2. The result may be extended to the case in which the initial and final directions of the axis are different, say a and 6. For the axis may be restored to its original direction a by a subsequent movement in the plane ha; and this latter movement, which is a rotation about the normal to a and 6, leaves unaltered the angle that any diameter of the wheel makes with the plane ah. Hence the original movement, shifting the axis of the wheel from a to 6 * Emmanuel and other Colleges, Second Year Problems, Wed. Jmie 8, 1898 Question 11. Mr Bennett, The Rotation of the Non-Spinning Gyrostat 71 by any conical movement, alters the angle between the plane ah and any diameter of the wheel by an angle equal to the solid angle enclosed by the cone formed by the conical surface ab together with the plane ha. § 3. A geometrical integration of Euler's equation leads to the same result as § 1. The axis, with its direction given by spherical polar coordinates d and (/> (radial and azimuthal), generates a solid angle (7 = J(1 -COS^).f/(/.. (1) The equation of motion, being (^ COS ^ + i^ = 0, (2) with zero initial values for ^ and ^, has as its integral <f> + i/j = or. (3) If the axis of reference ^ = is supposed (conveniently) external to the cone then </> is zero finally as well as initially, and i/r is the angle of resultant displacement of the wheel and is equal to the solid angle a. If, more generally, the gyrostat has a constant spin Q. about its axis, the Euler equation becomes cf) cos 9 + i[i = Q (4) with 4> + ip = a + Qt (5) as its integral. And the final rotation of the gyrostat is then given by the solid angle of the cone described by the axis plus the time- integral of the spin. It may be noticed that the angle (f) + i/j, with a value independent of the choice of coordinates, gives in itself a natural measure of the total rotation of the wheel, as followed and estimated by projection on the plane 6 = tt/'I. For on that plane the circular disc shows as an ellipse, with ^ as the azimuth of the direction of the minor axis, and ^ as the eccentric angle, measured from the minor axis, of the projection of the revolving diameter of the wheel. A distant observer on the axis ^ = 0, able to distinguish the two faces of the wheel, would in this way precisely reckon the amount of rotation, whole turns and fractional. He does not give merely the ultimate position, by naming a plane angle to a modulus of four right angles, but assigns the multiple of the modulus neces- sary for a correct account of the movement intervening between the initial and final positions. A kinematic representation of the angle </> + «/' may be obtained by supposing the circular rim of the disc to have rolling contact with the rim of another equal disc whose plane keeps parallel to the plane 6 = 7r/2. The angle of rotation of this latter disc about its axis (which keeps the invariable direction ^ = 0) is then </> + ip. 72 Mr Bennett, The Rotation of the Non-Spinning Gyrostat § 4. For the special case in which d is constant, so that the axis of the gyrostat describes a circular cone, the rotation is stated by Sir George Greenhill* to be 27r — (conical angle described by the axle), as against the solid angle itself found above. The differ- ence of sign of the latter can be accounted for by a reverse sign- convention: but the term 27t is unnecessary if 27r is implied as a modulus, and it appears to be wrong if the precise angle of turning is intended. If, specially, the axis of the gyrostat described only a small cone, then the angle of consequent rotation is certainly a small angle, and not an angle nearly equal to four right angles. He adds the remark that the movement "can be shown ex- perimentally with a penholder held between the fingers and moved round in a cone by the tip of a finger appHed at the end." But the illustration is inapt; for the creep of the penholder occurs in the sense opposite to that of the conical movement. The body-axode is a circular cone and not a plane, and it rolls inside a shghtly larger circular cone as space-axode; and hence the reverse move- ment. § 5. The movement of the non-spinning gyroscope here con- sidered is not yet among those that are familiarly recognised, though it has important practical applications and deserves to rank as a dynamical commonplace. Bodies suspended from a point on an axis of symmetry behave in the same way and for the same reason. No matter how the point of suspension may be moved about, and no matter what complicated conical movement is consequently executed by the axis, the applied forces have no moment about the axis, and the spin remains zero if originally zero. The resultant rotation is then given, as above, by the solid angle of the cone described by the axis. Aeroplane compasses, in particular, are found to keep their cards practically parallel to the floor, under the combined action of gravity and lateral acceleration, during a banked turn of the aeroplane. Hence, from inertia alone, and apart from all other sources of control or disturbance, the compass-card would be rotated, as a consequence of the turn, through an angle equal to the solid angle described by the normal to the card. For an angle of banking a and a change of course ^ the solid angle is not much less than (1 — cos a)^ if the banking is taken and left quickly; and for very steep banking this angle is nearly equal to the change of course itself, and the card would almost appear to "stick." As compared with considerations of magnetic disturbance due to the vertical component of the earth's field, and of mechanical disturb- ance due to rotation of the bowl and liquid, the pure inertia efiect * Advisory Committee for Aeronautics. Reports and Memoranda, No. 146, Report on Gyroscopic Theory, p. 13, § 14. Mr Benneit, The Rotation of the Non-Spinning Gyrostat 73 of the conical movement seems to need more emphasis than it has hitherto been awarded. It is here expKcitly isolated. The gyroscopic compass, Uke the magnetic compass, may at times suffer disturbance from this same source, if the compass- position in the ship and the run of the sea are such as to produce a circular or elliptical movement of the binnacle. § 6. It would be hard to trace to its primitive source the know- ledge of the small piece of mechanics here discussed. It is really implicit in all treatises on Rigid Dynamics, but fails to emerge clearly amid the pressure of more important movements. Among empiricists it must be well-nigh prehistoric. The sailor in coihng a rope makes a winding motion of the feeding hand to remove the kinks from the overtwist of the piece which is to form the next turn of the coil. The circus clown, with the vertex of his conical cap resting on his finger-tip, or the end of a stick, easily makes it turn round and round; and the postman collecting his mail knows how to twist up the neck of his bag with a circular movement of the hand he holds it by. Later among empiricists are those who, accustomed to handle magnetic compasses, are very familiar with the rotation of the card produced so readily by giving the bowl a horizontal circular translational movement (without rotation). More lately still Mr S. G. Brown has noticed the conical motion and its effect. In the abstract of his lecture to the British Associa- tion* it is described as a "new phenomenon" and is stated as being "explainable mathematically." More fully in his lecture to the Royal Institutionf he states that in virtue of the "wobbling" {videlicet conical) motion, "the needles and. card would then have a force applied trying to carry the moving system round in the direction of the wobble." This mode of expression is of course entirely illegitimate. The rotational movement observed needs no "force" to explain it; the very essence of the inertia effect is that it occurs with no spin about the axis of rotation and no couple about that line either. Mr Brown announces also (but without demonstration) that if his compass-disc "is carried round in a horizontal circular path without any wobble the plate still goes round or tries to go round with the circular movement" and that this "should be of interest to mathematicians." It seems likely that the sheer paradox in angular momentum thus propounded will readily dissolve when all the relevant physical data are revealed: and meanwhile the interest is but that of a heresy resting on hearsay. * British Association, Bournemouth, 1919. Evening Lecture, Fr. Sept. 12, "The GjToscopic Compass." Abstract, 11. 9-14. t Nature, March 11, 1920, p. 45, col. 2. 74 Mr Kienast, Proof of the equivalence of Proof of the equivalence of different mean values. By Alfred Kienast. (Communicated by Professor G. H. Hardy.) {Received 12 April: read 8 May 1920.] If Oi, ao, ... an, ••• denote the terms of a sequence of complex numbers, and 8^^ = a, + ... + an, then lim 8^"^ i [ ) is called Cesaro's /cth mean* of the se- :.(0) quence 8\ Putting h^^^= ai + . . . + a^, A<:'=ii/,«;'+...+An, then lim A^^' is called Holder's /cth meanf of the sequence hj. In a paper "Extensions of Abel's Theorem and its converses |" I found it convenient to introduce the expressions 5f=ai-F...+a,^ (?i=l, 2, ...), ^'^-l[sT^-+sf_,] (.. = 2,3,...), and proved various theorems concerning the limits lim s^ . Several writers have proved Theorem 1. Whenever Cesaro's (Holders) Kth mean exists and is finite, then Holders (Cesaro's) /cth mean eooists too, and both have the same value. * Bromwich, Infinite series, p. 310. t Ibid., p. 313. t Proc. Cambridge Phil. Soc. , vol. xix, 1918, pp. 129-147. different mean values 75 I propose to complete the researches of my above quoted paper by proving the theorem: Theorem 2. Whenever Holder s {and therefore Gesaro's) Kth mean exists and is finite, then lim s]f exists too, and both have the same value, ayid vice versa. The demonstration of both theorems is based upon relations between the mean values which it is possible to calculate com- pletely, as I have found, in a most simple manner. In §§ I to VI I determine the expression of s^^"^ by hf {X = l,% ...{n-K)), in § VII the expression of h\l^ by 4'^ (X = l, 2, ...7i), in § VIII the expression of S^^^ by h^;:^ (\ = i, 2, ...7i), and finally in § IX I consider two more general mean values. I. From the definitions follow " A=i "■A=] ri ,f= 1 S' ^J A<'',= i W hf+ 1 (2Af ' - ;,f >) + . . . ^\=2 X ^^ w |_2 ^ 3 ^ ^ +»^j(„_2)e,-(„-3)esi Adding a term which is zero, (2)_ (n-l)(n-2) (2) _ 1 (1 , (2) , 1 . (2) J i-'^ j^2) \ etc. Now I suppose that, by continuing in this manner, I have arrived at the formula 1 n—K , . ''' A = l where c is a function of the indices n and k, and the coefiicients n, K <^a,k(X = 1, 2, ...) are, for each k, definite numbers which are the values of a function of X for \ = 1, 2, — 76 Mr Kienast, Proof of the equivalence of Proceeding to build up the expression for s[l'^^\ applying the same transformations as above, we find "kAC^ - ") K'-"- (X - « - 1) Att'-M n-1 „(«+!)_ 1 'V o(«+l)_. ^-'^-lz.('c+l) I'^v'^ 7,(«4-l) from which we conclude K't'l^-^ ^ d.,.+A''-'' (2); n K i- /ON (^n, K4-1 ^= Cn, K V"/) and a series of relations involving the numbers c?a,k and cIk^k+i- Equation (2) is of the same formation as (1), and therefore (1) gives the required expression of s^^''^ by the numbers h'-j^K II. Since Cn i = -^^ , (3) leads to n Cn, K = n~'' (n — K)(n-K+1) ... (n — 1), from which follows, for k = 2, Cn^= ~ ~ , which is in accordance with the expression for s^^^ above. III. c?A.,K may be determined in the following way. Putting I ai = a2= ...=aA-i = 0, a^ = l, aA+i=... = (4), we find s(0)= =,.(0) _A (0)_-, (0) _-, ^(1>- - o(l)_A Jl) _ 1 „(1) _ ^K ^' 6a4-1 — > , T> *A-4-S! +i~\+l' ^+2 X + 2' o(2). ^+1 ' ^+2 (\ + l)(X + 2)' ^+'' (\ + l)(X + 2)...(\ + /c)' different mean values 77 and /,;*»=... =/t(VO, ^^=1, h['i=l, ..., n^ ... ^_i ^, /i^ ^, z^+i ^^-^, ..., Writing equation (1) for n = X + K, and substituting these special values, we obtain f \(\ + l)...(X + «-l) V^ ] rfx,.-(^ + «)| (\ + Kr (\+1){X+2)...(X + k)\' which is, for /c = 2, in accordance with the above expression for s^^\ IV. Lemma a. The coefficient c^a,k is a positive number for X = l, 2, ...,/c = 2, 3, .... It is easy to verify the inequalities ^4^^—^'— (y^ = 0,l,2,...,.-l), X+ K X + K — fl from which results, by multiplication of all the left-hand and all the right-hand sides, X{X + 1)^. .(x± «-l) > '>^ (X + kY ' -{X + 1){X + 2)...(X + k)' which demonstrates the assertion. Lemma ^. The coefficient c?a,k satisfies the equation lini|<.-^(«-l)/c(« + l)i| = (5). To show this we expand c?a,k in the form and the proposition follows. A consequence of (5) is v. lim ]S f^A, K = 00 , )l-*-x A = 1 78 Mr Kienast, Proof of the equivalence of and therefore the conditions of Stolz's theoreDi are satisfied. Thus we can state: Lemma 7. //' lim h^"^ = H exists and is finite, then lim 2 d,^X'^/ I d^,. = H. V. Let ai = l, ao = a3=---=0 (6); then , sl'^=l (n = l, 2, ...), ^^=^^^ (n=2,3, ...); therefore lim s^^^^ = 1, and consequently lims;f=lims^^'>=... = l. Furthermore h^^^ = h^^^ = h'-^^ ^ . . . = 1 (n = l, 2, ...)• From (1) and (6) we obtain C ) \n-K Imi s\'^'= 1 = hm Cn,K - hm - S dx.K, n-^cc n-*-oo 7j->-Qo n 1 1 n—K or lim - 1 dK,K = 0. VI. Now passing in equation (1) to the limit ?i-^qo , we find , , , , /I n-K \ (n-K , . ,n-K ) limsf = lim/i2«-lim (- S < J lim 2 d.,,/ii''7 S d,,. \, and this equation leads to the theorems: Theorem 3. Whenever lim h^^\^ exists and is finite, then lim s^"' exists too and has the same value. n More generally Theorem 4. When the function lf^^_^ oscillates between finite limits, then s^^ oscillates between the same limits. VII. The reverse propositions can be established in the same way. From the definitions follow n h^''=- J^^{(XH-2)4!>,-(X + l).f>J + ^{(.+ 2).f>,-(. + 2)0} (n+^y J2) 1 I ^ + 2 (2) n(n + l) «+2 ^nKZiXiX + 1) ^+2" dijferent mean values 79 Continuing this process we find A;:^=^n,«^-;i+' ^aai in In the same way as we determined Ch,k by (2) and (3), we obtain here n + K-\-l Thus, since 6)1 K+] — 6j; n + 1 _ (n + kY ^"' " " n{n + l)...{n + K-l) ' Taking the values (4) for the numbers a^, we find from (7) f = X f (^ + l)(^+2)...(X + /c) { X+fcY ] ■■''~ \ V X(\+l)...(\ + «-l)J ■ The considerations in § IV show that Expansion of /a_x in descending powers of X gives A. = g(«-1)«(« + 1)- + ^, + ...; n thus lim S /a, K = 00 . W-*.cc \ = 1 Introducing in equation (7) the vahies (6) for the numbers a^, we find 1 = lim e„^« + lim - 2 /a,« lim 2/a,«sJ;''L/ S/a^^ ; which gives, on account of Stolz's theorem, lim- i/A,. = 0. Thus equation (7) is completely determined and leads to Theorem 5. Whenever lim s^^'L exists and is finite, then lim h^"^ exists too and has the same value. n Theorem 6. Whenever the function s^^^^ oscillates finitely, then h^n oscillates between the same limits. Theorems 3 and 5 together constitute theorem 2. 80 Mr Kienast, Proof of the equivalence of VIII. The relation connecting Cesaro's and Holder's means can be deduced in the same way. We have L\ = i n(n + l) n{n+l) = 2hf — -1—lxhf. « w(w + 1)a=i " Assuming therefore o(«) I n ^ =C„ h'^:^ ^ r^ td, h['^ (8), we fand \ "[ K J ri+i,^K ~^^j^^^.«+i^ • Hence c„, „+i =(« + !) Cn+i, « , or Cn,K= k\. Starting from the numbers (4), we have sr'=... = sr_\=o. «r=i. -. *1"'=(""^+''). ■••; ?"=... = si as is easily verified by the formula .ti ^ ™,roV « / V a: + i ;■ Writing (8) for 7r = X, we find d),^,= \(\+l) ... (X + «-l)-X«^0. Starting with the numbers (6), that is with the numbers (4) for X,= 1, we have and from formula (8) follows A = l V K different mean values 81 so that finally ^n + K-l\ " ^ -^7 K J A = l Analogous considerations lead to n A<'')=1 ^!L_^ + fl_ M ^Zl ^ ...(10), \ K ) ,V'''\ K Formulae (9) and (10) prove theorem 1. IX. By similar considerations it is possible to arrive at a state- ment about the equivalence of two means of the kind examined in Part II of my above quoted paper. Let 6k , Ck denote the terms of two infinite sequences of positive real numbers, which have, when we write n n 1 1 the properties (i) lim Bn= 'X^ , lim Cn = go , .... 1 ^ /c6, 1 « KC^ (n) - 2 ^- , - 2 rT tend to limits or oscillate between finite limits. Then putting as before (o) ^ the means ^1'^^= w^K^T-i C^^)' t?=livt. (12), ^n 2 are connected by two analogous relations. From (11) and (12) follow b s^'\ = B s^'^-B J'\ (13), c s^'\=G f^-C J'\ (14). n n-l n n n-l n-l ^ ' VOL. XX. PART I. 6 82 Mr Kienast, Proof of equivalence of different mean values Substituting (14) in (11) we find, on adding a term which is zero, „(!)_ 1 r^2^ ,(1) , ^sf^ .(1) p Ai)x , V ih_h±i] c t^^^ Since 1 VCa Ca+1 + ^n+1 ^n ,(1) -6. ^n+i G,= i^~\C\-C\_,) + 2 Ca = 5. we can write 1 _ '"'+1 zl a, - i "+ Ca Ca+i/ I 2 ( ^ — ^^^ ,Ca Ca+1 C, .(15). Now there may be distinguished two possibiHties : Theorem 7. If VCa Ca+t/ ^ ^ w /A Ca fA+] C. < K (fixed), 0„ 6,1-1-1 lim — = 1 ?i-*.Qo '^n+1 -'-'71 and if ^JJ' a'pproaches a finite limit {or oscillates between finite limits), then s^^^ approaches the same limit (or oscillates between the savne limits). Theorem S. If " /&A &A+A p (1) ■ i "-'a ''A Hm V /"A 3 VCa = lim i^ ^()a_6a+i\ ^ 2 VCa Ca-|-i/ b G -~~^<K (fixed), and ift^^^^ approaches a limit, then s^^ approaches the same limit This is a known theorem *. The second relation results by substituting from (13) in (12) and proceeding in the same way.' The same formula is arrived at by interchauging in (15) b^ and c^, B^ and (7,^, ^'^ and s^^^ . From it we infer two theorems analogous to (7) and (8).' * Bromwich, Infinite Series, p. 386, Theorem V. Mr Harrison, Notes on the Theory of Vibrations 83 Notes on the Theory of Vibrations. (1) Vibrations of Finite Amplitude. (2) A Theorem due to Routh. By W. J. Harrison, M.A., Fellow of Clare College. [Read 3 May 1920.] p^ (I) Lord Rayleigh in his Theory of Sound, Vol. i, has considered the effect of introducing terms depending on x^ and x^ into the d'^x simple equation of vibratory motion -j-^ + n^x = 0. He treats the added terms as small and employs the method of successive ap- proximatioD. The object of this note is to point out that exact integrals can be obtained in the form of the series of which Kayleigh determined the first two or three terms. The solutions now obtained are valid for any relative magnitude of the added terms subject to the motion remaining vibratory. (a) The Symmetric System. The equation of motion is d 00 -^-^ + n^x T '2Bx^ = 0, dt'^ where /S is positive, and the upper sign is taken in the first instance. A first integral is where a is the amplitude of the vibration. We have <dx\^ (^2 _. ^2) („2 _ ^^,2 _ ^^2)^ fdxV \dt) or Jt where ax^^ =- x, k^ ^ ^a^Kn^ — ^a^). Hence* X = axi = a sn {{n^ - Ba^ft, h], [x =-0 when t = 0) _ 277a ^ g^+^ . (2m + I) tt (n^ - ^a^ft ^ Kko 1 - q'""^' "''' 2E: * For the expansions of elliptic functions quoted in this paper see Whittaker and Watson, Modern Analysis, 1915, p. 504, and Example (5), p. 513; or Hancock, Theory of EUiptic Functions, Vol. i, 1910, pp. 486, 494, 495. 6—2 84 Mr Harrison, Notes on the Theory of Vibrations Let the units of length and time be chosen so that a= l,n= 1. doc It is necessary that ^ < |, otherwise -^ vanishes first f or < x < 1. The effect of the term 2^x^ on the vibrations can be exhibited by the results of numerical calculation given in the following table: ^ X V •01 1-0006 sin />( + '0006 sin Hpt 9924 •1 1-0074 sin pt + -0074 sin Spt •9214 •3 1-0335 sin pi -4- -0348 sin Spf + -0012 sin 5jjt -7309 •4 1-0632 sin pi + -0676 sin 3pi + -0046 sin 5pt + -0003 sin Ipt -5997 •45 1-0928 sin pi + -1028 sin 3pt + -0108 sin 5pi + -0011 sin 7 pi + -0001 sin 9pt •5063 ^ tanh{tl^^2) We proceed to consider the equation d?x dt^ + n^x + 2^x^ = 0. If a is the amplitude of the motion as before, we have f^f = (a2 _ ^2) (^2 + ^^2 + ^^2)^ or r^j = {n^ + ^a^) (1 - V) (1 + l^V), where ax^^ = x, and [x^ = ^a^/{n'^ + ^a^). Write 1 — a;j2 = z^, so that ^^y = (n2 + 2^a2) (1 _ ^2) (1 _ ^^2)^ where k^ = ij?/{l + jjl^) = ^a^j{n^ + 2^a^). Thus 2 = sn {{n^ + 2^a2)^^, A'}, (^ = when t = 0). Therefore dU CttJO-l = acn {{n^ + 2pa^)h, Jc} ,m+i 12m + 1) 77 (%2 + 2i8a2)^^ cos r I ^■na ~ q" v2m+l 2Z X^ 1 + ?' In this case there is no limit to the value of ^, the motion remains vibratory, but the period of the gravest mode decreases Mr Harrison, Notes on the Theory of Vibrations 85 as /3 increases. The results of calculation, with n -= 1, a = 1, are as follows: ^ X P •1 •995 cos 2)t + -005 cos 3pf 1072 •5 •9818 cos pt + -0179 cos 3pt + -0003 cos 5pt 1^318 10 •9742 cos pt + -0253 cos 3pt + •0006 cos 5pt 1-569 100 •9582 cos 2it + -0402 cos 3pt + •OOIG cos 5pt + -0001 COS 7pt 3-975 1000 •9555 COS pt + -0427 cos 3pt + •OOIS cos 5pt + •OOOl cos 7pt 12-03 n'^x + ^ax^ = 0, (b) The Asymmetric System. The equation of motion is d^x where a may be assumed to be positive, as changing the sign of a is equivalent to reversing the direction of the axis of x. Let the scale of time be such that n = 1, and the scale of length chosen so that the amplitude of the motion measured from aj ^ in the direction of x positive is unity. Then f , j =-■ {I — x) {I + a -\- x + ax -\- ax^) ^{l-x){b + x) (c + x), where 6 = i {1 + « - (1 - 2a - ^a^f}la, c=\{l + a+{l--2a- 3a2)^}/a. The limits of the vibration are x = \ and x = — b. It is necessary that a should be less than \, so that the greatest value of b is 2. Writing I — x = {b + I) y^, we, have |)'=i«(c+l)(l-?/2)(l where F= (6+ l)/(c + 1). Hence y= sn {|a^(c+ lft,l}, and x^l-{b-^l) sn^ {^a^ (c + l)^i, k} b+ I {, E 27r2 * mf " Fy^),. = 1 - /C2 1 ^2772* 2Z The results of calculation are as follows : 86 Mr Harrison, Notes on the Theory of Vibrations a b X P •1 •2 •3 1-1125 1-2680 1-5657 - -0838 + 1-0557 cos pt + 0284 cos 2pt + -0006 cos 3pt - -2059 + 1-1306 cos pt + -0712 cos 2pt + -0033 cos 3pt - -4634 + 1-2634 cos pt + -1783 cos 2pt + -0190 cos Spt + -0018 cos 4p< + -0002 cos 5pt •9855 -9477 •8152 The calculations have been performed for illustrative purposes only, and no special care has been taken to ensure the accuracy of the digits in the final decimal places. (c) The solution of the equation dt^ + n^x + f ax^ in the form of a Fourier Series requires rather more elaboration of the algebra. The motion presents one novel feature which does not appear in the previous solutions. If ^ be positive, however small, the motion remains vibratory for any finite value of a, and if a and a/^ be great, the amplitude of the motion on one side is approximately a/jS times its amplitude on the other. (II) Kouth has shown (vide Advanced Rigid Dynamics, 1905, p. 56) that an increase in the inertia of any part of a vibrating system will increase all the periods in such a way that the modified periods are separated by the periods of the original system. This is true in general if the inertia of only one part of the system be increased, the definition of a single part being that the effect of increasing its inertia can be represented by a single term i (/^i^i +/2?2 + ■■■f in the expression for the kinetic energy, where g'j, q2, ... are the normal coordinates of the original system. For example, the theorem is applicable to the case of an additional mass attached at a single point of a stretched string, but not to the case of an increase of mass spread over a portion of the string, or to the case of two or more masses attached at different points. The theorem may be simply proved as follows. Let the modified kinetic energy be i iii + ?2^ +...) + ! (/ii?i + Mi + •••)^ and the potential energy be i(AiV + A2V +•••)• The equations of motion are typified by 'ir + W + Mr (M*i + Mi +...) = 0. Mr Harrison, Notes on the Theory of Vibrations 87 The determinantal equation for the periods is A2 (1 + ^j2) _ ;^^2^ Aii/^A2, /^i/^A2, .. flltl2^^, A2 (1 + IL^^) - Ag^, iUg.UgA^, ., = 0. ;i) Let Aj2, Ag^, ... be arranged in ascending order of magnitude. If A^ = 0, the left-hand side of (1) is (— 1)" as regards sign. If A^ = Aj^, the left-hand side of (1) is equal to fh' V and this is (-- 1)"-^ as regards sign. Hence all the roots in A^ are decreased and they are separated The validity of this proof depends on (1) the non-equahty of any of the values of A^^, Xo^, ..., (2) the non-evanescence of any of the constants /Aj, /Xg, .... In case of (J) one period at least of the modified system is equal to a period of the original, but the theorem may be held to cover this case. In case of (2) the theorem does not remain true. Suppose the ju.'s are all zero except /x^, /Xg, f^t^ •••• Then only the periods corresponding to q,., qg, q^, ... are changed. The periods belonging to these coordinates will be increased and their new values will be separated by their old values. But these new periods bear no relation to the periods belonging to the remaining coordinates and can occupy any position in regard to them except as specified above. Hence the theorem does not seem to indicate where the modified periods must lie in regard to the complete system of periods of the undisturbed system. An example is afforded by the modification introduced into the periods of a stretched string by a load attached at a point dividing the string into two lengths which are commensurable. Rayleigh's argument (vide Theory of Sound, Vol. i, p. 122), which serves to maintain the validity of the theorem in this case, is acceptable owing to the strictly defined relations which exist be- tween the periods in both states. But in an ordinary dynamical problem the theorem must be held to break down in the excep- tional cases under consideration since it fails completely to indicate the position of the modified periods in relation to the original periods. 88 Dr Searle, Experiments with a plane diffraction grating Experiments with a plane diffraction grating. By G. F. C. Searle, Sc.D., F.R.S., University Lecturer in Experimental Physics. [Read 3 May 1920.] Part I. Parallel Light. § L Introduction^ . When a plane grating is employed in accurate measurements of wave length, the ruhngs are set per- pendicular to the direction of the incident beam of parallel hght. When these two directions are not at right angles, the diffracted beam is no longer parallel to a plane containing the directions of {a) the incident beam and (6) a hne intersecting the ruhngs at right angles. The formulae apphcable to this general case are obtained in §§ 4, 5, 7 ; they are tested by the experiment of §§ 8, 9 for the restricted case in which the directions {a) and (6) are at right angles. § 2. The grating axes. It is necessary to specify the three axes of a plane grating and the origin from which they start. For a transmission grating, the origin is a point on the centre hne of one of the openings. In a reflecting grating, would he on the centre line of one of the reflecting portions. The axes are (1) The normal ON to the plane of the grating. (2) The transverse axis OT, a line through cutting the ruhngs at right angles. (3) The longitudinal axis OL, a line parallel to the ruhngs. The grating interval, i.e. the common interval measured along OT from centre to centre of the openings, will be denoted by d. § 3. Diffracted wave front and ray. At a distance of thousands of wave lengths from the grating, the wavelets due to the separate openings will merge into practically a single wave. For the mathe- matical purposes of this paper we shall speak of this wave as the diffracted wave front and of a normal to it as the diffracted ray. We may speak of the diffracted wave front passing through the origin 0, if we understand it to be a surface through cutting at right angles the normals to the distant wave fronts. The normal through may be called the diffracted ray through 0. In the case of reflexion or refraction at a pohshed surface, the time of passage from an incident wave front to a reflected or ' * 1 have to thank Dr J. A. Wilcken of Christ's College, and Mr C. L. Wiseman, M.A. of Peterhouse. Dr Wilcken took the observations of § 12, Part T, and assisted in other ways. Mr Wiseman gave valuable help and criticism in the mathematicaV parts of the paper. Dr Searle, Experiments with a plane diffraction grating 89 refracted front is independent of the particular ray. But, in the case of a grating, the time of passage from an incident to a diffracted front increases or diminishes by ir as the point of incidence of the "rav" is moved from the centre of one opening to the centre of the next. Here r is the periodic time of the vibration and i is a positive integer. § 4. Diffraction of a plane wave; general case. Take the axes of X, y, z to coincide with the axes ON, OT, OL of the grating, as in Fig. 1. Let J? be a point on the centre hne of the qth. opening and let the coordinates of R be 0, qd, h. Let the direction cosines of the forward di- rection OP^ of the incident beam be l-^, m^, n^, and let those of the forward direction OP^ of the diffracted beam of order i be l^, m.^^ n^. Through draw planes perpendicular to these two directions. The distance of R from the first plane, counted positive when the incident wave front reaches before it reaches R, is 7yi-^qd + n^h. The distance of R from the second plane, counted positive when the diffracted front leaves before it leaves R, is 7n^qd + nji. If Vq is the velocity of Ught and \q the wave length in a vacuum, and if yu^, /Xg are the refractive indices of the media on the two sides of the grating, the times corresponding to the two distances are /xj {mT<}d + nJi)lvQ and ijl^ {m.^qd + n^h^VQ, and these differ by qir. Thus, since tVq = Xq, we have 1^2 {rn^qd + n^h) — /Aj {m-^qd + n^^h) = T qi^Q. This result must hold good for all positions of R on the grating, for which q is integral. We thus obtain fjL^m^ = [iiiniT iXo/d, (1) H2n2 = fJ-ini (2) These equations completely determine the directions of the diffracted beams of order i. Let the incident and diffracted beams make angles e^, 63 with OT and angles rj^, 172 with OL. Then cos ei = /%, cos 63 = W2, (3) cos r]■^^ = n^, cos ''72 = '^2 (^) Hence (1) and (2) may be written jLt2 cos €3 = fii cos ei =F iXJd, (5) 1x2 cos rj2= [J-i cos t7j (6) 90 Dr Searle, Experiments with a plane diffraction grating Since m^^ + Wg- cannot exceed unity when the direction cosines are real, the condition that a diffracted beam may exist is mg^ + n^^ 5 1, or cos^ 62 < sin^T^g- I^ ^2 ^^^ V2 li® between and In, this requires that ')72 + ^2 > i""- It is noteworthy that eg depends only upon e^ and iXJd, and that 772 depends only upon tj^. We shall not further consider the case in which fi^ and /Xg are unequal, but shall confine the work to the special case of /lij = /Xg- The reader will find no difficulty in making the necessary modifi- cations. § 5. Diffraction of a plane wave; single tnediunfi. In practice each medium is air, of refractive index n, relative to a vacuum. If A is the wave length in air, Aq = /xA. We then obtain the simple equations m2 = m^ T iXjd, or cos e^ = cos e^ T iXjd, (7) '^2^*^15 O^ cos 172 = cos T^j (8) Since -q may be restricted to lie between and tt, we have 'n2 = Vi = 'n (9) The direction of the diffracted fay is easily constructed on a spherical diagram. Let the axes of the grating intersect a sphere about as centre in N , T, L (Fig. 2), and let NON' be a diameter. Let the continuation of the incident ray _L through meet the sphere in P^. The ^\ great circle arc TPj measures e^. Calculate e, . .-^^--^^N by (7) and take TQ = €^ on TP^. About f 2 y\ \ and L as poles draw small circles through N'f^ /IH"^ ^andPj. Theni>Pi=-7;. If the small circles T / / do not intersect, there will be no diffracted beam either by transmission or by reflexion. If the small circles intersect in the points Pgj Fig. 2. P,', then OP2, OP2 will be the directions of the two diffracted beams. Of the arcs NP2, NP2' of the great circle NP^P.^N' , one is greater and one less than Itt. If NP2 is less than \tt, it corresponds to the transmitted beam, and then iV^P2' corresponds to the reflected beam. When ej and i are given, there are two values of eg, and hence there are two points Q_ and ^+ on TP^. Thus there will be two directions {OP2)- and (0P2)+ for the transmitted diffracted beam, and similarly for the reflected beam. It may happen that only one of the two beams {OP2)- and (0P2)+ exists. Unless P^ is on the great circle LN, there will be two distinct values of mg^, and the condition ^2^ + ^2^ ^ 1 may be satisfied by the smaller value of m^^ but not by the larger. Dr Searle, Experiments with a plane diffraction grating 91 Since rj^ tas the constant value r], while m^ or cos eg depends upon A, it follows that, if white light is used, to each A there will correspond a position of Pg on the small circle through P^ with L for pole. § 6. The deviation. If D is the angle (< n) between the forward directions of the incident and the transmitted diffracted beams, cos D = l-J.2 + ^1^2 + n-ffi^. If the plane ZOP^ (Fig. 1) cuts OXY in OHi, where XOH^ ^ ^i, then, since P^OZ = rj, l^ = sin 7] cos ^1, m^ = sin 17 sin <^i, Wj == cos rj, and similarly for Pg- Thus 1 — 2 sin^ ID = cos D = sin^ 77 cos (^^ — </>2) + cos^ 77, and hence sin |Z) = siniy sin | (^1 ^ ^2)' (1^) as can also be shown from the isosceles spherical triangle P^LP^ in Fig. 2. In the case of the transmitted diffracted beam, l^^ is positive. Noting that n^ = n^^ cos 77, putting m-^= a + h, m.^ = a — b, and substituting for l^, l^, we find 2 (sin2 17) - 62) _ sin^T^ - a^ _ 52 _ ^,^^^,^2^ _ ^2 _ ^2)2 _ ia%^f. Thus sin^ ID is greater than b^ except when a = 0, and then the two are equal. When a = 0, ni^ = — m^. If we take m^ positive, we see, by (7), that, since i is positive, m^ = — m2 = iA/2(^. Hence b = iXj2d. Thus, if Dq is the minimum deviation, sin iZ)o = iA/2«; (11) Since rj does not occur in (11), m^ = — m^ gives a minimum of D for any given value of rj — a minimum having the same value for all values of 77. § 7. The sloped grating. For the experiment of §§ 8, 9 it is convenient to use axes differing from those of § 4, Now let OY (Fig. 3) coincide with OT and let OL make an angle 6 with OZ. Then the direction cosines of OL are sin 6, 0, cos 9, those of OT are 0, 1, 0, and those of ON are cos d, 0, — sin 6. Let ^1, m^, ^1 and I2, m^, n^ be the direction cosines of the forward directions OP^, OP^ of the incident and the transmitted diffracted beams. Let W2 = sin 0, so that the diffracted ray OP^ makes an angle i/f with the plane OXY, counted positive when P2OZ < In. Let the plane ZOP.^ cut OXY in OH^ and let XOH^ = m. Then, if -q is the common angle between OP^ or OP^ and OL, 92 Dr Searle, Experiments with a plane diffraction grating ?2 sin 6 + 712 cos 6 = cos 17 = l-^ sin 6 + n-^co&d. ...(12) We also have, by (7), if P^OT = e^, P^OT = e^, cos 62 = ^2 = m^ T i^^jd = cos e^ T iA/(? (13) Hence m^ is known at once. Using ^2^ = 1 — '^^^ — ^2^, we have, by (12), sin^ ^ (1 — m^ — ^2^) = (cos r) — n^ cos ^)2. Solving for n^ and taking the negative sign in the ambiguity, we have sin i/f = ^2 = cos 6 cos 7^ — sin ^ (sin^?] — m.^^)^. ..-(14) Using this value of Wg in (12), we find ^2 = sin ^ cos ?7 + cos ^ (sin^ 17 — ^2^)* (15) Since cos NOP^ = I2 cos 6 — n^ sin 6, we find from (14) and (15) that cos iYOPa = (sin2 77 - mg^)* (16) Thus the negative sign has been correctly chosen in (14) for the transmitted beam, since for this cos NOP2 must be positive. If the positive sign is used in (14), cos NOP2 is negative, corresponding to the reflected diffracted beam. In terms of i/j and co, the direction cosines of OPo are cos i/j cos to, cos i/j sin 60, sin i/j. Hence Wg = cos j/f sin co, and thus sin to = mg/cos i/j = cos e2/cos i/f (17) The two angles ifs and co completely determine the direction of the diffracted ray. In the experiment of §§ 8, 9 the incident rays are parallel to the axis OX of Fig. 3. Hence ?j = 1, m^ = 0, n^ = 0. We then have cos 62 = W2 = T iX/d, (18) cos Tj = sin 9, (19) and thus, since sin^ 7] — cos^ e^ = sin^ eg — cos^ 77, sin ijj = n^ = sin 6 cos 6 — sin 6 (sin^ ^2 — sin^ 9Y, . . .(20) ^2 = sin^ ^ + cos ^ (sin2 62 - sin2 ^)* (21) sin 60 = ma/cos ifj = cos eg/cos ijj (22) Since e^ and 9 may be restricted to be less than hr, we see that no diffracted beam will be formed if 9 exceeds its critical value e^. § 8. Apparatus. The general arrangement is shown in Fig. 4, The grating G is attached to a horizontal shaft A, with its plane parallel to A and its ruhngs perpendicular to ^. A horizontal colhmator L has horizontal and vertical cross-wires intersecting at C in its focal plane ; these are illuminated by a sodium flame S. The straight Hue joining C to the appropriate nodal point of the Dr Searle, Experiments with a plane diffraction grating 93 lens is the line of collimation, or axis, of the collimator. The parallel beam defined by C is parallel to this hne. After the light has passed through the grating, it is received by a goniometer K, and -^ S an image of the collimator wires r nC is formed in its focal plane. To fix the line of collimation, cross- wires are placed in the focal plane; they intersect in D. The u'^ goniometer is carried on a hori- zontal revolving shaft B, and its line of colhmation is perpen- dicular to the shaft. One cross- wire is parallel and the other perpendicular to the shaft; the E latter is also perpendicular to the U/i shortest distance from D to the axis of the shaft. The shafts are provided with divided circles E, Fig. 4. F, which are read by aid of the pointers U, U', V, V. A balance weight W is attached to the circle F. The point of intersection of the hne of colhmation of K with the axis of B should lie approximately on the centre of the grating, and the line of colhmation of L should pass through the same point. The angles 6 and i/j are measured by the circles E and F. § 9. Experimental details. The shaft A is set horizontal by aid of a level. The colhmator is adjusted optically by an auto-collimating method. The plane of the grating is set horizontal by a level, and the shaft is then turned through 90°, as measured by the circle E, so that the plane of the grating is vertical. Light from a flame is then reflected by a plate of glass held at 45° past the cross-wires and through the lens on to the grating, and the colhmator, pre- viously set for "infinity," is adjusted so that the image of C, the intersection of the wires, coincides with C itself. If the coincidence is recovered when the grating shaft is turned through 180°, the plane of the grating is parallel to the shaft. The line of colhmation is then both horizontal and also perpendicular to the grating shaft. The line of collimation of the goniometer is set perpendicular to the goniometer shaft B by an optical method. An auxihary col- limator, set for "infinity," is placed so that it is approximately perpendicular to the shaft. A plate of plane parallel glass is attached to the circle F near its centre so that its faces are approximately parallel to the shaft. It is convenient to make the shaft vertical; the glass plate can then be supported on a small levelhng table resting on the circle. By adjusting both the colhmator and the 94 Dr Searle, Experiments with a plane diffraction grating plate, the faces of the plate are made parallel to the shaft and the axis of the colUmator is made perpendicular to the shaft. In this case the image of the collimator wires can, by turning the circle, be made to coincide with those wires, when either side of the plate faces the colhmator. If the plate has been suitably placed, it will be possible, by turning the goniometer on its shaft, to receive the image of the collimator wires on the focal plane of the goniometer. The "vertical" cross-wire of the goniometer, i.e. the wire perpen- dicular to the shaft, is now adjusted so that it coincides with the image of the corresponding wire of the colhmator. The line of colhmation is then perpendicular to the shaft. The goniometer is then put into position and its shaft is levelled. The axes of the collimator, of the grating shaft and of the goniometer shaft are adjusted to be approximately in the same horizontal plane. The plane of the grating is made vertical, and the goniometer stand is adjusted in azimuth so that one of the diffracted images of the collimator wires can be made to coincide with the goniometer wires by turning the goniometer on its shaft. When the adjustments already described have been effected,, and when the plane of the grating G is vertical, the diffracted beams are parallel to the plane OTN. If OT is inclined at an angle 8 to the grating shaft, the direction of OT will be changed by 28 if G is turned through 180° about the axis of the shaft from Position 1 to Position 2, when the plane is again vertical. The goniometer is turned to receive a diffracted beam when G is in Position 1. If, when G is turned into Position 2, the inchnation of the beam to a horizontal plane is changed, the grating must be turned in its own plane until the inclination is the same for both positions. Since ijj is always small, cos i/j is nearly unity and hence, by (22), sin oj has a nearly constant value, for Wg is independent of 6. Hence, if the image of C, the intersection of the collimator wires, lies on D, the intersection of the goniometer wires, when the plane of G is vertical, the image of C will always lie very near the "vertical" cross-wire, and one setting of the goniometer stand will suffice for all values of 6. The plane of the grating is made horizontal and the index reading is taken. It is then turned through 90°; it is now vertical and in its zero position. The goniometer is next adjusted so that the image of C lies on the horizontal wire of K. The goniometer is then in its zero position. The grating is now turned through 10°, 20°, ... on sither side of the zero, and the goniometer is turned on its shaft to bring the image of C on to the horizontal wire of K in each case. If the grating circle has tivo indices, the grating is turned through 10°, 20°, ... as indicated by one index. In reducing the observations the mean of the angles furnished by both indices is used. Dr Searle, Experiments with a plane diffraction grating 95 Since e^ = ^tt, cos eg = Wg = iX/d. From the values of the interval d and the wave length A, cos eg is found and then the values of ip corresponding to the mean values of 6 are calculated by (20). These values are compared with the mean values of j/f given by the goniometer readings. § 10. Distortion of the image. As 6, and consequently i/r, increases, the observer sees that the angle between the images of the col- limator wires undergoes great changes. When ^ = 0, the images are at right angles, but the angle diminishes rapidly as 6 reaches its critical value. The theory shows that they are actually tan- gential one to the other when 6 has its critical value, but, as no light is transmitted in the critical position, the phenomenon cannot be observed. If the collimator wires are stretched across a small circular opening, the image of the edge is distorted into an oval, which is practically an ellipse having the images of the wires as conjugate diameters. When, however, 6 approaches its critical value, the oval begins to deviate from an ellipse. In Fig. 5 let OX, OY or OT, OZ, OL, ON meet a sphere described about as centre in X, T, Z, L, N. Let OJ be the diffracted ray corresponding to the incident ray OX; the ray OX corresponds to the line of collimation of the collimator and OJ to that of the goniometer, when the image of C is brought to the intersection of the goniometer wires. Let OP^ be a ray nearly parallel to OX and let OP^ be the corre- sponding diffracted ray. Let the great circles through Z and P^, J, Pg cut the great circle TXS in H, K, M. Let *S be _ the pole of ZJK and let the great circle Fio- 5 SJ meet ZP.^M in Q. Then ZJQ = ^tt. If the goniometer is mounted as described in § 8, and if its line of collimation coincides with OJ, its horizontal cross-wire will correspond to SJQ and its "vertical" wire to ZJK. The rays parallel to OP2 will come to a focus in the focal plane of the gonio- meter at D', whose coordinates referred to the horizontal and vertical wires through D (Fig. 4) are/ x angle QOJ and/ x angle P^OQ, where/ is the focal length of the lens. If points on a curve CC in the focal plane of the collimator give rise to diffracted rays whose directions are shown by points on the curve JPg o^ ^^^ sphere, and if the image of CC in the focal plane of the goniometer is DD' , the angle between the "vertical" cross- wire and the tangent to DD' at D is equal to /, the angle between the great circle JZ and the tangent at J to the curve JPg. 96 Dr Searle, Experiments with a pJune diffraction grating If JK = ^, Pa^ = 'A'' ^^ = ^' -^'^^ = ^'' ^lien tanZ= limit of 75-^ = cosi/f f-y^ ) (23) If XH = a, PiH = y, the direction cosines of OP^, OP^ are ?j = cos y cos «, m^ = cos y sin «, n-^ = sin y, ?2 = cos if}' cos a>', m^ = cos j/(' sin to', ng = sin 0'. Since ZOL = d, the direction cosines of OL are sin 6, 0, cos ^. But P^OL = P2OL = 7], P-PT = ei, P^OT = 62, and thus the funda- mental equations (13) and (12) become cos iji' sin oj' = cos 7 sin a =F iXjd, >. . . (24) cos ijj' cos ii)' sin d + sin ip' cos ^ = cos y cos a sin ^ + sin y cos 6. (25) The vertical coHimator wire corresponds to the great circle ZX, and for this a = 0, but y varies. If ly is the inclination to JZ of the corresponding path described by P^, tan/, = cos^(|^y(|^)^. Differentiating (24) with respect to y and then putting y = 0, so that 0', co' become ifs, co, we have — sin i/f sin co {di/j'ldy)^ + cos ijj cos oj {dw'ldy)^ = 0. Hence tan /r.= sin i/j tan co. The horizontal colhmator wire corresponds to the great circle"' XT and for this y = 0, but a varies. Differentiating (25) with regard to a and then putting a = 0, we have — sin iJj cos CO sin 6 {di/j'/da)Q — cos i/j sin co sin ^ {da}'/da)Q + cos i/f cos ^ (difj' lda)Q =--= 0. Hence, if Ih is the inchnation to JZ of the corresponding path ofP,, tan /^ = cos cos xjj cos ^ — sin ifj cos co sin 6 .(26) sm 0L> sm u Multiplying numerator and denominator by cos a>, and replacing cos^ CO by 1 — sin^ to, we find T , T U cos d — Uo sin 6 .^„, tanZjj= tanZ^+ A ^ . „ 27) sm CO cos CO sm Since the direction cosines of ON are cos 6, 0, — sin 6, and I2, Wg in (27) refer to OJ, l^ cos 6 — n^ sin d = cos iVOJ. In the critical position, J lies on the great circle LT, which corresponds to the Dr Searle, Experiments with a plane diffraction grating 97 plane of the grating. Then cos NOJ = 0, and the difference be- tween the tangents vanishes, i.e. the two curves touch at J . Since the distance of ZLX from T is constant, the curve corresponding to the vertical cross-wire is a small circle passing through J with T for pole. At J the small circle is perpendicular to the great circle TJ and the value of tan 1^ can be verified by spherical trigonometry. The horizontal cross-wire is represented by the great circle XT, but now both e^ and t] vary and no simple construction is available for the v:liole curve through J corresponding to this wire. The curve touches at ./ the small circle passing through J and X with L as pole, and cuts at right angles the great circle LJ . Hence tan III = cot LJZ, and then (26) can be verified by spherical trigonometry. If we find cos LJZ and sin LJZ and divide, we obtain the alternative form cos 6 — sintff sin 6 tan iu = . : : 7P • cos i/f sm o) sm v If the angle between the two small circles which intersect in J is A, then A is the supplement of LJT. But LJ = ^tt — 6, JT = eg, LT = \tt, and hence cos A = tan 6 cot eg. In the experiment cos eg has the constant value =F iXjd, and thus cos A depends only upon d. The angle A will vanish when cos A = 1, and this occurs when d = eg, i.e. when 6 has its critical value. We can make visible a finite arc of the small circle w4th L for pole. If we illuminate with white light the small opening across which the wires are stretched, the position of J on this small circle will be different for the different colours. The short length of cross- wire will correspond for any colour to a small arc of a curve touching the small circle at practically its middle point. The envelope of these small arcs will be the small circle itself. The image of the horizontal wire will thus be a dark curved line running across the spectrum. § 11. The critical values. The critical position of the grating is reached when 6 = eg, and we have, by (20), (22), the critical values cos e, sm ijjc = sm eg cos eg, sm oj^ = i , (1 — sin^ egcos^eg)^ cos e., ,^ J , cos^ eg tan a;. = — 9 , (tan i v)c == —■ " • ^ sm^ eg sm eg For the grating used in § 12 and with * = 1, cos eg = 0-33568, sin eg = 0-94198, eg - ^tt - 19° 36' 49". Then ^1 - ]8°26', co^ - 20°43' 18" and (7^.)^ = 6° 49' 17". VOL. XX. PART I. 7 98 Dr Searle, Experiments with a plane diffraction grating When ^ = 0, and therefore i/j = 0, sin o) — cos €2, oi —■ 19° 36' 49". Thus the maximum change in a> is only 1° 6' 29". § 12. Practical exatnple. The following results were obtained by Dr J. A. Wilcken, using a grating having 14,468 lines per inch, ' and, hence, an interval d = 1-7556 x 10"* cm. The adjustments described in § 9 were either effected or tested. The plane of the grating was not quite parallel to the grating shaft, but as both images of the first order were observ^ed, the mean results will be hardly affected. The calculated values of yjr were found on the assumption that the axis of the collimator is perpendicular to the transverse axis of the grating. Sodium light of mean wave length X = 5-893 x 10~^ cm. was used. Then, since the images were of the first order throughout, _ X _ 5-893 X 10- 5 ^^~ d~ 1-7556 X 10-* Thus e, = cos-i mg -= 70° 23' 11", and m^^ = 0-11268. Each of the observed values of yp' given in the table is the mean of four. Each of the first order images was observed, and for each image two values of d, one on either side of zero, were taken. The grating circle, which was printed on card, was a little eccentric relative to the shaft (it was a "home- made" affair), and, consequently, although the settings were made to integral degrees by one index, the other index did not always read integral degrees. Some of the mean values of 6 are, therefore, not integral degrees. The calculated values of -^ are those found from equation (20), which for con- venience is written sin t//' = ?i2 = J sin 26 - sin 6 [(cos 6 + m^) (cos 6 - m^)]^. For the sake of interest, the calculated values of w, /,- and I^ have been added. The last line in the table gives the critical values as found by calcu- lation. = 0-33568 = sin 19° 36' 49' d ^ V' OJ Ir In mean obsd. mean obsd. calcd. calcd. / // or'/ / // a t n 19 36 49 90 9 54 36 34 51 19 36 52 12 25 86 38 28 19 54 1 12 45 1 12 33 19 37 7 25 51 83 1 5 29 50 155 30 155 36 19 37 34 4113 78 53 41 39 51 2 5145 2 5018 19 38 19 1 46 73 42 30 49 48 4 7 30 4 7 25 19 39 58 128 20 66 31 55 59 49 6 2615 6 22 31 19 44 27 2 16 55 54 29 53 64 44 8 3215 8 29 8 19 5024 3 2 53 44 114 67 45 10 54 10 5113 19 59 10 3 55 4 33 20 15 69 44 14 18 14 10 3 20 15 19 5 9 40 20 20 49 70 2311 18 26 20 43 18 6 4917 6 4917 The agreement between the observed and calculated values of ■^ is satis- factory. Dr Searle, Experiments with a plane diffraction grating 99 Part II. Non-parallel Light. § 1. Introduction. In the following experiments the incident light does not form a parallel beam. The diffraction now not merely changes the direction of the axial ray of the beam, but also, in general, introduces astigmatism into, or changes the astig- matism of, the incident beam. The exception is when the incident and diffracted axial rays are perpendicular to the rulings and the deviation is a minimum. The diffracted rays will, in general, pass through two focal lines when the aperture is small. If the aperture is increased, aberration will appear and all the rays will not pass accurately through the two lines. Aberration can be minimised by keeping the aperture small, but astigmatic effects are inseparable from the diffraction in the general case. The formulae for the general case are easily obtained, but are complicated. We shall, therefore, consider only the case in which the axial ray of the incident beam is perpendicular to the rulings. § 2. Diffraction of an astigmatic hea^n. In Fig. 1 let OX, OY , OZ coincide with ON , OT, OL, the axes of the grating, as defined in Part I, § 2. For convenience, OZ will be taken as vertical. Let a beam, which started from a luminous point and therefore has a wave front, fall upon the grating near 0. Let OP^ be the continuation of the ray through 0, which has been restricted to lie in the plane OXY, and let OP^ be taken as the axial ray of the beam. Let P^OX = d^. Take OP^ as the axis of r^ in a new set of axes Or^, Os^, Ot^, of which Osi is in the plane OXY and Ot^ coincides with OZ. Let the equation to the incident wave front passing through be r, = i>SiV + W,s,t, + iT,t,^ (1) Let R he a point on the grating and let its coordinates referred to the grating axes be 0, qd, z, where d is the grating interval and q is an integer. Then the coordinates of R referred to the axes of the incident beam are ri = qd sindi, Si = qd cos dj^, t^ = z (2) If a line through R parallel to OP^ cuts the wave front O-^^ in F^, the second and third coordinates of F^ are qd cos ^j and z. By (1) and (2), the distance of F^ from the plane r^ = 0, which touches the wave front at 0, is ^S-^qH^ cos^ 6-^ + W^qdz cos ^^ + ^TjZ^, and the distance of R from the same plane is qd sin ^j. Hence F^^R - qd sin 6^ - ^S^qU^ cos^ d^ - W^qdz cos 6^ - \T^z^....{^) 7—2 100 Dr Searle, Experiments with a plane diffraction grating 1 When R and F^ approach 0, F^R becomes more and more nearly the normal at F^, and, for a small aperture, may be treated as the normal in the estimation of distances. Thus, ultimately, F^R is the ray distance from the wave front OF-^ to R. Let OP 2 be a diffracted ray of order i. By symmetry, OP2 is in the plane OXY, since OP^ is in that plane. Let P^X = 6^. Take the axial ray OP2 as the axis of ^2 in a set of axes Org, Os^, Ot2, of which OS2 is in the plane OXY and 0^2 coincides with OZ. Let the equation to the diffracted wave front passing through be _ 1 "~ 2 S2S2^+W2S2t2+hT2t2' (4) Then, if F2R, parallel to OP2, cuts the diffracted wave front OF^ in #2, the distance F2R is ultimately the ray distance from F2 to R. We then have F2R = qd sin ^2 - i>52?^^^ cos^ 62 - W2qdz cos 62 - lT2Z^....{b) The optical condition is that F2R differs from F^R by qiX, where i is a positive integer. We thus obtain F2R = FJl ± qiX. Since this holds for all values of z and all integral values of q, we have, by (3) and (5), sin 62 = sin ^^ ± iX/d, (6) S2 = k^S„ T4^2 = '^^i, T2=T„ ..(7) where k = cos ^^/cos $2. Since 6^ and $2 both lie between — ^tt and ^77 for a transmitted beam, k is positive. The direction of the axial ray of the diffracted beam is given by (6) and is independent of the constants S^, W^, T^. Equations (7) give the form of the diffracted wave front which passes through 0. If the deviation of the axial ray is a minimum, it follows from Part I, § 6, or otherwise, that sin ^^ = — sin ^j. Since d-^ and 9^ both lie between — ^tt and ^tt, cos ^2 = cos ^^, and thus k==\. Hence, in the case of minimum deviation, the form of the wave front is unchanged and the diffraction merely turns it through 2d about OZ. The restriction stated in § 2 must be noted. If the planes of the principal sections of the incident wave front are OXY and ZOP^, or, what is the same thing, the planes Or^Si, Orjtj^, then W^ = 0. It follows, from (7), that IF2 = 0, and thus the principal planes of the diffracted wave front are OXY and ZOP2. When TFj = 0, the section of the incident front by the hori- zontal plane ^^ = is r ^ = ^S^s^^, and the distance of the centre of curvature of this section from is Sj~'^. The vertical focal line of the beam passes through this centre of curvature. Similarly, the horizontal focal line is at a distance T^~^ from 0. The distances Dr Searle, Experiments with a plane diffraction grating 101 from of the vertical and horizontal focal lines of the diffracted beam are S2~^ and ^3"^. If the incident beam is stigmatic, T^ ^ S^ and W^ = 0. Then S2 = k'^Sj^, Tg = S^. Hence S^ = k^T^, and so the diffracted beam is astigmatic, unless k=l, i.e. unless the deviation is a minimum. § 3. The principal curvatures. The principal curvatures of the diffracted front can be found in terms of those of the incident front. Let the principal planes of the incident front intersect the tangent plane at in Orj^^, 0^-^ (Fig. 2). Take these, with Of^ along OPi, as axes for the front. Let the radii of curvature of the sections by 0^^171 and O^^^i be Bi~^ and C\~^. The equation to the incident front is then ii = hB,rj,^ + iC\l,^ (8) Let Orji make an angle tpi with Osi, as in Fig. 2. Then ^1 "= ^1' Vi "" '^1 ^^^ "Ai + h ^i^ 'Ai' ^1 = ~ ^1 ^^^^ ^1 + ^1 ^°^ 'Ai' and hence (8) is equivalent to ^1 = 2-^1 (^1 cos i/ji + ^1 sin 0i)2 + iCi (- Si sin tp^ + i, cos i/j^f. ... (9) Comparing (9) with (1), we find Si = i {B, + C,) + I {B, - C) cos 2<Ai, I If 1 = 1 (^1 - C'l) sin 2eAi, (10) T, = i(5i+Ci)-i(5i-Ci)cos2^,. I Then /Sg, Tf 2, ^2 can be found by (7). If the equation to the diffracted front referred to its principal axes is ^2 = ^^27)2^ + ^02^', (H) and if T^gOsg = «/'25 ^^^'^ ^2, O2, ^2 ^^^ related to S2, T2, W2 by equa- tions similar to (10). Solving for B2, Cg, 'A2' we have B, C2 tan 2^2 = 21f2/(.S2 - Tg) (13) The ambiguity in (12) has been settled so that, when i/j^ = 0, B2 = k^B^, C*2 = G^. Apart from mere reversals of direction, (13) gives two values of j/^a differing by hir, and corresponding to the axes 0-r]2, Ot,2- The arrangement of signs in (12) implies that when tjj^ = 0, ^2 = 0- Since, by (7), W^ and W2 vanish together, ijj^ and 02 niust reach \tt, tt, ^tt, ... together, and it follows that, for intermediate values, ^2 must lie in the same quadrant as ifj^. } = 1 (^2 + T2) ± [i (^2 - T2f+ W2^]\ (12) 102 Dr Searle, Experiments with a 'plane diffraction grating ^ Equations (12), (13) with (7) give B^, C^, i/fg in terms of >Si, T^, W^, which are given in terms of B^, C-^, i/j^ by (10). § 4. ^ simple case. If we take C^ = 0, the incident wave front is cylindrical. We then obtain ^2 = Pi {F + 1 + (P - 1) cos 2i/ri}, Cg = 0, ...(14) or tan i/(2 = A;~^ . tan ^1 (15) The maximum difierence between i/j^ and 1^2 occurs when tan i/j-^ = k^, and then sin (i/^i — iff.^) = {k — 1 )/{k +1). Since C*2 = 0, the diffracted wave front is cylindrical. There is therefore only one focal line at a finite distance from and this distance is -Bg"-^. If the principal planes of the incident front are turned round, i/j^ will change, so causing B2 to vary, and the distance of this focal line from will vary. § 5. Measurement of wave length. The results of § 2 can be applied in the determination of the wave length of sodium light by measurements made on an optical bench. On the bench slide three carriages Z), H, K, as shown in plan in Fig. 3; D carries a horizontal glass scale divided in mm., H carries the grating G*- (with vertical rulings), whose centre is 0, and K carries the con- verging lens system L, of focal length/. At the end of the bench is a vertical slit illuminated by a sodium flame F; to identify a point E on the slit, a wire may be stretched across the slit. The divided face of the scale faces the slit and this face and the plane of the grating are perpendicular to the bench. The line through the nodal points of L is parallel to the bench and passes through E. The scale is first placed in the position A^B^, at a distance from E exceeding 4/ by about 30 cm., and the lens is adjusted to form a real undiffracted image on the scale at C^. The axial ray of the incident beam is normal to the grating, and thus, if </> is the angle between the normal and the axial ray of either diffracted beam of order i, we have k"^ = sec^ <j). Since the incident beam Dr Searle, Experiments with a plane diffraction grating 103 corresponding to the point E of the slit is stigmatic, the vertical focal lines of the diffracted beams of order i will be at a distance OCJk^ from 0, along lines OPj, 0^^, each making an angle cf> with 0C\. Sharp images of the slit will pass through P^, Q^ and can be focussed on the scale if it is moved to A\By' . With the grating used in § 9, the two sodium lines can easily be separated. Then sin (/) = ^PiQi/OPj^. Since OP^ is difficult to measure, we suppose OX-^ known, where Z^ is the mid-point of PiQi- If OX^ = x-^, P^Q^ = 2;yi, tan <^ = iPiQJOX, = yjx^. The glass plate protecting the grating prevents an accurate measurement of OX^. We therefore move the lens carriage along the bench so that the undiffracted image is focussed on the scale at C'g. If the scale is moved further towards 0, the difPracted images can be focussed at Pg? Q2- If OX2 = x^, ^2^2 ^ '^Ih' tan (j) = y^l^o,- Hence tan (j,^{y^- y,)l{x^ - x^) (16) Putting 6-1^ = 0, $2 = (f) in (6), we find X = d sin (jiji, (17) where i is the order of the image and d is the grating interval. From (16) and (17), A is determined. Since it is an angle we measure, small errors of focussing will be of little account, for, in spite of them, the point in which the axial ray cuts the scale in each case will be correctly estimated, and this is all that is necessary. § 6. Test of laiv of obliquity. Let OCj = u^, OC^ = ^2' ^Pi = ^'i) OP2 ^ V2- Then v^ — v^ = [{x^ — x^)^ + (y^ — y^fY- But, since ^2= sec^^, we have, by § 5, u^ = u-^ cos^ (J3, v^ = U2 cos^ ^, and thus i\ — V2= (w^ — Mo) cos^ cf). Since Mj — Wg is known from the bench readings, we can test the law for the vertical focal lines by comparing the two values of v^ — ?',• As we are now concerned with the positions of images the focussing must be accurate. If the slit is not too narrow, the diffracted images of order i of the horizontal wire stretched across it may be focussed on the scale. If these are at jOj, q^^ when OC = ^^ and at ^03, q2 when OC = ^2, and if p^q^^ = 2r]^, ^2^2 = ^772, then ViP2 - mi - Q' + (^1 - ^2)']^. Since, by § 2, 1^= T^, we have p^2 = '^h ~ ^'2- The two values of PxP2 are compared. It is difficult to obtain satisfactory readings for x and ^. This is largely due to the fact that the diffracted rays in the horizontal plane through do not meet in a point but touch a caustic of large 104 Dr Searle, Experiments with a plane diffraction grating radius. If Q^Ri, drawn perpendicular to Q-fi in Fig. 3, cuts OC^ in i?i, the radius of curvature of the caustic at Q^ is ^QJH^. The length of the caustic between the points of contact of the tangents from Ml, iVi, where M^N^ is the width of grating actually used, is 3 (^iMi ~ ^1^1 ) approximately. § 7. Adjustment of the lens. The lens, a converging system, is adjusted optically. Let its focal length be / and the distance between its nodal points be t, where t is positive when the distance between the principal foci exceeds 2/; for a projection lens as shown in Fig. 3, t will be negative. When the distance of the luminous point E from the scale ACB exceeds 4/+ t, there are two positions of the lens for which an image of E is formed on the scale. Let M, N (Fig. 4) be the nodal points corresponding to the principal • iB ^^ m' M 1 -— -3r^ ~^3r^ -=:,,^^__ C A s' R' S R E Fig. 4. foci to the right and left of L. In Fig. 4 let EC be the horizontal line through E parallel to the bench and let the other Unes be pro- jections upon the horizontal plane through EC. Let MR, NS be the perpendiculars from M, N on EC. Let RE = p, SO = q. Then in the second position of the lens, R'E = q, S'C = p. When the angles are small, RS = R'S' = t. Let I, I' be the images of E in the two cases. Take CE as axis of a; and horizontal and vertical lines through C as axes of y and z. Let the second and third coordinates of M, N, I, r be y, z, 7), I, r, Z, Y' , Z' . Since /iV is parallel to ME, and I'W to M'E, 7 = 7y + yq\p, Z = ^ + zqjp, T' = t^ + yp\q, Z' = ^ + zp\q. Since {p + q)\pq = II f, Y-Y' = y{q-p)lf, Z-Z' = z{q-p)lf. Hence, if the image has the same position for both cases, then y = 0, and 2 = 0, and thus M lies on EC. The emergent ray through N will then be parallel to EC for all positions of the lens carriage. The lens is best mounted so that it can turn about a vertical pivot whose axis passes through M. If the support to which the pivot is fixed is moved through the distance MR at right angles to the bench, M will be brought on to EC. Then by turning the lens about M, N can be brought on to EC. Dr Searle, Experiments with a plane diffraction grating 105 To identify C, a pin is mounted on a carriage so that its tip coincides with E. The carriage is moved along the bench so that the tip touches the scale AB. The point of contact is C. If L is adjusted on its carriage so that / coincides with C for both cases, then M, N lie on EC. § 8. Other experimental details. The scale ^ B is set perpendicular to the bench. A set square XYZ, with the right angle at X, is held with XY in contact with AB. A pin is held close to XZ. If, when the carriage D is moved along the bench, the distance from the pin to XZ is constant, AB is correctly placed on its carriage. The scale must be horizontal and the slit vertical. The plane of the grating can be set perpendicular to EC optically. The lens L is removed and a small triangle of white paper is fixed to AB so that a vertex coincides with C. The grating G is placed midway be- tween C and E and is adjusted on its carriage so that the image of C by reflexion at G coincides with E. To allow a close test of parallax, a few grains of lycopodium may be placed on AB when the image of the slit does not fall on a dividing line. § 9. Practical example. Using a grating with d = 1-7526 x 10~*cm., the following results were obtained: The image of first order was used; thus i = \. Bench reading 118-50 cm., glass scale readings 97-52, 73-84 cm. 141-21 cm., „ „ „ 89-64, 82-18 cm. Hence ih = \ (97-52 - 73-84) = 11-84 cm., y^ = | (89-64 - 82-18) = 3-73 cm. Also x^- x^^ 141-21 - 118-50 = 22-71 cm. Hence tan cf) = {y^ - y2)l{x^ - x^) = 8-11/22-71 - tan 19' 39' 7". Then X = dsin (p/i = 5-894 x 10-^ cm. § 10. Experiment with an astigmatic incident beam. The experi- mental test of the results of § 4 is a good exercise in optical manipu- lation. Fig. 5 is a plan of the apparatus. Two cross- wires, inter- secting in E, are fitted into a tube turning about a horizontal axis 106 Dr Searle, Experiments with a plane diffraction grating in a hole in the board D. A circular scale is attached to D and P has a pointer J which indicates its angular position. Only one wire is used in the measurements, but the second wire is useful as identifying E. The wires are illuminated by the sodium flame F. A lantern projection lens L is placed so that E is at its principal focus; for the best results, that end of L should face E which faces the lantern slide. Beyond i is a cylindrical tube Q, resting in two F's, V, V, and against a stop C/, and thus having only one degree of freedom. A piano-cylindrical lens A is attached by its plane face to one end of Q. A lens of about + 2-5 dioptres, such as is used in spectacles, is suitable. The grating, centre 0, is placed at G. A ground glass screen H can slide on the main optical bench, which also carries G, Q, L; if possible, D should be carried on the bench. On a short auxiliary bench slides a second screen K; the angle between the benches is cf), where sin cf) = i\ld. The ground sides of the screens face 0. Suppose, for a moment, that E \^ & luminous point. Then E^ at the focus of L, gives rise to a parallel beam falling on the cylindrical lens A. This lens converts the plane wave front into a cylindrical front. If the "power" of ^ is + i^ dioptres, a "real'' focal line, parallel to the generators of A'b surface, will be formed' 100/i'' cm. from A. This focal line can be received on the screen H, By § 4, the diffracted front is cylindrical and there is only one focal line at a finite distance. This focal line can be received on the screen K. If A is turned by turning Q on its axis, the focal line of the diffracted beam will turn about the axial ray and the distance of the focal hue from will change. The experiment tests the relation between the linear displacement of K and the angular displacement of A. When a wire is used instead of a luminous point, images of the wire will be formed on H and K when the generators of A are parallel to the wire. If the pointer J is set in any position, a sharp image can be obtained by turning Q. When the adjustments of § 11 have been made, H is set to receive the image of the wire, and HO is measured. As a correction we may add tj ^i, where t is the thickness and ix the index of the plate covering the grating. A line ruled on H is made vertical by aid of a set square and a level, and P and Q are adjusted so that the image is vertical and ^j = 0. Then K is set so that the diffracted image is in focus on it, and the reading of K on its bench is taken. Then P is turned by steps of 10° or 15°, Q is turned in response, and K is adjusted in each case so that the image is focussed. When P has been turned through 90°, so that ifj-^ = ^tt, the image is hori- zontal, and, by (14), since i/r^ = \tt, its distance from is equal to the measured distance OH. When the image is vertical, j^i = or it. Since B-^-^ = OH, B^-^ = OK, we have, by (14), Dr Searle, Experiments with a plane diffraction grating 107 ^^^ = F+l + (F-l)cos2<Ai' ^^^^ where k ■= sec cf) and sin <f) = iX/d. To compare theory with experiment, we may plot the value of OK given by (18) against the bench reading of K. If the zero of this bench is at the end nearest 0, the points will lie about a straight line equally inclined to both axes. An alternative method is used in § 12. § 11. Experimental details. The cross- wires should be mounted so that E is as nearly as possible on the axis of P. The lines joining the nodal points oi L to E are made coincident and parallel to the bench by the method of § 7. The axis of Q is set approximately parallel to the bench; optical methods are available. The cylin- drical lens A is adjusted optically. For a given direction of the wire at E, there are two positions of Q, 180° apart, in which A forms a sharp image of the wire on H. If the positions of these images are not identical, the error can be corrected by moving A at right angles to its generators across the end of Q. To set the lens L so that E is at its focus, a plane mirror is substituted for H, Q and G are removed, the cross-wires are illuminated and L and the mirror are adjusted so that E coincides with its own image. The plane of G is made perpendicular to the bench by the same method, the plate covering the grating serving as the plane mirror. The bench on which K slides is adjusted optically. First set P and Q so that a vertical image of the wire is formed on K. Then slide L along the main bench and readjust K. If the position of the image relative to K is unchanged, the auxiliary bench is correctly placed. If a micrometer eyepiece is used in place of the screen K, two images will be seen except when the wire is horizontal, since sodium light has a double spectrum line. Unless the wire is very fine, the images will overlap. The doubling of the images causes no inconvenience. § 12. Practical example. The following results were obtained with a grating of 14,493 lines per inch. For this grating, d = 1-7526 x 10~* cm. The wave length was 5-893 x 10~^ cm. The image of first order was used; thus i = 1. Hence sin (p =^ 0-33625, k" = sec^ </> = 1-1275, (/> - 19° 38' 55". A cylindrical lens of + 2-5 dioptre was used. The corrected value of OH was 38-91 cm. The angle ^|^^ was varied from 0° to 180° by steps of 15°. The bench readings in columns 2 and 4 arc theoretically identical, and their mean is given in column 5. The values of OK calculated by ( 18) are given in column 6. To facilitate comparison, the mean difference between columns 5 and 6 has 108 Dr Searle, Experiments with a plane diffraction grating been added to column 5, as suggested by Dr Wilclcen, and the results are entered in column 7. Bench Bench Mean OK OK -Aa ^1 reading ^1 reading reading calcd. obsd. cm. cm. cm. 18-52 cm. cm. 18-52 180 18-51 34-51 34-36 o / 15 18-62 165 19-10 18-86 34-77 34-70 14 10 30 19-54 150 19-90 19-72 35-51 35-56 28 32 45 20-68 135 21-21 20-94 36-58 36-78 43 17 60 21-87 120 21-96 21-92 37-71 37-76 58 29 75 22-74 105 22-89 22-82 38-58 38-66 74 7 90 22-91 22-91 38-91 38-75 90 The observed value of OK is a little low at 0° and 90° and a little high at 45°. Probably the incident wave front was not accurately cylindrical. The last column gives ^^ ^^ calculated from tan ^^ — ^~~^ ^^^ ^i- The difference between \//-2 and yjr^^ is too small to admit of measurement with simple apparatus. Mr Whiddington, The Shadow Electroscope lOQ' The Shadow Electroscope. By R. Whiddington, M.A., St John's College. [Received 15 June 1920,] A simple form of Electrostatic Voltmeter of low capacity is frequently useful in the laboratory. The instrument under descrip- tion is of the gold leaf type designed primarily for class instruction and while not capable Of the highest precision is yet sufficiently accurate for many purposes*. All leaf electroscopes with which I am familiar require some sort of optical system such as a microscope to view the leaf. Attempts have been made to use a scale placed near the leaf for measuring purposes, but when too near, disturbing electrostatic effects are encountered, placed too far away parallax errors become obtrusive. It occurred to me that the difficulties might be overcome by simply throwing a shadow of the leaf on a semitransparent scale some centimetres away, using a small 2-volt lamp as a source of Hght. The first instrument made on these lines consisted of a tin cigarette box with the lamp at one end, a transparent scale at the other end and the gold leaf system with its insulation in the middle. It was found as expected that quite a sharp shadow could be obtained when the lamp filament was nearly parallel to the leaf. The final design of electroscope is shown in section in the figure, the photographically reproduced scale, graduated in volts, being shown below. It will be seen that the scale is practically even from 100 to 500 above which the leaf becomes unstable. The quadrant shape of metal box was chosen as being most likely to give an even scale and a constant capacity over its working range. The tube (T) carries a well fitting sulphur plug fitted centrally with a quartz tube down which passes the rod (R) which carries the leaf within the case and a small cup at the top. The metal arm (A) is for clamping and tilting purposes and carries an earthing terminal (E). Just below (A), a short side tube is arranged carrying an ebonite block (B) in which a small lime coated spiral is fitted. When B is pushed home the spiral finds a place behind R. Its object is, when heated from a 2-volt cell, to provide a source of ions for * The original instrument, of which this one is the final form, was designed in 1919 for the Naval officers under instruction in Physics at the Cavendish Laboratory. 110 Mr Whiddington, The Shadow Electroscope experiments on ionization, its position behind the leaf precluding the possibility of disturbing convection currents. Cup for Condenser To Earth. To £ l/bLTS. The voltage range of the instrument is from 100 to 500 volts and with a good leaf it is possible to estimate to 1 volt, an accuracy sufficient for most purposes. Mr Whiddington, The Shadow Electroscope 111 The scale was graduated by applying known voltages from a small direct current generator*, measuring them by a standard Weston Voltmeter. I have found that with this instrument and the scale repro- duced above, it is sufficient, when no more than approximate results are required, to register the shadow of the leaf for two positions only — ^zero and one other, say 200 volts. To effect this it will generally be necessary to alter the sensitiveness somewhat by adjusting the height of the sulphur block in T. This is no doubt due to the non-uniform aluminium leaff available. Charging the Electroscope. After connecting E to earth, the leaf may be charged positively by induction from a rubbed ebonite rod. If a negative charge is required care should be taken not to overcharge the leaf. If an appreciable leak is observed a small piece of smooth silk rubbed lightly over Q will almost certainly cure it. Insulation troubles are nearly always traceable to hairs and dust particles attracted under the comparatively high voltages used. It is therefore best to conduct the experiments in a dust free room. The following are a few of the experiments which can be carried out with this instrument. Experiment 1. To determine the capacity (Cg) of the electroscope hy comparison with that of a sphere of radius r cm. Method. Charge the leaf to a voltage F^ as indicated by the scale reading (with the case earthed), and then share the charge on the J leaf with the insulated sphere thereby causing a drop in potential to Fg. Then since q = Cg F^ = (Cg + ?") Fg, ? being the original charge, r The following table shows a series of measurements taken on * Kindly lent by the Electric Construction Company, Wolverhampton. t Cut with scissors from leaf approximately -0004 cms. thick. X It is here assumed that the capacity of the sphere is equal to its radius. This is only true when the sphere is far removed from other conductors, a condition which can be approximately realised in practice if a long thin stiff vertical wire be inserted in the cup of the electroscope (or stalk of the condenser as the case may be) and the sphere touched to the top of the wire. If this precaution be neglected the results obtained will be too small. Further, it must be remembered that when bringing up the sphere to the electroscope for charge sharing, any charge on the insulating handle will affect the leaf by induction and spoil the results. This effect may be got rid of by passing the handle through a flame occasionally, merely touching the ebonite is often sufficient to produce a charge. 112 Mr WJiiddington, The Shadow Electroscope these lines using an insulated brass sphere of radius .3-25 cm.*; Fj and V^ are the scale readings in volts. Fi V, V^/V, 490 330 235 160 330 235 160 no 1-485 1-405 1-470 1-452 From the above readings the mean value of Fj/Fg = 1'450, whence C^ = 7-2 cm. Experiment 2. To determine the capacity of a parallel plate air condenser by the method of Experiment 1 . The readings tabulated below were obtained with a specially designed circular plate air condenserf . The diameter of the central plate being 4-25 cm., and its distance from two outer earthed plates being 0-15 cm., the capacity C'a can be calculated from the formula for a parallel plate air condenser, viz. 2 — -^ cm. Inserting the proper values for the present case leads to the value 60-2 cm. The experimentally determined value may be expected, if anything, to be rather greater than this calculated value owing to the extra capacity of the edges of the central plate. The method is essentially the same as in Exp. 1 but in this case the formula is vjv. 1 Using the same sphere as in Exp. 1 the following results were obtained, the insulated central stalk of the condenser fitting in the electroscope cup (see figure) and the outer plates being con- nected to earth. * An ordinary bedstead knob mounted on an insulating ebonite rod. t The main point in the design is the protection of the central insulated plate from dust, small hairs etc., which under the comparatively high potentials em- ployed would be attracted to it with resulting insulation troubles. The central plate is therefore sandwiched between two outside parallel plates, one of which is provided with a peripheral spacing ring which in butting up against the other outer plate completely encloses the inner insulated one. Insulating grooved buttons of ebonite form the insulation. Mr Whiddington, The Shadow Electroscope 113 Fi V, vjv. 350 332 1-054 320 306 1-046 500 480 1042 460 441 1-043 420 402 1-044 From these readings the mean value of VJV2 = 1*045, whence C^ + Ce = 72-2 cm., and since C, = 7-2, C'^ = 65-0 cm. Experiment 3. To determine the Specific Inductive Capacity of Ebonite. This can be readily carried out by using a second condenser exactly similar to the one used in the previous experiment but with circular plates of ebonite separating the plates instead of air. Then if this condenser (capacity C'j,) is placed on the electro- scope in the manner of the previous experiment, and charged to a potential (F^), and the sphere is used in the manner previously described, the resulting collapse of the leaf will be so small as to be hardly readable owing to the large capacity of the ebonite condenser. It is therefore more convenient to use the air con- denser of measured capacity Ca in place of the sphere. It is suffi- cient to hold Ca by its outer case for earthing purposes, touching its central plate momentarily to the corresponding plate of the condenser Cj, on the electroscope. The potential resulting from this sharing of charge (Fg) is noted. We then have that Ca+C, + Ce _ Fi whence C, + C, Fo a Fi/F,-1' in which both C^ and Cg have been previously determined by Fa V, vjv. 330 250 1-320 340 262 1-296 258 192 1-.341 480 362 1-336 350 265 1-322 260 195 1-318 VOL. XX. PART I. 114 Mr Whiddington, The Shadow Electroscope experiment. The above table gives the results of an experiment. From which the mean value of V1/V2 comes out to be 1'322. By calculation from this value Cj, = 194-7 cm. Assuming the identical dimensions of the two condensers* the Specific Inductive Capacity of Ebonite is just the ratio fi iri _ 194-7 ^' '^ 65-0 = 2-98. A value not far removed from the accepted value which ac cording to the table of Kaye and Laby will usually lie between 2-7 and 2-9. Experiment 4. The comparison of two capacities by the ioniza tion leak method. It is convenient to illustrate this method by giving as an example the results of an experiment using the same two con densers as the preceding experiment. Method. If when the lime coated spiral is glowing steadily the slow leak of the electroscope be observed firstly with C'^ in position and then with Cj, in position, the capacities can at once be compared, for if Ta and T^ be these times it can be shown that ^g + ^e _ Tg Cb + ^e Tj, The following table gives some results obtained with this method. In order to eliminate as far as possible any variations in the amount of ionization (which depends very greatly on the temperature of the filament and therefore on the e.m.f. of the power supplying cell) the readings for T^ and Tj, were taken alternately and as quickly as possible. It will be seen that under Times in seconds j ■^ a T, 9-2 9-2 9-6 9-4 25-0 24-6 25-8 25-4 Mean 9-35 25-2 * This can easily be tested experimentally. Mr Whiddington, The Shadow Electroscojpe 115 the conditions of this experiment, in which a well charged 2-volt lead accumulator was used, there is very fair concordance between the various readings. Leak observed from 400 volts to 200 volts. If now in the above-mentioned expression we assume the pre- viously determined values of C^ and Cg, viz. 65-0 cm. and 7-2 cm. respectively, the value of C^ comes out to 188-0 cm. leading to a value for the specific inductive capacity of ebonite of 2-90. This value is in as good agreement as is to be expected with the deter- mination of Exp. 3. 8—2 116 Professor Baker, On the Hart circle of a spherical triangle On the Hart circle of a spherical triangle. By Professor H. F. Baker. [Read 9 February 1920]. This note is concerned with the problem, given three arbitrary plane sections of any quadric, of finding a fourth section which shall be tangent to four of the tangent planes of the three given sections. If the three given sections are concurrent on the quadric they have only four tangent sections, and the fourth section is unique, the projection of the figure on to a plane (from the point of concurrence) giving rise to Feuerbach's theorem of the nine- point circle. In general the three given sections have eight common tangent planes; in fact any two of these sections lie on two quadric cones, and the six vertices of the cones so obtainable lie by threes on four coplanar lines; the three cones whose vertices are on any one of these lines have a pair of common tangent planes, which thus touch the three sections. The eight tangent planes of these are thus accounted for. There are now fourteen ways of selecting, from these eight tangent planes, four which all touch another section ; six of these ways, in which the four tangent planes selected are tangent to a fourth section passing through the point of con- currence of the three given sections, are easy to recognise, and do not need further consideration. There are however eight ways of choosing four from the tangent planes which shall all touch another section lying in a plane w forming with the planes of the three given sections a finite tetrahedron. § 1. We are thus lead to the problem of the condition necessary and sufficient in order that the sections of a quadric by the four faces of a tetrahedron should have four common tangent planes; and the main object of this note is to state this condition in a form which in fact leads to great simplification of what is generally presented as a somewhat intricate theory, and to point out several results, apparently new, which follow from this. Let the tetra- hedron be 0, X, Y, Z; denote the intersections of the quadric with OX hy A, A', those with OY by B, B' and those with OZ by C, C"; similarly denote the intersections with YZ by V, U', those with ZX by F, 7' and those with XY by W, W. In general, if each edge of the tetrahedron be joined by j)lanes to the two points in which the quadric meets the opposite edge, the twelve planes so obtained touch another quadric. But it may happen that this new quadric degenerates into two points, say S and S' ; then, with a proper choice of notation, the four lines AU, BV, CW are concurrent in a point S, and the four lines A'U', B'V, CW con- Professor Baker, On the Hart circle of a spherical triangle 117 current in another point S'. That this should be so is a necessary and sufficient condition that the four sections of the quadric by the faces of the tetrahedron should have four common tangent planes. The condition may be stated in another form; take on the edge OX, the point Ai separated harmonically from A by and X, and the point Ai separated harmonically from A' by and X; in the same way take on each edge of the tetrahedron the harmonic conjugates, with regard to the vertices of the tetra- hedron lying on that edge, respectively of the intersections of the quadric with that edge. The twelve new points so obtained lie on another quadric, which we may describe as the harmonic conjugate of the original in regard to the tetrahedron. The condition in question then is that the harmonically conjugate quadric should break up into two planes, say a and a'; these will be the polar planes of >S and S' in regard to the original quadric. We may illustrate this condition by applying it to the (Feuer- bach) case of three sections of the quadric which are concurrent on the quadric, say in 0. The fourth section of the quadric touched by the four common tangent planes of the three given sections OYZ, OZX, OXY is then constructed as follows: on the plane YOZ take the line p through 0, harmonically conjugate with respect to OY, OZ, to the line in which the plane YOZ is met by the tangent plane of the quadric at 0; let this line p meet the quadric again in P; obtain the points Q, R of the sections ZOX, XOY in a similar way. The plane PQR is the fourth plane required. In this case one of the planes a, a' is the tangent plane at 0. § 2. We may obtain a direct verification of the sufficiency of the condition in general by using it to obtain any one of the eight (Hart) sections m which can be associated with three given sections YOZ, ZOX, XOY, so as to form four sections with four common tangent planes. Let the quadric, referred to YOZ, ZOX, XOY and the polar plane of 0, have the equation ax'^ + by^ + cz^ + ^fyz + 2gzx + 2hxy = t^ ; with an arbitrary choice of the signs of Va, Vb, Vc, take u = h (/ -f Vb Vc), V = -| {g + Vc Va), w = | {h + Va Vb), and then I, m, n so that mn = u, nl = v, hn = w; the eight planes required are then expressed by Ix + my + nz — t^ = 0. It is at once seen that this follows from the condition stated above. If we introduce A, fi, v so that / = Vb Vc cos X, g ^ Vc Va cos /x, h = Va Vb cos v. 118 Professor Baker, On the Hart circle of a spherical triangle a plane of this latter form is /- cos la cos |v /r Va -.s + y V6 cos |A - cos l-v cos ^A A cos |-jLt /-cos iA cos ia ^ ^ ,TT\. + Z-VC — — -2r_ t = 0, (H) ; cos |v on the other hand a common tangent plane of the three given sections in x = 0, ^ = 0, 2 = is at once found to be X Va cos {s- X) + yVh cos (s — /x) + 2 Vc cos {s — v) — t-^ = 0, (I) where s = ^ (A + yu, + v) ; and it is easy to see that the section (I) touches the section (H) at the point of the plane (H) which lies on X Va : y^/h : zVc = p (q— rf : q (r — p)^ : r (p — qf, where, for brevity, p, q, r stand respectively for sin {s — A), sin (s — fj,), sin {s — v). I The four planes (I) which touch the section (H), as well as the original sections inx = 0,y = 0,z = 0, are obtained from the above equation by replacing A, yu,, v by ± A, ± /x, ± v, respectively. The eight sections (H) are obtainable from that above by re- placing Va, Vb, Vc, A, /x, V respectively by 1 {Va, Vb, Vc, A, fji, v), (— Va, Vb, Vc, A, 77 + ix,7t -\-v), {Va, — Vb, Vc, X + TT, ix,v + tt), {Va, Vb, — Vc, X + tt, jx + tt,v) together with those obtainable from these by changing the sign of ^1. § 3. The following result gives a construction for the position, upon the section, i, of the quadric by the plane (I), of the point in which this section is touched by the plane (H). Upon i we have three points, its contacts with the sections in a? = 0, ?/ = 0, s = 0; we also have two points, namely those in which i is met by the plane from to the intersection of the planes ABC, A'B'C , which plane is at once found to have the equation x Va + y Vb + s Vc = 0. The point to be constructed is the apolar complement of the two latter points in regard to the three former points. This result may be made clearer perhaps by stating it for a sphere in Euclidian geometry: If D, E, F be the mid-points respectively of the sides BC, CA, AB of a spherical triangle, the planes of the great circle arcs EF, BC give a diameter, and the three diameters so obtained are coplanar; let I, J denote the intersections of their plane with the inscribed circle of the triangle ABC; let P, Q, R be the points of contact of this inscribed circle with the sides BC, CA, AB. Then, Professor Baker, On the Hart circle of a spherical triangle 119 upon this inscribed circle, the point of contact with the Hart circle, which touches this and certain other three tangent circles of the sides of the triangle, is the apolar complement of I, J in regard to P, Q, R. For the particular case of the nine point circle of a plane triangle the result has been remarked by Prof. F. Morley, as was pointed out to the writer by Mr J. H. Grace, Bulletin of the American Math. Soc, I, 1895, 116-124 ("Apolar triangles on a conic"). § 4. Another result may also be stated here. To introduce it and render its meaning clearer we state it first for the Hart circle of a spherical triangle in Euclidian geometry. If this circle meet the sides of the spherical triangle ABC respectively in U, U' on BC, V, V on CA, W, W on AB, then, with proper choice of notation, the arcs AU, BV, CW are concurrent, say in S, and the arcs A'U', B'V, CW are concurrent, say in S' . The result in question is that S, S' are the centres of similitude of the circumscribed circle of the triangle ABC and the Hart circle. It is a direct generalisation of the corresponding familiar fact for the nine point circle of a plane triangle. Stated in the more general way here adopted, which is also the more precise way, the theorem is that the lines OS, OS' are each the intersection of two planes through which touch both the section m and the section by the plane ABC. If PQR, P'Q'R' be two sets of three points lying respectively on two plane sections of a quadric, such that PP' , QQ' , RR' are concurrent, the sections lie on a quadric cone having this point of concurrence for vertex; thus a plane through touching the section [x by the plane ABC equally touches the section /x' by the plane A'B'C . Now S, the point of concurrence of AU, BV, CW, is the vertex of one cone containing the sections //,, xd; and S' is similarly the vertex of one cone containing the sections jx' , rn. The line OS', joining the vertex of one cone containing the sections fx', m to the vertex of one cone containing the sections fj., fj,', passes through one of the vertices of the two cones containing the sections, /jl, m; as OSS' are not collinear, this line OS' passes through the vertex other than S of a cone containing /x and w. The two cones containing fx and m thus have their vertices on OS and OS'. Now to each of these cones there can be drawn from two tangent planes, which intersect in the line joining to the vertex of the cone; the four planes so obtained touch the sections fi and w, and thus are the four common tangent planes of the cones with vertex standing on the sections /x, ta. Two of these planes therefore intersect in OS and two in OS'; which is the result we desired to obtain. There are as we have said eight sections m each touched by four of the common tangent planes of the sections in YOZ, ZOX, XOY. These tall into four pairs, the planes of a pair intersecting on the 120 Professor Baker, On the Hart circle of a spherical triangle polar plane of 0, being harmonic conjugates in regard to this plane and 0; for the pair associated as above with the two planes ABC, A'B'C the lines OS, OS' are the same. There is another pair associated similarly with the planes AB'C , A'BC, a third pair with the planes BC'A', B'CA and a fourth pair associated with the planes CA'B', CAB. And it may be remarked that the sections by the planes ABC, AB'C, BC'A', CA'B' are all touched by four planes, as follows at once from the fact that AA', BB', CC are concurrent; so also the sections by the planes A'B'C, A'BC, B'CA, CAB are all touched by four planes. § 5. Another remark may be made, relating to a property which appears in Euclidian geometry as Salmon's theorem that the tangent of the radius of the circumcircle of a spherical triangle is twice the tangent of the radius of the Hart circle. Let P be the pole of any plane section of a quadric, upon which any point A is taken, and be any other point; denote by p the Cayley separation of the lines OP, OA in regard to the quadric, and by S the Cayley separation of P from 0. It can then be shown that p is independent of the position of A upon the section, and is indeed symmetrical in regard to P and 0, being connected with 8 by an equation sin S sin p = ± 1. Calling p the radius of the section in regard to the point 0, it can be shown that if p, p' be the radii of any two sections a, a' whose planes intersect in a line I, and the planes joining I to and to the vertex of one of the two cones containing a and a' be respectively denoted by co and y, then tanp/tanp' is equal to the homography (y, a>; a, a') or to the negative of this. In particular when the planes y, a are harmonically separated by to and a', this leads to tanp = 2 tan p'. In our figure the plane a, which is the polar of S in regard to the quadric, passes through the line of intersection of the planes ABC and w, since S is the vertex of one of the cones containing the section by ABC and the Hart section zu, and this plane a also contains the vertex of the other cone containing these sections; it can easily be proved that the plane co which joins to the line of intersection of the 'planes ABC and m is harmonically separated from m by the planes ABC and a; thus the planes a, ABC, oj, w have the relation of the respective planes y, a, w, a in the general description just given. It follows tbafc if p, R be the radii of the sections m, ABC, we have tan -R = 2 tan p ; which is what we wished to prove. § 6. A last remark may be added bringing into relief the con- nexion between the present point of view and that of the Euclidian geometry. As hitherto, let OXYZ be a tetrahedron whose faces meet a quadric in sections having four common tangent planes. Denote by ia, i^, iy the Cayley separations OX, OY, OZ in regard Professor Baker, On the Hart circle of a spherical triangle 121 to the quadric; by ia, i^' , iy' the Cayley separations YZ, ZX, XY] by (A), (B), (C) the Cayley separations of the pairs of planes meeting respectively in OX, OY, OZ; and by (A'), (B'), (C) the Cayley separation of the plane XYZ respectively from the planes YOZ, ZOX, XOY. Each of these separations is ambiguous in sign and by additive multiples of it, unless we enter into further detail. There are however equations by which all of them are deducible from a, P,y; and these equations may be represented, when proper regard is paid to the ambiguities, by a' = ITT + ^ — y, ^' = iiT + y — a, y' = iV + a — /3, ... sinha , , .,. sinh (^ - y) tan (A) = — 7~ r , tan (A) = — , ' , q ^—. ^ ' cosh (e + a) ^ ' cosh (e + ^ + y) where e is such that 2 tanh e = tanh a tanh ^ tanh y — tanh a — tanh ^ — tanh y. And these lead to (A') = {B) - (C), {B') = (C) - (A), (C) = (A) - (B), which may be used to define the Hart section. § 7. In what has preceded we have stated a sufficient condition for the Hart section, namely that AU, BV, CW are concurrent. It can however be proved that this is also a necessary consequence of the existence of the four sections of the quadric all touched by four other planes, provided we exclude certain particular possi- bilities which are easily stated. Precisely, given three arbitrary plane sections of a quadric, no one of which degenerates into two straight lines, so that the equation of the quadric referred to these and the polar plane of their point of intersection is of the form {abcfgh^xyzf = t-^, in order that these with a fourth section (also not two straight lines) should form a set of four sections all touched by four planes, if no relations are assumed to hold among the coefficients a, h, c, /, g, h, it is necessary that the condition in question (that AU, BV, CW are concurrent) should hold. In order that the sections by a; = 0, ?/ = 0, z = 0, ^ = of the quadric {abcdfghuviv\xyztf- = should have four common tangent planes, the cones enveloping the quadric along these sections must be concurrent; if A be the four-rowed determinantal discriminant, and ^4, 5, ... the minors therein, it follows that the necessary and sufficient condition for this is that the equation {ahcdfghuvivjVA, VB, VC, VDf = A should be satisfied for four choices of the signs of VA, VB, VC, vD. It proves to be possible to examine all the ways in which this can happen, and the result is as stated. 122 Professor Baker, On a projperty of focal conies On a property of focal conies and of bicircular quartics. By Professor H. F. Baker, [Read 9 February 1920.] The property of focal conies referred to in the title is the well- known one that if P, R be any two points of the focal hyperbola of a system of confocal quadrics, and Q, S be any two points of the focal ellipse, then the distances PQ, PS have the same difference as the distances RQ, RS. The theorem remains true if every one of the distances be replaced by the Cayley separation of its end points in regard to an arbitrary quadric of the confocal system, and the original theorem is then obtainable by making the para- meter of this arbitrary quadric increase without limit. It is shown that the generalised theorem is equivalent to the geometrical theorem that two enveloping cones of the arbitrary confocal exist, each of which touches the four lines PQ, QR, RS, SP. The theorem that the sum of the two focal distances of a point of an ellipse is constant may similarly be replaced by the theorem that the sum of the Cayley separations of a point of the ellipse from the foci is constant, in regard to an arbitrary confocal conic; a theorem is obtained which includes both this last result and the former. It is unnecessary to point out that this last result is equivalent \^'ith Chasles's theorem that a variable tangent plane of a quadric cone makes angles with the planes of circular section whose sum is constant (Chasles, Geom. Super., 1880, § 812, p. 517). The property of bicircular quartics referred to is that the angles which a variable bitangent circle of one mode of generation makes with two fixed bitangent circles of another mode of generation, have a constant sum (Jessop, Quart. Journ., xxiii, 1889, 375). This is shown to be equivalent to the former theorem. There exist much more general theorems in regard to the generation of a quadric with the help of a thread of constant length, whose systematic investigation is in connexion with the theory of hyperelliptic functions (Chasles, Liouville, xi, 1846, 15; Darboux, Theorie des surfaces, Livre iv, Ch. xiv, 296-312; Staude, Math. Ann., xx and xxii, 1883; Finsterwalder, Math Ann., xxvi, 1886; Maxwell, Works, ii, 156 or Quart. Journ., 1867). I have added some lines in regard to this general point of view. § 1. Ii P, Q, R, S be four coplanar points of a quadric, and through the lines SP, PQ, QR, RS be drawn four arbitrary planes, respectively, a, ^, y, 8, the lines a^, ^y, yh, 8a meeting the quadric and of bicircular quartics 123 again respectively in P', Q', R', S', then (1) the points P', Q', R', S' are equally on a plane, (2) if by the angle between the sections of the quadric by the planes a, ^ be understood the Cayley separation of these planes, measured by the homography of these planes in regard to the two tangent planes to the quadric drawn from their line of intersection, then the sum of the angles at P, R, determined respectively by the sections a, ^ and y, 8, is equal to the sum of the angles at Q, S, determined respectively by the sections /3, y and 8, a. That P', Q', R', S' lie in a plane follows from the fact that the four quadrics consisting of (i) the original quadric, (ii) the planes a, y, (iii) the planes ^, 8, (iv) the planes PQR, P'Q'R', have seven, and therefore eight points in common. For the relation between the angles, denote by 6 the section by the plane PQRS, and in general by (a, B) the angle between the sections (a, ^). Then we have 77 ^ (d, a) + (a, ^) + (^, d) = {d, y) + (y, 8) -^ (8, d), and therefore (a, /3) + (y, 8) = 277 - {6, a) - (9, ^) - {d, y) - {d, 8), which is also the value of (^, y) + (8, a), the ambiguities of inter pretation being properly settled in each case. In a plane we have the theorem that if P, Q, R, S be concyclic points through which pass pairs of four circles a, ^, y, 8, namely a, B through P, /?, y through Q, y, 8 through R and 8, a through S, then the two angles (a, /3), (y, 8) have the same sum as the two angles (^, y), (8, a); and this, not depending on the Axiom of parallels, may well be regarded as a fundamental theorem. Further if P' be the other intersection of a and ^, etc., the points P', Q', R', S' are concyclic. The connexion of this result with the theorem of the angles is incidentally remarked by Prof. W. McF. Orr, Trans. Camb. Phil. Soc, XVI, 1897, 95. § 2. Regard the bicircular quartic in question as the projection on to an arbitrary plane of the section of a quadric by a quadric cone of general position, the centre of projection being an arbitrary point of the quadric. An arbitrary tangent plane of the cone cuts the quadric in a section projecting into a conic having two points of contact with the bicircular quartic, and this conic, passing through the nodes of the quartic, is for us a bitangent circle, of one mode of generation. The other three modes are obtained by considering the other three quadric cones through the intersection of the quadric and the first cone. Take then two bitangent circles of the bicircular quartic of the first mode of generation, say a and y; their points of contact will be on another circle, say/?, as appears from the three dimensional figure. Take also two 124 Professor Baker, On a property of focal conies bitangent circles of a second mode of generation, say ^ and S, with points of contact on a circle, a. We shall prove that the eight points of intersection of the pairs of circles (a, ^), (/S, y), (y, §), (8, a) lie on two circles 6, 6', four on each. These circles 6, 6' pass through the two intersections of the circles p, a, and separate these circles harmonically; the circle p is orthogonal to the principal circle to which the bitangent circles of the first mode are all ortho- gonal, with a similar statement for cr. For the proof, let a = 0, y = be the equations of any two tangent planes of a quadric cone, whose generators of contact lie on a plane p = 0, so that the cone has the equation ay — p^ = 0. Let ^S — o-^ = be another quadric cone, whereof ^ = 0, 8 = are tangent planes touching the cone on ct = 0. Then a quadric E = through the curve of intersection of the two cones has an equation of the form E = ay -p^-m^ (^S - a^) = 0, so that the four lines a = 0, j3 = 0; ^ = 0, y=0; y = 0, 8 = 0; 8 = 0, a = 0, in which the two first planes a, y meet the two latter planes ^, 8, intersect the quadric ^ = in eight points lying in the two planes p + ma = 0, p — ma = 0. We have then a proof of Jessop's theorem in regard to the bicircular quartic curve*. § 3. Eeciprocally let any two conies be taken in space, not intersecting one another. Consider a quadric touched by the common tangent planes of these two conies. Then if A, C be any two points of the first conic, and B, D any two points of the second conic, it follows from § 2 that the pairs of tangent planes to this quadric from the lines AB, BC, CD, DA touch two enveloping cones of the quadric, say F and G. Or, as a line lying in a tangent plane of a cone is a tangent line of the cone, there are two en- veloping cones of the quadric which touch the lines AB, BC, CD, DA. And, comparing the equations of § 2, the vertices of these cones lie on the line joining the points R, S, in the planes of the conies, which are the poles respectively of AC, BD in regard to these conies, and separate R, S harmonically; the positions of the vertices depend on the quadric taken to touch the common tangent planes of the conies. Moreover, as the reciprocal of the * The direct analytical proof is, of course, simple. Let the fundamental quadric be x^ + y~ + z^ + I" = 0, and bitangent circles of two modes be obtained by pro- jection of the polar sections respectively of the two points [{a-d)x, (b-d)y, {c-d)z, 0], [{a-c)^, {b-c)v, 0, (c^-c)t]. Then the angle between these circles, being the Cayley separation of these points, is the angle, in rectangular Cartesian coordinates, between the two lines X/px = Y/qy, X/p^ = Y/qr], where p^ = {a- d) [a - c), q^=(b- d) (b - c). This generalises at once to the Cyclide; cf. Jessop, Quartic surfaces, 1916, p. 106. and of bicircular quartics 125 theorem in regard to the angles, if we consider the homography of A,B in regard to the quadric, say a, and take the corresponding homographies for the pairs B, G; C, D; D,A respectively, say b, c, d, we have ac = bd, or a/d = b/c. In words, the difference of the Cayley separations of A from 5 and D, in regard to the quadric, is the same, for unaltered positions of B, D on the second conic, when A is replaced by any other point C of the first conic. This result includes the particular case of the focal conies of a confocal system, for which we may also consider the further particular case of actual Euclidian distances between the points. (Cf. § 10 below, where the relation between the separation and the distance is given.) § 4. If we assume that the sides of the skew quadrilateral ABCD in § 3 touch an enveloping cone of the quadric, we can deduce the relation between the Cayley separations in another way. In fact if the sides of a skew quadrilateral touch any quadric having ring contact with a given quadric, the sum of the Cayley separations belonging to the sides of the quadrilateral, each taken in proper sense, is zero, the separations being measured by the latter quadric. For if ^T be a tangent to a quadric V, which has ring contact with a quadric U, drawn from a point A, the Cayley separation AT in regard to U is independent of T. If A be (^, 77, ^, r), T be (x, y, z, t), so that, with usual notation, V^. = 0, 7^^ = 0, and Z7 be 7+ P^^ 0, then TJ^ = PJ^, U^^ = PJ"^, and hence which is independent of x, y, z, t; and U^^KUJJ^Y is the cosine of the separation in question. Therefore, if the ^iAqsAB, BC, CD, DA of the skew quadrilateral touch 7 respectively at L, M, X, Y, we have the following relations among the separations (AB) = (AL) - (BL), (BC) = (BM) - (CM), (CD) = {CX) - (DX), (DA) = {DY)- (AY), {AY) = iAL), {BL) = {BM), {CM) = {CX), {DX) = {DY), leading to {AB) + {BC) + {CD) + {DA) = 0, or {AB) - {AD) = {CB) - {CD). In the application of this result above, 7 was a cone. § 5. We may however make an application in which C/ is a cone, and 7 not a cone, U being an enveloping cone of 7. Namely, if the sides of a skew quadrilateral touch a quadric, the sum of the four Cayley separations of the vertices, each in proper sense, in regard to any enveloping cone of the quadric, is zero. The reciprocal theorem, is that if two plane sections a, y of a quadric 126 Professor Baker, On a property of focal conies be both touched by each of two other sections ^, S — and if, taking a fifth arbitrary section, co, of the quadric, we measure the angle between the planes of two sections a, 13, which touch one another, in the usual way, by considering the homography of these planes in reoard to the tangent planes drawn from their line of intersection to the section co — ^then, with proper sense of measurement, [a, /3] denoting the angle between these planes, we have [a, P] + IP, y] + [y, 8] + [S, «] = 0. Now take one of the two quadric cones containing the sections a, y, and regard this cone, and the section co, as fundamental; speak of a, y as circular sections of this cone, of opposite systems because each has two points common with the other and with a». Then we have ChasJes's theorem that a variable tangent plane of a quadric cone makes angles of constant sum with two planes of circular section of the cone, of opposite systems. § 6. The reciprocal theorem is that a generator of a quadric cone makes angles of constant sum with two conjugate focal lines of the cone, that is, considering the conic in which the plane of w cuts the cone, and the quadrilateral formed by the common tangents of this conic and oj, makes angles of constant sum with the lines joining the vertex of the cone, to an opposite pair of intersections of two of these common tangents (Chasles, loc. cit., § 827, p. 528). Projecting on to an arbitrary plane we have the theorem that if P be a variable point of one of two conies having S, H as common foci, the Cayley separations PS, PH in regard to the other conic have a constant sum. An elementary proof can be given depending on the fact that if PS meet the other conic in >Si, S2, and PH meet the other conic in Hj^, H^, then, with proper notation, each of S^H.,, S^H^ passes through a fixed point of the line SH. § 7, This theorem for conies is a particular case of the following : Two conies V, W, have both double contact with a conic U, and also both have double contact with another conic K. From a point P oi K a tangent PX is drawn to F, and also a tangent P Y to W; then the Cayley separations PX, PY, taken in regard to U, have a constant sum (or difference) as P varies on K. Two tangents are possible from the point P to the conic F; but the separation PX is the same for both. If F degenerate into the pair of tangents to U from a focus S of U, and W into the pair of tangents to U from the conjugate focus H, then the conic K, touching these four tangents, will be confocal with U, and the tangents PX, PY will become the lines PS, PH. Thus the theorem includes that of § 6. and of bicircular quartics 127 § 8. The proof of the general theorem of § 7 is, analytically, identical with that of the following theorem, of three dimensions, which leads, in § 9, to the theorem of § 4, and may thus be regarded as summarising all the analogous theorems here obtained: — If two quadrics V, W both have ring contact with a quadric U, and also both have ring contact with a quadric K, and PX, PF be tangents respectively to V and W from a point P of K, the sum (or difference) of the Cayley separations PX, PY, in regard to U, is independent of the position of P upon K. When V and W coincide the difference of the separations is zero for all positions of P and the quadric K is unnecessary. The theorem is easy to prove. In order that two quadrics V = 0,W = should both have ring contact with another quadric U = 0, they must, if P = 0, Q = be suitable planes, be capable of the forms V = U - P^, W ^ U - Q^ and thus F, W must have two points of contact, there being an identity of the form V-W = pq, where p = 0, q = are two planes. Any quadric having ring contact with both V and W is then capable of either of the identical forms V +1 {a-'^p - aqf =0, W +1 {a-'^p + aqf = 0, wherein a is a constant, and two such quadrics can be drawn through an arbitrary point. We. may then suppose U=V + l {a-^p - aq)\ K=V + 1 (b'^p - hq)\ where ii = is the quadric of the enunciation, and 6 is a constant. Thus we have the identity U - K = l (a-2 - 6-2) (^2 _ a^j^Y), involving in particular that U, K have two points of contact on the line joining the points of contact of V and W. Putting P = I [a-^p — aq), ^ == | {a-'^p + aq), this is the same as (1 - a2) (C/ - A') = P^+Q^+ 2aPQ, where a = {a^ + b^)/{a'^ — 62). This again, if U is not zero, is the same as (P2- U) (Q2 - U)- {PQ f aUf = (1 - (t2) UK. We remarked however above (§ 4), that if 6, (f> be the Cayley separations PX, P Y, taken in regard to V, P Q cos 8 = — 7 , cos (b = —- 128 Professor Baker, On a pro;perty of focal conies where the coordinates in IJ, P, Q are those of the point P. If this point be on the quadric K ^ 0, but not on U = 0, we thus get cos 6 cos <f) + a ^ ± sin ^ sin (f> showing that 9 ± (f) ^ constant, as was stated. § 9. Now suppose a skew quadrilateral ABCD of which the sides AB, BG both touch the quadric V, say in X and Y, respec- tively, while the sides CD, DA both touch the quadric W, say in Z and T respectively. The quadrics 7, W are supposed to have two points of contact, so that quadrics can be drawn having ring contact with both. Let TJ be one such; let K be another such passing through C, and let A be on K. Then, considering Cayley separations in regard to V, we have {BX), {BY) equal because F has ring contact with U, and also {DZ), (DT) equal because W ■ has ring contact with U. By § 8 we also have (AX) — (AT) equal to (GY) — {GZ), if a proper sense be assigned to the separations involved. We infer therefore that (AB) - (AD) = (AX) + (XB) - [(AT) + (TD)] = (AX) - (AT) + (XB) - (TD) = (GY) - (GZ) + e{YB)-l (ZD), where e, ^ are each ± 1. Without making the proper detailed examination, we shall put both e and ^ equal to 1, so obtaining (AB)- {AD) = {GB)- (GD). This is verified (§ 4) in the particular case where the quadrics F, W coincide, there being then no need for the condition that A, G lie on the same quadric K having ring contact with F and W. § 10. A line joining a point of one focal conic to a point of another focal conic of a confocal system of quadrics is a particular case of a line touching two confocals of the system. And such a line is part of a continuous curve which on either of these two confocals may consist partly of arcs of the line of curvature which is the intersection of these two fundamental confocals, and partly of arcs of geodesies touching this line of curvature. As was recog- nised by Chasles this continuous curve has everywhere the geo- metrical property that if we take two other confocals of the system, the homography of the tangent planes drawn to one of these from a tangent line of the curve, in respect of the tangent planes drawn to the other, is the same for every point of the curv^e. That the analytic formulation may equally be regarded as uniform for all parts of the curve seems often to be unnoticed; it is recog- nised however by Staude in the papers above referred to. Let us consider the system of confocals and of bicircular quartics 129 x^ y^ z" + A-.+ ^^ = h a + X b + X c + X where a > b > c, using A for ellipsoids, ju. for hyperboloids of one sheet, V for hyperboloids of two sheets. Suppose that the straight portion of the curve touches the confocals for which X = 'p,X=^ q, and denote by dw the Cayley separation of two consecutive points of the curve taken in regard to the confocal of parameter A = ^. Putting F {x) = ^{x + a) {x ■i-b){x + c) {x - p) {x - q), L^=f{X), 3P=f{iJi), N^=f{v), the curve is such that (A -p)dX ^ r (/x - p) d^i ^ ( {v - p) dv ^ ^^ L ' J M ' ] N (A -q)dX [ ifx - q) dix ( {v - q) dv _ L +j — li ' J iT^-^' while, with S^ = F (6), [{X -p){X- q) dX ^ f (/z - ?)) ( ^ - q ) dfx , ^{v -p ){v- q) dv J {X-d)L } {yi-6)M ' j {v-e)N e where tv = jdw. By supposing 6 to increase indefinitely, and re- placing 6^dw by ds, we have the corresponding result when Euclidian distance is used. In the notation of hyperelliptic functions (see Multiply -periodic functions, Cambridge, 1907, pp. 35, 36, using the p, q as a^, ag ^re there used), we have p {x — p) {x — q) dx 2{d-p){d-q)]^^ x-e y y , e,x,s, x,Xo,y /,,^. =o\ ,, x,xo, 1 1 „ J (w ' + A;) where ((f)) denotes the place conjugate to (6), Ic is such that '^ {h) vanishes identically, and when d is large the significant terms of the functions l,-^, l,^ ^^^ d~^pq and d~^ {p + q). Along a straight portion of the curve, joining, suppose, the points (Aq, /Xq, Vq), (A, IX, v), the places (A), (/i), {v) of the hyperelliptic construct are coresidual with the places (Aq), {ixq), [vq), and we can satisfy the identities F {x) - [0 {x)f = 4 (X - A) {X - ix){x- v) ix ~f,) (x -/a), F (x) - [ipo {x)f =4:{x- Ao) {x - (Xq) {x - Vq) (x -/J (x -f^), VOL. XX. PART I. 9 130 Professor Baker, On a property of focal conies, etc. where ip (x) = 2 {Ix^ + mx+ n), iJjq (x) = 2 {l^x'^ + m^x + n^). With this notation we find, for the Cayley separation of the two extreme points, „ = tanh-(^)-ta„h-(^^), leading in particular, if r be the distance of these points, to r = I — Iq, and to r^{d-X,){d~f.,){e-v,) 2 tanh w Q + ipQ (d) tanh w ' The character of the symmetric functions of the places (A), (fx,), (v), regarded as functions of w, along any portion of the curve, seems eminently worthy of investigation. And it appears that the total value of tv, along any closed portion of the continuous curve, is expressible by an aggregate of the periods of the integral Q [ {x-p){x-q ) 2{d-v){d-q)] {x-d)y ^' where y^ ^ F {x), with integer coefficients; these will then be un altered by any continuous small deformation of the arc of the curve. This remark appears to lead to all the known results. In conclusion I should like to refer the reader to a most inte resting note by Mr A. L. Dixon, Messenger of Mathematics, xxxii, 1903, 177. Professor Baker, Ninth point of two plane cubics 131 On the construction of the ninth point of intersection of two plane cubic curves of which eight points are given. By Professor H. F. Bakee. [Read 3 May 1920.] Cayley has collected, in a paper reprinted in Vol, iv of his Papers, pp. 495-504 {Quart. J., v, 1862), the various solutions given of this problem, regarded as a problem of plane geometry, by Pliicker, Weddle, Chasles and Hart, depending for the most part on the generation of a plane cubic curve (two points at a time) by the intersection of a pencil of lines and a homographic pencil of conies. So far as I have been able to notice, geometrical con- ceptions present themselves to an unbiassed child in the first instance as three dimensional, and he feels it to be an abstraction to regard plane geometry as self-contained; the discussion of the most natural Axioms of geometry seems also to point in this direc- tion; and the most valuable part of a training in geometry would seem to lie in the cultivation of a faculty for visualisation of relations in space. However these things may be, it appears to me always to be an interesting extension when a property of space is shown to follow from a property in space of higher dimensions, this being generally accompanied by the removal of some artifi- ciality. Thus, I regard the very simple example which now follows as being logically at least as fundamental as a proof in the plane. Let A, B, C, M, N and P, Q, R be the eight given coplanar points. Take a point D outside the plane of these. There are qo ^ quadric surfaces containing A, B, C and the lines DM, DN; let Q be one of these (other than that consisting of the planes ABC, DMN). Let DP, DQ, DR meet this quadric again in P^, Q^, Rj^. A definite twisted cubic curve can be drawn through D, A, Pj, Q^, R^ to have BC as a chord (see below). This cubic curve, meeting D. in D, A, Pj, Qi, Pj, meets O in a further point, say 0^. If DO^ meet the original plane in 0, this is the ninth point required. For the space cubic is the intersection of two quadric surfaces drawn through D, A, P^, Q^, Pj, both having the line BC as a generator; denote these by U and V. The quartic space curve of intersection of U with O contains D, A, B, C, P^, Qi, R^, and meets the generators DM, DN of Q; this curve then projects from D on to the original plane into a cubic curve containing the eight given points A, B, C, P, Q, R, M, N. The curve of inter- section of V with Q. projects from D into another cubic through these eight points. The point 0^, on the space cubic, lies on U 9—2 132 Professor Baker, Ninth point of two plane cubics and V, and on Q, and so projects from D into a point common to the two plane cubics. This justifies the statement. Incidentally any two cubic curves in a plane are shown to be the projections of two quartic curves in space lying on the same quadric; and the plane problem is put in connexion with the space 1 problem of finding the remaining eighth intersection of three 1 quadrics with seven common points. To construct a twisted cubic curve with five given points D, A, Pj, ^1, Ri to have a given line BC as chord, we may for instance first construct a quadric surface by the intersection of corresponding planes of two homographic axial pencils with DA, BC as axes, three pairs of corresponding planes being those con- taining Pj, Qi, Ri, and then construct a quadric surface by the intersection of corresponding planes of two homographic axial . pencils with DP^^, BC as axes, three pairs of corresponding planes being those containing A, Qj^, Rj^. These quadric surfaces intersect in the cubic curve required. It is seen that analytically each step requires only the solution x)f linear equations. Indeed, if the conic through A, B, C, M, N be written (referred to ^, P, C, D) as Ayz + Bzx + Cxy = 0, the line 31N being x + y -\- z = 0, we may take for Q, the quadric t {x + y + z) = Ayz + Bzx + Cxy. The general plane cubic curve through the five points A, B, C, M, N may be taken to be {Ayz + Bzx + Cxy) {Ix + my + nz) + {x + y + z) x [qy + rz) = 0, and two cubics through these and P, Q, R may be found by solving for the ratios of I, ni, n, q, r in the three equations obtained by substituting the coordinates of P, Q, R. Corresponding to two sets of ratios l-^ : m^ : n^ : q^ : rj, and l^ '. m^ : % • % '■ ^2 ^^ chosen, there are two quadric surfaces t {l-^x + nijy + fijz) + X {q^y + r-f^z) =-- 0, t {I2X + m^y + n^z) + x (q^y + r^z) = 0, which intersect in a cubic curve containing D, A, P^, Qj^, Pj and having BC for chord. The combination of these with the equation of O will lead to a linear equation for Oj, from which is found. Or the solution may be stated, naturally enough, without reference to three dimensions. Professor Baker, On a proof of the theorem of a double six, etc. 133 On a 'proof of the theorem, of a double six of lines by projection from four dimensions. By Professor H. F. Bakek. [Read 9 February 1920.] The theorem in question is that if five lines in three dimensions, of which no two intersect, say a, b, c, d, e, have a common trans- versal, say / ', and we take the five transversals other than / ' of every four of these five given lines, the five new lines so obtained have also a common transversal. Namely if a' be the transversal, beside/', of b, c, d, e, and b' be the transversal, beside/', of a, c, d, e, and so on, so that we have the scheme a b c d e a' b' c' d' e' f in which every line intersects those not occurring in the same row or column with itself, but not the others, in general, then there is a transversal/ of a' , b', c' , d' , e' . We see that the theorem is that if we take eight lines a, b, c, d and a', b', c' , d' , so related that a' meets b, c, d, while b' meets a, c, d, and c' meets a, b, d and d' meets a, b, c, and if e',/' be the two transversals of a, b, c, d and e, f be the two transversals of a', b', c', d', then the meeting of one of the two former, / ', with one of the two latter, e, involves the meeting of the other, e', of the two former, with the remaining one, /, of the two latter. But the original relation of the eight lines a, b, c, d, a', b', c', d' has a certain artificiality; the object of the present note is to show that there is a simple figure in four dimensions, possessing perfect naturalness, being determinate when four arbitrary lines of that space are given, from which the figure in three dimensions may be derived by projection ; and that the condition for this derivation is precisely the intersection of the two transversals e and/'. The naturalness of this figure lies in the fact that three lines in four dimensions have just one transversal. § 1. In order to show this, it is necessary to enter into some detail in regard to the elements of the geometry of four dimensions; this appears worth while for its own sake; and in order not to over-emphasize the importance of the theorem in three dimensions which is here made the excuse for this, we first give an elementary proof of this theorem, employing only three dimensions (Proc. Roy. Soc. A, Lxxxiv, 1911, 597), 134 Professor Baher, On a proof of the theorem of a double six With the notation above, denote the respective intersections (6', c), (6, c'), {c', a), {c, a'), {a', h), {a, h'), (a,/'), {bj'), {c,f') by 4, A', B, B', C, C, W, r, W. Let/ be the transversal other than e of a', b', c', d', which we may represent by/ = {a', b', c', d')le, 'f V, Pig. 1 and denote the points (a'J), {b',f), (c'J) respectively bv U, V, W. Similarly let /^ be the transversal other than d of (a'," 6' c' e') which we may denote by/^ = {a', bi, c', e')/d', and let'the points («',/i), (6',/i), (c',/i), be U^, Fi, W,. ^ Now take the lines a, b, c,f; e a',b',c'J'-d''' The two quadric surfaces defined respectively as containing {b, c e) and {b', c', d'), have, both of them, the two generators e and 'd' which are intersecting lines. The other common points of these two quadrics are then coplanar. Such points are A and A' respec- tively {b\ G) and {b, c'), and U' or (a,/') and U or (a'J). Thus U les on the plane A, A', V). So, by considering the quadrics (c, a, e), [c , a , d ), we find that V lies on the plane (B B' V) and by considering the quadrics (a, b, e), (a', b' , d'), that W lies on the plane {C, C , W). By taking the hnes ■ ^, b, c,f^\ d\ <&',c',/';4' and considering the pairs of quadrics {b, c, d), {b\ c', e'); (c, a, d), {c' , a', e'); {a, b, d), {a', b', e') we similarly show that V, V, W„ lie respectively on the planes {A, A, U), [B, B', V), {C, C, W), and therefore coincide of lines by projection from four dimensions 135 respectively with U, V, W, being the intersections of these planes respectively with the lines a', b' , c' . Thus/^ =/is a common transversal of the lines a' , b' , c' , d' , e' ; as was to be shown. § 2. Now take four arbitrary lines a, b, c, d in four dimensions, of which no two intersect. Two of these lines, determined by four points, two on each, determine a threefold space, defined by the four points, and this meets a third line in the four dimensional space in a point. From this point, in the threefold space, can be drawn an unique transversal to the two lines spoken of. Thus three lines in four dimensions, of which no two intersect, have an unique transversal. Let then a' be the transversal of b, c, d, and similarly 6', c', d' the transversals respectively of c, a, d; a, b, d and a, b, c. Denote the points (6', c), (6, c'), (c', a), (c, a'), (a', 6), (a, b') respectively by A, B, C, A', B', C' and the points (a, d'), (6, d'), (c, d'), {a', d), (6', d), (c', d) respectively by P, Q, R, P', Q\ R'. In general use the word plane for the planar twofold space which is determined by three points, and the word space, or threefold for the planar threefold space determined by four points; as above remarked two lines determine a space, each line being determined by two points; reciprocally two spaces, in the most general case, intersect in a plane, there being a duahty of properties in four dimensions wherein a space is reciprocal to a point and a plane to a line. The points A, A', being respectively on the lines C'Q', BR', are in the space {a, d), and evidently are in the space (6, c); the points P, P', being on the lines QR, B'C respectively, are in the space (6, c), and are evidently in the space {a, d). Thus the four points A, A', P, P' lie in a plane, which we 136 Professor Baker, On a proof of the theorem of a double six may denote by a, namely that common to the two spaces (6, c) and {a, d). We see how much more naturally this arises than the statement, to which it is evidently analogous, in the three dimen- sional figure considered in § 1. It follows that the lines A A' and PP' intersect one another, say in L. Similarly the plane, ^, of inter- section of the spaces (c, a), {b, d), contains the lines BB' and QQ', which then intersect, say in M; and the plane, y, of intersection of the spaces {a, b), {c, d), contains the lines CC and RR' , intersecting, say, in N. The points L, M, N are however all in each of the spaces [a, a'), {b, b'), (c, c'), and so in a line, the intersection of these spaces. For instance the line A A' joins a point {A') of the line b, to a point (A) of the line b', and so is in (6, b') ; and joins a point (A) of the line c, to a point (A') of the line c', and so is in (c, c'); thus L, on the line AA', is in the spaces {b, b'), (c, c'). But the line PP' joins a point (P) of the line a to a point (P') of the line a'; thus L is equally in the space {a, a'). Similarly both M and N are in the line of intersection of the spaces {a, a'), {b, b') and (c, c'). Thus the space {d, d') passes through the line of intersection of the spaces {a, a'), (6, b'), (c, c'); for we similarly show that each of L, M, N is in the space {d, d'). We denote this line by e; evidently its relation to the lines a', b', c', d' is exactly similar with its relation to the lines a, b, c, d; the plane, a, for example, defined as that common to the spaces (6, c), (a, d), is equally the plane common to the spaces {¥, c'), {a', d'); and so on. It is usual to speak of e as the line associated with a, b, c, d; examination of the figure of fifteen lines and fifteen points which we have constructed will show that there is entire symmetry of mutual relation, and that we may speak equally well of any one of the five lines a, b, c, d, e as being associated with the other four; further e is also associated with a', b', c', d'; and indeed, taking any line of the figure, the eight lines of the figure which do not intersect it, consist of a set of four skew lines and their transversals, and the line in question is associated with either of these two sets of four. There are then 15'2 ^5 = 6 ways of regarding the figure as depending upon a set of five associated lines. § 3. Consider now what planes exist meeting the lines a, b, c, d. In four dimensions an arbitrary plane does not meet an arbitrary line; two such elements which meet lie in a threefold space. It can be shown that a plane meeting a, b, c, d can be drawn through two arbitrary points, one on each of any two of these four lines, so that there are oo ^ such planes. Further that every such plane also meets the associated line e. Further that two planes meeting a, b, c, d can be drawn through an arbitrary point of the four dimensional space, and, for instance, an infinity of such planes can be drawn through any point of the Une e. Also, if the two of lines by projection from four dimensions 137 planes through an arbitrary point 0, to meet a, b, e, d, meet the line e in T and U, then the two planes which can similarly be drawn through to meet the lines a', b', c', d', meet the line e in the same two points T and U. In general two planes in four dimensions have only one point in common; when they have two points in common, the join of these points lies in both the planes which then both lie in the same threefold space. By what we have said there is a plane through OT intersecting a, b, c, d and also a plane through OT intersecting a', b', c', d', with a similar state- ment for planes through OU. Namely considering the two planes through which meet a, b, c, d and also the two planes through which meet a', 6', c', d' either one of the former meets one of the latter in a line. To prove these statements we may proceed as follows. The joining line of two points arbitrarily taken respectively, say, on the lines b and c, will meet the space {a, d) in a point, from which, in this space, a transversal can be drawn to a and d. Then the plane of the original join and this transversal is a plane, say w, meeting the four lines a, b, c, d. The point of intersection of these two lines determining this plane m is evidently on the plane, a, common to the spaces (6, c) and {a, d). Similarly the point of intersection of the plane m with the plane, ^, common to the spaces (c, a) and (6, d), is a point from which two transversals can be drawn respec- tively to the pairs of lines c, a and b, d; and the plane of these transversals is a plane through this point meeting the four lines a, b, c, d; conversely the join of the two points where the plane C7 meets the lines c and a lies in the space (c, a), and so intersects the plane /S, namely in the supposed unique point common to w and j8; this join is thus identical with the transversal drawn from the point (m, ^) to the lines c, a. There is thus an unique plane m, meeting a, b, c, d, passing through any general point of the plane «, beside the plane a itself. It will follow from the general result enunciated above, to be proved below, that the plane w', drawn through the same point of the plane a to meet the lines a', b' , c' , d' , meets C7 in a line intersecting the line e. Take now any general point 0, and a varying point P of the line d; a plane can be drawn through OP to meet the lines a, b, this being the plane containing OP and the common transversal of OP, a and b. Let this plane meet a, b respectively in P^ and P^. Thereby any position of P, on the line d, determines the position of Pi on the line a. Conversely given and Pj, a plane can be drawn through OP^ to meet 6 and d, which, being unique, coincides with the former. Thus any position of Pj on a determines the position of P on d. The correspondence being algebraic, it follows that Pj, P describe homographic ranges respectively on a and d. Using the line c instead of b, we obtain another range (P') on d. 138 Professor Baker, On a proof of the theorem of a double six also homographic with (Pj). Thence the ranges {P'), (P), on d, are homographic ; and, if not coincident, they will have two common points, which may coalesce. When P has a position in which it coincides with P', there is a single plane containing 0, Pj, P2, P3, P, where P3 is the point of c on the plane OP, P' . Thus through the point can be drawn, either an infinity of planes meeting all of a, b, c, d, or else two, which may however coincide. When is on the line e, the plane Od' meets a, b, c, and it also meets d because, as we have shown, e, d, d' are in a three dimensioned space. Equally the planes Oa', Ob', Oc' meet a, b, c, d. As there are thus more than two planes through meeting a, b, c, d, it follows, by what we have shown, that there is an infinity; this is when is anywhere on the line e. The aggregate of planes so obtained, by taking to be every point of e, is identical with the aggregate of all planes meeting a, b, c, d, namely any plane meeting a, b, c, d can be identified with one of these; for taking on e, and P on d, this P determines Pj, P^, respectively on a, b, when regarded as belonging to one of the coincident ranges on e, and determines Pj, Pg, respectively on a, c, when regarded as belonging to the other range on e. Thus every plane meeting a, b, c, d also meets e, or more generally five associated lines are such that every plane meeting four of them also meets thefifth"^. In general, as we have seen, from any point on a plane meeting a, b, c, d (and e), there can be drawn another such plane. If the point be on the conic through the five points in which the first * The reader may compare the proofs of this result given by vSegre, Circolo Mat., Palermo, n, 1888, 45, Alcune considerazioni....The elementary theorems here given for the geometry of four dimensions are of course well known; but I have thought that it was necessary for the purpose of this Note to supply demon- strations. The reader may consult Bertini, Introduzione alia geometria projettiva degli iperspazi, Pisa, 1907, a volume of 400 pages, p. 177. In English there is Mr Richmond's paper On the figure of six points in four dimensions, Quart. Journ., XXXI, 1899; Math. Annal., Lm, 1900 (see also Trans. Camb. Phil. Soc, xv, 1894, 267), which deals with a diagram intimately related with that of the text, and CooMdge, A treatise on the Circle and Sphere, Oxford, 1916, p. 482, etc., where the lines of four dimensions are replaced by spheres. The origin of the five associated lines seems to be a result given by Stephanos, Compt. rendus, xcm, 1881, p. 578. I have not seen it formally remarked that the property of the double six follows from the geometry of four dimensions ; indeed the argument given in § 1 was invented in ignorance of this. The fifteen points and lines of our figure (Fig. 2) are the diagonal points and transversal lines of the figure considered by Mr Richmond. See also Hudson, Kummer's Quartic Surface (1905), Chap. xii. of lines by projection from four dimensions 139 plane meets a, b, c, d, e, the second plane coincides with the first. It is not necessary for our purpose to prove this, § 4. The theorem that two planes can be drawn from an arbitrary point to meet the lines a, b, c, d is obvious from the theorem in three dimensions that four skew lines have two trans- versals, the proof of which also depends on the fact that two homo- graphic ranges on a line have two common points. For, if we project a, b, c, d from 0, on to an arbitrary threefold space E, the planes joining to the two transversals of the four lines of S so ob- tained, all meet a, b, c, d. And, we now see, e projects into a fifth line meeting these two transversals. When is on e, the projections in 2 of a, b, c, d are all met by the projections in S of a', b', c', d' ; for the plane Od' , for example, meets a, b, c, and meets d because e, d, d' are in the same three dimensional space; thus the pro- jections in S of a, b, c, d are four generators of the same system of a quadric surface of which the projections of a', b', c', d' are generators of the other system. The planes from each meeting a, b, c, d intersect the space S in lines all meeting the projections of a, b, c, d; that is, in lines which are generators of this quadric of the same system as a', b', c', d'. The planes from meeting a', b', c', d' similarly give rise to generators of the system (a, b, c, d). Thus any plane from the point meeting a, b, c, d meets any plane from drawn to meet a', b', c', d' in a line through 0; and every line drawn from in a plane of the former system is the intersection with this plane of a plane of the second system. If be a point of e lying on a plane drawn from a point H, not on e, to meet a, b, c, d (which therefore also meets e), the line HO lies in a definite plane meeting a', b' , c' , d' . Thus either of the two planes of the first system, those meeting a, b, c, d, drawn from a point H, not on e, meets one of the two planes of the second system, those meeting a', b' , c' , d' , drawn from H, in a line; and the two lines so arising intersect the line e. § 5. Hence we can obtain from the four dimensional figure a figure in three dimensions with the characteristics of that used in proving the double six theorem. If, in the four dimensions, p, a be the planes drawn from an arbitrary point to meet a' , b' , c' , d' , and p , a' those meeting a, 6, c, d, and if p and a' meet in a line, as also p and a; and if we consider the intersections with an arbitrary threefold space S, of these four planes, and also of the planes joining to a, b, c, d, a', b', c', d' , denoting these twelve lines respectively by (p), ...,{a), ..., then, arranged as follows: {a) (b) (c) (d) (p) (a) {a') (6') (c') id') ip') {a'), 140 Professor Baker, On a proof of the theorem of a double six these form a double six, any one of the lines meeting the five which do not lie in the same row or column with itself. § 6. Conversely we now proceed to show that if a^ h^ Cj d^ a{ h{ c-l d^ be eight lines in three dimensions such that no two of a^, h-^, c^, d^ intersect, while d-l intersects a^, 6^, c^, a{ intersects h-^, Cj, d^, etc., and if one of the two transversals, say I, of a^', hy, c{, d^, intersects one of the two transversals, say m' , of a^, \, Cj, d-^, then these lines may be obtained by projection from four dimensions; namely «j, &!, Cj, d-^, a{, bj', Cj', c?j' are projections of four lines a, b, c, d in space of four dimensions and of the transversals a', b', c', d' of threes of these, respectively, while I and m' are the intersections w^ith the original three dimensional space of planes in four dimen- sions meeting respectively the set a', b' , c' , d' and the set a, b, c, d. We give an analytical proof of this. And for this purpose first explain an analytical view of the theorems which have been given in §§ 2, 3, 4, which indeed renders these very obvious. It is fundamental that a point may be represented by a single symbol, say P, the same point being equally represented by any numerical multiple of this, say mP, where m is an ordinary number. Then a space of r dimensions is one in which every y + 2 points, Pi, P^, ..., Pr+2^ are connected by a sy2ygy, w^Pi + m^P^ + ... + m^+2P^+2 = 0, where m^, ..., m^+2 are ordinary numbers; thus the space is deter- mined by any r + 1 points of it, themselves not lying in a space of less than r dimensions ; and, in terms of such r -f- 1 points, say ylj, ..., ^y+i, every other point of the space may be represented by a symbol Xj^A-^^ + x^A^ + ... -f x^+i^^+j, where x-^, x^, ..., x^+i are ordinary numbers ; whose ratios may be called the coordinates of this point, relatively to Aj^, ...,Ar^j^. Thus any point of a line determined by two points A, B, is representable by a symbol mA + nB, in which m, n are numbers; and any point of a plane determined