Fear} , ite One Base * leteknks SS tel ae Ha sea ccd pastes: SUAS NERS it Ea ASSIS eats ESS Rowse ON = ane nthch ee . macy ten so wakes Atri ue rset By 5 oe Poe ae mee fester et $650 G Be ates te AEE Oey eer Se Liles ee eure te ie ; ua ARs at The te pee ag baer + "ee 7 iy pate ot aT ree, Faded pat tes a teats aoe oo PEG Cid ae peta gate 2" ee Pret eee Pia en ee ee eee ee! a ae as pe Digitized by the Internet Archive in 2009 with funding from University of Toronto http://www.archive.org/details/transactions22camb TRANSACTIONS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY VOLUME XXII. (1912—1923) er \ a CAMBRIDGE AT THE UNIVERSITY PRESS AND SOLD BY DEIGHTON, BELL AND CO. LTD. AND BOWES AND BOWES, CAMBRIDGE. CAMBRIDGE UNIVERSITY PRESS, LONDON. M.DCCCC.XXIIT ADVERTISEMENT Tue Socrery as a body is not to be considered responsible for any facts and opinions advanced in the several Papers, which must rest entirely on the credit of their respective Authors. THE Society takes this opportunity of expressing its grateful acknowledgments to the Synptcs of the University Press for their liberality in contributing towards the expense of printing this Volume of the Transactions. PRINTED IN GREAT BRITAIN XVI. XVII. XVII. CONTENTS OF VOLUME XXII. On Compound Denumeration. By Major P. A. MacManon;R.A., Se.D., LL.D.,F.R.S., Hon. Mem. Camb. Phil. Soc. : ; . A Class of Integral Functions defined by Taylor’ s Series. By G. N. Watson, MA. . The Hydrodynamical Theory of Lubrication with special reference to Air as a Lubricant. By W. J. Harrison, M.A., Fellow of Clare College, Cambridge . The Superior and ae Indices of Permutations. By ee P. A. MacManon, E.BS. . The Domains of sielaly Motion i a Liquid Ellipsoid, and the Oscillations of the Jacobian Figure. By R. HARGREAVES, M.A. . On the Fifth Book of Euclid’s Elements (Third Paper). By J M. J. M. Hitt, M.A., LL.D., Se.D., F.R.S., Astor Professor of Mathematics im the University of London The Invariants of the Halphenian Homographic Substitution—to which is appended some investigations concerning the Transformation of Differential Operators which _ present themselves in Invariant Theories. By Major P. A. MacMauon, F.R.S. Vector Integral Equations and Gibbs’ Dyadics. By C. E. WEATHERBURN, Ormond College, University of Melbourne a x6 . On certain Arithmetical Functions. By 8. RAMANUJAN - . On the Fifth Book of Euchid’s Elements (Fourth es: By M. J. M. Hit, M.A., LL.D., Sce.D., F.RB.S. . The Character of the Kinetic Potential in ltromagati By R. HarGREAVES, M.A. . The Field and the Cordon of a Plane Set of Points. An Essay in Proving the Obvious. By Eric H. NEVILLE On certain Trigonometrical Sums and their applications in the Theory of Numbers. By S. RamanuJan, B.A., F.R.S., Trinity College, Cambridge . Asymptotic Expansions of Hypergeometric Functions. By G. N. Watson, Sc.D., Trinity College, Cambridge ’. Asymptotic Satellites near the Equilateral-Triangle Rauilitréum Points in the Problem of Three Bodies. By DanteL Bucuanan, M.A., Ph.D., Queen’s Univ ersity, Kingston, Canada Terrestrial Magnetic Variations and their Connection with Solar Diplo a5 which are Absorbed in the Earth's Outer Atmosphere. = S. CHapman, M.A., Sc.D., E.R.S., Trinity College, Cambridge 3 a sie On the Representations of a Number as a Sum of an Odd Number of Sue es. By L. J. MorpELL, M.A. (Cantab.), Birkbeck College, London ‘ The Hydrodynamical Theory of the Lubrication of a Cylindrical Hone under Variable Load, and of a Pivot Bearing. By W. J. Harrison, M.A., Fellow of Clare College, Cambridge 309 341 361 313 XXIL XXIV. XXYV. XXVI. XXVII. XXVIII. CONTENTS . On Integers which satisfy the Equation # + x* +} i +2°=0. = H. W. RicHMonD, M.A., King’s College, Cambridge . On Cyclical Octosection. By W. BuRNsIDE, M.A., F.R.S., Hon. Fellow of Pembroke College, Cambridge . Congruences with respect to Companies Moduli. By Major P. A. MacManon, R.A., F.R.S On the Stability of the Steady Motion of Viscous Liquid contained between two Rotating Coaxal Circular Cylinders. By K. TAMAKI, Assistant Professor of Mathematics, Kyoto Fae University, and W. J. Harrison, M.A., Fellow ot Clare College, Cambridge On a General Infinitesimal Geometry, in reference to the Thay ae Bata By WILHELM WIRTINGER (Vienna) . oe On the Fifth Book of Euclid’s Elements (Fifth Paper). By M. J. M. Hiut, Se.D., LL.D., F.R.S., Astor Professor of Mathematics in the University of London The Influence of Electrically Conducting Material within the Earth on Various Phenomena of Terrestrial Magnetism. By S, CHAPMAN, M.A., Trinity College, and T. T. WHITEHEAD x ae nee at x The Escape of Molecules from an Atmosphere, with special reference to the Boundary of a Gaseous Star. By E. A. MILNE, M.A., Trinity College, Cambridge Some Problems of Diophantine Approwimation: The Analytic Properties of Certain Dirichlet’s Series associated with the Distribution of Numbers to Modulus Unity. By G. H. Harpy, M.A., Savilian Professor, University of Oxford, and J. E. LirrLewoop, M.A., Trinity Colleve: Cambridge Free Paths in a non-uniform Rarefied Gas with an Application to the Eee of Molecules from Isothermal Atmospheres. By J. E. JoNEs, 1851 Exhibition Senior Research Student, Trinity College, Cambridge = : GENERAL INDEX PAGE 389 405 4135 425 439 449 463 483 519 535 557 I. On Compound Denumeration. By Mason P. A. MacManon, R.A., Se.D., LL.D., F.R.S. Honorary Member Cambridge Philosophical Society. [Received May 1, 1912. Read May 6, 1912,] Art. 1. I propose to examine the subject of compound denumeration, otherwise the partitions of multipartite numbers, by a direct application of the Theory of Distributions which was developed by me in the Proceedings of the London Mathematical Society *. It will be shewn that the actual denumeration may be made to depend upon the theory of the symmetric functions of a single system of quantities. Such a system is Ce Bee Che, Seep and I write in the usual notation (1 —a,2) (1 — ax) (1— 4,2) ... = 1 —a,¢+ ane? —a;a*+... = 1 — 1+hya+ hoa? + hye? +...’ so that the quantities a are the elementary symmetric functions and the quantities h the homogeneous product sums of the quantities a of the system respectively. With these functions are associated the differential operators ds; =0a, + HO, (d,s) Sie + as0a,,. + Sho Ast D,= where (d,°) denotes that the linear operator d, is raised to the sth power in symbolic manner so that it denotes not the performance of d,, s times in succession, but rather an operator of the order s. I first consider the partitions of a bipartite number (pg) and note, as observed long ago by me+, that the partitions are separable into groups which depend upon the partitions of the unipartite numbers (p), (q) respectively. Thus the partitions of the bipartite number (22) nine in number are separated into four groups: Gr (2, 2), Gr (2, 1), Gr (1, 2), Gr (22, 1), (22) (21 01) (12 10) (11 11) (20 02) (20 01 01) (10 10 02) (11 10 01) (10 10 01 01), * Proc. Lond. Math. Soc. vol. x1x. 1887, ‘‘ Symmetric Functions and the Theory of Distributions.” + American Jour. of Math. vol. x1. 1888, p. 29, “‘Memoir on a New Theory of Symmetric Functions.” Won, .O40h oy ak 1 2 MAJOR MACMAHON, ON COMPOUND DENUMERATION. where it is to be observed that supposing the partible number to be (pq) (here p=q=2) the p number is in partition (2) in the two first groups and in partition (1°) in the last two; while the g number is in partition (2) in the first and third groups and in partition (12) in the second and fourth. In fact if the numbers p, q have P, Q partitions respec- tively the partitions of (pq) are separable into PQ groups for every partition of p may be associated with every partition of q. We will now study the enumeration of the partitions appertaining to a given group. Consider the group Gr {(p.™ p™...), (HG ...)}, where (p,"p.™...), (@¥qX?.--) are given partitions of p and q. The most extended partition of the group contains =7r+Zy parts, while that which is least extended contains a number of parts equal to the greatest of the integers =z, Ly. No generality is lost by the supposition =a > Xx. In a partition of the group the biparts may be ordered so that, as regards the partition of p, the first =a biparts are (p..) 7, times (p..) m. times, &e., and this will be the case for every partition of the group. The second element of any bipart may be either zero or one of the parts qj, qs, ... of the partition (qq...) of g. There may be also biparts of the form (0q,), (Oq), .... The biparts are therefore of one of the three forms (p,q), (ps0), (Oq:), and their number has lower and upper limits =a and =a+ Xx. We now suppose there to be m, parcels of one kind Ts Fs a second kind &e. y parcels of another kind differing from those above. Altogether =++Zy parcels of a specification which may be denoted by the partition (1,7... Xx) of the number = + Zy. We also suppose there to be xX. Objects of one kind X2 i a second kind &e. X7 objects of another kind differing from those above. Altogether 4+ Zy objects of a specification which may be denoted by the partition (x:X2--. 27) of the number 7 + Zy. The number of objects is equal to the number of parcels and we may consider the number of ways of distributing the objects in the parcels so that each parcel contains one object. When, as in the present case, the number of objects is equal to the number of parcels and one object goes into each parcel the notion of the parcel is not essential and we may consider two sets of objects of specifications (m7... =X), (Xi Xe++- Lar) respectively ; MAJOR MACMAHON, ON COMPOUND DENUMERATION. 3 and the problem is the enumeration of the sets of two-fold objects that can be formed by making =2+y pairs of objects, each pair consisting of an object from each set of objects. This problem is precisely the same as that of determining the number of partitions of the bipartite number (pq) which appertain to the group Gr {(p.™ p™..-), (Qik qk? ...)}- To explain this consider the partitions of the bipart (33) which appertain to the group Gr {(21), (1°)}. Here m=1, m=1, fr =2, Y.=3, 2y=3. We consider objects specified by (113) as being the first elements of biparts; these are aeeeelan Ot Ome O): or if we want to exhibit the fact that they are first elements we may write them 2x, 1x, Ox, Ox, Ox. With these consider objects specified by (32) as being the second elements of biparts; these are or as they may be written *1l, *1, xl, x0, x0. Combining the two sets of objects in all possible ways so as to form a single set of two- fold objects we obtain the four sets 2 LE OF. 200; 700; 2 O ee OLS Ole OO: 2, UT? OI 200s 100: AD, 10), Wik, il, Oi corresponding to the four partitions (21, 11, 01), @Qiy 10:7 (Oley 700); (20511; OL OU COS 10: 0b 01 08); of the bipartite number (33) appertaining to the group Gr (21), (1°). It is clear that there is in every case a one-to-one correspondence between the distri- butions as defined and the partitions under examination. The number of the distributions was shewn (loc. cit.) to have either of the two expressions DE DED S les Is, DS Dives: Deo Mae. Wee t MAJOR MACMAHON, ON COMPOUND DENUMERATION. The expressions are equivalent and may be evaluated by means of theorems given (oc. cit.). The whole number of partitions of the bipart (pq) is im consequence pip iy 1b eee Dee ol Peel (ee ™ x the double summation being for every partition (p," p.™...) of (p) and for every partition (gx gX...) of (q). We apply the method to find the number of partitions of the bipartite (33). We have Group We 2 Na Ne Partitions in Group (3),Qy) LO 2 1 0 2, De <= 2 (QD) 1 0 -2 Ae DDR 3 aay ~ a 0% ss lone ip pin 2 (21,3) 2 1 Sl a heme epran. —— 3 (21), (21) P15, 2°.) ae Ree eee 7 (21),(1)} 1 1 3 38 O 2 D,Dehhy= 4 (ayy. S01 = ke Oe, Dae = 2 (2p 3 0° 2 1 2 3° a pee 4 (iy 3 0 3 8. Ouse eh ere 4 Total 31 No calculation is required if we are given Tables which express the h products in terms of monomial symmetric functions, Thus since a Table shews hgh?=... +7 (212) +... DID ilee =. The above resulting numbers are all shewn in the Tables which proceed as far as the weight 6. In the above case where p=gq, it is not necessary to consider all the groups because the two partitions that define the Group may be interchanged. Thus the two Groups {(3), (21)}, {(21), (3)} are identical and the whole numbers of partitions might have been written D2h? + 2D,D,h? + 2D; D,hsh, + D. D2 hgh? + 2D; D2hgh, + Deh. We now remark that, formally and algebraically but not operationally, this expression may be written in the factorized form (Dh, + Dy2h, + Dghs) (Diy + Dh? + Dshs); for the multiplication gives Deh? + D, Dh? + D,Dshihs+ DShgh, + De Dyhgh? + Dy Dghghs + Ds Dy hgh, + Ds Dahgh? + Deh, and, observing that, by the well-known theorem of reciprocity Di Die = Dis hele, D;,Dehshg = Ds Dahsh?, the truth of the statement is verified. MAJOR MACMAHON, ON COMPOUND DENUMERATION. 5 In fact, formally and algebraically but not operationally, the double sum > 2 Dp Dy, .-. Ds hig hy, «++ lees =X Xu Xa be written at may y {2.Dx Dz, -2- hse}. {2 Dah hy, ---}: ™ x The factorized form may be regarded as a symbolical expression. By the above method the following numbers have been calculated faeces bdo 15). | a ig he oe ae es aaa =*= | seat Oe: On eetie |e LOm eo a 4 | 12 | 29 | 57 | 109} — 5 | 19 | 47 | 97 | 189/336 i iL Thus, from the table, to find the number of partitions of the bipartite (43) we take the row commencing 4 and the column headed 3 and find at the intersection the number 57. The numbers agree with those obtained by expansion of the generating function 1 (1—2)QA—y)( —2*)(1—ay)1 —y*) (1-2) (1 — a*y) (1 — ay?) 1-¥)... Art. 2. The distribution of =x + y objects of type (m7... Sy) into =r+Zy parcels of type (x: x2--. 27) one object in each parcel has necessarily resulted in our obtaining the whole of the partitions of the group under view. Remembering that =3w>Zy we may if we please make a distribution of Xw+s objects of type (m7...s) into Sats parcels of type (yixX2--. 227 — =x +s) where s is any number included in the series On, din 2ihereey? We will thus obtain a number for the enumeration which is Dred D om WD ben coc lass eB and this number also gives the number of partitions of (pq) which contain =x++s or fewer parts and also appertain to the given group. Hence also the number of such partitions which contain exactly 7+ 9s parts is DIED ye ae glee Mpa en lesesy tg — Deve Deny) lly, Nye cae Nag sy ta 15 or as it may be written De De ines) Dg lig Mit Msc a 4 pe Oaealbg Ni, os Ne Sa'pa—a} The whole number of partitions which contain =7-+s or fewer parts is 5 > DDE see Delis eee hisn— sy 45) Ne 6 MAJOR MACMAHON, ON COMPOUND DENUMERATION. the double summation being for all partitions (pi™ po™...), (GX'gX ...) of the numbers (p) and (q); and a similar summation gives the whole number of partitions which contain exactly =7+s parts. The expression for the number of partitions which contain Xa+s or fewer parts can be given the factorized symbolic form ; (C= DED eee hsn—sx+s) . (= Dsh,, hy, Bo) J x As an example let us consider the partitions of the bipartite (44) which appertain to the group {(211), (211)}. Here s may have the values 0, 1, 2, 3. m=1, m=2, ty¥=3, W=L m=2, 2r=3. For s=0, we have D,D,h.h,=2, and the two partitions into 3 parts are (22> LIES sli): (21, 12:- 11): For s=1, we have D,D2h,h?=7, shewing that there are 7 partitions into 4 or fewer parts; in addition to the 2 which have exactly 3 parts already written down we have 5 which contain exactly 4 parts; these are (22, 11, 10, 01), (21, 20s 01)s (2A Ey 10) 102); (20, 12, 11, 01), (20)) Ae 02): For s=2, we have D.2D,h2h;=11, and we find that in addition to the 7 forms already written we have 4 which contain exactly 5 parts; these are (22 10 10 O01 O01), (QE 10> 10) 40201); (20) 2," 10) 018 100); (20 11 10 02 01). Finally for s=3 we have D,D,D,hsh,h, = 12, and we have 12—11=1 partition which contains exactly 6 parts; this is (20 10) 10°02) OL “01). *To explain the general method of calculation it is to be noted that Dlom, = lone and that when operating upon a product, D, acts through each of the partitions of s. Thus Dyhyhan = y-ghin + ghia + Qa hn-1 Dghihinhn = hag hmhn + hilm—slin + lalimln-s + hy gin alin + Rp_ehinlin—y * fia ltgn—aln— + yy Am—alin + Apalan hn—a + ign in—a at [hess yg oy an * Vide Proc. Lond. Math, Soc. vol. x1x. 1887, pp. 127—128, ‘The Algebra of Multi-linear partial differential operators.” MAJOR MACMAHON, ON COMPOUND DENUMERATION, 7 The calculation of D,D,D,.h;hzh, therefore proceeds as follows :— D;DsD, . hghghy.= D;Dz (heh, + hgh? + hghz) = D,(2hgh, + he + 2hoh, +hs + hg + 2hohy + hohy + hz + heh) = D, (2h,3 + Shah, + 2hs) =2+8+2=12. Art. 3. In the next place we examine the effect of employing the elementary func- tions ;, My, a3, ... Instead of the homogeneous product sums fj, ho, hs, ..... The distributions enumerated by the number De De mice iD: Gy, Ay, «+» Wsn-Sy+8> are those of objects of type (7:72...s) into parcels of type (yi: x2... 2r7—Zy+s) one object being placed in each parcel subject to the restriction that no two similar objects are to be placed in similar parcels. The corresponding partitions of the bipartite number (pg) are those which appertain to the group {(p,"p.™...), (qq ...)}, which contain exactly =4r+s parts, the zero bipart 00 not being excluded as a permissible bipart, and in which no particular part (including the bipart 00) occurs more than once. Of course the double sum = = DDE eee IDE +++ USn—Sy+s> Py enumerates such partitions for the totality of the groups. To see the meaning of this result consider again the partitions of the bipartite number (44) which appertain to the group {(211), (211)}. For s=0, we have D,D,a,q,=1; since =7=3, this means that of all the partitions of the group which contain exactly 3 parts, the zero part 00 being admissible, there is but one in which there are no similarities of parts. This partition is in fact (21 12 11). For s=1, we have D,Da,a,2=5; for a Table which expresses @ products in terms of monomial symmetric functions gives TER pepe) (PAE) eee giving De ea — Thence we conclude that there are just 5 partitions which contain 4 parts involving no similarities. These are (21 12 11 00), (22 11 10 01), (21 12 10 01), (21 11 10 02), (20) 2) On); the set including the one previously found with the part 00 added. 8 MAJOR MACMAHON, ON COMPOUND DENUMERATION. For s=2, we have D2D,a2a,= 5, since C207 = ae D221) sos and the 5 forms indicated are found to be (225 110018 100); (2 A2 SLO Ot 00); @i 11 120 02) 00); (20% AZ Ads 01900); (PAN ab Tk 2 Ko): the parts in each involving no similarities. Finally for s=3, we have D,D,D,a;a,a,=1, since A3Mo, = ... + (821) 4+... and the form indicated is (20 11 10 02 O1 00) containing no similarities of parts. We obtain information concerning the partitions of the group which contain different parts when 00 is excluded as a part; for denote by Q, the numbers of partitions of the group which contain exactly s different parts, the zero part being excluded, we have Q=1, 8+ Qa=5, W+Q=5, Q=1, whence Q,=4 and Q,+Q,+Q;=6. This number 6 which enumerates the partitions of the group which possess different parts is either D, DoD, aa. + D. D?Za.a,2, or D2 DyaZa, + Dz Dya.a,. In general we have the relations Qs, = Dz, Dz, ... Ay, Vy, »++ Azzy, Qse + Qsr41= Dz, Dz, ... Dyay, Oy, «+. Ue—sy 419 Qsn+3y—2 + Qsn+3y—1 — De De wee Ds, Qse+3,—1 = D,,D,, ... Ds, aya yy, Ue +++ Use: xe W215 Hence the number which enumerates those partitions of the group which have different parts, the zero part being excluded, has two expressions; for Se + Qsn41 Sg) ei i Qsn+2,-1 = D,, Dy, ++: By, Oy, »++ Ute—ty + D,, Dy, '... Daly, thy, «1 Osea + Dy, Dy, na Dy dyg, Ogg «in Wig Sy 4:4 =e = D,,D,, ... Didy, dy, ..+ Aae—3y41 + Dg, D,, ... Dy, a, --« Azn-Zy43 HD, Diy os: Dye Qigg dvs Oger sy 5 F x05; where if Sy be uneven both series extend to $(Sy+1) terms, whilst if Sy be even the first and second series extend to 42y and 43y—1 terms respectively. MAJOR MACMAHON, ON COMPOUND DENUMERATION. 9 That these two series are equivalent may be shewn algebraically as follows. For brevity put =+— Ly =@ and note that Dy iy, Pyy »++ Dor = Ay, Ay, +++ Ug + Vly, 1 Ay, +++ Uo+1, Dy dy, Ay, +++ U94-2 = Vy, —1My, +++ Bon, + Vy, 1 by, 14x, +++ Caras = S Ss Dy Gy, Gy, «+» Ag4-3= Uy, —1 By, —1 Uy, «++ Vora + Willy, 18x, 18y,—1 By, +++ Boss, &e. Directly we operate upon the relations with D,D,,... the equivalence is obvious. Art. 4. There is no difficulty in extending this theory by fillmg up the gap between the elementary functions and the homogeneous product sums. For suppose hy, ky, ky, ... be functions derived from the homogeneous product sums by deleting therefrom all terms which involve quantities of the system (from which the symmetric functions are derived) to a higher power than &. Then the distributions enumerated by the number DE Dea DCE ele cs. X1°"X2 * are those of objects of type (m7,...s) into parcels of type (xi:x2-..=r7—Zy+s) one object being placed in each parcel subject to the restriction that more than & similar objects are not to be placed in similar parcels. The corresponding partitions of the bipartite number (pq) are those which appertain to the group {(p,"p.™...), (qq ..-)}, which contain exactly =w+s parts, the zero part 00 not being excluded from being an admissible part, and in which no particular part (including the part 00) occurs more than k times. Art. 5. I pass on to consider the similar theory of tripartite partitions and it will be found to shew what the theory is for multipartite partitions in general. Consider the tripartite number (pqr) and the partitions appertaining to the group {(p:™ po™ ...), (Qi gar? -..), (71772 ...)}, wherein we will suppose =7>Zy > Xp. The partitions involve at least =a and at most r+ y+ p parts. Reasoning as in the bipartite case we find that for partitions into =7+s parts, where 0= (1); B, — (1),4, C; = (1); B, M, = Qs L, UG Catslels A,=(2)+(1), Be=(@42+(042 O=(2) B+ (Be My= (na Let (na Le? &e. &e. &e. where if the multipartite number be n-partite, Z, M are the n—2th and n—JIth letters of the alphabet. We have then to find the coefficient of (Gai saa Ea (QOH 00 th) (GaGa ooo tH) a0 in the development of M,,M,,... M;,. We have the sought number equal to DY REMI DAs el DENY SY OSI OAS Ieee Oat 825170) CR Mi Me reorient and observe that D,;® . Cm = BsCm—sy D,® «Din = C;Dm—s, IDC) Mit =I sl oe. relations which enable the regular and progressive calculations of the sought number. In all cases the theory of the partitions into dissimilar parts is reached by substituting the elementary functions a,, a, d3,... for the homogeneous product sums A,, Ay, As, .... The totality of the partitions into =x +t, or fewer parts is given by the expression Se eee Dy Ol DH ey PDE)... D8) DEO DEO. De Ooo... MM, MMe coe, x po the summation being in regard to all the partitions (pi pst ...), (GaixiqxX*...), (Try? ...), (S17 55%...), «.. of the numbers p, q, 7, 8, ... respectively. ay a) =." ScOyrA Sea ee cee Te Oke es ake eee sa 8 = “iors ‘atcln “qa eeryas ‘ont ree aT iot aoa Wy cial riz “lt nan oy ni; a < 2 4 be oping aad 10 Skee sida 2 Ye = ib a) {yee EN ; ; ; A i? ad ny : ‘5 See he O “ ’ n \ Se ae ‘var Peer eat NPE NY ee Pere re lt ad Ge | yar 3 S74 1 pea She er sero Gee 2 rl i) Lyre) 2 OBC pee , ét ie fe I “4 j ’ ’ r Ftitppas ‘ = iti i you rae | ne Md at af 7 pees 4 & » Y ’ i zen Wt Peat ng - ‘ , ps id bs ahh CE) Ot trang omy hal @ * Puree ™ = . : 7 ven 1) f bh Senne (ayaa . . 2 a t mite j nigia fag Dad att’ 64 eae i “ Ai ? ' on oe i a, ee ee A + aa : . ~ ; » t \ . t i - 7 : J ( ° 1 a arnt a uN Ye ) ‘ - ‘ =? sD fu bd » ' 7 , Ip . fod a . vars ai ay (eC Vig. Sua « i ‘ i) wi ity aul : ’ _ 1? a 7 ? AL. ¢ " s rf ] = : 4 ' " one @ at Fe ne som Il. A class of mtegral functions defined by Taylor's series. By G. N. Watson, M.A. [Received Sept. 20, 1912. Read Nov. 11, 1912.] 1. Ir is sometimes possible to determine the complete asymptotic expansion of an integral function by making use of asymptotic expansions of more simple integral functions. An instance is afforded by the deduction* of the complete asymptotic expansion of the function 2 a". (n+ 6) F,(2;@)=> OEY Sea n= W(n+ OP [where x (y) is analytic} in the vicinity of y= ] from the asymptotic expansions of functions of the type a a” G3@30)=> ———— Tp ( U 0) ee n(n aE @)8 In this memoir, I propose to obtain the asymptotic expansion of a class of integral functions of a more general nature than the function Fg (x; @) defined above. I am inclined to think that the integral functions which will be considered are the most general integral functions which possess the two properties (i) that the coefficient of the nth term in the Taylor's series which defines the function is a simple function of 7, and (11) that the asymptotic expansions of the functions involve only powers and exponentials of the variable. 2. Let f(#) be a function defined by Taylor’s series HMC) SO GTA | Gee aS capky) doapnecocsesenaadoseocoetonebosece (1), and let it be possible to define a function ¢$(s), which is analytic im certain regions (to be specified presently) of the plane of the complex variable s, such that when s is equal to any positive integer 7, eo” b(n) P(an +1) =Cn where $ (7). exp {9 log n} —>a finite limit as »—»%0; in order that f(#) may be an integral function we must take Ri(a)>0; * Barnes, Phil. Trans. Roy. Soc., vol. ccyi. A. (1906), x (y) possesses an aaynploke expansion of the form pp. 273—278. x (y)=bo + by 1+ boy +..., + There is a large class of functions Fg (x; @) which are when y is large and real; to such functions Barnes’ results such that x (y) is not analytic for large values of y, although do not apply. Wei, O00 UNiGs 10 3 16 Mr WATSON, ON A CLASS OF INTEGRAL FUNCTIONS apart from this restriction, a, 6’, g are any constants, real or complex; it is convenient to determine a so that | arga|<437. The hypotheses* which we shall now make concerning the function ¢(s) are as follows: (i) That a number, /,, exists such that $(s) is analytic on the right of the line R (Gs) =h,. (ii) That numbers y’ and 2 exist such that 7 >0, 0vy’, | arg (s/a)|~y’; the expansions are valid and the series on the right converge when R(s+0)>0; w=—R(@) and Bisa number® such that R(8)<0; and if B be the integer such that 0>R(8+B)>—1, then the coefficients in the expansions satisfy the inequalities , Se a 1 ( ©) 9GI)= Grey E fide |< TT (GR flop (es) Beam. seve cannes ae seaguceamate (26), Hig | cake 1" (Ke D8 in ceing cccmsweccaacoerceeee tee (2d), when &>1 and H is some number independent of /; Mis any number less than M, where M, is a positive number depending on p and X. In this memoir, I propose to investigate the asymptotic expansion of f(a), defined by the series (1), for large values of « when the conditions stated above are satistied by ¢(s/a), so that * These conditions are satisfied if ¢ be a member of a very large class of functions which can easily be constructed. Some examples are given below in § 15. + A comparison of equation (8) below with Part v. of Barnes’ memoir (loc. cit.) shews that, in the special case when the expansion for ¢ (s/a) is not asymptotic but con- vergent, the asymptotic expansion of the function / (:r) can easily be obtained by Barnes’ methods. t+ These terms are introduced in a memoir by the writer ‘*A theory of asymptotic series,” Phil. Trans. Roy. Soc., vol. coxt. a. (1911), pp. 279—313. § See Rendiconti del Circolo Matematico di Palermo, t. xxxiv. pp. 65—84, || © must not be purely real. ‘| If R(s’)<0 we take 8=8'; if R(8’) > 0 we choose 8 so that 8’ —£ is a positive integer. DEFINED BY TAYLOR’S SERIES. IN7/ the expansions (2a) and (2b) are valid when R (s +@) > 0; these two expansions will be utilised in obtaining the asymptotic expansion in question. When |a—1/>1, it will be shewn that the asymptotic expansion of f(#)can be obtained for all values of arg @. When |a—1|<1, the asymptotic expansion of f(x) can be obtained for a certain range of values of arg; in the part of the plane not included in this range the asymptotic expansion of f() depends on the behaviour of f(s) on the left of the line R (as) =h,. The analysis to be employed is so much simpler when we may take M=1 than when it is necessary to take M <1, that we investigate the case MJ =1 separately in Parts I and II of the paper; the case when M<1 is investigated in Parts HII and IV. The symbols* A, O and o will be employed throughout to mean ‘a definite constant,’ ‘of the order of, and ‘of order less than, respectively ; thus f(«)=0 (g (x)) means that Lt sup {|f(#)|+g (x)} is finite; while L>o ft (@)=0(g(#)) means that Pos sup {|/(x)|+g (#)} =0. In Parts I and III te the paper, A will be supposed to be faaerendeats of the variables « and y and also of a variable integer &. Part I. Preliminary asymptotic formulae. 3. Let us define the integral function t E(x) a the series, an Seoaesmatechacstesemtccoscer en adatesstsanscesetes ( E, (2)= om 2 T (w+k+1) (3), where & is a (large) integer ; we shall obtain asymptotic expressions for 4, (a) for all values of «. (i) Let |2|<1; then | ~ v a | eres Et + E+ 1 SD EBay | < an t= ee n(k+1) K+ 1 Sk) J 2 — : > 1 Thus, when |w|<1 and />1, |B @)l1, |argxv|<3r—6 where §5>0; then by multiplication of series Pr ie aes a a (—)" kan a pl S 1 hk React. Ce) maa(GERI ce AEE) a! meaD So SES along a contour which may be taken parallel to the imaginary axist passing through the point s=—J; where 7 is some fixed integer, and /& is taken to be such that 4>/+1. On the contour | /(s+%)-!|<7+1 and {|P(-s)a*ds|<2m7K where K depends on 7 and 6 only when \e|>1. Thus, when |#|>1, |arga|<3nr—6, £>7+1 K Gall : | By (2) |< (ae (3b). * The use of the symbols K and O is explained by + Some properties of this function have been given by Hardy, ‘‘Orders of Infinity” (Camb. Math. Tracts, No.12); | Hardy (Proc. Lond. Math. Soc. ser. 2, vol. 1. pp. 404—405). the symbol O was introduced by Landau (Primzahlen, + See Barnes’ memoir (loc. cit.), Part 1. Bd 1. p. 61). 3—2 18 Mr WATSON, ON A CLASS OF INTEGRAL FUNCTIONS To discuss the function A,() for other values of 7, we need the following Lemma*: Lemma. Vf 1>u,>uw>...>0, and if s, denote the sum of n+1 terms of the series Tu, 6 + use" +... @ being a real angle, then | s, | <| cosec ($a) |. [For (ql —e) Sr= 1 + (uy a 1) e' +(u2— uy) enw ore +(Un — Uy - 1) is —Up efit. so that | (1—e') S,|<1+ (1-24) + (& — ty) +--+ (Up 1 — Un) + Un <2, ie. | S,|<|cosec ($o)|.] (iii) Let a=re” where rk, 82. As in the portion of Barnes’ memoir+ which deals with the function Gg (#,@) we may shew that Gp (x; @,k) = = | | - ee (1 _ ah (1 i Ve Ex (w—y). dy, Qin © the integral being taken along a contour starting from the point a, encircling the origin in a positive direction and returning to the point 2 This contour is marked with double arrows in Fig. 1. The many-valued functions are specified by taking arg (1—y/x)=0, arg {— log (1 — y/x)} =0, when y lies on #O before the cireuit of the origin has been made. Now let us deform the contour into that marked by single arrows in Fig. 1; the four parallel lines make a non-zero angle with the line Ox, and they make an angle less than 4a with the real axis; the circle surrounding O is the circle y =1; the lines PQ are * Cp. Bromwich, Theory of Infinite Series, § 20. by term, of the series + We have © 2B-1 p—(e+n)2 yn b> ~ — ul e-7iB T (1-8) : n=o ['(n+%+1) ’ as pes 1, «P and OQ are less than 6k/2|exp|z). We note that on the loop from Q round the origin, when | y|>2)|x|, R(«—y)<0, since OQ is inclined at an angle less than 17 to the real axis. Fig. 1. We may now write the formula for G3 (x; @,k) in the form Gp (05 8,8) = C—O] | tog (1-4) | (1-2) Be e— yay Qrix = log z(t A) (1-2 Be —wvay a @ler|-e(-2) (-3- FEA ant Ren ae ae (4), where C denotes the contour starting from Q, encircling the origin, and returning to Q: in the second and third integrals the many-valued functions are specified by the values which they had at P in the original loop integral before the circuit of the origin was made; since we have written y=7+2, OP’ is a line equal and parallel to #P. We write this formula for Gg(z;0@,k) m the form GAGA ON) = 1s UREN ered ons cain se secceeceesaeeeeeense at (4a), and we proceed to find inequalities satisfied by | J}, |Z.|, |Z; |. Let us first consider J,; on PQ we have the following inequalities satisfied: log (1 - ") arg - log (a _ Yh arg @ = 2 | IK L also R(a—y)<0, so that by (8a), (3c) and (3d) we have | A, (~—y)|< K+ (k+1); and it is easy to see that the length PQ is less than 2) zx}. Ke \-4|>x, 4{|logr|+) |}, it is easy to see that K fe R(®)-1 dp = re! {| log r| + | |}2-2@(k +r] a |} Ts We now divide the path of integration into three parts, viz. (i) from r=0 to r=h,, where k, =k, Gi) from r=k, to r=k, (ii) from r=k to r=p. There are two cases to be considered, the first when 02: x | and hence (56) is still true even when R(@)= 1. We next have to estimate the value of the integral We divide the path of integration into two portions, the first portion being such that on it |y|<2|z|, the second being such that on it | y|>2|«|; we call the integrals along these portions J,’ and I,” respectively. On the portion for which | y|<2| |, we notice that hog ( _ | SG larg” + arg {=Hos (1 - 2) eI \y x} | y Z | /y\B-1 / e-1 so that in’|1, and {1— Y) < K, since R(@)< 1. y| ‘ a / K It follows that ID’ < pen | | Ey (w@ — y).dy|. Now all the asymptotic expressions for Z;,(«—y) are comprised in the single formula : ey | a es YC (e+1)| Pe&+1)’ and on the portion of the contour under consideration |a«—y| >K\« 1 f yee a aol Hence INi2||; on these portions of C, R(x—y)<0, so that by (8c) and (3d) K Bs (@—9)'< Dey fer ey} Putting 1—y/r=re (r real) we divide the path of integration into three portions*, viz. (i) from the pomt where y=2/2z/|to the point where |1 —y/e|=h; (ii) from this point to the point where | |1 —y/x| =k; (iii) from this last point to the pot where y= OQ; noticing that |\dy|< K \a |.dr, we may prove+, in precisely the same way as we proved (5)), that Kala: |B) 1+1, |x >2, and 0< R(@)<1, then ’ (x: OI K |e” K \a)-8| K .-2® : | Ga (x; 0,k) < etl) Tee) +raapt FUER) (oe iz (6), where K is independent of / and z. We wish to extend this result, so as to cover a greater range of values of 6. First, let 02= N+ ¢, where WV is a - integer and 0< R(¢)<1. at gn-N qa —N).(n—N+6)8° We easily deduce from (6) that, when V< R(@)N+4+/+4+1, TGs |alee ed ct) oe K .& 8) WHIFB-1/ 7. 7. += lokReccte (6a). |aN+t+B T(k+l)~ P(k+1) ° C(k+1). {log k}-28 Similarly, when — V< R(0)<—N+1, WN being a positive integer and k>—N+1+1, Ga (a;0,k)|< eS pe e’| ae Riper ee | K \ xX kR® Tp T(k+1) jalt@-N-1) P(N +k+1) T&+N41)+ T(E +1). (log &) =e) Then Ga (a; 0k) = 4 Ga (e: o,k— N)- Ga(w; 0,k)| < It is convenient to quote the complete asymptotic expansion of G@s(a; 6,k) when |! is large and & is fixed; we a ae n—k Ga(«; 0, k)= r = aati OE) = 2 Fo w ee me = rac (n—k+ 09? consequently}, if [log (1 — y) P71 — y)?* #7 = (- yf S dy (—y)", n=0 = sn—k x N n Gp (0; 8,8) = . et [ (cd TB+2) | og: Cea h)= ‘i > OGLE aero tam | 2, E(Bya" 7 ane = - e N _\ntk rm a= k & +(-aytflog (a) | 3 Ne aye yp tollee(—o)] Co, where |arga| <7, arg(—a)|—R(@) so thath+1—p>0.] Now if (7 @) is satisfied we may shew that J, is O( « |") where Z is the value of R(as)/R (a) on the contour D. Further, we may shew, by Cauchy’s theorem, that J, is equal to minus the sum of the residues of the integrand at those poles which lie on the right of D. We thus get Tees | SOM I Seamer) (ie) 7S ev D(ans 1) ~~ a-ay1 SiGrn/a) Ge 1) Remembering that g = 2p + 1, we get IMs (weI ma, 2 p(n).e” 2 iD ad ig! (n/a) ; E ee > > expi(Qz7 oo (OI) eect ees : =. h@ean a=1,+ Se, exp (27rint/a) T'@a) (axel) (8) We are thus led to consider the behaviour of the integral function 2 o (n/a) PERG Napa, Of reer, BSS econ cURL eBe ae Re RSET clo De > Tage) (9), where y = (ae9)"!* exp (27rit/a). But, for the values of n involved in this summation, we may expand $(n/a) into a conver- gent series of inverse factorials (by the result of § 2), so that 1 ay As A Ds (n+ 0) | a1 4 (n+ 1) (n+ 2) 3 # |: * We take the contour to lie on the right of the line R(as)=y'; y' being defined as in § 2. Vor, XXII. No. OI, 4 24 Mr WATSON, ON A CLASS OF INTEGRAL FUNCTIONS (On comparing both sides of this equation as n—» 2, we see that the first 8’ — 8 of the coefficients a, d, ... vanish.) From this expansion, we may shew that F'(y) can be expressed as the sum of a series of functions of the type Ge(y; 4, ee for we have F Ls Am af ae S by. n Of @)= ee nai(nt+ OP U(ntm+l) y=h41 (n+ OF C(m+1)’ = er Ue G+2 ss wens bin = (n+1)(n+2)...(n+h+ i (vn +1) (n+2)...(n+h+2) e Also, since » >h +1 bene)! | ___“ IC Ey C+ | an “[(A+h+2)! 0 (h+tk43)! UP and, since | d;4,|< Kk! b+ \log (k + 2) , we easily find that, gua function of k and og (k+2))8 flog (k+3))4 ban < Ga ix | REE Pelee dieses | (k | I} aese flog (t+ 2)}? au ie fhe at| . B -0|*8 | khh—e — 0, as kom since h+1—p>0 Further, if by, be the greatest value of | by, al for n>h+1, we have ke Ss >, ny” f S . |y ia =) n=ht+1|(n+ O08 |.n! oO \n Since R(n+©)>0 throughout: the summation Ly » —*AG is finite for any n=ati |(n+ OF |.n! J assigned value of y. And consequently k F(y)= = any’? Ga(y; O+h41, m+h+1)+J, m=0 oo | gy (m where |J;| y Ra ra) Tee ,— 0, as k—> x when y is assigned That is to say, for every finite value of y, F(y= S any" Ge(y; O+h+1,m+h+1). m=0 6. We shall now shew that when arg y|< 4}, F(y) possesses the asymptotic expansion ev v Sue rd- B) 2 = ay \—¥ F(y)= yp E = y" +o(ly| | Sector Oeaueae nee (10) where S, = = On + pie m ; th offi heal if ART ear ee e coefficients ,,c, are given by the expansion {log (1 — w) PP (1 — w)?-™ = (— a > mCn(— 2)", n=0 and @=@+h+1 while v is any fixed integer bo or DEFINED BY TAYLOR’S SERIES. We shall also shew that when \arg(—y) <47, F(y) is O( y) where X is determinate We have F(y)= = any'"Ge(y; O+h+1,m+h+1)+ = any Ga(y; O+A4+1, mth+)). m=v+l D m=0 On substituting the result (6c) into the first series in this equation, we get at once when largy|0 that y any’ Ge (y; O+h4+1, m+h+1) m=v71 Kier] 3 fant h+I| agen) & |Mn(m+h+l) [yA 8 | marti | Um +h+ 2) mari] Pim +h +2) S Om (m +h +1)FOrh+) DP (m +h +2) {log (m+h+1)p-*8 Ste | 2/72) |S) m=vt1 where WV is the integer such that V V +1 Remembering that, for the values of m under consideration, a,,, qua function of m, is such that a,,= O(T' (m) {log (m+ 1)}? m*), we see withcut difficulty that Cn (m +h + Ly / {log m+} P(m+h+2) ae eee Om (m+h+1)FOthH) aa 7 : * and 1f (m+ h+ 2) {log (m+h+ 1)p- Rp) Now h+1—N—yp>0 since R(O+h+1)>WN, and R(8)+ B<0; and hence the series = S ee aly S a are absolutely convergent. Hence it is evident that a nf Ge(y; O+h41, m+h+1))| < ee rte +E | yh | 4 | yht Consequently, when | arg y|<7, Fy)=" ole, a — sally + Oly Mo) ] + OC y'), , 2—R(B)+h. where X is the ee of the numbers h+1, 1— R(@) This result may be written in the form vy [¥H=h-1 8, g ( =8) 5 o(\y ea *)]+00y a), mG) (Y= | — y" * u=-F Oh 4—2 26 Mr WATSON, ON A CLASS OF INTEGRAL FUNCTIONS and v may be chosen in such a way that V +/—h—1 is any fixed positive integer; for v is any fixed integer and / is subject only to the condition that » > V +. Writing v in place of V+ /—h —1, we see that this result may be written in the form: ev |2 S,-P(1— 8) F(y)= = (y) af rn=0 y" +o yh)] Oy when |argy| <7. This is, effectively, the result stated at the beginning of this section. 7. If we substitute the asymptotic formula of § 6 for F’(y) in the equation (8) we see that if one or more of the numbers larg 2"" 4 ang {exp(2mip/a)}|_ (w=—p, 1p, 2—p,-- +) is less than $7, and if none are greater than $37, the asymptotic expansion of f(z) is given by the formula Pp. ex gle eglat2ripja v Sn . ip if = B x f@) Tae f : x 1 (ate gaara 18 o(ja)h | sees (11), (alla egla+2minja yp where v is any fixed positive integer. 8. In order to appreciate clearly the range of validity of the asymptotic expansion (11), we have to consider the inequality (7a), viz. | {sin y log | z| — cos y arg z}|< 42d +7 cos y—7|(qcosy—A)), in some detail. (A) In the first instance let us suppose that A <2 cosy; this inequality may also be written |a—1|<1. When 2 lies in the portion of the plane* defined by the inequalities —araAn0 and the inequality (7a) is satisfied if q=1. In other words, if |a—1|<1 and |sin ylog|z|—cosyargz|< 4A, then f(x) possesses the asymptotic expansion exp (a*e9*) ( » S,.T(1—8) rae . Q= sia 13 a “+ o(|e a Jatefis’ oh deter eeae (13). (B) If, however, « does not lie in the region of the plane which we have been consider- ing in (A), we may choose arg (— z) so that —(m cos y — $7rA) < sin y log | z| — cos y arg (— z) << reosy—47rA ......... (14). And, when z is such that (14) is satisfied, we may shew that #(a)— ZL p(n) emma” 1 7. p(s) (—2)° —s Tr (an Tu 1) = onan 5 sin (7s) P (as ri 1) GUS” eww ns.choetus'auislecwas ciety If the subject of integration be a uniform function of s and if it tends to zero as s > x» when —h, < R (as) 2 cosy, ie. |a—1|>1. Let us choose g to be such an odd integer that AK —ICOS|¢y)-<19): COS |) PAW I COSIry sna norjas cia a\eciae cae sclussneeelalet (16). Whatever be the position of « in the plane, we can always choose argz so that the inequalities —47A 0, |arg (ate9!")| <4; and then (7q@) is satisfied since cos y — | (gq cos y — A)| >0. Further, we have to ensure that the arguments of all the expressions 2!/*e9/*¢e?""#/« lie between +37. But this is certainly the case, for | arg (al/« eagle e2riule) | < dor + arg (e?rP/s) <4ar+ pA cosy <4$7{1+(q—1)A~™ cosy} <7. Thus, when |a—1/|>1, the asymptotic expansion of f(z), valid over the whole plane, is given by exp (aVeegiet2miviay ( » S,.0(1— 8) aa ot] — a: - eee - ST (@) a jeep | 8/2 egBla+2mipB/a Wee (allo egla+2miuja)n 28 Mr WATSON, ON A CLASS OF INTEGRAL FUNCTIONS where arg x is determined by the inequality [sin y {log |#| + R(g)} — cos y {arg « + I'(g)}]| < 377A, q¢=2p+1, ¢ is defined by (16) and the coefficients S, are those defined in connection with equation (10). Since the first 8’ — @ of the coefficients a, a,, ... vanish, so also do the first 8’ — 8 of the coefficients S,, S,, ...; it will, therefore, be possible to modify the above expansion slightly when §’— 8 is a positive integer and not zero; we shall not carry out this modification, as a more convenient form of the expansion will be obtained in Part IV. Part II]. Asymptotic formulae required when M <1. 9. The fundamental function to be investigated in place of &,(%), when J/<1, is the integral function «2 a yal {M (n+e)+1} {iM (n+e)+2}... {Ml (n-+e)+h} The asymptotic formulae are required for this function when ¢ is an assigned positive number, & is a (large) integer and z has any value. E,. (2; M, c)= a” T (Mn+ Me+1)T (hk) It is evident that Ey (x; M, e)=" ae neon! T(din+Me+k+]) * Now = eM (»+¢)(1—¢)§-1 is uniformly convergent and each term of the series is continuous (when n=0 . k>1) for the range 0<¢<1 for all values of ~; consequently* we may integrate term-by-term and we get 1 We (ee May FUE i (1—2)F-1 2 exp (2t®) dt. If R(x) >0, | exp (wt) | <|expx| on the path of integration. Hence, when & (x) >0, ju | By (x; VU, e) 0, (1 ae (2). (!—1)}, qua function of ¢, has one (and only one) maximum when O0/. Consequently if R(x)2>/ and £>/+1 we have =a\q (1=eFexp (2 (x). (1-1) < (Fe ) when 0/+1 and R(x) >/, then ee K T(k-l) 1 (Me+k+1) T(Me+1) eS cs 1, E, (x; M, )=- 275 |. ee ) §— (k=1) (1-2-2 eMe- +14 (Me — M41) (1 —2)F-1 ea a, Since | exp (z#)| <1 when R (x) <0, we see that |B (@3 Ih 0)| < are vare!. {(k—1) (1 —t)F-2 eMe- +14 (Me — M41). (1— 0-1 2e-IN t or (di c— M+2) ~M\a2\T (Me-M+k+1)° The reader will easily see, on making the necessary modifications in the work; that this is true when /=1 Hence when (x) <0, we have, in addition to (17 se: 2 (Me— M+2) ae | Ey (x; H, c)\< <7 zr (Me — TEE) BEEN ) Pon CC Ee (17 d). The results (17) are all true when M=1. 10. We now have to discuss the asymptotic behaviour of the integral function a” « Ce = = Pai (n+ OF {M(n+c)+1} {M(n+c)+2}...{M(n+e)+k}’ when || is large, k is a (large) integer, 0< R(@)< 1, R(8)< 0, c=R(O) and Ml As in § 4, we may shew that —BT(]— 98 B-1 ay\ 9-1 Ga(a; 6, k; M, jee |- log (1 = 2) | ( — 1)" By (ey; M, 0) dy, round the contour marked with double arrows in Fig. 1; and, as in § 4, we may deform the contour into that marked with single arrows in Fig. 1, so that, preserving the notation of § 4 we have Ga(z; 0,k; M,c)= =) 5 log (1 Be: ale (1 a vy PP for 1 i B-1 ~ «T(8)J pq - log (1-4) ( =) E,(x—y; M, c).dy = 1 n 8-1 n e-1 q z1'(8)! op |- log (-2)| (-2) E,(—7; M, ¢).dy ...... (18) Sif, tid heh 30 Mr WATSON, ON A CLASS OF INTEGRAL FUNCTIONS \ |B8-1 6-1 On PQ, as in § 4, |- log (a - 2) | ( = 2) E,(a—y; M,c) |. Ey. {—; M, c) |. x) dr, where p< 6kexp|xz|; whence we deduce that re dr o {| logr|+)a|p-# |\J,)< K . Ey (— 7; MM, c)|. We now divide the range of integration into three parts, viz. QG) from r=0 to r=k, where ky =e, (i) from r= to r=k™% (mm) from 7 —™ to 7 =p); we observe that: on (1), {\logr|+/o/}F® < K, on (il), . {| log 7| +|o|}#8)— < K {log k,}#6)—, on (111), {\log 7|+|o!}*@— < K flog k}®#)> ; also in (i) and (ii) we use the inequality |£,.(—)| < K {Ul (Me+k+1)}~ derived from (176); and in (iii) we use the inequality | ;.(—)|< K {\n\ (Me — MW +hk+1)}~ derived from (17d); and we deduce that a ay I ( Mc+k+ 1) / J3|, RB) Mf (c—1) »)| Rig)—1 ok) EE flog eA, _ Wiel) PY (Mc+k+1) «@ V(Me—-M+k+1) since 0< ¢<1; on substituting for /, and making use of Stirling’s formula, we may write this result in the form 1G ’ Obs 1 (k +1) {log k}-*®) a (196). Next we have to estimate the value of the integral ih / 68-1 e— | |- log ( 1- | (1 - ¥) Ey (a@—y; M, ec) dy. JC \ @ L DEFINED BY TAYLOR'S SERIES. 31 We divide the path of integration into two portions, the first portion being such that on it |y|<2 ||, the second being such that on it |y|>2|«|; we call the integrals, along these portions, J,’ and J,” respectively. As in investigating J,’ in § 4, we may shew that (deill<< a | { Ey, (cc NUM C)\LY |e oateecrscescaeere ee sesceee (19¢). We now have two cases to consider, according as R(x) <0 or R(x) >0. (A) If R(x) <0 then R(x —y) <1 throughout the contour and the length of the contour is less than &|x); so that by substituting for #,(#—y; M, c) from (17a) and (175), we get naa K\a| 11S (38) TP (Me +k +1)’ A K ie: | J, |< jae | T(Me+k+1) Bolstelpfarreimuleinielets cieisetelteialotere iets (19d), when R(x) <0. (B) If R(x) >0, let arg x= where |Q|<47. If O is not actually equal to +47, we can divide the path of integration into two parts, on one of which R(#—y)>4R(z), and on the other R(a—y)<4R(«); and the length of each of these parts is less than K | «|. On the first part we have from (17c), if k>/+1, Bee Me) < — eo Le., since R(w#—y)>4R(2), | Bi, (a@—y; M,c)\< On the second part, by (17a), K (sec 0)! | e*|.| e| |ja!/V(Mce+k+1) ’ and |e¥|0 and k>1 +1, see K (sec 0)! | e* | K |é| Oh SE ee jaP =) PD (Me+k+1) |a®@|D(Me+k+1) A modified form of this result is desirable when © is nearly equal to +47. To obtain it, we notice that, on the path of integration, R(y)>—1, and hence by (17a) and (176), when R(x) > 0, PEs. P(lc+h+1)° From this result we derive. by (19c), the following inequality when R(x) >0: K\é@| 0 (eee (19/). Lastly, we have to estimate the value of the integral -r \ 76-1 OS n= |[-toe (1=Z)]" (1S) Bae 95 Moy, iWoOnsexexehI= No, UE: | Ei, (a@—y; M, c)| < | Jr |< or 32 Mr WATSON, ON A CLASS OF INTEGRAL FUNCTIONS taken along that portion of the contour C for which | y| >2 |; on this portion of the contour the formulae (176) and (17d) are applicable. By putting 1 — y/x = re™ (r real) and dividing the path of integration into three portions, viz. (i) from the point where |y|=2|a| to the point where 1 —y/x| =k; (ii) from this point to the point where 1—y/x|=k™; (iii) from this point to the point where y= OQ, and on making use of (176) in (i) and (ii) and of (17d) in (iii) we find, in just the same way as that in which we obtained (5d), that yey SE) Ds bw Oh (199) /+ 1, and let |w|> 2, argv =. Then 7 | 1-8 |Ge(a; 0,k; M,c)\< KU K\a K 1 |T(Mc+k+1) o I (Mc+k+1) i TP (k +1) {log hk} -®®) --.(20), where U is defined by the following inequalities : (a)) “Gwhenel (ZO! Oia... Sapeereeete es cents Aenea ne meee een ce eee nee eee (20a), (i) when R(x) >0, both the inequalities, U< Saad oy betes phat at (oe (20b) OT | yen co atavt adel. camonsccind is cosh Svante eee (20c), are true. Part IV. The asymptotic expansion of f(x) when M <1. 11. The analysis of § 5 down to equation (8) still holds when M <1, so that, as before, if ry =e%r, a=AeyY, gq=2y+1, and {sin y log | z| — cosy arg z}|<4a2A +7cosy—m/|(qceosy—A)|, bar Pp. h pee Me SF (we) exp (2rip/ then = TCO Sn me F ((we?)"* exp (2 7iw/a)), where I,=O(a") and Ly' and | arg (s/a)| <\ + 47, P (sia) is analytic and # (s/a) = exp (—8' logs). | a’ + + 4g a" + Ba . where Qn’ |< A,p".n!, | Ras" |< Aoo,”. n!. Let h be an integer greater than y’ and let @ be such that h<—R(@)h, te i eae ae 8 aa = i= = qj, ea, oS alee s—h [mr e+ + nt Rn where | @,|< Asp”. 7!, | Ry s"* |< Ayo,".n!, and A;, A, are independent of n. (This follows, without difficulty, from Stirling’s formula.) Now if we have a finite number of asymptotic expansions with the same ‘characteristics, their product may be represented by an asymptotic expansion with the same characteristics *, that is to say that, when |s|>h,, |args| 0, (s/a) i 21 b, b, 1 eS 6=) 1G.oy) |" MG=.) Ll ME=p ew ela +a =| 2) where | },|< H’ T'(k) {log (k+ 1)}", (k>1, H” independent of /), and R(b)<0, while V is the integer such that 0>R(b+V)>—1, phw=—R(O). Using the notation of § 7, we put h+1+0=0, R(h+1+®0)=c, so that 0< c< 1. It may be noted that the first b-.8’—h-—1 of the coefficients b), b,, bn, ... vanish; this is evident when we consider the effect of making s— 2x in (21). From (9) and (21) it follows that — arti S y” a. ie Efi*, b, b, B= 2 nl (at OP Eeerc +o)+1 (M(ntc)+ 1} Mmteye a} h+1 : . : 2 R+1 < Bin y” OS aa where | | Dias Der |Baol=|1 09 =a 7 + Good | } {M(m+e)+1}...jM(n+e)+k+1} {M(nt+ce)4+1}...{M(n+c)+k+2} | bea | PY (Me + 1) fa Desa | T (Me + 1) ST (Me+k+2) * F(Mc+k+3) Qua function of k and m, berm is O (log (k +m +1)}" (k+m+1)-“), so that 2 flog (k + m+ 1)}” By is O( 3 ), tel By Eas mai (k+m+1)" Lee S10. ( {log (A: + 1)" (e+ 1) 7}? by Cauchy’s condensation formula, since Mc > 0 and {log (k + m+ 1)}" (k+m+1)-%> diminishes as m increases when k+m-+1> exp |V(Mc+1)"!. * See Phil. Trans. Roy. Soc., vol. ccxt. A. (1911), + This is legitimate since the former function satisfies pp. 279—313, § 4. the requisite conditions. 59 a 34 Mr WATSON, ON A CLASS OF INTEGRAL FUNCTIONS : ‘ log (kK +1)F Since Brn is O (ee) it follows that, for any fixed value of y (i.e. any value not depending on &), an < By, e y” ons eynt° Sg uhde that is to say, F(y)=y" S Om Gy (y; 8,m; M, c). m=0 Taking v to be any fixed integer, we have the equation = bm Gs (y; 8, m; M,c). 1 F(y)=y"" 5S Din Ge (y; 0, m; M, c) + yO m=0 m=v Now, by the theory of partial fractions, we may write : = == < Ps, m {M (n+) + 1}{M (mw +c) + 2} Be {M(n+ce)+m} es M(n+c)+s’ and also — A ae! 1, 1 1 1—c—sM M(n+c)+s M\n+1 ore —c—sM~)(2—¢c—sM) \ (n+1)(n+2) (n+1)(n + 2)(n +3) eh a this series converging absolutely when n +1 >1—c¢—sM~; this condition is certainly satisfied when n > 0. That is to say, when n>0, 0< mv (v fixed), we may expand 1 {M(n+c)+1}{M(n+c)+2}...{M(n +c) +m} into the absolutely convergent series d, »m (les n+1 (w+1)(n+2) °°” where dy,m, qua function of k, is O([' (k—e— M™)); we may now justify the rearrangement of the series S binG (y; ,m; M,c), m=0 in the form by Gi (y, 0) + (Did, + ba di,y +... +bydh,») Go(y, 8, L) + (Db, doy t+ bodoo+.. .+b, as +) G (y, @, 2) ae as in the somewhat similar work of § 5, so that we may write sees >> bin Gy (y; @, my, M,c) = > din Gy (y; 8, m)y+ dim Gy (Y; é, m), m=0 m=0 m=v+1 where d’,, qua function of m, is O(T(m—e— M-)), Consequently F(y)=y'* y dn Gy(y; O,m)y+ yh {d’, Gy(y; 0, m) + bin Ge (y; 0, m; M,c)}. m=0 m=v+1 of n, is [Since the coefficient of y"'*” in the general term of the second summation, gua function 1 \ 0 | et : (ce acer : DEFINED BY TAYLOR’S SERIES. 334) it is not difficult to see that the coefficients d,, are those which occur* in the formal expansion : g (“TE - Ge j d, ds ) a ni(n + 0)? n+1 FERC) Wea even though this expansion does not converge for any value of n.] From the equation F(y)= Pe S am Gs (y; 9, m) + 2 Sy yl {din Ge (y; 8,m) +b Gy (y; 8, m:; M, c)}...(22), we may deduce the mei expansion of F'(y); ; for we have, by (6c), S LnmT 1— elie OOM) = pia I 3, “7 ” +0( yy] +0¢ y |S + |g), where ees — m—n a= =o i n)’ and the coefficients ¢,,,, are defined by the expansion [log (1 os Dir el as gq) Pes = ( -—yy > Cn x | yy” Also from (6a) and (20) we have SE {d’nGs(y; 0, m) + dm Gy (y; 8, m; M, 0)}| m=v+l1 os id j K |e ge Up K.m ) ee ee tle F Get nye T(m#1)* T(m +1) {log mp-* J an i i= =) r APRS roma zs KU es K\y neds = Ve ee ye | DP (Me+m+1)* P(Me+m re De T'(m + 1)(logmy*®| Since re ree it follows that 7 din | i a = => O(m*) 0, m= 2 22) m=v+1 ( ) ie = | dm | me a 2 1 m=v+1 D(m +1) {log mp-#O Ee Rear sii ro ea {log mp =F < K, since R(b) <0. Also since },, = O {['(m) (log m)"}, it follows that ~o | b 2 ; [=m = > —i—Me { Re: Renal) co a O(m {log m}") < K, since Mc >0 and S oe [Onl aS mav+1 [' (m +1) {log mp-# a 3 yl (m— {log m}"*¥—) < K since V+ R(b) <0 Consequently, if l0, U, <(secargy)|y|—| ev +| 2! and U,<|e < * The coefficients can be obtained successively by a limiting process 36 Mr WATSON, ON A CLASS OF INTEGRAL FUNCTIONS That is to say, if |argy|<$7—A, A>0, > {dm G;, (y; 0, m) + Din G, (y3 0, m; M, c)} | = 0( ee aa ) ceeseeeee (23 a), m=v+1 while if t>argy>47—A, Stn Go(y; 8m) + bn Gr (ys 8, m; M,0)} |= O(\e y#|) + O(y2) «.-.-.(28D), | m=vt1 : where J is determinate. We deduce that, if | arg y¥|/<4a7—A, Fy) possesses the asymptotic expansion Set RE) 3) has ; PY) = a 3," +0(\y| *| Te eae fe a (24), while if w>argy >47—A, By) = Oe) Oeics 2s s.ntache nw eeeenatenseaee (24 a), where / is any fixed integer and J is determinate. 12. From equation (8), quoted at the beginning of § 11, we can now see at once that if any one or more of the expressions («e?)"* exp (27it/a), where t=—p, 1—p,...p, has its argument <}7—A, the asymptotic expansion of f(z) is given by the formula: : 2 | exp (areesetite) » T,T (1-6) i) ee Ne rere > Se [Sie Dh [apeccaococ : t (2) toep L(a* egiatanitiayb—h-a i (a eyatemitiayn +0(|2 ) (25), The choice of the numbers p and g with the corresponding determination of the value of arg # is now made in precisely the same manner as in § 8, subsections (A)...(D). Remembering that the first 1+h+’-b of the coefficients )), b,,.... vanish, we see that the first 1+h+'—b of the coefficients 7,, 7,,... vanish, and so it is convenient to take yp >1+h+ 8 —-b. 13. It is easy to see that the nth of the coefficients 7), 7,,... (commencing with the first which does not vanish) is a linear combination of a)’, a,”,..., @’n-, Where ” ’ Pe ¢$ (s/a) =exp (— 8’ logs) & + : + uh and the only other arbitrary elements in 7, are 8’ and 6; and it is evident that the expressions for 7’, will be unaltered in form it we supposed that the development were not asymptotic but convergent. That is to say, we may obtain the asymptotic expansion of f(x) when ¢ (8) satisfies the conditions specified in § 2, by treating the development for (s/a), viz. : (re Joa exp(— 8 log 8) jo $245 tea, as if it were not asymptotic but convergent; and, in particular, by taking a=1, if the function x (n+ 0) of § 1 be not analytic in the vicinity of n=, but possess an asymptotic development, with grades equal to unity, valid for the range jargn| <}mr+X,(A>0), Barnes’ develop- ment* is still valid. The asymptotic development which we have obtained seems to be an interesting example of the fact that in certain cireumstances it is permissible to treat a subsidiary asymptotic development as if it were a convergent series. * See § 1. DEFINED BY TAYLOR'S SERIES. 37 14. The theorem which we have proved may be enunciated formally as follows *: Let f(x) be an integral function defined by Taylor’s series, F(@) = +60 + ,27+ ..., and let it be possible to define a function ¢(s) such that when s is an integer _ b(n) Boe T(an+1)’ (\arga|y’, | args|<$a+X, (A>0), $ (s/a) is analytic+, and possesses the asymptotic expansion : (s/a) = exp (— 8’ log s) eee + +S +R] > where ja,” |< A,p”.n!, | Ry s"**| +’ and let h be the integer such that h<—R(@) ——————— ————____ 4 “ (z) r F = (ae eglatent it/a.) B” tek (ave eviatanitia\n 0 ( | 2 | Df ) where y is any fixed integer and the coefficients U, are given by the series U, aes < Cn—m n n= 0 i: ( 1 = b —m) m,n—im> and the coefficients d»,, are given by the expansion [log qd De yl (1 = ca = (= De > en (- yy”, m=) so that the coefficients U, can be calculated with sutficient labour, should they be required. 15. It is not difficult to shew that in the case of the generalised hypergeometric series pla (a1; G23 --- Gp} Piy P2s---- Pas Z) (¢+1> p), the coefficients satisfy the conditions laid down for the functions discussed in this memoir. We thus obtain an independent proof of the results contained in Barnes’ memoir on the generalised hypergeometric function f. Another class of functions which satisfy the conditions of this memoir is obtained by replacing the gamma functions involved in the generalised hypergeometric series by reciprocals of G-functions ; it will be remembered§ that the G-function is an integral function satisfying the difference-equation G (n+1)=L (nm) G(n); and it is not difficult to construct other functions whose asymptotic expansions are given by the result stated in § 14. * We have changed the notation slightly, in such a ~ Proc. Lond. Math. Soc., ser. 2, vol. v. (1906), pp. way that the first coefficient in the asymptotic expansion of 59—118. J (x) does not vanish. § See Barnes, Quart. Journ. Math. yol. xxxr. (1899), + c, is supposed to be finite when x is such that ¢(s) is pp. 264—314. not analytic near s=n. foe ‘ywrwe fn 4 ia aaa nb ” sed ' “so : > te Ae pe - . Poul ee, at. Weg Ul. The Hydrodynamical Theory of Lubrication with Special Reference to Air as a Lubricant. By W. J. Harrison, M.A., Fellow of Clare College, Cambridge : Lecturer in Mathematics, Liverpool University. [Received 11 May 1913. Read 19 May 1913.] THE theory of the lubrication of surfaces moving relatively to one another and separated by a thin film of oil or other lubricant is one of considerable practical interest. The cognate problems are essentially hydrodynamical in their character, and have this interest, that they are among the few problems in the motion of viscous fluids which can be solved approximately for the case of large velocities. The theoretical work of Osborne Reynolds* and of Pétroff+ is of extreme complexity, partly because Reynolds considered the case of an incomplete cylindrical bearing, and Pétroff introduced further complications into its form. It must be admitted that the forms of the bearing considered by these investigators are those which occur most frequently in practice, but their analysis is so complex and methods of approxi- mation so laborious, that even the mathematician may fail to grasp the essential character of the results obtained, or to be expected. This fact alone is a valid reason for treating a simpler form of bearing. All results become simple, and the theory of lubrication can be elegantly illustrated by considering the case of a complete cylindrical bearing. It was not till after I had completed my investigations in this case that I came across the very elaborate treatment by A. Sommerfeldt of the same problem. Our resulting formulae are identical. But the present treatment of the problem being somewhat different from Sommerfeld’s, shorter and in one or two points more direct, will perhaps appeal more directly to experimenters. A question is raised in the course of this paper as to the validity of experiments which have hitherto been made to determine the moment exerted by the traction of the lubricant on the journal. This point is of importance, as by means of this moment the nominal coefficient of friction of the journal is obtained. In the latter part of this paper the method is extended to take account of lubrication by means of an elastic viscous fluid such as air. It was stated as long ago as 1885 by Hirn§, that under suitable circumstances air is the most perfect lubricant. In 1897 a series of very beautiful experiments was carried out by Prof. Kingsbury|| on the lubrication of a cylindrical journal by air. The results he obtained, which are apparently accurate to a fair * Phil. Trans., 1886. § Engineering, Jan. 30, 1885, p. 118. + Mémoires de VAcad. de St Pétersbourg, vur Sér. || Jour. Amer. Soc. Naval Engineers, Vol. rx, 1897, Classe Phys.-Math., Vol. x, 1900. p. 267. + Zeits. fiir Mathematik, Leipzig, 50, pp. 97—155, 1904. Vou. XXII. No. Il. 6 40 Mr HARRISON, THE HYDRODYNAMICAL THEORY OF LUBRICATION degree, exhibit in certain details wide variation from those to be looked for in the case of lubrication by an incompressible liquid. I have not succeeded in obtaining an explicit solution of the differential equation determining the pressure in the film of air, but I have integrated it numerically by Runge’s method, using the data of Kingsbury’s experiments. The degree of approximation of theory to experiment is quite satisfactory. I have integrated the differential equation in the case of plane surfaces, and give some results below which exhibit more clearly the very marked effects of the compressibility of the air on the magnitude and distribution of the pressure. But apart from the new results obtained, this paper will serve the useful purpose of recalling attention to Sommerfeld’s work. A subsequent paper by A. G. B. Michell* is also worth attention and will be referred to below. It might be in place to remark here that I have obtained some results and have work in hand treating of cases in which the influence of variable speed and variable load on the lubrication of a cylindrical bearing is taken into account. Case of Incompressible Liquid. In proceeding to determine the equations which give the motion and the pressure of a film of liquid separating two surfaces moving relatively to one another, it is to be observed that the inertia terms can be neglected as well as the effect of gravity, since forces depending on these terms are negligible compared with the internal stresses arising from the rapid shearing of the liquid. Again, on account of the thinness of the film its curvature can be neglected, and therefore the same equations hold whether the surfaces are plane or cylindrical. Sommerfeld has transformed the equation V‘y=0, which is satisfied by the stream function ¥, from Cartesian coordinates («, y) to polar coordinates (r, @). He proceeds to use essentially the same method of approximation as employed by Osborne Reynolds. The only result of this transformation is to introduce relatively unimportant terms, as will be seen below. The coordinate # will be measured along the moving surface in the direction of motion, the coordinate y normal to this surface. The motion is steady and will be assumed to be two-dimensional. If w, v be the component velocities at any point in the liquid, p the pressure, the equations of motion are . a) Ss SVB acs Si ced, Ra ee (1), Op X oa HON AU ss waicergeiein s 54 So acs vans > GRRE aE (©) where yw is the coefficient of viscosity, and the equation of continuity is Ou Ov __ ant a URE acer So istnccttie. opcccl celinaneenoenuae (3). * Zeitschrift fiir Mathematik, Leipzig, 52, 1905, pp. 1283—137. WITH SPECIAL REFERENCE TO AIR AS A LUBRICANT. 41 The boundary conditions are u=U, v=0, when y=0 u=0, v=0, when y=h where hf is the variable distance between the surfaces and is a function of 2. Since the surfaces are nearly parallel v will be small compared with wu, and the rate of variation of w in the direction of « will be very small compared with its rate of variation in the direction of y. Accordingly equations (1) and (2) become op Cu = ay? mterefalatavaroisisiate afeleisielerelers)e cioletail-ia]aictnieiaieteiee nierietets (5), Op gy TO cercctttrrntesreseeeeeeeeeeectenscccetesnnsensee (6) From (6) it is seen that p is independent of y, and (5) is then integrable, giving Lop , h— = 5 and Y—Wt U ie y in which the constants of integration have been adjusted to suit the boundary conditions (4). : hou r qh Now from (8) — dy =— I =(0) U 0 Ox 0 8 (,.9P\ _ 6,4 oF | Hence 4 an (” 2) = 6uU ap? which gives he = = 6uU (h—h,), where h, is the value of h at a place of stationary pressure. Fig. i. In the case of a cylindrical journal, let O be its centre, a its radius; O’ the centre of the outer bearing, a+ 7 its radius. Let O00’=cn, 0 Bes Tee ee AS (1—¢)(1 +c0086)+4(1 +¢00s 6,)(1 —¢ +3[1 + ceos 6) +(1-2)"# 2 -2)—(1 + ccs &) +e) tax! ips 5t|- Now p must be a single-valued function of 6, hence 3¢+-(2- 2) ens 6,0 22255 ..-... nce (7). This equation determines the positions of the max.-min. values of the pressure. The remaining constant can be determined if the pressure is known at any one point. It is to be noticed that equation (7) does not restrict the value of c except fo the range —1 The component of force exerted on the journal by the traction 7 (measured in a direction opposite to that of the rotation) is S acting along 00’. 5 ou N y =— " f=-0G),., —|#" 34,0? k tat | 9 _4U + 6(1-¢) 7 |1+ccos ~ (2+ )(1 + cos OF | Hence S=|"fsin 6adé = 4ruUa [2 fs = 6 ne (2+e)a—e)% WITH SPECIAL REFERENCE TO AIR AS A LUBRICANT. 45 The couple exerted on the journal is dru Ua? (1 + 2c*) Mu =| fade = ee 1 a+ey—eF being measured in a direction opposite to that of the rotation. Now if f’ be the tangential traction on the surface of the bearing, measured in the direction of rotation, =) al { el if P\ ay) oe U G) =« |Z: Hence on the outer surface the corresponding forces and couples are Ff’ (acting upwards through 0’) = R, ge feu E _2a- ag ne 2+¢ ,_ 4orpUa? (1 — 02 | i a ae a 7 (2+) In the first place it is to be noticed that S and S’ are not equal and opposite, but these components are of a smaller order than R and FR’, and will therefore be neglected; a closer approximation would establish their equality—in fact, in Sommerfeld’s work they are shown to vanish. The expressions for R and M given above are the same as those obtained by Sommerfeld. But the inequality of M and M’ is essential. Taking into account the fact that R acts at the point O, and R’ at the point 0’, it is easily verified that the force system (R, M) is balanced by the system (R’, M’). The ratio of WM to WM’ for different values of ¢ is as follows: c= 0 oF 4 6 3 M/M’= 1 1-03 151 2°69 138 Now this difference between M and M’ is of very considerable importance, and so far as I am aware has not been previously noticed. The following conventional terms are usually adopted: __load per unit length of bearing < diameter ; _ couple due to traction Nominal friction = ; : ‘ radius x diameter Nominal load E Lake nominal friction Coefficient of friction = ———_____ nominal load Hence the coefficient of friction X for the journal is given by o°8 _ 1 “ke” “aaee for the bearing N= a — — ) 2/2n 3a When X has its minimum value, X= 4X’. X has a minimum value when c=1//2; XN’ has no minimum value except zero. 44 Mr HARRISON, THE HYDRODYNAMICAL THEORY OF LUBRICATION It is immediately obvious that, if the coefficient of friction be determined by experiment, it is important to know what is the couple which is measured. It may be safely assumed that in experiments hitherto conducted the object has been to measure the couple exerted on the bearing, the direct measurement of the couple exerted on the journal being out of the question. But if this is so the actual coefficient of friction for a given journal is greater than that derived from experiments made upon it, unless the speed be sufficiently great to make ¢ comparatively small. It is important to observe that there are in reality three force systems under consideration which are equivalent but not identical. The force R and couple J exerted on the journal by the liquid, the force R’ and couple M’ exerted by the liquid on the bearing, the force R” and couple MW” applied to the bearing to keep it in position. Now R=R’=R’, in magnitude, but M and M’ are always unequal since the lines of action of R and R' are not the same, and the magnitude of WM’ depends on the line of action of R”. It is necessary that M” should be equal to M, if the correct coefficient of friction is to be obtained by experiment, and accordingly the line of action of R” must be adjusted so as to pass through the centre O of the journal, which is a variable point depending on the load and velocity. Since the relative magnitudes of Ra and M are so disproportionate, the slightest error in this adjustment, even if the line of action of R” be only zg45 of an inch out, causes a considerable percentage error in the measured moment. In the case of incomplete cylindrical bearings, for which the are of contact may be 120° or less, M and M’ will be more nearly equal. But there is the same possibility that M’’ will be equal to neither the one nor the other. Suppose now the line of action of R” to act at a constant distance « from O’ towards O, i.e. we suppose the line of action of the applied force to be independent of the velocity of the journal and of the load applied to it. Then M’ = M” — R's, hence ” =X + a/a, where 2” is the apparent observed coefficient of friction («/a may be very small and yet comparable with 2’, as seen previously for «= cn = 00’). Coefficient of friction. Velocity of bearing. Fig. 2. > * WITH SPECIAL REFERENCE TO AIR AS A LUBRICANT. 45 The variations of X, X’, X” with the speed are exhibited in figure 2. The graphs are not drawn quite accurately to scale, but they represent closely the behaviour of the three coefficients. Curve 1 shows the variation of \ with its minimum value at ¢=1//2; curve 2 exhibits X’, and this curve displaced a constant distance along the ordinates into the position of curve 3 exhibits X”. The displacement of curve 2 to produce curve 3 may be either upwards or downwards. Cc Fig. 3. In figure 3 a typical curve is drawn showing the variation of the coefficient of friction, as observed, with the speed. It would appear that the part of the curve AB represents the effect of imperfect lubrication owing to insufficient speed. The remainder of the curve BC might very well be a part of any one of the curves shown in figure 2, or roughly approximate to one of them. But there is a good reason why it cannot be associated in general with eurve 1, which exhibits the true coefficient of friction. For it is a fact worthy of attention that the observed minimum value of the coefticient of friction may be less than the minimum calculated value of X; this is so in the case of Kingsbury’s data. It is, moreover, clear that he measured the value of M’, and not that of M. He measured the couple exerted on the journal, while it was kept at rest and the outer bearing rotated, by the torsion of a wire fastened to a point on its axis. It may be considered as moderately certain that experiments hitherto conducted cannot be relied on to give more than a rough estimate of the true coefficient of friction at fairly low speeds. It is hoped that some attention may be given to the points here raised in future experiments. Sufficient is now known about the theory of lubrication for all practical purposes, so that such refinements as are here proposed have for their object merely the possibility of obtaining data on which to base a closer comparison between theory and experiment. If it is affirmed that the comparison which is presented in Osborne Reynolds’ paper is sufficiently close, two objections to that comparison may be raised. (a) Pétroff has pointed out what seems to be a serious mistake in sign, which is carried on in subsequent operations. (6) Osborne Reynolds was compelled first to estimate the distance between the bearing surfaces by means of his theory and then make a final comparison between theory and experiment. 46 Mr HARRISON, THE HYDRODYNAMICAL THEORY OF LUBRICATION Such a course as that indicated in (0) is practically mevitable except in the case of a closed cylindrical bearing; in other cases the difficulty of obtaining by observation the relative positions of journal and bearing is great. Now this relative position of the two surfaces has been determined by Kingsbury in the case of an air-lubricated journal. Thus his data allow of a complete comparison. To a consideration of these experiments we now proceed. It may be added that Sommerfeld gives a very complete treatment of the lubrication of a cylindrical journal by an incompressible liquid, which is further illuminated by a number of curves showing the relations between the various quantities W/, R, »,c, U which enter into the theory. One such curve is reproduced in curve 1 of figure 2. Case of Elastic Fluid. Some of the most accurate experiments which have been made on the lubrication of a complete cylindrical journal were those conducted by Prof. Kingsbury, using air as a lubricant. He investigated the distribution of pressure round the journal and along its length, the point of closest approach and the magnitude of the shortest distance, and also the moment of the friction exerted on the bearing. But, regarded generally, the results show a marked divergence from the state of affairs indicated by the theory for an incompressible lubricant. This divergence is of course due to the elasticity of the air, and presents a somewhat interesting problem for investigation. The main points of this difference will be found exhibited in the table below, in which the air is treated as inelastic, having the density of atmospheric air, and average pressure equal to atmospheric pressure I. All quantities are expressed in foot, Ib, second units, The diameter of the journal was } ft., and the difference of the radii 2? x 10~ ft. U is the velocity at the surface of the journal. The pressure p is of course given in poundals per square foot. The first row of the double sets of data comprises those obtained by obser- vation, the second row those calculated from the theory already developed in this paper. Revs. per min. 230 805 1730 U 602 | 2107 45-29 : 39 1625 091 112 £3 £3 m o- (Pans. — Hl) 1 1-16 1-44 169 ‘ .35 | =~+«3 | 24s (Poin. — 11) 10 -1ié | —144 ~1469 ee = 197 131 1-36 (Poms. — Prmin.) VO 2-32 288 338 = _23° _ 43" 43° (®.xx — 180°) _57° _ 76 _33 51° 96" 129° (Grin, — 180") 37° 76" 82" a 73° 139° 172" ae 114° 152° 162° WITH SPECIAL REFERENCE TO AIR AS A LUBRICANT. 47 It is clearly seen that the facts which need explanation are these: (1) the range of the pressure decreases and seems to approach more or less to a definite limit as the velocity increases; (2) the position of the maximum pressure is displaced nearer to the point of closest approach ; (3) the position of the minimum pressure is displaced further from that point. Steady Flow of Air under Pressure. In the same way that the steady flow of liquid under pressure between two parallel planes is introductory to the problems of lubrication, so the flow of air in the same case serves as an introduction to our extended theory*. Consider the flow of air in two dimensions between the parallel planes y=+d. It will be assumed as an approximation that the equations of motion and continuity are op ou sip aye corte eeene nee encanee (8), ap ay SH VV necccsc crc ven renee esses ssrssssesscssceseseseses (9), SUT I aa pak mh (10). 0x The conditions are such that the relation between the pressure and the density is p=kp. Hence from (10) pw or pu is a function of y only. Also from (9) p is independent of y, and therefore from (8) taking into account the condition ~w=0 at y= +d. It follows that pe is a function of y only; accordingly put p*?=asa +b. Let p, be the pressure at «=O, p, the pressure at w=/. Then Pp? = po° — (Po — pr’) #/L. Thus the velocity at any point is given by (po? = pr) (B= y’) yl {ps'—(pat — pr?) /} For a circular tube of radius a and length /, it is easily shown that? uu = pe eB =p) (Oe) Spl pu’ — ( po’ — pr") 2/l5* r being the distance from the axis, z the distance along the tube. * In addition it has some bearing on the researches of Searle by a different method, Proc. Camb. Phil. Soc., Prof. A. H. Gibson on the flow of air through pipes. The Vol. xvn, Pt mu, 1913, formula obtained below has also been arrived at by G. F. C. + Cf. Lamb, Hydrodynamics, §§ 338, 339. Wor sexchi No: TT: ~I 48 Mr HARRISON, THE HYDRODYNAMICAL THEORY OF LUBRICATION The mass of air crossing any section of the tube per unit time is fe Pe | (a? — 7) 2ar dr / J aoe } 0 | Po — pr ll ra pu 2arr dr “- 0 _ mas (po = pi°) 16pnkl _ TA (Po— Pi) Po + Pr ad Sul rade This is the formula assumed by Prof. Gibson *. It remains to investigate the order of the terms neglected in the equations (8), (9), (10). Using the data of Gibson’s experiments for velocities below the critical value, I find the following approximations, taking 2 as unity: Ou Ou ou 10 pe pe 10 Coma). Came Pua It would appear therefore from the order of the quantities neglected that in all ordinary cases of stream-line flow in pipes the approximation assumed is quite accurate. In fact, long before the velocities are such as to cause the approximation to break down turbulent motion supervenes. In the same way the use of these approximations in the theory of lubrication could be justified. But it must be pointed out that, as a matter of fact, the approximations necessarily break down immediately in the neighbourhood of stationary pressure. But the effect of this failure is probably negligible, since it is confined to a very small part of the lubricant. Motion of a Film of Air between Two Surfaces, one of which is in Motion. It is necessary to solve equations (8), (9), (10), subject to boundary conditions (4). Equation (11), above, becomes 1 dp U = Wa de y(y—h)- h (y —h). yoh ra Hence | »| i Fe (pu) dy y ¢ JO 0 h® dp z as |? (“age ae + 374 | But v=0 at y=0, y=h, and accordingly UN 0... es eee (12), pw dx p where / is a constant. It may be as well to remark that for a gas mw is independent of the pressure. * Proc. Roy. Soc., Vol. uxxx, p. 114, 1908, WITH SPECIAL REFERENCE TO AIR AS A LUBRICANT. 49 Lubrication of a Cylindrical Journal by Air. Using the same notation as in the previous part of the paper, equation (12) becomes 4p pee |-meesayp| oe (18). dé 7?{1+e cos 6P n(1 +c cos 6) p This equation I have not been able to integrate in any form convenient for calculation. But equation (12) is easily integrable in the case of inclined plane surfaces. I have solved a number of cases of motion between such surfaces to illustrate the effect of elasticity. These results have an interest of their own, and I shall present them later. But since making these calculations, which were intended to explain roughly the discrepancies between theory and experiment in the case of Kingsbury’s data, I have integrated equation (13) numerically by Runge’s method of numerical integration, using the data of Kingsbury’s experiments. These numerical solutions will be found shown by curves 2 im figures 4, 5, 6. The ordinate represents pressure (6°83 x 104 is atmospheric pressure), the abscissa is 6, the angle POO’ in figure 1, Curves 3 give the distribution of pressure round the journal observed by Kings- bury; curves 1 give the pressure on the supposition that air is inelastic. It is seen that the extended theory goes far towards the explanation of the discrepancies referred to above. It will be noticed that in figures 4 and 6 the observed and calculated maxima and minima of the pressure are in good agreement. As regards the maximum pressure the agreement is not so good in figure 5. In connection with the differences which still remain it needs to be pointed out that the theory is based on the assumption of an infinitely long bearing. In particular the differences in position of max.-min, pressures in curves 2 and 3 are to be attributed partly to the finite length of the bearing. Michell, whose paper has been referred to above, has investigated the influence of finite length in the case of inclined plane surfaces. He states that he was unable to solve the same problem for a cylindrical journal. It might be as well to add a few words on the numerical solution of equation (13). It was found necessary to divide the range of 360° up into intervals of 10° or 20°, according to 7°83 230 REVS. PER MIN. 5-83 50 Mr HARRISON, THE HYDRODYNAMICAL THEORY OF LUBRICATION 7°83 805 REVS. PER MIN. 5°83 1730 REVS. PER MIN. 5+83 Fig. 6. the position in the range. Not only did the number of intervals make the calculation very laborious (a single complete solution involved the writing down of 12,000 digits), but it was further necessary to provisionally assume the values of two constants before an evaluation WITH SPECIAL REFERENCE TO AIR AS A LUBRICANT. 51 could be commenced. These two constants were practically the pressure and its gradient at the beginning of the first interval. The tests as to the correctness of these assumed values are (1) the pressure at @=0° must be equal to that at @=360°, (2) the total mass of air must be constant, or ie = 31) (cosa = 0: 0 The assumed values of the constants have to be varied until these conditions are satisfied. To give further details here would take up too much space, but it is believed that the curves obtained are quite accurate. Case of Inclined Planes. Consider two plane surfaces of which the upper is fixed and is of unit breadth, inclined so that their greatest distance apart is d,, and their least distance apart d,. In the first place let the motion of the lower plane be as indicated in figure 7. This case has been treated by Osborne Reynolds for incompressible liquid, and he virtually obtained the following results : Position of maximum pressure &, = dy/(d, + dy), Distance between planes at this point h, = 2d,d,/(d, + d,), he Bye U (d, —d,) 2dod, (dy + d,)” In the extension to the case of elastic fluid we write )=d,—- bz, where b=d,—d,, in Maximum pressure given by Pax, — I equation (12), which can then be written Put ph=w, and we have b —wdw eee ig mes —w—6Uw +k bE The form of the integral depends on the sign of pkb—9U?y?= K*, and for our present purpose this will be found to be positive within the required range of U. Hence $ log (bw*/w — 6Uw + k) + ue tan gs a = lee BS O ceodocdosdse (14). Now p=II for c<=0, h=d,; x=1, h=d,. Hence substituting and subtracting the two equations so obtained, we have an equation from which to determine k; C can then be determined. To find the value and position of oP 2; the maximum pressure, we notice that when oe aL CYT HID” scoucccacoonSndonsnbonosadsansbass00cq0dee (15). Hence the position x, of the maximum pressure is given by the equation d,—ba, _, 2(k/6U)? eee 02 Sapuisdain 60 Wea k k/ 6 ie if 1h —; [ 3 + eu tan a Ke us ae — tan? old, ga osdonas0c00" (16) K K K 52 Mr HARRISON, THE HYDRODYNAMICAL THEORY OF LUBRICATION Having found z, from this equation the value of the maximum pressure follows at once from equation (14), The following tables exhibit the results of calculations, for the purpose of which it was taken that I] =7 x 10+ (15 lbs. per sq. in. approximately), 4 = 10~. The second row of figures in each case comprises those obtained from Osborne Reynolds’ formulae. d, = 10-4, d,—4x 105 | U 5 | 10 20 a 80 1-58 2 (Prax: 10> ‘80 1-61 5:22," | 2 75 ral ‘81 “max, | ‘71 ‘71 “71 | U | a || 10 20 | 29 | 4 “80 i =a | 24 | = © (Pmax. Il) 10 +29 | 47 | “29 - 61 | Ga | 69 ae ‘57 ‘57 ‘D7 Thus the effect of the elasticity is to foree the position of maximum pressure nearer to the narrower end, and to decrease its value. id id, H : x-0 —U e=1 x=1 —u XL=0 Fig. 7. Fig. 8. We proceed to consider now the case of motion indicated by figure 8. Similar equations to (14) and (16) are obtained, except that the tan~ is replaced by a logarithmic form. The following are the results obtained by calculation. d,=10-4, d,=*4.x 10-4 U 5 10 20 “s viva ees 2-33 (Pin. — TH) 10 — 80 | —1°61 — 3-22 | | : 67 63 53 “min 71 Py fl 71 WITH SPECIAL REFERENCE TO AIR AS A LUBRICANT. 53 10 20 | | < — 22 — 37 — 75 | (Pmin. = Il) LOpe ay) | tr — +89 | 58 52 49 | Xin. 57 Dif yi : F be | | we The effect of elasticity is to displace the position of minimum pressure in the direction of motion and to decrease the total range of pressure. It is interesting to notice that, in accord- ance with expectation, although for an incompressible liquid with the reversed velocity the fall of the minimum pressure below atmospheric pressure is the same as the rise of the maximum pressure above, yet the effect of elasticity is to make the rise of the maximum pressure above atmospheric pressure greater than the fall of the mimimum pressure below it when the velocity is reversed. i As these cases of the lubrication of inclined plane surfaces were solved with a view to illustrating the differences between Kingsbury’s data and calculations based on the theory for an incompressible liquid, it was necessary to obtain some solution in which both a maximum and a minimum pressure should appear. The case indicated in figure 9 suggested itself as instituting a fairly close comparison, the dimensions being chosen so as to agree as closely as possible with the dimensions of the cylindrical bearing. In fact the solutions of this case agree very well with the observed facts, and although this comparison is no longer necessary, the following solutions throw further light on the behaviour of an elastic lubricant. Consider now the case of two inclined plane surfaces placed as in figure 9, with respect to a third moving plane surface. For an incompressible liquid the pressures at A, B, C are all A Cc Fig. 9. equal and the max.-min. pressures and their positions are the same as in the two separate cases, which have been already discussed. But this is not so if the fluid be elastic. In this case, for instance, the pressure at B may exceed the maximum pressure in AB in the absence of BC; and, also, there may be no position of minimum pressure, if the velocity be sufficiently great. The same integral forms are obtained for p between A and B, B and C as in the separate cases; but whereas the constant & had different values in these cases, it is easily seen (by reason of the continuity of velocity at B) that / must have the same value for a given velocity in both sections. This value will be intermediate between the two already obtained. Let @ be the pressure at B, then for the motion between A and B we have the conditions p=U,2=0; p=oa,c=1. Substituting these in equation (14) and subtracting, we obtain ee bid? /u —6UId, +k ° © ba*d?/u — 6Uad, +k Bu U E a bild, — 3uL fae bad, _ 3uU i K . ae — log (dy/dy) a Om SEGH): Kk : K 54. Mr HARRISON, THE HYDRODYNAMICAL THEORY OF LUBRICATION, ere. Treating the equation for p between B and C in the same way, we have bIEd?2/p + 6U Td, — hk _ bad, [n+ 6U ed, —b — 8 elt) 3BuU, (K’ + blld, + 320) (K’ — bard, — 3nU) 3K? 18 (KT bIld, — 3uU) (K’ + bad, + Bn) 3 log =e) a where K’= pkb + 9U*p*. The equations (17) and (18) determine the value of the constants kh, a. From the previous numerical calculation, if we take d,=10~, d;="4x 10, U=5, it is known that o=7 x 10!, k= 129-2 satisfy (17), and oa =7 x 104, k=111°3 satisfy (18). Now assume w = 7°5 x 10* (say), and find the values of & which satisfy (17) and (18), respectively. These two sets of values can be plotted, and the intersection of the lines joining them gives approximately the common solutions of (17) and (18), some slight adjustment being necessary. The following tables give the results of the calculations. The corresponding calculations in the case of incompressible liquid are not repeated, since they are the same as in the two separate cases, and have already been given. a,—10e- d,=-4 x 10-4 | 0 Bia | 10 20 | (Pmax. — 1) 10-* | ‘98 | 2-08 3-98 | (Pmin. — TH) 10~* | J aacog — 72 =67 | (~—T) 10-4 | 47 1-54 3-43 max. 80 “85 90 Emin. 60 “49 34 d,= 8 x Ome d, = 6 x 10-4 U 5 10 20 (Pinas LL) LOSS -28 | 60 1:12 (Pmin. — 11) 10-* =o) eee 20) --18 (aw — II) 10-* ara] 41 ‘95 Lenax ‘68 75 ‘84 yin 47 37 24 It will be seen from these tables that the effect of placing together the two portions AB, BC is to increase both the maximum and the minimum values of the pressure. It is clear from both tables that the minimum pressure decreases and increases again with increasing velocity, while its position approaches continuously to C'; ultimately the pressure reaches a maximum at B and then falls continuously to atmospheric pressure at C. The behaviour of the air, therefore, for large velocities is in marked contrast with that of an incompressible liquid. IV. The Superior and Inferior Indices of Permutations. By Mayor P. A. MacMaunon, F.R.S., Hon. Member Camb. Phil. Soe. [Received 12 January 1914.] REFERENCE is made to the paper on Indices of Permutations*. I here define indices of a new kind and subject them to investigation. Let any assemblage of letters be a,1a,'2 ... a,'s and consider any permutation of them. If any letter precedes p,’ letters which have a smaller subscript we obtain the component p; of the Superior Index. The Superior Index of the permutation is defined to be =p, = P the summation being in respect to every letter of the permutation. On the other hand if any letter precedes 7,’ letters which have a larger subscript we obtain the component 7,’ of the Inferior Index. The Inferior Index is defined to be ee the summation being in respect to every letter of the permutation. Ex. gr. Consider the assemblage a‘@*y°6 and the permutation BaadyaaBBy 4 64 , 4+6+4+4=14 the Superior Index 355 3311 , 34+54+54+34+3+4+14+1=21 the Inferior Index. If the permutation be reversed yBBaarydaa8 74433 , 74+ 44+443+3=21 the Superior Index 122331 11 , 14+24+24+343+1+4+1+1=14 the Inferior Index, and we see that the Superior and Inferior Indices of the permutation are respectively equal to * “The Indices of Permutations and the Derivation therefrom of Functions of a Single Variable associated with the Permutations of any Assemblage of Objects.” American Journal of Mathematics, Vol. xxxv. No. 3, 1913. Vor XGXil, No. LV. 8 56 Mason MACMAHON, THE SUPERIOR AND INFERIOR the Inferior and Superior Indices of the reversed permutation. This is obviously true in general so that we can assert in regard to the assemblage a, a,!2 300 ais that the collection of numbers which specifies the superior indices of the permutations is identical with the collection which specifies the inferior indices. Hence, in regard to the permutations of any assemblage, Lea It is also readily established that, for every permutation, p +r = i,; for consider that part of a permutation which involves two letters a, am. Suppose it to be Selig ok ge os. Cg cS OE ce The portion of the superior index due to these two letters is, if k > m, UK nl Ebi” Hi” + ed) EH Ha”) Cin!” + «) and the portion of the inferior index is (lim ttm ) (ye ie +...) + tm ((s--)- Adding these together we find that the two letters contribute to the sums of the two indices the number lan Uy Thence obviously Dp +r = Ste, leading to the relation 2a =o Se? The maximum value of p’ is clearly 21,7, and its average value 4 2%. The function =z” is of degree 7, in w and if it be divided by w*% it is unaltered by the See 1 substitution of - for x because s rP - s2hy — Sa 2hie-p', A function of « which satisfies these conditions is sda) =) Sean ht 8) (1—a)(1—2)...(1—2"), (1—2) (1—a*)... (Ia)... (1—2)(1—a*)... (1 —a's) and it will be shewn that this is in fact equal to La?’ In the first place consider the assemblage a‘Q’, and write 2a” = F, (i, )). All permutations which terminate with 8, contribute Fr(t,j —1) to F, (i, j), and those which terminate with a, contribute wa F(t—1,)). Hence the difference equation F, (i,j) = @ F, (t—1,)) + Fr (i,j — 1), INDICES OF PERMUTATIONS. 57 the solution of which, satisfying the above conditions, is (1) (2)... (i+j) (1) (2)... (i). (1) (2)... D) where (m) has been written to denote (after Cayley) 1 — 2”. F, (2, p= Similarly for the assemblage a‘Q%y*, write Sa?’ = F, (i, j, k). The permutations terminating with y, 8, @ respectively contribute F,(t,j, k—1), 2 F,(4,j7-—1,k) and «+*F,(¢—1, 7, k—1) to F, (i, j, 4), leading us to the difference equation F,(¢, 7, k) = aF** F, ((—1, 7, k) + & F, (7-1, k) + Fr (4, j, k—1), the solution of which, satisfying the conditions, is (1)... 4@+j+k) I) cde o cca Nu @iynoncs) Similarly we reach the difference equation Fz (ti, ta, t3 «+. %) = gattiet +t Fr, (=I, ty, «.. 1%) iis Ca aa sfogetamae SW (gy Mealy Ao aay eee Ne (Ga tay neete = 1): the solution of which, satisfying the conditions, is "(1)... G, +i, + -.. + is) NsnckCR eC yese Cayeadace (U1) yoca (tH) This result is remarkable because it establishes that Sa? = TaeP, Lied Pr to, ec where p is the Greater Index of a permutation (vide American Journ. Math. Vol. xxxv. No. 3, 1913). In fact the whole collection of Superior Indices coincides with the whole collection of Greater Indices, but it is not easy to establish this by a one-to-one correspondence. Observe the permutations of aaBy. Permutation Greater Index Superior Index aapy to) (0) aay8 3 1 agay 2 1 aya 2 2 apya 3 2 ay8a 5 3 Baay 1 2 yaaps 1 3 Baya 4 3 ya8a 4 4 Byaa 2 4 y8aa 3 5 Sa ser - NAC)4 _3)4) (1)(2).(1)-(2) (AP io 2) | bo 58 Mason MACMAHON, THE SUPERIOR AND INFERIOR The particular result Sui (1) (2)... (i+j) (yee)... G) establishes that the permutations of the assemblage a'8’ which have a superior (or greater) index equal to p’ are equinumerous with the partitions of the number p’ into parts, not exceeding 7 in magnitude and not exceeding 7 in number. The property of =z? that is before us leads to interesting relations between the functions F, (i, j)- Write the assemblage a8 in the form ai" B5-D g@ QP wherein a, b are any two numbers, such that a $7, b $7. It is to be shewn that F, (i, j)= a" F, (i —a, j — b) F, (a, b) wherein the summation is in respect of every composition a, 6 of the constant number a+b. The number zero is not excluded so that if for instance a+6=4, the summation will be in respect of the compositions 40, 31, 22, 13, 04, it being understood, as above stated, that a+ 7, b >}. It will be admitted that when the permutations admit of representation in the form Some permutation of a'-*Q7~? followed by some permutation of a%Q?, the expression 2/~* F, (i —a, 7 —b) F(a, b) denotes 2?’ for the permutations in question. If we sum this expression for all values of a and b which give permutations involving i7+j7 —a—b letters followed by permutations involving a +6 letters we must arrive at the expression of a?’ for the whole of the permutations of a’ B). Hence F, (i,j) = =a ¢ F, (i—a, j —b) F, (a, b) where a+b = any constant number. This interesting relation between the functions « has a very interesting particular case. If a+b=co a constant, we have F, (i,j) =27 F, (¢, 0) F,(i—o, j) + 2° F) F,(¢ —1, 1) F,(i— oo +1,j7-1) +... + F, (0, o) F, (i,j -¢). Putting o=7 we obtain F, (i, j)= a" F,(j, 0) F,(i-j, j)+ © F,(j -1, 1) F,i-j +1, j -1) +... + Fe (0, j) Fy (i, 0) and if we now put 1=7 Fy (j,j) =2" {Fz (j, 0)P + 29 (F(7- 1, IP + 29 [Fy (9 — 2, 2)? +... + [Fe (0, )}% This result is a generalization of the theorem in regard to the sum of the squares of the binomial coefficients, for putting «= 1, it becomes 8)-@l+@+Ge-+0): INDICES OF PERMUTATIONS. 59 In general the reader will see that we have the relation TIE (Gane Uh) as Salis — 4s) (@y +g + ... +51) + (ty—1 — Ag) (4, + QQ +... + Mg_2) +... + (ig — a9) ay is (Gy, ty — Aa, --- Up — Gy) le Lee (Aya «ne Ly); the summation being for every composition of a given number a, + a, + ... +, into s or fewer parts a, a,,... as; such that a, +7, for all values of s. The Superior Index as defined is obtained by adding several numbers together. This is the simplest way of obtaining the index, but the numbers so added are not the most interesting that come up for consideration. Ifv>vu the letter a, adds a number to the index if it precedes one or more letters %,. Denote by p’, the number added to the index due to the positions of the letters a,, a. Moreover a, may precede 1, 2, ... or 2, letters a. Denote by p'yu,¢ the number of letters a, which precede exactly o letters a,. Every time an a, precedes exactly letters a, the number o is added to the index. Hence Dee = onus or 27 vu, oat BP 'vu, get ty Pu, ae Also if p”, denotes, in regard to the whole of the permutations, the sum of the numbers added to the indices by reason of the relative positions of the letters a,, a and py, ~ the number of times in the whole of the permutations that a letter a, precedes exactly o letters a,, u P vu =D ose i) oF 2p" yu, a+ BP ou, gti + UD oni, hie Now we know the value of p”,, from the following consideration. In any permutation consider merely the letters a,, a. If 7’,, denotes the number added to the Inferior Index by the relative positions of these letters we see that Mana Foam Coy for any one letter a, contributes to the sum of the Superior and Inferior indices the number 7, and therefore the total of 7, letters a, contributes the number 7,,7,. Hence the average value of py, in a permutation is 47,7, and thence the number contributed to the Superior Indices of all of the permutations by the relative positions of a, and a, is Si)! >)! of ba Pim = Bute TT al al It will now be proved that (Deas has a value which is independent of the number co. Consider the permutations of the assemblage q ig eat oy tlt I te is GP Gadt tO Oe nC, which is derived from the original assemblage by adding an a, and subtracting an a,. In any permutation fix the attention upon the 7, +1 letters a,. Call the one on the extreme right the last au, the one nearest to it the last but one a,, the next one again the last but two a@,, andso on. Now delete the last a, but o from the permutation and substitute for it the letter a,. We have thus an a, followed by o letters a, and the assemblage is the original assemblage of letters. We thus construct a case of an a, followed by exactly o letters a, from every one of the sie atv): AES Ga 1) | (=e ees 60 Masor MACMAHON, THE INDICES OF PERMUTATIONS. permutations of the assemblage a, « . . = Oe. a, s Sy)! : u (21)! Thence ' = ~~~ : P osc Oe Ly ae ell a value which is independent of co. a” samy ” per a ” Therefore P vu,0 =P Wien 25° P UU, Ty? 3 } tu + ba (32)! leading to DP oc=\~ or g Piu=( 09 )Pomwe= Bute ay : an a 21) ! and ie = SS Eee Ore res Us SA We deduce that the average value of p”,,, in a permutation is ty ty + 1° To illustrate these results take the assemblage aa@y wherein 7,=2, %=1, 7;=1. Ba ya 7B aay 0 0 0 aay 0 0 1 aBay 1 0 0 aya8 0 1 1 aBya 1 1 0 ayB8a 1 1 1 Baay 2 0 0 yaas 0 2 1 Baya 2 1 0 yaBa 1 2 1 Byaa 2 2 0 yBaa 2 2 1 I 4 ” pe ” = ” — ed e. Here Pao=Pai=P 12S orqrdin and we verify that in the first column the numbers 0, 1 and 2 each occur 4 times. Also p a= (3) pate and we verify that the sum of the numbers in the first column is 12. . ur ” ” 1 Again Pa0o=P n,1 =P a,2=3-12=4, F 3 P’a=(5)-4=12, and we verify that in the second column the numbers 0, 1 and 2 each occur 4 times and that the sum of the numbers is 12. Again P 30=) a= A 12=6, 2° P's = (3) 6 = 6, and we verify that in the third column the numbers 0 and 1 each occur 6 times and that the sum of the numbers is 6, V. ‘The Domains of Steady Motion for a Liquid Ellipsoid, and the Oscillations of the Jacobian Figure. By R. Harereaves, M.A. [Recewved 8 February 1914.] ONE of the oscillations of ellipsoidal type for Maclaurin’s figure of equilibrium has, at the junction with Jacobi’s series, a period exactly one-half that of rotation; i.e. if a day means a period of rotation, the natural equatorial tide is here semi-daily. This isolated result was reached some twenty years ago, and seemed of sufficient interest to stimulate enquiry into the course of the periods of oscillation of the Jacobian figure. As the present work is of recent date the stimulus has been tardy in operation, The scope of the investigation has been extended to cover other matters which, like the question of periods of oscillation, require for their complete discussion much laborious calculation with transcendental equations. The results are made accessible by the use of diagrams to represent the domains of steady motion for a homogeneous liquid ellipsoid under its own gravitation, and an inspection of these is sufficient to shew what kinds of steady motion are possible for an ellipsoid of given shape. Special attention is given to the Jacobian form where a full series is treated with reference to shape, angular velocity and momentum, and kinetic energy; while the periods of the ellipsoidal oscillations are added for a smaller number of cases, sufficient to make the course clear through the entire range. In respect to the Jacobian an interesting feature is the connexion of the movement in values of angular velocity and momentum along the series, with the quantities on which secular stability depends, With respect to motion about two axes the most interesting point is that the conditions laid down by Riemann for his Case II are entirely superseded by the condition of positive pressure. It is proposed to describe the main results in general terms before proceeding to the analysis on which they are based. For the spheroids the oscillations may be called polar and equatorial; in the former the equator remains a circle but its radius and the polar axis are subject to periodic change, in the latter the polar axis is unaltered, the equator suffers a periodic elliptical deformation. The oscillations of the Jacobian near the opening of the series differ little from those of the spheroid; as the form moves away from the spheroidal the terms polar and equatorial fail to describe them, but in the ultimate position the oscillations become respectively equatorial and Vou. -X XII. No. V. ) 62 Mr HARGREAVES, THE DOMAINS OF STEADY MOTION FOR A LIQUID polar if the word polar is now applied to the long axis, equator to the nearly circular ellipse containing the short axis. At the initial point of Jacobi’s series as stated above the equatorial oscillation has exactly a semi-daily period, the polar oscillation has a slightly shorter period. The periods diverge as the ellipsoid is elongated, the former increasing the latter decreasing. In the limit of extreme elongation the former period is daily, the latter has a finite value while the period of rotation is indefinitely long. For a certain range from the initial Jacobian, and also for a range of Maclaurin’s spheroids on each side of the junction, the periods differ little from the half-day in excess or defect, both cases being represented. The position for spheroids is that the frequencies », and n, are finite for the spherical form where the rotation is indefinitely slow; the former, at first the greater, falls the more rapidly with increase of oblateness, and equality is attained for c/a = 5892 = cos 53° 54’, when n?/@* = 44. The Jacobian junction is reached when c/a = "5827 = cos 54° 21'27", and n.= 20, n,7/w? = 41182. The value n,=2o is reached for c/a = 5612 = cos 55° 52’. With these values may be compared the entries in the table for Jacobians for values of a up to 30°. Now any external body in the presence of which a liquid ellipsoid is rotating, will shew a period something more or less than the day according as the relative orbital motion is direct or retrograde, and its quasi-statical tidal influence will have a period near the half-day. Accordingly ellipsoids of a shape deviating to some moderate extent from that at the Maclaurin- Jacobi junction’ will be specially sensitive to the tides induced, in consequence of the closeness of periods of the natural and forced oscillations. The first calculations of periods were based on material provided in Sir George Darwin's paper on ‘Jacobi’s Figure of Equilibrium*, Some irregularity appeared in the succession of values for these periods, and a new series of points was determined, generally in close agreement with Darwin’s paper. In one point of some importance there is disagreement. Darwin found a maximum for the velocity of rotation at some little distance from the beginning of the Jacobian series, I find an uninterrupted fall. ‘This question and that of the rise in value of the momentum appear to be connected with quantities occurring in what is called the test of secular stability, as restricted to deformations consistent with ellipsoidal shape. I find this restricted test to be satisfied through the whole range of Jacobians, and in connexion with it a regular fall in velocity of rotation, a rise in angular momentum, and for kinetic energy a rise to a maximum situated, as Darwin found it, in the range where elongation is considerable. Ellipsoids of given shape may be represented on a diagram by coordinates 2, y the ratios of one axis to the two others. If we take a standard order a>b>c and write a=c/a, y=c/b, then with «<1, y<1, ab>c and coordinates «=c/a, y=c/b. ELLIPSOID, AND THE OSCILLATIONS OF THE JACOBIAN FIGURE. 65 Case I, in which the greatest and least axes are those of rotation, has for field a narrow lune (fig. 2) skirting the lme OS and bounded by the hyperbola y(1+«)= 2 representing the equality 2b=a+c. A vertex V of the hyperbola (e=V2—1, y=2-—‘2) lies symmetrically with regard to OS, the boundary has at Sa tangent + 1=2y in common with the boundary for rotation about a mean axis, but the fields lie on opposite sides. Case II, in which again the greatest and least axes are those of rotation, is defined in a preliminary way by OP, an are OQ (fig. 3) of the hyperbola y(1—#)=2 representing 2b=a—c, and an arc QP of the quartic ¥?— 42° = y?(y?— 2"). The intersection x=2—N3, y=Nv3-1 is the point where the loop of the quartic has 2 a maximum. P (G) 2 4 Fig. 3. Field of steady motion about two axes. Riemann’s Case II. In this connexion an interesting feature emerges. For all other cases of steady motion the condition for a positive value of the pressure is satisfied as a necessary consequence of other conditions. In this one case it is not; it proves to be more stringent than either algebraical relation, and the line of zero pressure therefore supersedes them as the effective boundary. If we suppose motion possible between the pressure line and the algebraical boundaries, this is the one region in the whole range of steady motions where there is a manifest occasion for the separation into distinct masses. Case III, in which the greatest and mean axes are those of rotation, has a field O7'P (fig. 2). The part OT is the hyperbola y(1—2)=« representing 2c=a—b, the part TP is given by a transcendental equation. It will be noticed that the boundary of I is single and algebraical, that of II single and transcendental, that of III composite with sections of each type. 66 Mr HARGREAVES, THE DOMAINS OF STEADY MOTION FOR A LIQUID With the aid of the diagrams it is easy to examine all the cases where overlapping shews that more than one type of motion is possible for certain shapes of ellipsoid. We may note that there is a small region in the neighbourhood of «='22, y=°38, not a case of extreme dimensions, for which no state of steady motion exists. [PR of fig. 1 crosses OT and OV of fig. 2 above their point of intersection.] P = Ss 8 6 2 “4 “6 8 10 Fig. 4. Field of Roche’s steady motion. Contour shews extreme cases. Roche has made use of equilibrium forms* in which the attraction of a distant body is taken into account, with the limitation that the rotation and relative orbital revolution have the same period. The connexion with the above seems sufficient to justify the inclusion of this case in the diagrams. The case represented is that in which the liquid mass is extremely small in relation to the attracting body, and the interval between the smooth curve SP and the broken line SJP corresponding to the other extreme where the distant body is small, will be bridged by a M O 7) 4 6 8 K Fig. 5. Graph of w? for figures of Maclaurin, Jacobi and Roche. succession of intermediate forms. The graph for w*/4mp is also set in the diagram (fig. 5) for the same quantity in relation to Jacobi’s ellipsoids and Maclaurin’s spheroids. * I have only seen the reference in M. Poincaré’s Hypotheses Cosmogoniques, p. 54. ELLIPSOID, AND THE OSCILLATIONS OF THE JACOBIAN FIGURE. 67 §1. Riemann’s* equations to determine the frequencies » and the ratios of amplitudes da:db:de may be written \a?Baq — n° (a? +¢*)} = + (aby — nc?) db = Ta a ene a (1), da | = db {bE — n? (b? + c*)} ac (abEw — n°c*) a= 0 giving a quadratic in n® when the ratios are eliminated. The total energy is mH/5, and E,q denotes the second differential coefficient of EH with regard to a, when a and 6 are independent variables with which c is connected by the condition of constant volume. The steady motion for which the oscillation is considered is given by _Aa?—Ce 27? 27” a: ah Ga nthe daaaaees acer (2) ie Gs eo | Dt ae Gb Ge EOPe in which 7 and 7’ are Riemann’s constants of integration, and A = 2rpabe | dr|V (a? +r) (+X) (2 +A). na, Differentiating (2) we get #,, and substitute the values of 7%, 7? from (2), so that the final expressions are in terms of a, b, ¢ only. It is convenient to use integrals F,G,H= 2rpabe | Cees aia die Goiikbits seco (3). 0 {(a2?+X)(B?+ A) (2+ A)}? When a, }, ¢ are treated as three independent variables, 0A é and similar equations, while 4d +B+C= 4p gives po G (ne) 42 =O) 0a and a£, is oo By this method we derive from (2), 2 =o 222 22 972 — 72h2) — 9 Aig? 2 a Te | @E gq = 4Farbc? + G (2a2e? + bc? — a*b*) — 2Ha? + 6a on aoe ; abEqy = 2Fab'c: + G (ae? + be! — ab") — He? — bab } ane’ or on clearing 7° and 7° WE ag = 4F arb? + G (5a2c? + bc? — a®b?) + Ha? | (4) ab Eq = 2Fa2b?c? — 2G (a2c? +b? — ab?) — 4H?) £ The quantities A, B, C are connected with F, G, H by SAR =—IRG2C? = (Ge (BP. C2) ted ema teeeth Sees ce cela se raielees steels (5); * Riemann, Abh. K. Ges. Wiss. Géttingen, vol. 1x. 1860; Collected Works, p. 168 sqq. 68 Mr HARGREAVES, THE DOMAINS OF STEADY MOTION FOR A LIQUID and the relation Aor p = FF 2atG? = 2G 20? + 3H. ree. nes canens anode =-heeteeeenese (6) - represents A+B+C=47p. These results apply to the whole domain of rotation about a single axis. § 2. For Jacobi’s ellipsoid 27/(a—b)’= 27'/(a +b)’=@, and in terms of @ equations (2) are wo? = (Ala? — Gc?) a2 = (Bb? = Ce?) [b* 2 < yg. Vou. XXII. No. V. 10 70 Mr HARGREAVES, THE DOMAINS OF STEADY MOTION FOR A LIQUID : : p> d (A2(A+7) : An equivalent* of the integral o= | Pere peor dx is = A(B.—4y) dr, [73M (A+y) [, MPSS + |, SE 201 + 97-28) —9 +0) =0; T 2 in virtue of which N38 [eee +t 08 ae a pcan (20). ee 4 Pp: ON and then an at a bles 2 a dn = Thus WV also is positive and L>2N, ee N. Since Z, M and LM—WN? are positive the energy-expression (17) is a minimum for the Jacobian figure. § 4. The relation between g’ and y' which corresponds to movement along the Jacobian line is got ae Pee the variation of the equation of condition and is = -[ a {yy eae +979'} == : = {y(g +) — 48} [gg (y+) +7’ fy (g +A) — 28h), ns Tha or O= =| a a (org +2087) 5 | ; } : 3 as modified = the equation of condition, Le. the relation is My’ + Ng’ =0. The variation of w? is given by Aw?=—g'Q+2Ny'/g where Q=39{ AGB aL 0 P: so that L/g=o?— Q/2. The variation along the Jacobian line is Ao* eis ean g (Q+ =e): or if we prefer, oe) = 7 (9+ tts) = =( 4-34) Seals akrcsls « cose ereaeeer (21). As M and Q are both positive, @* falls continuously as g increases, 1.e. in passing along the Jacobian line from the junction. From the second form of (21) we get hares a eon? (z- 4) i a (22). Also for this particular variation along the Jacobian line 10 ie ae ae A. (T+U)=9 (£47), or A2(2'+ 0) =O using Ah==A (gw). It is clear then that increase of energy, and of angular momentum both follow from the condition LM>N* But yer Ree fy A N? eel AS aaron - (0+ 5 aa) 7 mar (4-57) - wg Betas. cbinnscvaraceten (23), the sign of which is not determined by the inequalities stated. The sign is at first positive * This step occurs in C, O. Meyer’s paper, Crelle, t. xx1v. (1842), where the continuous fall of w? is established. ELLIPSOID, AND THE OSCILLATIONS OF THE JACOBIAN FIGURE. 71 but ultimately negative, the kinetic energy having a maximum for a somewhat elongated Jacobian (v. infra p. 76). § 5. The transition to the spheroid is interesting. With a(1+qa), a(1+0’) for the varied axes, g =v +b'+4(a24+b"%), and y=—-14+1/(1l+ay1+0'P or yf =—2 (a +0')4+3 (a? 4+ b*) + 400! = — 29 + 3g? + (a —b’). In this case the independence of y’ and g’ is only realised in terms of the second order. The part of the variation (18) which remains of first order is 2a°q’ |- w* + 21p (a — c*) c| x CEM) >] : 0 Pz giving the value of »* appropriate to the spheroid. This value of »* appears in the second line of (18), and then the variation of second order in which now y =— 29 =—2(a +0’, takes the form (1+ 4M —4N) 9? 4+ 2mpa'e | Nd 0 a {c? (2a? +r) — a4} {92 +(a —b'y} or on reduction Pn 5 rAdr 2mrparc (a +b’)? : 0 (+2) (+A)? + 2mparc (a’ —b’)? | == {c? (2a? +r) — a4}, ~ 0 2 ES tet 4c? (a? = c’) 3c? (a? a 2 C+r (+r) a result verified by direct treatment of Maclaurin’s case. The first term is always positive, the second positive from the spherical end to the Jacobian junction and thenceforward negative. The term which changes sign is a term carried over from the original first variation to what in terms of a’ and b’ is second variation, and the necessity for change is due to the circumstance that a=b gives a relation between g’ and y’ to the first order as the expression of the condition of constant volume. , ert: m Te Tie §6. If we use for kinetic energy = ta =o (Geen write 7/(a—b)=7'/(a+b)?=@/2, the first variation agrees with (18), the second shews a positive increment, and the minimum property is more easily assured than with the ! and after forming the variation above formula. The greater stringency of the dynamical minimum condition is in the case of the oblate spheroid represented by the withdrawal of the limit of validity from Riemann’s point to Jacobi’s junction. The minimum theorem may be stated as follows: a homogeneous body with kinetic energy due to rotation about a principal axis, and potential energy due to its own gravitation, will, when restricted to constant volume and ellipsoidal form, shew a minimum of total energy, if for values of the momentum below a certain limit the ellipsoid is an oblate spheroid, and for values above the limit it has the Jacobian form, and in each case the velocity is that given by hydrodynamical theory. The juxtaposition of kinetic and potential energy in the minimum problem postulates some mobility or capacity of accommodating shape to stress, but not the complete mobility of a liquid, for that demands Riemann’s variation method; or if such mobility exists, then 10—2 72 Mr HARGREAVES, THE DOMAINS OF STEADY MOTION FOR A LIQUID a frictional action is postulated sufficient to suppress, as they arise, all motions involving departure from that of a rigid body. In pure hydrodynamics an existence theorem is established, the argument of secular stability extends the scope of its application. If an external body is taken into account the problem is more complex, but it seems a probable forecast that the tendency will be to transfer angular momentum and with it energy, from rotation to relative orbital revolution; the historical order suggested is a passage from the direction of P towards the Jacobian junction, and thence to the spheroidal form. §7. We proceed to methods of approximation which it is advisable to use near each end of the series. Near the Maclaurin junction Jacobi’s figure approximates to an oblate spheroid, at the other extreme to an elongated prolate spheroid. We can in these cases avoid the elliptic functions, and calculate F, G, H directly. Thus for an approximation to the oblate spheroid write a@=ai(l-+e), b=a(l —e); so that (a? +d) (+A) = (a2 +A)? — eas, where e? is small. Then es dx Beat 15eta® 2, be | - i ~~ ——_—— al (a2 + A} (2+ A)? To (e+ ny 8(@+AS | ; or with @+r=(@—C)(v?+]1), C+A=(@—-—C)v?, B=(P—C)(v+1), c=(@—c*) vr, _ 4arpabe i dy’ | 3 (v2+1) 15e (v2 +1)! | F @_ot y? (v? Tay| )s 2 (v2 + 1) ar 8 (v2 +1)! SP osc 4xpN1—é@ | ieee 3e(2 +1)? 15e*(v2 +1)! aL Aa (ta een (+1) (a? — c*)? y(t )|. py? (v2 + | oF 2 (v2 + 1: =e 8? + 1 a The integrations are effected by the use of - a d ‘ Qn =v (+ | FiGtaily a 1D?’ in which Q@=1—veot»y, and (2n+1) (2241) gn—2ngnyr Hl -reeeereeee ene (24) is a sequence equation. Thus _ 4apJ1-@ Se. 15 et cia eran 2 et | | ,_ 4npV1—e 3é 5 G= os |e a+ F(a a+] ge BREE (25). eal 3e2 H =4rp V1 —e(v2+1) [a- 242+ 4s + za (9s — 241+ 4s) + | Since ac? + bc? — a*h? = (a? — c® P (v? + 1) fp? —1 +e? (v* + 1)}, ELLIPSOID, AND THE OSCILLATIONS OF THE JACOBIAN FIGURE. (63: and c?=(a?—c*)v*, the Jacobian condition (9) written to the order e* is 0=7 (4 — G)—(@—-G) +e E {v? (Qs — 9s) — (Gs — 9s)} + (* +1) (Qo — »)| +e | 2 (8-9) — (— a) +3 + DG—@) 26 rave qs Ge) (Ge Fr) gE GP=Of))|\ SE 260. Senc0d (26 a), - 3e ; and (4) + et Gt | LP (Gs— GW) + G9 — WF GE] + -- 3 F ) 15 Se si} =a-a+e}, e— 9) - +I. 9) HEL Oo q) = 3 + 1)(qi— 9) + --. Siawidees (26 5), on using the equation of condition. The term in (26a) not containing e vanishes for y=y, the value at the initial point ; é and the fal! in w? are of the order »,—v. The condition of equal volumes makes apie; = a2b?c? = atc? (1 — e°), while o:c?=v?+1:v*, so that @ and c® differ from a,* and ¢,? also by quantities of order vy—v or e, but a and b} differ by a quantity of order e. For a single calculation a value of y in slight excess above 54°21'27'-45 may be taken, then ha=l—-wcot ~=1—vcot», while g,... are given by the sequence equation. A method applicable to all forms deviating only slightly from the initial is to find differential coefficients, and calculate their values for v=». Differentiation is made by dq < dg y 2 = (2n —1) (Gn — Gn), Die id @ and we may note that with the Jacobian condition d . ae hai! 4 4 i {7 (qi — 92) — (G2 — 95) = HC —VY/(L +), Fe (a-)= CES {—1+q, (42+ D}. When evaluation is carried to the second order of Aw and fourth order of e, & = 3-445813Ay — 2°78454 (Ay)... ae ee) = Np, eR Ra 0 ra 1 3k ce 28). ea = 09355743 — -1294005Ay — ‘087816 (Aw)... - ' Here Ay is in circular measure, and values up to 30” carry us approximately as far as the position a=15° in Darwin’s notation (v. infra p. 79). § 8. It may be of service to add expressions in terms of qg, and »* for various quantities used in dealing with a spheroid a, a,c where a*:c?=y?+1:0% Thus =P {15 2 2 Saas ee) Y= J (@-eF@ +1) {15 (+ 1) n DV is G 5v?+ 3 —3 (2+ 1) (Se? +1) qi}, =o 2(a@—e) He TO) a4 4 62? — Wg = bo 4 I], 74 Mr HARGREAVES, THE DOMAINS OF STEADY MOTION FOR A LIQUID which satisfy a relation DP arc —sGa(20? 107) - SM eeeeen oe eeeee neces meneame eee (29). Also A=B=2m7p {1—(v?+1)q}, C=4rp("4+1)4, wo? = 2p {1—(3v7+1)q}, and — EL = Aa? + Cc?=27pa? {1 + (38v?— 1) mh. The equatorial oscillation of the spheroid has 6a +6b=0, and n= Ea — En = 47 p (38 + 8 +1) —v —- 1}; the polar oscillation has da = 6b, and .-.(30). ny? (Bv? + 1)= (v? +1) (Lua + Ew) = 277p (v? + 1) {(27 4 + 18y?— 1) — 9? + 1} The Jacobian condition in the form (7) or (9) leads to eB (Gitc bE SR Gh. cdédasaresnaceoccsc6csdacced 150006 (31), and it may be verified that this makes n,?=4*. Or to establish this in a way which shews more clearly the points on which the exact relation turns, we may take the value of Eyq— Eq in the general form (10), which for ¢=b is {2 Fate? + 4@a*c? + H (a? + c*)}/a* If we now combine the Jacobian condition at the junction Ga? (2c? — a*) + Hc? = 0, with the relation (29) true for all oblate spheroids, we have 2Fatc? = H (3a? — c*). This makes Bua —Ewm=4(Ge+H) or n2= 40%. §9. The treatment at the other extremity depends on its approximation to the form of the prolate spheroid. For the latter with axes a,c,c and @:c=vp?:y—1, and pi =z loge ae 1, (Gn 1) = 1) pn 2npnya =U vie. eneeceaseeees (82) ; the formulae for F, G, H are 4app; 4mp(P.— Ps) 4mp (pi— 2p. + Ps) @ c? (a? — c?)’ ve-—Ce e-¢C For the elongated Jacobian write B=y(l+e), C=yP(1l—-e), *®=(-Y)v, C=(@—y*) (v1), then proceeding as for the oblate spheroid _ 4ap Vv1—e (a? — y?)? (vp? — 1) We are concerned with the case in which py approaches the limit 1, say v*>=1+&, and note that while p, becomes infinite p,, p;... approach limits 4, }, 1... and approximately oT 6 tos 47rp e? ’ re dap _@ os : ve 8 . ae Pag ep (n+), 6-22, (m-m- 2), H=tnee (n—2n.+0-£(9-D 15 Be es Pst > Pot —s- Prtet, G.... ELLIPSOID, AND THE OSCILLATIONS OF THE JACOBIAN FIGURE. “I oO 2 With L= loge g we have as sufficient approximations Hi; 1 1 5E* p=L-1+8(5+ i) ps= =5- = (=), P=] ( - =) Soc teA aoa (34). The Jacobian condition G (a*c? + bc? — a*b?) + He? =0 is then ai : 4 oy Mat GW ape (p.-p.- 7g.) (— 2e + & — 2e&*...) + \.- pa + Pa — 5 ( Pr —8)f &(1—e)=0, the solution of which gives e= F(QL—8) 4+. (814 20L4 5) tone csecceseessenceeee (35 a), and then ow = Ge? + A =2rp& {2L—3+4+ &(BL—1)+...} 20.0... eceee eee es (35 b). The first approximation uses only the main terms of G, H, the second involves an advance of one power of e, so that only the main terms of @ and H are here required. Values of Eq, ... Sufficient for our purpose are: /9 61 BG 3): @ Bog = 4p (@— 7) & gE —— b* Ey, = 4p (a? — 9°) & E — }. ; iby ab Eq, = 4p (a? — 9) & [p-f (ls Ik One root of the quadratic in n? to which (1) leads is then n,* = 4p &Z, the other is ng? = 2p (1 _ =F Thus n2:@? falls from the value 4 to 1 in passing through the whole Jacobian series, ie. this oscillation changes from a semi-daily to a daily oscillation. For n, the ultimate ratios of amplitudes are ous SaaS for np, — =0 to the first order, and fle. OF iE DO b a higher order = — &(2L aa approximately. These ratios answer to the description given above. The Jacobian algebraical condition a*l? > c?(a?+ 0?) is expressed by 2 +y7°<1. In the limit #2=£ and y=1l—e=1-—£(2L—3), from which 2+y°=1—&(4L —7), ie. the circle is approached: an approximate equation to the curve near c=0, y=1 is l-y= (2 log, - 3) . The positions for = 05, and €=-02 are entered in the table (p. 80). Darwin pointed out that there is a maximum value for kinetic energy which falls in this range. Assuming that it does lie here, the value of (a*+ b*) w is to be made a maximum as dependent on the single variable & With abe=1, @&+b=€- ale +75) and we have to make £ [2x - 38+& cee 6) | a maximum, the condition being L-3+e(-D\=0 76 Mr HARGREAVES, THE DOMAINS OF STEADY MOTION FOR A LIQUID As L =3 roughly €is not very small, on the margin of applicability of the formula. But the correction given by the second term is a small fraction of the prime value, and & = °13791 giving e="051624 is probably near the mark: it corresponds well with the place indicated by Darwin’s table (D in fig. 1). § 10. We proceed to examine the boundaries of the domain of steady motion about one axis. The condition 7=0 is expressed by (Aa? — Ce®)/a = (Bb? — Ce*)/b, or G (abe? + atc? + bc? — a*b*) + H (ab + c?) =0...(36). The algebraical limit required to make the coefficient of @ negative with c/a=a, c/b=y is @+ay+y?<1; the form (ab+c*)(H—abG)+c(a+by G@=0, shews that abG > H for this line. The ellipse 22+a#y+y?=1 is in fact much nearer to the Jacobian line. The inequality 2° + y+ y*>1 which appears to hold for the latter is evidently true if abG@ > H, but I have not found any proof of either inequality apart from the calculations on which the diagram is based. We accept Riemann’s calculation for the terminus of r=0 on OS, and treatment similar to that for the Jacobian gives =f CL —3) He 1) (2) =3) (37), for the terminus at P. The first approximation to its equation near P is 2. 1—y=2(2log.--3). For intermediate positions the expression of (36) by elliptic integrals has been used, viz. (F — E) cot? acos 8 (1 + cos 8) — E {cos? B + cos*y(1 — cos B)} + cos 8 sin vy cos y (2 — cos 8) = 0 At the boundary r’=0 the mean axis is axis of rotation, and if we retain the order of magnitude a>b>e, the condition is (Aa? — Bb*)/a = (Bb? — Cc*)/e or = G (ach? — a*b* — bc? + a’c*) + H (ac — b*) = 0 ...(39). The first form of the condition shews that b the axis of rotation is intermediate between a and c. If the axis of rotation is taken for numerator of the fractions used as coordinates in the diagram X =b/a, Y=b/c the field for this case is above the line SP where it adjoins the field for least axis as axis of rotation. In the form (ac—b*)(acG+H)—(a—cyPG=0 it appears that ac>* or 1> XY, and the original form then involves b? (a? +c*)> ac (ac +6?) or X:—XY+Y?>1; the boundary therefore lies between XY=1 and X?— XY+ Y?=1. The graph shews the inequality Y + Y <2, or 6 less than the harmonic mean of @ and e. Near the terminus at P, working as for the Jacobian, we find =F (2L—3)\— FG —1) (QL — 8) ak. Aoacinccnen de cldaesiacecnaesee (40), and the equation to the curve near P is Y=1+X (210g. = -3). The expression of (39) by elliptic integrals (F' — E) cot? a !cos y + cos? 8 (1 + cos y)} — EF \cos* y + cos? 8 (1 + cos y)} + cos 8 sin x cos y (2 + cosy) = 0...(41), with a> 45°, has been used for intermediate positions. ELLIPSOID, AND THE OSCILLATIONS OF THE JACOBIAN FIGURE. 77 Near the spherical terminus of the boundary r’=0 we may write a?=1+4a’,... and then P=(1+)A)+(1 +A)? Sa’ + (142) Sa’h’ + 0'6'y’. Proceeding to the second order of small quantities and using Ya’ + Sa’B’=0, P=(1+A)>—2(1 +2) Sa’B’ =p — p(w — 1) Ee’f' where ~ Ir "> du { Ch Gi 1 1 =1+,. Thus A=20 | ee ee 1—= +5 +(5-- 55) 22 ‘of or f Pdi (@+rxyVP P| 1 ue | Be Qe 2p : Ihe GIP SG /8y : A 4rp=5—5 + Tt QR creeeeereteeteeeeeetreeeeeees (42). The condition (39) then gives eee ,/f== 73 the first approximation to the curve near S is X + Y=2, the second is 3(X?+ Y?)+ NY =7, and the initial value of « for use in the elliptic functions is 45°. § 11. If we wish to shew what shapes of ellipsoid admit of rotation about either axis, we transfer to coordinates 2, y having «=c/a, y=c/b so that Yy=1, c=X/Y, X =2/y. The curve 7’=0, SH P in fig. 2, starts from S along the line # +1 =2y which corresponds to X+Y=2. Near P its equation is 1—y=«(2 log, 2/a—3) corresponding to Y=1+ X (2 log, 2/X —3); Le. the transformed curve approaches P touching the line t=0 but above it as shewn by terms of the next order. On this transformed boundary 7’ is not zero for the motion in which the least axis is that of rotation. The region SHP represents ellipsoids capable of either motion, but with different values of 7 and 7’ except along the line SP: in particular the Jacobian ellipsoids of the range HP are all capable of steady motion about the mean axis. At H we have Aa*— Bb?= (a —b*) (Bb? — Ce*)/b? as Jacobian condition, while for 7’=0 in the alternative motion da? — Bb? =a (Bb? — Cc*)/c, and therefore c(@—b)=ba or P=arc/(at+c), Le. y2=a2(1+2) or cosytan?8=1. [Working through the G and H relation we get also a factor b?—c? pointing to intersection at P.] The position of H is near «= 4026, y=-7515, or a=67° 17, y=66° 15’ 24”, least axis 2 of the greatest. The 7 of the one motion and @ of the other are connected by t?=@?a?(a—c)*/2(a+c), the kinetic energies are in the ratlo a—c: a+ 2c, the squares of momenta as (a—c)?(a+c):2a*(a+ 2c). Numerically the kinetic energy of the motion about mean axis is 33°1°/, of that for the Jacobian, the angular momentum 212°/,; so that the disparity is considerable. In the region of rotation about mean axis lies the case of irrotational motion to which Sir Alfred Greenhill called attention. The condition is (Aa? — Bb*)/a? (a? + 3c) = (Bb? — Co?)/c? (Sa? +c?) or c? (a? — b?) (3a? + @) (Ge? + A) = a? (6° — c*) (@ + 3c?) (Ge + A). With respect to this I find that near S it agrees with r’=0 to the second order of small quantities. Near P I get 2e[po— ps + & {5 (ps — ps) + pi — po} = & [3 (pi — po) — 3e(p — ps) + 4&* (p. — p2)], leading to e=38 (202 —- 3)-& (2424 —56L+ >) sies's iViOra XeXchi= No; 1, ll 78 Mr HARGREAVES, THE DOMAINS OF STEADY MOTION FOR A LIQUID This gives a contact with SP at P similar to that of the Jacobian, below SP in the standard system, above it in the other, and the sign of the second term suggests the point of inflexion necessary to secure this tangency. § 12. The table of Jacobians is calculated for 5° intervals of the variable a, those for a=5°, and a=10° by a method which will shortly be given, the rest by elliptic integrals; beyond these are added values for the place of maximum kinetic energy, also for &=-05 and £=-02 in the extreme range. The object in choosing a for the regular intervals rather than y, as in Darwin’s table, is to secure a safe interpolation. Gauss’s form may with advantage be applied to tabulated integrals in which the integrand is calculable or tabulated more fully than the integral. Where Taylor’s theorem holds good within a short interval a to a + b, 2 par b (gym | Ly, | tale =b | a t5 Hel +5 te” +35 Me +~]| - 2 2 6 b a \ 43 = OUigiis + 4. + Bears Way otarela\reisaerearaane ( ), or = 5 (Marae + Uara’o) + 4320 too. | where X+A’=1, A? 4A? =2, Le. A='211325..., A’ ="788675.... The form buq+» 1s sufficient b : i he ae for a short range, 3 (Maras + Wasp) covers a much wider range. Thus requiring an elliptic integral with 60° 28’ for y we should write 9 =60° 14’, use sin§’=sinasiny’, and then with Ay for the circular measure of 28’ the simpler corrections are Aycos®’ for EH and Ay/cos 8’ for F to be added to the tabulated values for y= 60°. In the columns giving (a?+6*)o?, (a?+0*)*, and E, use is made of abe=1; the transfer to any other scale of magnitude is well understood. On completing the work I was anxious to discover the source of the discrepancy between Darwin’s results and my own as to the maximum velocity, a discrepancy occurring where the use of elliptic integral tables is troublesome and uncertain, viz. for smaller values of a; and this paragraph is a sketch of work undertaken with that object. The forms in (15) and (16) are expanded in powers of sin*a, the method is therefore closely allied to, though not identical with, Darwin's treatment given in a long footnote. ‘lhe expansion of the left- hand member of (15) has the form C,sin‘a+ C,sin'a+..., where C,= sin‘ y tan y {q, (3y* + 140? + 3) — v? — 3}/8, with y=coty and g,=1—ycoty. The bracket vanishes at the junction by (31); near it Pry mea, : eae : e the value of C, is e Ay, and a first approximation gives oa Ay+C,;sin?a=0. We find 1C : : = 2q, sin® y (cos* y — sin? y), and C,= sin" y tan y {q, (80° + v4 + 33v? + 3) — (v2 + 1) (v2 + 3)}/32, ELLIPSOID, AND THE OSCILLATIONS OF THE JACOBIAN FIGURE. 79 which at the junction is reduced to Cs = — qSin* y cos y (cos? y — sin? y)/2. Thus the first approximation is Ay = g Sin’ asin ¥y cos y, which admits of immediate interpreta- : / iy. : tion. For x= cos (y+Ay)= cosy ( 1— goin a sin® 1) , and abe ot = cos y € + join’ a sin*y) ; and so «+ y=2cosy=2, ie. the first approximation to the line near the junction shews a direction perpendicular to OS. If the mght-hand member of (16) is expanded as D,+ D,sin?a+... then 2D, cosy=1—q,(3»7+1), 16D, cos y= sin? y {5v? + 3 — 39, (v2 + 1) (5v? + 1)}, or when simplified by the condition at the junction 4D, cos y = sin? acos* y {1 — g, (3v? + 1)}. Thus as far as sin? a A wo sin? a Pe Le ry (cos* y — sin? y) {1 — g, (By? + 1)}, and the first term is Ay iD, cosy) with the value of Ay just found. We have d dy which at the junction = 4y {q, (38v?+2)—1}; and therefore 2 D, cos y = z [q: (27! + 30v? + 3) — 9v? — 3], re = sin? a sin? y {g, (B8v4+ 14v? + 3) — v? — 3}/12, which vanishes at the junction. Thus the first term in Aw? depends on sin‘a. The details of this method are very tedious, and I have not applied it to obtain the coefficient of sin‘a, The position is that w* with respect to the variable a is a sustained maximum, with respect to y an ordinary maximum, the variation depending on sin‘a or on (Ay) [Aw is of order e? or sin*a or (Ay)*] Connecting this with the method of approximation in § 7, with y for the value at the junction ; ; : sin? @ sin* (1 + sin? a + : sint a) - Su aLaTyicos ia == sIn?4 sin? =sin?a+—s eee ae 2 UND 4 4x 3445813 : : sin‘ @ sin ¥ cos : sin? @ sin y cos y + Zale et (3 cos? y + sin? y) sin’ @ sin y cos y sin‘ asin‘ y + 4x 3445813 = ‘1183906 sin? a + 0564942 sin‘ a+°03562sin'a+..., (15 cost y + 10 sin? y cos* y + 3 sin‘ y) {1 +S Seosty +8sint)| ca or in minutes of are = 407-00 sin? a + 19421 sint‘a+ 122°45 sin’ a+... | @- and = 0935574 — 0040948 (sint a+ sin’ a) — 0009615 sin’ a.. | TP 80 Mr HARGREAVES, THE DOMAINS OF STEADY MOTION FOR A LIQUID The last pair give a solution in terms of the angle a@ for all small values, more trust- worthy I think than results derived from the direct solution* of (15) by tables, up to a=15°. Schedule of solutions for the Jacobian form of equilibrium. ] | | a 1 x y e/azp (a2 + 1*)Pu?/4zp|(a>+)er/4xp, —2E/4zp ne | n,?/o* 0° 54° 21’ 27-45 | 582724 | -582724)| -0935574- T68872 | -268203 1681975 | 4 4-1182 at 54° 24’ 33-6 58199 -58346 -0935572 768876 268205 10° 54° 33’ 54-6 + | -57978 | 58567 -0935536 768933 | -268210 | ks 54° 49’ 40” | 5760 5894 -09354 “7692 | -2682 16818 3°9873 | 41316 20° | 55° 11’ 40” -DTOT -59AT “09349 ‘7699 -2683 25° | 55° 41’ 5638 | 6016 -09339 7714 2684 30° | 56° 17’ 40” “BDAY -6102 09322 “T745 -2687 1-6800 | 38887 | 4-2447 SSre ors 212" D441 6207 “09291 “T7197 -2692 40° Hie o) “5309 6331 “09243 “T887 -2700 } 45° | 58° 58’ “5156 “6481 “09162 “S017 -2710 | 16707 3°6876 | 4-5190 5O~ 1 60° 11’ 30” 4971 “6654 “09043 "8225 2727 Die Olma! oO. | -4752 -6856 “O8868 “8543 2753 60° Gamo 1o- | 4493 7092 “08610 9031 “2789 | 1:5385 3°1626 | 5°3315 65° | 65° 16’ 48” 4182 7367 “08238 ee -2841 | 70° | 67° 38’ 30” “3804 7689 -O7694 | 11054 2916 Lue 60529 41" | -3339 “8074 “06896 Jeikspayl -3023 1°2444 2-894 6-821 80° | 74° 5’ | -2742 8540 05694 17685 = 85° | 78° 59’ =6” | -1911 9127 “03773 3°0055 -3368 | | 1330 -9496 -02360 4-9407 *3412 (max.) |-0497 | -9887 00220 16-36 -3000 | 0200 ‘9975 “00124 | 42-23 -2291 S00 90° 0 1 0 L 0 0 al | 1 § 13. We now propose to discuss the delimitation of boundaries for steady motion in which two axes are involved. Riemann curiously enough omitted to state the order of magnitude of the three axes for his several cases, and dealt only slightly with boundaries of transcendental type. In other respects his statement of the analytical conditions is so clear and precise that the most judicious course appears to be, to state his results in his own notation, then transfer, when the order is established, to a standard order a>b>ec and add what is necessary to complete the delimitation. Riemann uses 6 and c for axes of rotation in all cases. ; : . : b Case I. Greatest and least axes those of rotation. Riemann gives boa>—. Here b b : : : a, ~ +< <2, and qz lr therefore < 1; thus the order is b>a>c with a possible equality in the limit. For standard order we interchange a and b when the condition is 2b>a+e, a and c being axes of rotation. The boundary 2b=a+c is represented by the hyperbola * The difficulty in working with the tables at this point is that a smal) alteration in a or > gives a much less altera- tion in the residue of (15) than in (16) or in the individual terms of either. Thus asmall error in the adjustment of a and ¥ involves a larger error in w*. ELLIPSOID, AND THE OSCILLATIONS OF THE JACOBIAN FIGURE. 81 y (1+) =2z, the part relevant is the are from O to S, and the vertex V of the hyperbola a=V2—1, y=2-—V/2 lies symmetrically with regard to the chord OS. The region concerned is a band skirting Maclaurin’s line, and the state of motion is continuous with Maclaurin’s for a=). Case II. Axes of rotation, greatest and least. Riemann takes b>c and then b—c>2a, 2a, b?—4a? and b?—a? are both positive and ca>c. The transfer to standard order as in Case I gives a—c>2b and c*(a?—b*) 2a, y?— 4a°>y°(y?—*). The part of the quartic which is relevant is half of a loop which sets out from O along the line y=2z, touches SP at P, and has at Q a maximum value for # of value 2—V3 where y=V3—1. The region is to the left of these lines, viz. within O@P; near O the hyperbola imposes the more stringent condition, and this is the boundary till the point of intersection is reached at Q where z is a maximum. Beyond this point the quartic imposes the more stringent condition and is the boundary to P. But this proves to be merely preliminary. The loop boundary is 0= 40-8 (a? +0?)+ eC= D, and D is a factor of the determinant of Riemann’s equations for ¢, S, 7. These equations have right-hand members, and with a zero determinant are not unconditionally consistent. The condition is 2Gb?+ H = 0, which is an impossible relation. The loop section of boundary is certainly not valid as a boundary up to which steady motion is possible. The intervention of a pressure condition, which in fact supersedes both algebraical conditions as the effective boundary, will be discussed below. Case III. Axes of rotation, the greatest and the mean. Riemann takes b>, and then > 1) (Ag? 102 2 2 3a

2a the integral inequality holds, and will continue to hold for values of ¢ less than 2a but not so small as a unless b becomes infinite. There is certainly a range within which the order is b>c>a, which is converted to standard order by a cyclical change, which makes the conditions 2c <(a—b), and [XP ea) Geb +e) (C2 =) 0 eee (45). 0 Pz , The first boundary 2e- =a—b is y(1—2a)=«; the inequality y (1 —2zx)>~ is observed to the left of the hyperbola up to the point 7’ at which the integral equality is satisfied, for which we know that 1>y>3. If this equality holds for a line 7P proceeding to P, the order remaining a>b>c, this line will complete the boundary. The equality (a2b? + bc? — a®c? — 4c) G — (4c? — &*) H=0 meets the curve 7’=0 or (acb? — a*b? — bc? + a*c?) G + H (ac — b*) =0, 11—3 82 Mr HARGREAVES, THE DOMAINS OF STEADY MOTION FOR A LIQUID for values making the eliminant ac (6 — c) (a—b — 2c) (a +b — 2c) vanish, ie. for b?=c* giving the point P, and for a=b+ 2c which is precisely the point T where the two boundaries meet. Near P we get 2e (ps — ps) + 5e& (p, — ps) = 3& (p: — po), 5) 9) Near P the curve lies below the Jacobian, the type of first term is the same, the sign and approximately e=3F (2L —3)— 38 (sz — 240+ of the second approximation opposite. The Jacobian is met at the point P, and again for @ (3°?—Bb)=c(4e—6) or 2 (4y—1)=37-1 giving an approximate position y=°622, x ="5417; and the curve P7 must have a point of inflexion in order to cross the Jacobian. Solutions were obtained of the equation reduced to elliptic integrals, viz. sin? a cos 8 sin y cos y (sin? a+ sin? 8 + 4 cos? y) + 2 sin* a cos* y (2 — sin? a) =(E— F cos? a)[cos* y (4— sin* a) + cos? 8 (sin? 8 + 4 cos’ y)]...(46). Along the whole length of PT the transcendental boundary this case is continuous with - the case of rotation about mean axis only. Here Riemann’s S=0 makes uw and w’ vanish, and gives also 47(4c?—b*)=G(b?—c*); the relation between G and H gives a ratio for vy: v* in the theory of mean axis which is clear of transcendentals, viz. =(a+2cP—b?: (a —2cyP —B*, and the values of v, v’ for the two cases agree. § 14. The condition of positive pressure (or o positive) is one that is little in evidence, for the reason that in all cases of rotation about one axis as well as in Cases I and III the condition is satisfied as a simple consequence of other conditions. In Case II it restricts the field within narrower limits than the composite algebraical boundary, and the equation p or «=O supersedes other conditions as the effective boundary of the case. The pressure is positive so long as F+3(2G?+ H)/D>0, D meaning 4b — b? (a? + c*) + ate? with standard order. The condition D=0 is represented by the quartic loop, D being negative between OP and the loop. The line p=0 may be expected generally to lie well within the loop, but may meet it at P or O where other quantities than D are small. Near P the equality is represented by Ps(2e — 3&* — GeE*) = 3&*[2 (p.—p;) (1 + €) +p: — 2p. + ps] which treated as before gives e = 6&(L— 1) +96 (6L - ) +e The curve near P lies below the Jacobian or the boundary PZ’ of Case III, and turns more sharply downwards. . ELLIPSOID, AND THE OSCILLATIONS OF THE JACOBIAN FIGURE. 83 We examine now the position near 0 where ¢ is small. The reciprocals of (5) give A B Cc @_A (eA) @_H_-A *'@_-AC-eS - i ad while —G and H are got by writing Aa*, Aa‘ respectively for A,.... Thus if ¢ is small the condition is reduced to P= C 3{A (@ — 26) + BE} a (a> — 5?) (a* — 46) Comparing the right-hand members of (14) € has a term with cosy in the denominator and is relatively great when ¢ is small, so that the positive character is assured unless @—4* is small. But near the boundary a—c= 2b leads to a? —46*=2ac. and in fact makes the above negative, but we can adjust 7 in a—fe=2b so as to satisfy the condition which > 0. is then : , 3A (a*— 257) + BE} des 27 (a® — 6) cosy i ; age 2A : : or as a= 2b approximately, the condition is (> oF aay” Now with y nearly 90° and 8=60° Seis) | sin? a cos @ 3 or cos y Scosy and therefore a= 60°, C is proportional to , and 24 +B to 1(F,+22,) where E, and F, are complete elliptic mtegrals for a= 60°. Hence the condition is F>4(F, +22, > 152619, using F, = 2715652, E,=1-21106. The pressure line near 0 is therefore a —fe=2b or y — 22 = fry = 27z* = 3 05242" approximately, whereas the hyperbolic boundary is y= 2x2 +227 near 0; the curve then lies above the hyperbola but has the same tangent y=2r at QO, with a radius of curvature reduced as 2 : 3. For a general position we use FP \(@—F) (F —¢)—3%}| —3 (26 + H)>0, or the boundary is Fa* (sin? 8 sin? y cos a — 3 cus‘ 8) — 3 (2Ga* co 8+ H) =0. From (47) it follows that Fa‘, —Ga?, H are respectively proportional to Acota—B+Csin*a, Acos*a—Beos8+Csin*acosy, A cos*a—Bcos 2 + € sin* acos y. In this way three points were determined in a middle range by use of elliptic integrals. § 15. Attention may be called briefly to some features of the steady motion about two axes, say 5 and c m Riemann’s notation. The angular velocities w,, a, and momenta hs, h, are constant, and moreover h.:h,;=@.:,: in fact we find mes o 10 the positive value of the radical applicable to the case in which w, w’ have like signs. (+ —2a* + V(2a +b + c) (2a —b —c) (2a +5 — cc) (2a —b +), The frame of the ellipsoid moves im such a way that any line belonging to it describes a cone with uniform angular velocity about a Hne fixed in space. The axes in most parts 84 Mr HARGREAVES, THE DOMAINS OF STEADY MOTION FOR A LIQUID of the fields for I, II and III are very unequal; the irrotational element in the motion must therefore be very influential in order to equalize the effective moments of inertia. In general the kinetic energy is not expressible in terms of o’s only, for h; being us ee ‘(a—bPwt(atbyPw'} o and w, being w+w’, the kinetic energy m == ((a—bf w + (a + bw? + (a — cf + (a +ey v} — [a —bpwt+(atbypw'} (w+w')—2(€+0*) ww’...] 2m ., ; =h,o, + hs; — = ‘(a2 +°) vv’ + (a? + B?) ww. But when w, w’ have like signs, v and v’ have unlike signs and the bracket may vanish, in which case the kinetic energy would be double that belonging to a rigid body with the momenta and velocities. In Riemann’s notation (a? +c?) vv’ + (a? +B?) ww’ = {(a? + &*) T + (a? +0?) S} x V(2a + 6 +c) (2a—b—c) (2a+b—c) (2a—b +0), the positive sign attaching to Cases II and III where S is negative, the negative sign to Case I where S is positive. The condition (a? + 6?) 7+(a°+c?) S=0 is represented by 6a‘ (Ga? + H) + H (a? — b*) (a? — cc?) =0, which cannot be satisfied for III. But for II where in standard order the relation is 6b (Gb? + H) =H (a? —b*) (b* —¢), it is satwfied for a line from O to P within the domain of II. The line leaves 0 within the region of positive pressure (angle a slightly less than 74°), quits this region in the upper part, but remains below the quartic in approaching P. To the left of the line the kinetic energy is less than the amount specified above, to the right it is greater. The existence of the line is a curious feature, which rather emphasizes the characteristics of this motion, but the line does not appear to have any true dynamical significance. A consequence of the influential part played by irrotational motion in Case II is that the status would be greatly altered by a very small amount of friction. The reduction of mechanical energy would involve a movement towards the pressure boundary where cohesion ceases and the conditions become disruptive. If this approach takes place in the upper part of the diagram where there is approximation to the spindle shape, separation into two or three less elongated bodies seems probable. If the approach takes place in the lower part of the diagram where the form is that of a thin elliptical disk, the formation of a globe and ring seems more probable. That is we postulate transverse lines of weakness for the first case, an annular line (or lines) for the second. Without professing any special con- fidence in the application of a homogeneous fluid theory to cosmogony, it seems permissible to set down the above suggestions as those which arise most naturally from this part of the subject. ELLIPSOID, AND THE OSCILLATIONS OF THE JACOBIAN FIGURE. 85 § 16. The figure of equilibrium discussed by E. Roche, where a distant attracting body M on the line of the greatest axis moves in a circle with an orbital period equal to that of rotation of the liquid ellipsoid, is represented by the equations a? (A — @ — 2v@*) = b? (B— w + vo’) = c (C + v0”), where y= M/(M+m) may range from 0 to 1. For v=0 the distant body has an inappreciable influence, and the course of solutions is represented by the curve SJP representing spheroids and Jacobians. The other extreme y=1 is the case which has been specially examined. The general course for intermediate values is easy to forecast, and some help is given by the treatment near the spherical terminus after the method of § 10. Thus (1—«#)/ — y) = 3 +1 =sec? a near S, ie. the tangent of the inclination of curves to SP near S varies from 1 to 4, and the initial value of the angle « used in the elliptic functions varies from 0° to 60°. The curves lie above these tangents and have one point of inflexion after reaching the position wo .4(1—2) for which y is a minimum. The first approximation near S also makes =a ; 4p 15(38v+ 1) For v=1, the equation of condition is sin? a cos B sin y cosy (6 + cos? y) — 2 sin? a cos* y — (H — F' cos? a) {cos? y + cos® B (3 + cos? y)} = ei @” cos’ asin? 8 sin x cos fF and '=(E — F cos?a)cos?8 + Hsin? a cos? y — 2 sin? acos BP sin y cos yi; o/ 9/ 4arp cos B Paes (48) The table gives values for four positions a Sy x y w?/4arp 70° 43° 22'3 ‘72688 “95155 016205 (i Bye 2 Y5y 59131 94298 021162 80° 63° 174 *44945 94528 “022400 85° omelon ‘28875 ‘96069 “017705 The maximum value of @/47p inferred by interpolation is ‘02255, the mimimum value of y ‘9424, the place of the former nearer to P than the latter. VI. On the Fifth Book of Euclid’s Elements (Third Paper). By Ms Jeo Me Hirer: M-A., LL.D. se.Ds F.R:S:, Astor Professor of Mathematics in the University of London. [Received 1 September, 1914.] I, PRELIMINARY. 1. THE Cambridge Philosophical Society published two papers by me “On the Fifth Book of Euclid’s Elements” in their Transactions, vol. Xvi. part Iv. (1897) and vol. XIx. part 1. (1902). These will be referred to in what follows as my first and second papers. I shall also have occasion to refer to my two editions of the Fifth and Sixth Books of Euclid published by the Cambridge University Press in 1900 and 1908, and my Theory of Proportion published by Constable and Co. (1914). Il. OBJECT OF THIS PAPER. 2. The special object of this paper is to study the Fifth Book of Euclid from the point of view of its relation to the principle afterwards known as the Axiom of Archimedes. “I purpose to set out the results which can be obtained (a) by considering this principle in connection with the Fourth and Fifth Definitions of the Fifth Book; (b) by considering how far this principle is necessarily involved in the proofs of properties of Hqual Ratios given in the Fifth Book. Ill. THe Axrom oF ARCHIMEDES AND THE FourTH DEFINITION OF THE FirrH Book. 3. Though the principle is now always known as the Axiom of Archimedes it is very clearly assumed in the Fourth Definition of the Fifth Book, which Sir T. L. Heath translates as follows: Magnitudes are said to have a ratio to one another which are capable when multiplied of exceeding one another. Thus it is assumed that if A has a ratio to B, or B to A, then it is always possible to find integers n and p, such that nA >B and pb> A. This plainly assumes the Axiom of Archimedes, and it reads like an anachronism to call the axiom by its usual name, but I shall conform to the usual practice throughout this paper. Won, JON aos Var 12 88 Dr HILL, ON THE FIFTH BOOK OF EUCLID’S ELEMENTS. IV. THe Axtom oF ARCHIMEDES AND THE FIFTH DEFINITION OF THE FIFTH Book. 4. Euclid did not consider the bearing of the Axiom of Archimedes on the conditions enumerated in the Fifth Definition of the Fifth Book, but important results can be obtained by doing this. Sir T. L. Heath translates the Fifth Definition as follows: Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, if any equimultiples whatever be taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order. Hence AS Beep, if when we take any equimultiples whatever rd, rC of A and C: and any equimultiples whatever sB, sD of B and D, then the only relations which are simultaneously possible are (Il) rA>sB, rC>sD; or (Dl rA=sB, rC=sD; or (II) rAsB, it is necessary that rC>sD; and also that if we find rC> sD it is necessary that 7A >sB. So that to express the full meaning of (I) we may say that if r, s are any integers what- ever such that 7A > 3B; then quust! eC > 2D. 22s,..3357.5- 2 ee eee (1). But if r, s are such that rC>2D, then aust 9A > sB ....5<5s.5-cn-tonsenepene eee (2). In like manner the full meaning of (II) is expressed by saying that if r, s are any integers whatever such that rA=sB, then must 70 =D) ...<..sccscescsceueenpon---noeeseeee (3), but if 70 =sD) then: must 7A = sB 2. ..-6.22/Gesesceee-- oe (4). Similarly the full meaning of (III) is expressed by saying that if 7, s are any integers whatever such that TA < sB, then anus) 70 < sD 2.2 .%tyc8. soe eee ee (5), but if rC < sD, then must ré'sD. Then r@—sD is a magnitude of the same kind as D, and now introducing Archimedes’ Axiom we can assert the existence of an integer n, such that n(rC —sD)>D, . nr > (ns +1) D, but since rA =sB, *. nrA =nsB <(ns + 1) B, so that the integers (mr) and (ns+1) are such that (nr) A <(ns+1)B but (nr) C > (ns +1) D, which is inconsistent with condition (5). Considering next the combination (y) with (6), i.e. rA >sB, rC=sD, it can be shewn in the same way by interchanging A with C and B with D in the preceding proof that this is inconsistent with condition (6). It has therefore been proved that the combinations (a) with (6), (a) with (c), (a) with (8), (a) with (+), (8) with (c), and (y) with (db) are inconsistent with the conditions (5) and (6). Dr HILL, ON THE FIFTH BOOK OF EUCLID’S ELEMENTS. 9] Hence, if the conditions (5) and (6) are satisfied, the only combinations which are possible are (a) with (a), (8) with (b) and (y) with (ce), which are exactly the combinations permitted by Euclid’s Fifth Definition. Hence, if Archimedes’ Axiom be assumed, the satisfaction of the conditions (5) and (6) is a sufficient test for the proportion Al g Uae SOIR ID), If we take the Fifth Definition as translated by Heath and strike out the words “alike exceed, are alike equal to, or”, the remaining words would embody conditions (5) and (6), though the meaning would not (I think) be very clear. A similar demonstration to the above will shew that conditions (1) and (2) are a sufficient test of the proportion Al GIB 28 (016 10) These, with a like reservation as to the obscurity of the meaning, would be represented by striking out from Heath’s translation of the definition the words “are alike equal to, or alike fall short of.” 7. It will now be shewn that (1) is equivalent to (6). Tf (1) holds, then all values of 7, s which make rAd >sB also make rC'>sD. Now suppose that r,, s, are such that 7,0 s,B, le. we have either (a) mC5,B. In case (a) s, D—7,C is a magnitude of the same kind as C, and therefore by Archimedes’ Axiom an integer n exists such that n(s,D—r,C)> C, ~. (nr, +1) C0 < ns, D. But since 7, A =s,B, Seu WAG SWE . (nr, +1) A >ns,B. Hence the integers (n7,+1) and (ns,) are such that (vr, +1) A >(ns,) B, but (nr, + 1)C <(ns,) D, which contradicts the condition (1). In ease (b) TA >s,B, but 10<-s,D. This also contradicts (1), for if 7,4 >s,B, then (1) requires that 7,0 > s,D. Hence neither (a) nor (b) can hold, and therefore if 7r,C s,B, to prove that 7,C >s,D. For, if not, either rC=s,D or 7,0 B, *. mr A > (ns, +1) B. But since r,C=s,D, >, nr,C =ns,D, *. nr, C <(ns, +1) D. Hence the integers (nr,) and (ns,+1) are such that (nr,) C< (ns, +1)D, but (nr,) A > (ns, +1) B, which contradicts (6). In case (d) mCs,B, which also contradicts (6). Hence neither (c) nor (d) can hold. Hence if 7,A >s,B, then must 7,C>s,D, and therefore condition (1) holds, Hence by the aid of Archimedes’ Axiom it has been shewn that if (1) holds, then (6) holds; and if (6) holds, then (1) holds, Hence (1) and (6) are equivalent. By interchanging in the above proof A with C and B with D, it follows that (2) and (5) are equivalent. 8. To sum up, the six conditions (1)—(6) involved in Euclid’s Definition can be reduced by purely logical processes only to a smaller number in three ways, viz. to (1), (3) and (5); or to (2), (4) and (6); or to (1), (2), (5) and (6). If in addition to purely logical processes the truth of Archimedes’ Axiom is assumed, then the six conditions can be reduced to two in the following ways, viz. to (1) and (5); or to (2) and (6); or to (1) and (2); or to (5) and (6). Further, (1) is equivalent to (6), and (2) to (5). The reduction to the pair (5) and (6) possesses certain advantages in dealing with some propositions over the other forms (see Arts. 11 and 14 below), Dr HILL, ON THE FIFTH BOOK OF EUCLID’S ELEMENTS. 93 V. THE PROPOSITIONS IN THE FirrH Book oF Euciip. THEIR DEPENDENCE ON THE AXIOM OF ARCHIMEDES. 9. The next step is to classify the propositions of the Fifth Book. G) The first group consists of propositions dealing with magnitudes and their multiples. These are Nos, 1, 2, 3, 5 and 6. With these should be included the following proposition : “Tf A, B, C are magnitudes of the same kind, and if A be greater than B, then integers n and ¢ exist such that nA >tC >nB.” The proof of this forms the greater part of Prop. 8, and it depends on Euclid’s Fourth Definition, so that Archimedes’ Axiom is involved. This proposition belongs properly to this first group, because it does not deal with ratios. In order to distinguish it from Prop. 8 I will refer to it in what follows as the principal part of Prop. 8. The only place in the Fifth Book in which the Fourth Definition is used explicitly is in the proof of this principal part of Prop. 8. (1) The second group consists of propositions dealing with Unequal Ratios. These are Nos. 8, 10 and 13. The proofs of these depend on the Seventh Definition, the test for distinguishing between Unequal Ratios; whilst the proof of Prop. 8 (as has been already noted) requires also the Axiom of Archimedes. The propositions in this group are used sometimes singly and sometimes all together in the proots of Props. 9, 14, 16 and 18—25. (iii) The third group consists of propositions dealing with Hgqual Ratios, which depend on the Fifth Definition and do not necessarily require the Axiom of Archimedes. These are Nos. 4, 7, 11, 12, 15, 17 and 18. Euclid’s proofs of all of these except the last do not require the Axiom of Archimedes. His proof of Prop. 18 assumes not only Prop. 8, and therefore the Axiom of Archimedes, but also the existence of a fourth proportional to three magnitudes, of which the first and second are of the same kind. Simson gave a proof free from either assumption. It is essentially the same as that in Art. 154 of the 2nd Edition of my Euclid V. and VI. Another proof of Prop. 18 is given in Art. 14 below to illustrate the power of the proposition in Art. 6 above, but this assumes the Axiom of Archimedes, because that Axiom was employed in proving Art. 6. (iv) The fourth group consists of propositions dealing with Hgqual Ratios which require both the Fifth Definition and the Axiom of Archimedes. These are Nos. 9, 14, 16, 20—23. 94 Dr HILL, ON THE FIFTH BOOK OF EUCLID’S ELEMENTS. Euclid’s proofs of these propositions are made unnecessarily indirect and therefore difficult by his use of Props. 8, 10 and 13 in their proofs; thus bringing in the idea of Unequal Ratios to prove Properties of Hqual Ratios. It is shewn in the works mentioned in Art. 1 that these propositions can be proved by » the aid of the Fifth Definition and the principal part only of Prop. 8; and it is shewn that then Props. 14, 20 and 21 can be treated as particular cases of Props. 16, 22 and 23, whilst the Euclidean method requires that Prop. 14 should be proved first as a stepping-stone for Prop. 16, and in like manner Prop. 20 for Prop. 22 and Prop. 21 for Prop. 28. (v) The fifth group consists of propositions dealing with Hgual Ratios which depend on propositions in the third and fourth groups. These are Nos. 19, 24 and 25. Euclid in his proofs employs only the propositions in the third and fourth groups. He does not make any direct use of the properties of unequal ratios with which the second group is concerned. Inasmuch as proofs of the propositions in the third and fourth groups can be given which do not necessarily depend on the properties of Unequal Ratios, it is possible to regard the propositions in this fifth group as not depending necessarily on the properties of Unequal Ratios. They do however depend on the Axiom of Archimedes. Enclid’s proofs are I believe the simplest which can be given. The proofs given in the works mentioned in Art. 1 of the propositions in the third and fourth groups are such that each proposition is deduced directly from the Fifth Definition, those in the fourth group requiring also the Axiom of Archimedes; but the proof of each proposition is independent of all the others. In my second paper I attempted to obtain similar proofs of Props. 19, 24 and 25, but these, as will be seen on reference to Arts. 70—73 of that paper, are very complicated and indirect. I asked my friend Mr Rose-Innes if he could find something simpler. He has sent me those which now follow, Arts. 10—12. It is possible that no further simplification can be attained, but they are not as direct and the steps do not follow so automatically as those which I have given of Props. 16, 22 and 23 in my Theory of Proportion. I give also (Art. 14) a proof of Prop. 18 which will illustrate the power of the proposition in Art. 6. Buc: v. 19. 10. Let A, B, C, D be four magnitudes of the same kind such that Ai =O sarang eA 0.) “Bis: to prove A-—-C:B-D=A:B. Take any multiple of A, say rd; and any multiple of B, say sB. There are three possibilities, (i) rAsB. Consider (i). Since rA < sB, Dr HILL, ON THE FIFTH BOOK OF EUCLID’S ELEMENTS. 95 therefore sB—rA is a magnitude of the same kind as A, B, C, D, and since A>C, A —C is a magnitude of the same kind as A, B, C, D. Hence by Archimedes’ Axiom an integer p exists such that p(A—C)>sB-rA, -. pA >(sB—rA)+ pC. Hence* an integer q exists such that pA>q(sB—rA) > pl. Now since pA >q(sB—rA), -. (p+qr) A > (qs) B. But A:B=C:D, “. (p-+4r) C> (qs) D, OHTA (GT) Ce (QS) LD aceneer aeons ae scc ore shcckonsevanie cect (1), but Gj (GLE IED ey NON, Weteococonoodud ase ccencese ea nea aRRS A (II), ere gl Gls. —a7Al)) (Gir) C>(qs) Dee eseee sees: from (1) and (II), * sB—-rA+rC>sbD, ~. r(A —C)sB. Then since TALS therefore integers p, q exist such that pA >q(rA —sB)> pe. Since pA >q(rA —sB), (qs) B >(qr— p) A, therefore we have provided that p< qr), (qs) D>(qr—p)C because A: B=C: D. 5%, (OOS GRE Dea (010) Oe coceonacndodcatnoadcnteecsepneceeee (III). But q(rA —sb)> pC . g(rA —sB) +(gs) D> (qr) C............ from (III) and (IV), 7A — sB-- sD >7-C: . r(A—C)>s(B-D). * If X>Y+Z, an integer g exists such that X>qY>Z. Wor 2050015 ISG WA 13 96 Dr HILL, ON THE FIFTH BOOK OF EUCLID’'S ELEMEN'S. If however p (qr—p)G, for the left-hand side is positive and the right negative or zero, . pO + (qs) D > (qr) C. But q(rA —sB)> pC, w. g(rA —sB) +(qs) D > (qr) C, - rA —sB+sD>rC, *. r(A—C)>s(B—D) as before. Hence if rA > sB, then r(A—C)>s(B-D). It results from the three cases considered that A—C:B-—D=A:B. Eue. v. 24. [This demonstration illustrates the power of the theorem in Art. 6.] 11. To shew that if A:C=X:4Z, and if Hija Ch We then A+B:C=X+Y:Z. (1) Let us suppose r(A+B)D because PD is the least of the quantities. Hence, assuming Archimedes’ Axiom, integers p, q exist such that pB >q(C—D)> pD, -, g(C—D) >pD, *. gO>(p+q)D. But PAu oi Ones . 595 OL Sak (0 BIC )W oy apanconnton se basooo pegpooRceracBeHasencsed (1), but DBS Gi (CD) ™ zeeeresasces sees s He Seeneee ence (11), therefore from (I) and (II) gA >q(C —D)+qB, -. A>C—D+B. Since D is the least, this inequality shews that A exceeds B by C—D, and A exceeds C by B—D. Therefore A is the greatest. Moreover the inequality shews A+D>B+C. 13. There is a certain resemblance between Mr Rose-Innes’ proofs of Euc. v. 19 and Euc. v. 25. If we compare the inequalities marked (1) and (II) in v. 25 with those similarly marked in v. 19 or with those marked (III) and (IV) in v. 19 this resemblance will in part appear. It rests on the basis that either proposition, having first been proved independently, can be used to prove the other. Euclid’s proof of v. 25 depends on vy. 19 and other propositions. I will now shew how to use v. 25 and other propositions to prove v. 19. 13—2 98 Dr HILL, ON THE FIFTH BOOK OF EUCLID’S ELEMENTS. Starting from A:B=C:D, -, rA:sB=rC:sD (Vv. 4). Suppose, as in v. 19, that A>ZJ, B>D. These give rA>rC, sB>sD. Ga) If now rA >sB, then since rA :sB=rC:sD, ~rG >1sD: Hence rA is the greatest and sD the least of the four magnitudes in the proportion rA :sB=rC:sD, .. TA+sD>sB+rC by V. 25, *. r(A —C)>s(B— D). If therefore rA>sB, then r(A—C)>s(B—D). qi) If rA=sB, then rC =sD, *, r(A—C)=s(B—D). If therefore rA=sB, then r(A—C)=s(B—D). (iii) If rA rC, and sB>sD, therefore sB is the greatest and rC the least of the four magnitudes in the proportion rA:sB=7rC:sD, .sB+rC0>rA+sD by Eue. v. 25, *. r(4 —C)sB. In discussing this inequality we should have been obliged to discuss the three cases FSG PBR Pee This justifies the remark at the end of Art. 8 that the use of conditions (5) and (6) may sometimes be more convenient than that of (1) and (2). VIL. The Invariants of the Halphenian Homographic Substitution—to which is appended some imvestigations concerning the Transformation of Differential Operators which present themselves in Invariant Theories. By Major P. A. MacManon, F.R.S., Hon. Mem. Camb. Phil. Soe. [Kecetved 16 June, 1914.] INTRODUCTION. THIS paper follows naturally the one published in these Transactions in 1908 under the title “The Operator Reciprocants of Sylvester's Theory of Reciprocants.” The particular object in view is the study of the invariant operators of the theories of invariants and reciprocants and the transformation of those operators. There is great advantage in adding operators to the invariant material dealt with. It was not at first recognized that the operators were effective because they themselves possessed invariant properties. The relations which establish those properties shew the exact conditions under which the operators are effective either as generators or annihilators. In certain cases homogeneity or isobarism or both may be necessary in the algebraic forms; in others the forms must possess properties in regard to other differential operators. The two simple substitutions of Sylvester and Halphen, both of period 2, suffice to disclose and elucidate the invariant properties and to discover the relations that exist between the two theories. What I have called the b transformation, that was brought to light in the first paper, is herein further examined in regard to the special logarithmic case and two new transformations, the h and the s, are discovered and examined. Transformations are shewn to exist which bring the seminvariants and pure reciprocant defining operators to the simplest possible forms, and shew instantaneously a complete system of ground forms in each case. The paper is divided into two sections—the first deals entirely with the Halphenian substitution, and the invariants, algebraic and operational, are exhibited in their categories. Attention may be directed to the symbolic method of Art. 8. Section II treats of the transformation of linear operators in general, with special reference to the subject-matter of Section I. Vout. XXII. No. VII. 14 102 Mason MACMAHON, THE INVARIANTS OF SECTION I. ON THE INVARIANTS OF THE HALPHENIAN HoOMOGRAPHIC SUBSTITUTION il ve = ¥ ; y = xX Ss 1. If we consider the binary quantic (do, h, oy ses) (46, v)", and y any function of #, we may suppose that : 1! ds**y = ——— ee * (s+ 2)! das If we now make the substitution 1 ¥ “x Y=xX the invariants which are such that they are homogeneous and isobaric functions of L£ OG (8 aoe are, as is well known, seminvariants of the binary quantic, and conversely every seminvariant is an invariant of the Halphenian substitution. They satisfy the well-known partial differential equation Og = 0a, + 2,00, + 320g, + +..=0. In the paper communicated by me to the Cambridge Philosophical Society in 1908 I considered Sylvester’s substitution RN, GX the invariants of which, when homogeneous and isobaric functions of a, d), d2,-.., have been called pure reciprocants. Such satisfy the differential equation Va = 4. 4$4;70q, + 5 (doa,) Oa, + 6 (Apa, +40,2) Og, +-.. = 0. Certain forms arise from both sets of substitutions and are thus both seminvariant and pure reciprocant functions of a, a, a:,..... Such are invariants for the general homographic substitution _uX+mV+y, pa eX + mY +r AX+pV+p ’ i AX+uVY+yr’ and have been named by Sylvester projective reciprocants and also principiants. In fact principiants may be defined by the simultaneous partial differential equations OF 0 ya 0! Since moreover O.=— Vz, 1 where O=7; 0 b 2, =—-—, b," 3 Byeezos 3c, = — >-, Ost ion Oy THE HALPHENIAN HOMOGRAPHIC SUBSTITUTION. 103 ] by as 9 % ip b.=-2-, Co b SG oCre 2 Cot Cy ie Ac, i 30¢,c, 40c,* 7 Cy Co Cole it follows that principiants are seminvariants which remain seminvariants after the substi- tution of — 2a, —38a,+ 8a, —4a;+ 30a,a,— 40a,%,... for hh, (Ps. Qs, ... respectively, and also after the substitution of —ta, —4a.+2a°, —1ta,+ }a,a,—$a,',... for the same quantities. Ex. gr. The principiant Ay? Ay — BA) 4,0, + 20,5 is, to a power of a, prés, merely multiplied by —3 by the first substitution and by —+ by the second, as may be readily verified. 2. In this Section of the paper I discuss the general invariant theory of the Halphenian substitution with particular reference to the invariant operators. I write in a usual notation , dijery Lidty. dumeeeslcaar ey. dV 1 aY Te d A,», where s<¢ 2. dX Then, as shewn by Halphen, s! dx? p(T), \ & = X*A,, f Ay\ a=—X°(A, + Y)? er Ae fS\ Age S\ Agee A,) — (_\s VY 28+3 s s-1 s—2 oO, ay = ( ) xX 14, + (4) XY +(5) x2 + yal 14—2 104 Mason MACMAHON, THE INVARIANTS OF The Halphenian substitution, like that of Sylvester, has a period 2; it follows (i) that the substitution may be given the symmetrical form rd ve: Ye a (ii) that, in any relation involving the letters Ai@, (hy Og Cratare eee. Al. An wane we may interchange the small and capital letters. We may further consider the operator symbols OG Gs Gia Graces Gry Gre Grey Wiig ccc and then if TIGA G2 AGS CAs can Gy Gn hep Gey Cinco ») acpi ere XG His Ans A,, cee Oy, Cx, Or, Ons Wie ae) the interchange of small and capital letters shews that FQ, et, Ue; Gy, --- Oy; Ox; 03, On Oars oe VE PCY; Fb, Oy5 Cy; ..nOys Ore Gwe aed is an absolute operational invariant of the substitution of even or uneven order according as the upper or lower sign is taken, In fact every symmetric function of UF 2s ee PY a ieee) is an invariant of even order and every two-valued function of the two functions is an invariant of uneven order. Thus the relations (Rel. z2dp _ Xi A,, yield the invariants (1 +2°) a, ZO of even order and (1 —2*)a, of uneven order. 3. We assign to the letters y, «, t, a), @,...a certain degree 7 and weight w and deduce the characteristic 37 + 2w of the letter. We write » for the characteristic and then we have the following scheme: y 1 t iy ae (hy Oye Ox | Con On emcr One i 1 0 1 1 ] 1 -—1 0-1-1 -1 -1 w —-2 -1 -1 0 1 s 2 hl 1 0-1 —s vy —-l -2 1 3 5 2s+3 1 2-1-3 —5 ... —(2s+8) Vem Lae eee Ave Oy Oxy Op O04, Q4,.-- 04, i 1 Opal 1 Ras TD: 1 —-1l 0-1-1 -1 ... —1 w —i1 1-2 -—3 -4 ... —(s+3) 2 4... s+3 v 1 2-1 —-3 -5 ...—(2s8+4+3) =—1—2) ly 63) Vee ees It will be remembered that for Sylvester's substitution the characteristic was 3i+ w; here it is 3i+ 2Qw. THE HALPHENIAN HOMOGRAPHIC SUBSTITUTION. 105 The degree, weight and characteristic of a product of symbols are each of them formed by adding together the numbers which appertain to the symbols. The investigation is much simplified by an accented notation; we multiply each letter or symbol (except “) by # raised to the power of half of the characteristic of the letter or symbol; similarly for the capital letters. Thus we put a ty=y, NOSIS, pe ar. NG Sis x2 Uy =o XK A, =A,, eee Kid,=- 4), a 2s+3) q — (-)° ane X22s+3) 4. =(—) VARS ‘ The change of sign will be noted. Also that what we have written ye(2s+3) 4 s is first of all 2~ 225+) A, in obedience to the law of formation. The relations of Halphen become ye, t=—-T-Y*, a =A,, (= A aa On = ANIA Any, wherein there is no occurrence of w or _X. For the accented letters we have the scheme: 2 Os AR OR ino) ety Op Ga Os Ghe Grisso @ps v 1 0 i 1 il 1 —1 Oo -1 -1 —-1 —1l eas 10 =} —F =F = § i 0S meee 38 3 v 0 0) 0 0 0 0 0) 0 0 0 0 0) ji ay ee OE SR Des TOs! Woes On Ou ts HO a 1 0 1 1 ieee 1 -—1 0 —-1 —-1 -1 ...-1 eee Os = pe es 2 Onn ene 0 So we SO we 3 v @ @ @ One 20 0 so that each of the modified letters has the degree 1, the weight —% and the characteristic zero; while each of the differential inverses has the degree —1, the weight +% and the characteristic zero, Moreover every combination of accented letters and symbols has the characteristic zero. 106 Mason MACMAHON, THE INVARIANTS OF It is to be observed that in every relation in accented notation it is permissible to inter- change small and capital letters, both in symbols of quantity and in differential inverses. The relations a} =A, —A;, ag =A, —2A,;+A4;, &e. indicate that every function of Gas stare is a function of differences of the quantities JW, Ales Ais abeoe and thus satisfies the partial differential equation 04, +04, +04; +--. =0. In fact we will presently establish the formula of transformation Oas = 04, +04; +04; +---, which is the analytical statement of the observed fact. The Linear Invariants. 4. Making use of the principle of invariance above set forth, the relations yield the absolute invariants of even order : y ’ : My y ay — 2a, + 2az', ad — 3a, + 3d2, a, — 4a,‘ + 6a, — 4a; + 2a;, a, — 5a, +10a. — 10a; +5a,,, from which is derived the reduced set Accented Unaccented 1 - = Y, omy, ee ©? Ao, . . 5 z a, — as, — £20, — 27s, a. — 2a. \ 5 Ont i ls — ZO, + Q,, ZB Ae + 2272 As + U* Ay, MAN an : 9 Apes 13 1s a; — 3a; + 3a; —a,, — £2, — 3x7 a, — 3x% d;— ©? Ug, Similarly we derive absolute invariants of uneven order which may be exhibited in the reduced forms Accented Unaccented y +20, a ty — Qe t, a OT — a2 &— Qn ane a, — 3a, + 2a,', —x a, — 3a dy — 2 ds, — a, + 4a; — 5a, + 2a;, ~ at (ly — 4c ay —5a? a, — Qu? ds, ‘ ~ 4 ‘ an . ‘ 2 eee! 13 15 rire ds — 5a, + 9a; — Ta, + 2a, — v2 a,— 5a? a,— 9a as — Te? ay — 2e2 a, COCO K sera ence eee seen eseeeeseseseesss -§ 駧§ ee eseneereseeesceseneseseces THE HALPHENIAN HOMOGRAPHIC SUBSTITUTION. 107 The general form here in its unaccented aspect may be taken as ge (28+3) i +(s+2) ( LAg4, + $ (8 + 3) a) Lbs 49 + 4(s + 4) G) Pdgrs +... + 2ath dasa} ; 5. These invariants, linear in the quantities, may be exhibited in symbolic form by writing and then Psd and we derive the relations PCy ie (UP), G2 pa (yl — 22), shewing that ps1 — p)§ (1 — 2p) = (—)' P21 — Py (1 — 2P), which establishes that pd = py (1 — 2p) is an absolute invariant of even or uneven order according as ¢ is even or uneven. Moreover the identity (1 = 2p)? + 4p 1 — p)=1 shews that we may exhibit the invariants of even order by forms pd — py’, and those of uneven order by forms p*(1 — p)* (1 — 2p). The sets of linear invariants above set forth are given in these symbolic forms by the successive integer values of s. We have thus obtained the whole of the invariants which are linear in a, a, ds, .... 6. The theory of the invariants of higher orders in a, a, ds,... 18s very simple because we may take alternative symbols p, q, 7, ... on the one hand and P, Q, R, ... on the other, where p+P=q+Q=r+Rhe=...=1, and then (p- q)*2 (p- r)*3(q— De ong = (et as est --(P—Q)* (P— R)*8(Q— BR) ..., and we thus obtain the whole series of invariants which present themselves in the invariant theory of an ordinary binary quantic. In fact we obtain all forms which satisfy the equations 1g = Ay Oa, + 20; Og, + Bay Og. +... = 0. Ex. gr. (p—qyv =pg — 2p'¢ + pg = 2 (ay as — a”) = 2x? (aya. — a,°), and &? (Q)Q_ — a;*) = X° (A, A,— A{’), exhibiting the absolute invariance of «° (a). — dy"). The fact that the operator 2, is itself an invariant under the substitution will shortly be established. 108 Mason MACMAHON, THE INVARIANTS OF The Invariant Operations. 7. Let the functional equation y— jie) become by Halphen’s substitution Y=¢(X). Let.z, y receive simultaneous increments Er, 1Y; and let the increments received by X, Y in consequence be ERG (ane y + ny=f (a + &x), Y+HY=¢(X +24), where by reason of the substitution involved (+&)+5)=1, 1+” = 1+ H=——=(1 5); so that and we deduce the relation _ wr! 3S By Taylor’s expansion we find ny = téa + E70? + a, E%a + ..., Ee A eX AG ee eee = Z H Now, since 7 = Moreover £ = ee ; a xX 3 W _5¥) (HY 1Y so that aes cy ra (ey 3x): a 3 fis ny HY : Obtaining the expressions of Fe Sy from Taylor's expansion above, and substituting herein we find 1 es ey a ee a (35 + Mo Ea + a FFa* + 2) =- xi (7-5 y tAEX +4,5x"+...) ; but we (+-52) ee rk (7-5 S) so that we are led to the relation o} (aye + ayE%0? + asfa* +...) =—X4 (AEX + A\EPN? + A,EPX" + ...); or proceeding to the accented notation ap & — a, £* + a &*—... =— (ASB — A, 2+ ASB. (1+&)(1+8)=1. +), where THE HALPHENIAN HOMOGRAPHIC SUBSTITUTION. 109 From Halphen’s relations we deduce 0, = X0y + 07, 1 = xy? (—)'0_, = X-#8 (04, — X04, + X794,,,—---)5 and herein writing a Oy = Oy, xt Oy=0r, w4,=-%, X~tp=-dr, o 2(@8+3)9, = (-)°0,,., X-2O+e) 9 =A oe, we find Oy =0y —Or, Op =— Or, 1 Gy Oa = On, aF G F ) Clan, SF 5 *) Oa §+2 +. 8. To obtain these relations in symbolic form write f=a,, OD) iCh rp IP = A ny | DE Ss and then put symbolically (ae, =H (84, = Ks when we find that the relation (0,-8.4 (77 *\ Oat : K s+1 may be written fae =)" G a Zz) : Moreover, if s=—1, k°=K® yields 0g =—dr7; if s=—2, i= -z-¥ yields 0y =dy:—Or.. The important observation is now made that the symbols &, K are in fact related in the same manner as &, & for, from the relation (14+& (14+ 8)=1, — 8+1 we at once deduce gen (jn (=) 5 Hence we may regard € and & as symbols such that B=(-Yee» Ba 8e, This remarkable circumstance points to the important fact that in the relations ay & — a; £2 +a, — ...=— (A.B —A, B?+ A, BS — ...), (1+ &)(1+8)=1, Wor, XO:OUi, IN@; VWAUE 15 110 Mason MACMAHON, THE INVARIANTS OF as well as in any rational integral relation connecting the quantities \ \ A A [BAIR cits Ui Cee with OY EAR AGS we are at liberty to write Ef = (ones Bt =(—)" 4, where s may have the values al fee 0 hed Wes ee and (Ma an ssih, bone UAL Ue From the relation (1 +£)(1+ 3) =1 are obtained the useful relations (1+ &)0:=-(+2)éz, which indicate that, gudé invariant functions of &, £70: and (1+ &)0 are invariant operations of uneven order. . . : even : : When performed upon invariant functions of | oon order they produce invariant : uneven functions of order. even Invariant Operators of Zero-order in the Coefficients. 9. These are all obtainable from the relation 0+804+23)=1, for Se aS so that Est (— (4) is an invariant in symbolic form, of even or uneven order, according as the upper or lower sign is taken. We obtain the two series Even order Uneven order go, QE +E", ani : E’ fe = ; eaves Bae a - aac + Observe that by reason of the difference in sign, for a given value of s, in the relations &* =(—-) Oates Be =(- YH Oy : THE HALPHENIAN HOMOGRAPHIC SUBSTITUTION, 1a even order yields an unsymbolie form of “7©Y°? uneven z ven a symbolic form of order. We thus obtain the invariant operations Uneven order Or, Oay + Ons + 0ag + ..-, 20ay + 20a + 30ag + 40ay + -.-, B0ag + 60g, + 100, + 150, + ..., 20,2 + 405, + 100a¢ + 200c¢ + ..-, 2s 2s+1\ . Dis, 20 a95_, +( “ Oa'gg + ( ) ) Berm t ( 3 ) Oa'se 42 + /2s+1 2s+4+2)\ . 2s4+3 ( 1 ) Oarss ta a ( 2 ) Canta + ( 3 ) Oa'on4s +... From these we may derive a reduced set Or ‘) Cay t20a,’ + Cag 5 Oas + 200g + 30a. +.-., and, for s>0, 2s+3 2s + 4 » (28+ 5 ( 3 ; Caves Be 2 ( 4 ) CR oe 3 ( ~ ) Oo'se45 “Fisdiss The unaccented expression of the operations is = av 204; £2 (05, = a7 Ch WR hes = aon) =9 F & *(0,, — 24 Oo, + 34 Og, — -.-); In obtaining these it must be remembered that the characteristics of a, and Oa, are the same numerically but differ in sign. Even order 204 — Or, 209 + Gay + Cag +--+; 20a + 30ag + 40ag +++; 20ay + 30ag + 60a + 100g, + ..., 2s Qs + 1 (% 4-2 G ) ®t (9 ) Orsi t | 3 ) Oni, 2s+1 25+ 2 200, + ( 1 ) Owes 4s LE ( 2 ) Carat ses for s>1. 15—2 112 Major MACMAHON, THE INVARIANTS OF The reduced set is 20y — Or, 2Oqg + 0a, + Ong + ---> 20ay + 30ag + 40ag +--+, 20ag + 50ae + V0ag + ---> of which the unsymbolic forms are Qa? Oy + a *a, a ® (20q,— 210q, + ©? Oa,— ++); a * (20q,— 327 Oq,+ 402 0a, — ---); xc Ay (20, — 5a as ae Dita Oas as, -), To verify the first of these we have Qabdy +x 20, = 2X —2 (XOy +07) + X? (- r") = ON ep aaetan. We may also note that = = so that any is an invariant operator in symbolic form which denotes in unsymbolic form an operator of uneven order; it is s s+] Qe st (7) Bent (OS) Brana te and in unaccented form at f S\N el\ Hoa -(*) arte (2) arta al Operators of the First Degree in the Coefficients. 10. We have before us the two relations p+P=1, (1+8(+8)=1, and the established relation (ap €— a, &* +a, & —...)=—(A, 5 — A; B+ ASS —...). It yields the relation Ay Oa + 1, Oay + 1, Cay +... = Ag 04g + As 04; + As Oy + -- or, as we shall write it, Ty =Ly,. This gives us the invariant of even order Tq = Ay Oag + 4; Oay + Me Oag +...- In unaccented form this is Jip. = 0a + Gr0a, = a = A044, + A,04, +... =I4. THE HALPHENIAN HOMOGRAPHIC SUBSTITUTION. To obtain the operator in a form which is wholly symbolic we write Gs =p, AY = Re ae and we then have rT apes 14 Pe’ so that I a is the sought symbolic form. This can be obtained directly from the relations ; pt+P=1, (1+8&0+8)=1, for these may be written and by addition we get 1 p+g=—-(P+a), ee tee oF ieepey LPs: ’ :)- tie (=) ) Since (ee = (( (58 we find by expansion and interpretation / Ma s wes s+1 Ns ao 20,.+(5) ay Ba, + ( 9 \ ag GS. Sp oben = {4304.4 (f) tet ($7) araenu tf. This in unaccented notation is s+1 ot Lag + (2) ar20, + (251) ade, +} + 2 establishing that s+1 gst |e 4 @) AO a, Ve ( x ) 20 a5 44 FP at 113 is an absolute invariant of even or uneven order according as s is uneven or even. For s=2, we find Ay Oay + 20; Oa, + Bas ag + --- = — Ao 04+ 2A, O4y + 3A2 04+... or in Sylvester's notation Qe =-— 24, and 270, = — X704. Q, is the operator which causes all seminvariants of a binary quantic to vanish. We see that, in this theory, 2", is an absolute invariant of uneven order, The operation either causes the invariant operand to vanish or produces an invariant of contrary order. 114 Mason MACMAHON, THE INVARIANTS OF The absolute invariance of J, defined above clearly shews that every invariant is homo- geneous in the letters a), th, Me, .--- The above series of invariant operators, linear in a,, @,, as, ..., can also be obtained by repeated operation of £70; upon £/1+p£. The former operation gives invariants because E70: = B70z. The relation us ae aes yields Ay Op — Ap Oy = A, de — Ap Or, equivalent to ©? (,0; — Aydy) = X* (A,07 — AyOy), establishing that a? (4,02 — Gy 0y) is an absolute invariant of even order. The operation of (1 + &)é@: upon ae: gives eee (1 + p&)’ and yields the invariant of uneven order ay Og + (2a; — Ay) Oa, + (Bas — 2a,*) Oa, + (Aas — Bas‘) Oa, +... 5 equivalent to Ly 0 + (2a, + Ay) Og, + (Bad, + 2G,) Oa, + (Aad; + Bay) Oa, + ---- The operation may be repeated indefinitely. The operator Wa = 4; 0a; + 22 09; + 3d; 0a, + ++. - 11. The symbolic form is ae equal to (1 — P shewing us that We=—-—Q4+ We; and establishing that 2We- Oe is an absolute invariant of even order. + PEy’ It is equivalent to 2W,-—«#"0, because We = Wa. We see from this result that W, is not an invariant, so that every invariant is not an isobaric function of a), a), ds,...; but that this is the case, exceptionally, when the operand satisfies the equation O7— 0! 12. Two operators now present themselves for examination, viz. Accented Notation Unaccented Notation we =—tde + Wa, w, = — td, + Wa, wy = — 27 dy + We, Wy = — 2ydy + Wr. Since d=-T’-—Y"*, de=-0r, we find Wr =-(1°+ Y)0p—OQ44+ We = Wr — YOr -—Q4; so that 2W»—y'd—Q is an invariant of even order. Ss THE HALPHENIAN HOMOGRAPHIC SUBSTITUTION. 115 This is equivalent to 2W, + 210, — 27Q,4. Again, Wy=—2Y‘(0y-—dr)+ Wr —Ydr-O4= Wert Y'dr—-Qy, so that 2Wy ty or —Qa, equivalent to 2W, —2# 1 y6,—27Q,, is an invariant of even order. The operators Ie=toe +L, Ti —t0p-+- le. equivalent to Ty =y oy + Le, I, = yoy + I. 13. We obtain easily Ty, =Ip. + Y0r, I, =Ip— XY 07, equivalent to I, =Ty-, Mi —eligs establishing the invariants of even order 2 Sea & Io The fact that 7, is an invariant shews that every invariant is a homogeneous function of Y, %, Gy, A2,...; but since J; is not an invariant, every invariant is not a homogeneous function DEA g G5 Uns sa: Transformation of a General Multilinear Operator. 14. Write 1 ‘ F ; ; m (ao + a) U+ ds U+ 2.) HA ing HAUTE pe t+..., and consider the operator HQ mo Oa’, + (Me +) Oma O, + (p+ 2v) a p28 aa On On+2 Herein p, vy may be any real numerical magnitudes, zero included; m also may be any real numerical magnitude, but will usually be a positive or negative integer and more usually still a positive integer. The zero value of m has been shewn by Hammond to be connected with the function log (a. +a, u+a, w+...), and will be considered later. n may be taken to be zero or any positive integer. The operator under consideration may be briefly written (fw, v3; Mm, n)y =(—)” areN Goes ei) (,v3 Mm, N)q, a relation indicating the accented and unaccented forms respectively. Write as =a E—a,E?+a,&—-..., Ae — Ay = — Ay s+ A, Se —..., so that, as has been shewn above, ag =— Az. 116 Mason MACMAHON, THE INVARIANTS OF In the symbolism explained above, (—)"" (uw, vs mM, N)g: = = = v) esis ag” ae Te Tp Og ar, and to carry out the transformation we have the relations ag ee Ae E aaa ene ’ E*0¢ => = Oz We then find (—)"*) (pw, vs m, N)g = (-)"? & = v) Smt (i 2k a) at Aa 2: pen (1 + E)—ntm A=™ O= Az} r Write now (1+ 2)F= 30,23, so that C,, -(") if & be a positive integer >s. We then have (—)"7 (mu, v3 mM, N)q be pe ear Ton = > S =F) C;, Ss = m+s+1 Az m s \m ae >> vC,, ies Ein—m+s+2 Jilsy m1 0= As. s Comparing the general term herein with the symbolic form of (—)m7 (}4, 13 MM, 1) 4% viz. (2 == ») = n,—m4+1 Ag ™ VD; Sn—mt2 A = m,—i 0= As, we find by = ey C,, —nim + (u — mv) C,, —n+m—1> Le C;, —n+m> m =m, nm =n+s. Hence (m, V5 M, Nw = 3 (-)n** {we CE —n+m—1 + Mv (C,, —n+m — C,, nes VUs nim; MM, 1+ s} As s=0 shewing that the transformation produces a sum of multilinear operators of the same general form and of the same degree in the coeftcients. 15. Leaving out of consideration for the moment a zero value of m, we find that in some cases the transformation produces a single operator. There are two cases. Case I. If p=my, —n+m=0, (mn, 1; n, ng =(—)"*(n, 1; n, na, or at (n— 3) (n, 1; n, 2) =(-)" xXi(n-3) (n, 1; n, n)a, THE HALPHENIAN HOMOGRAPHIC SUBSTITUTION. 117 establishing that oi” —3) (n, 1; n, 2)aq is an absolute invariant of even or uneven order according as n is even or uneven. For n=1 we have Og =—-O,4 equivalent to c70,=—XO0y, a relation already met with. For r= Ds (2; 1 3 2, 2) er = (2, 1 ; 2, 2)a; which we write Jy=J4, . —— — equivalent to fa Bd fl, SoS Bl je where J, is the important operator which generates pure reciprocants from pure reciprocants and seminvariants from seminvariants, and causes the vanishing of forms which are both pure reciprocants and seminvariants. The proof of this was given by me in the paper* and communicated by me to Sylvester. The latter incorporated the theorem into his Lectures on Reciprocants delivered before the University of Oxford. In the present theory of the Halphenian transformation we see that 28 J, is an absolute invariant of even order. Case IJ. Let yv=0, —n+m—1=0. Then (1,0; n+1, n)g =(—)"(1, 0; n+ 1, n)4, equivalent to a2” (1,0; 41, n)q =(—)” xe (1, 0; n+1, n)y, a relation already met with in the form Qe = (-yn A 2 n+) There are no other cases. 16. We can also specify the conditions under which the transformation produces a sum of any given number of operators. Whatever the given number there are invariably two cases. Thus (—)"(w, v3; n+1, nr)y =(w,v; n+1,n)¢g—v(nt+1,1; n+1,n41)y4, equivalent to ae” (u,v; n+1, n)q =(—)" (xen (u,v; n+1,n)4t+ X2@-2) (n +1,1; n+1,n+4+1)y4}, yielding the special cases ely tvuWe=plytvWy—-—vQ,, Ve =—Vatds; establishing the invariant Quly +v(2W,y — QO.) of even order, and the invariant 2Vy—Jy of uneven order. * Proc. Lond, Math. Soc. Vol. xvi. p. 75, 1886. Won, SOr0Ul, IGS” WAU 16 118 Mason MACMAHON, THE INVARIANTS OF Also (—)" 1, 0; n+2, na: =(1,0; n+2,n)¢—(1,0; n+2,n+1)4. Again for three operators on the dexter (—)"(w, v3 N+2, Nae =(p,v; n+2, ne — {ut (n+2)y, 2; n+2,n+1hat+v(n+2,1; n+2,n+2)4; (—)" (1, 0; n +38, n)a =(1,0; n+ 3, n)¢—2(1, 0; n+3,n4+1)4+, 0; 2+3,n+2)y. Observe that if we multiply up by min the general formula and then put m=0, w= 1, v=0 we obtain a formula already reached, viz. m+1 +2 Sy Oa, = Oars AP C 1 \anas ie le 2 ) Ca ngs a The logarithmic case. 17. The case corresponding to m=0 will be best understood from a consideration of the two operators 2a) a, — a," 3a)°d, — 3a), a, + a3 Ses 1 | wa A) Scie ay 0 ay 2a, + ay eee a 2a a, — a BAy? Ay — 38) 0, d, + a8 R = wok 0, = o2 1 0 0 3 (Tt Boer 1 ete a dy Ca, + Ae? a a, Oa, ar We observe that a ly ; log (a Ry te ye os w+ oe Ay ly ay So : Ds Ga Ua : eit = Baath Me tithe ie ae dy 2 OS 3 (tg? so that putting a,+a,u+a,u°+...=U, we may briefly denote U S, by (0, 1; log—, -1) A Ay a 2) Lame o§ and R, by (0, eo —.'0') , Ay a just as we might have written : ‘ 1 (u,v; m, 2)q in the notation (u, vy; — U™ n) : m a Proceeding to accented forms we find 1 Sa =— 2" Si , 3 Re= te? R,. In symbolic forms \ ¢ ‘ ‘\ 9 o . ms hy Quy dy — a? 3dp 2d, — 3ay a, ay + a, Ss, = £ 2 hs ty E tp? f ay* Ry = ES, . Now, putting as on a previous page, ag =a &—a,E*+a,&-..., aia THE HALPHENIAN HOMOGRAPHIC SUBSTITUTION. 119 \ ay a Ay ate ones Sah)! \ Ao we have log ag = log (a, €) + log @ - and differentiating with regard to &, Di Gy Sth, Binoy = Baron GA seo va pny J_G , 2 dy =? Say *03 — 3a) y's +h® -, 0: log ag = & ay aa an? & a EP+ ob, or £0; log ag = & + Sq, £0; log as = &— Ry. Hence Sa = &0; log ag — E°= 7 rag — &, = Re =~ F0glog af + £=—* dag +6. € Transforming from & to Z, g E+E Niede Re= a aa: 25 ; from which Sy —Re= a =F ; => = Bdz log Az — Ge : a result which shews that Ry — (0a; + 0a; +0, +...) is an invariant of even order, and 2Sq — Re — (0a, + Oa; +a, + --.) an invariant of uneven order. Moreover, since 20g + Oa, + 0a, +---, Cay + 0a; + +++, are invariants of even and uneven order respectively, we find invariants Ry + 0a; of even order, 2Sy — Ry —0a, of uneven order. Equivalent to these we find invariants go? (R,+0,,) of even order, a 2(28S,+a7R, +2 0q,) of uneven order. The interest of these results lies in the circumstance that S, is what Sylvester has termed a Reversor in the theory of pure reciprocants. It was discovered by Hammond. The allied operator R, appears here for the first time. If we apply the operator «~ 2 Ra+0q,) of even order to the invariant pply P 0 5 3 2x7a,+ 2a, of uneven order, te—2 120 Mason MACMAHON, THE INVARIANTS OF we obtain an invariant of uneven order. The verification is 2 ogka + aig.) ate 2X4 +A z = (Rat ag) (2a? a, + 22 ay) = - Je = o- s) ny) The Invariant Reversors A; Cag + 42 Cay + As Cay + ++» = Pa = — ZPas As Cay + 2a; Oa + «+» = Qa = — FYa- 18. We have pe=- P= CaF = le— pes establishing that 2pq — La = — 2rpa—Lna, is an invariant of uneven order. aol qe = Ce oecee 14 PE} = (A, — 2A, + A, ) B?— 2 (A, —2A, + A,) B8+3(As —2A; + A,) Bt—..., =—gst+2We—-Q4; shewing that ga — We =—7Qa— Wa is an invariant of uneven order. Pa and gq are therefore generating operators for the transformation, the former when the operand is homogeneous, the latter when it is isobaric. If j be the weight of the highest letter in the operand, the !atter is a full invariant if jla—2Wa causes the operand to vanish. Also jPa — Ya =—(jpa — Ya) + Gla —2We)+ Qa, shewing that 2 (jpa — qa) — (Gla — 2Wa) =— 22 (jpa — a) — (jLa —2Wa) is an invariant of uneven order. Hence when the operand is a full invariant + @ (jpa — Ja) generates an invariant of uneven order. The Invariant Generators Py = (aay — a) Oay + (dp ds — Gy Ms’) Bag +--+, Qa: = (Ay dy’ — 2a,"*) Oa, + 2 (ao a; — 2ay‘dr') Oa + 3 (do Gs — 2a,‘a3') Ogg +++. 19. These operators generate invariants in the theory of the binary quantic. Here Py =a) pa — Ly; whence Py =A} (Ia—pe) =A) Te = = Asn Aes shewing that P= —aP, is an invariant of uneven order for the Halphenian substitution. THE HALPHENIAN HOMOGRAPHIC SUBSTITUTION. 121 Also Qn a Ao Ja’ = OLE Wa =— Ay qa + 2A . Wa + (A, = DAr) Og or Qa hae Qa oS (Ay aa 2A,’) Qe 3 establishing, since a, dq; is an invariant of uneven order, that Qa + a; Og = — @? (7Qq — D4) is an invariant of uneven order. The Pure Reciprocant Generator Gg = 4 (Ay as — @;%) Aay +5 (Ap ds — Gy a)) Og, + 6 (aay — Ay d;') Oa +--+. 20. We find that Go =Qr + 38Pe +a We =— Ga + Ao Ws —- AQ. Hence the invariant of uneven order 24¢—a Weta Oe. To verify this result take as operand Qs — Ons We have 2G» (ay. de — ay?) = 10 (a, 2a; — Baya; a, + 2a,'%) + 4a, (aa, — a,*), Ay Wo (dg Gz — Ay”) = Zap’ (Ay Ay’ — My, ”), Og (aya — a, *) = 0. Hence the result of the operation is 10 (a,?a; — 3a,a, a, + 2a,“*) — 2 (ay — 2a,') (a, ay — a,), which is an invariant of uneven order because p23 — 8) d, ds + 2a, and a, —2a,; are so, and a)a. —a,? is an invariant of even order, G, by itself is a generator also in the theory of seminvariants when the operand is a combination of seminvariants of weight zero; for then 2Gq — a Wa +a, Oo is equivalent to 2G,. Thus the reader may verify that ys — BAA, A, + 2a,* Gq (dys — a,2)2 is an invariant of even order; for the operand is a seminvariant of uneven order and of weight zero. 122 Mayor MACMAHON, THE INVARIANTS OF SECTION II. PARTICULAR CASE OF THE ) TRANSFORMATION. 21. Write as usual 1 = (Gp + Gy + Ayu? +...) =Aino + Am U + Amo? + ..., and consider the operator HOmo0an + (w+ v) Aman. , + (e+ 2v) Am20 ans Spaccs which it has been convenient to denote by (H, vy; m, N)a- As was shewn in a previous paper* the substitution 1 Ay = ba? 0 b t= — Fa it. b,' om °° bs, ~ bibs i Os pea ape —557 which is derived from the formule for the interchange of the dependent and independent variables in the differential coefficients Lh ue da 2Neas2 said converts (BH, v3 Mm, N)q ; 1 into a= {u (n—m+ 2), w—mv; n—m+2, n}p. An exception however occurs when m=n+2, for then n—m+2=0. I refer to Art. 18, p. 156 of the paper (Joc. cit.) where the transformation of (p, V; m, nde by the Sylvester substitution was considered. The symbolic form there given, viz. ‘b es = v) Basa Ti + permite my becomes, when m=n+3, tc —_ 1 n+3 yy Nt2 an! (85 v) en +n, which may be written (, = _ v) 0, log (E). 9" tn! + vn **7/. v * “The Operator Reciprocants of Sylvester’s Theory of Reciprocants,” Trans. Camb. Phil. Soc. Vol. xxt. No. v1., 1908. THE HALPHENIAN HOMOGRAPHIC SUBSTITUTION, 123 Now 0, log E=0, log (tm + Hy? + a7? +...) 1 / A, a \ =—+40, log (1+—n+=77 +... _ tonlog (1+ 2atcatt..), 1 ay 1 2ra,—a,? , 1 37°a, — 37a,0, + a,° sel = 895) 7 Tt 7 4.2), nee To — 0,7 37°a,— 38Ta% +4,? , seen 7 n+ 7 ie SP adorn so that the symbolic form becomes 37a, — 3Ta)a, + a,* 3 ; 5) 2 be are ( “ ) chy i CAG eel. === = + —yp = + — ~ + n+3" 4 n+3 7” 7 1 T n+5 iy a n pei Le and now writing —0@,,_, for 7" we find me ae A 270, — Ay” 37? a — 387A, a, + a,° ) 5%, (Sap v) ( cap ert oer g ee 7 Ont T 2 T for the transform of (uw, v; n+3, n) by the Sylvester substitution. We now make a unit increase of suffix throughout, writing Gi, Chis Gh ond OD ama, CAR Boas then write 6 for a and n—1 for n when we find that (uw, v3; n+2, n)q is transformed by the 6 substitution into in b, 2b,b.—b,. _, Bbytby— Bbybrby +B," a n+2 bn c SoD) ay v) (5 oy Fis Conte ate “a? bnis uF mS) n+1 b2 and we derive the particular cases (0, 1; n+2, n)a b, 2b,b. — b2 3b.2b, — 3b)b, b. + bf ; = iB Oba + be = Ong == bs Obnts oto hetste ts (+2; 1; n+2, n)z =_— Obns The h transformation. 22a Lf ae —__ =]+hu+h,u?+hjui+..., l—aqut+auw—a,uw+... then Gh Sltag Ae = h,? — he, a; = h? = 2hyhs + hse and, putting a,=1, I examine the effect of making this substitution upon the operator (4 v3 m, N)a: 124 Mason MACMAHON, THE INVARIANTS OF We have 1 (—yrus 7 Cag 1Saqutaw—... G—qu+avz—...P UW Oaghe + UH Og, Resi + --+5 or (—SO (1+hwthwt...P= Oa, hts + U0q, herr + U0a, ere t - So that comparison of the coefficients of w on either side gives us the value of 0g, hs:p as a quadratic function of h. and we find 0,,=(—)** ae + oe + “(he + 2h. Bik + (2h,h. + 2hs) Gey ee ale If we write symbolically On, =k’, Oa, = (A (1 + yk + hk? +h +...¥, and AmoOa, + Ama, 1 + Ame%an,. + =(—)"k" qd ats hk a h.k? sk: hk ar 300) (Gino oo Cink + enna = Qing ke? + eae) 5 1 but no — Orn lb -F Omsk? — ... =o (+ lak + hk? + hgh? + Son) ee giving us AmoOay, + Gm O + Ame an+1 anyo tT = (yt (bik thal! +g + J = (yr A ie mo + Paces b+ Iams? + Imai +.) leading to the relation (1, 0; m, n)g=(— yn m ze, 0; 2—m, ny. Again (0,1; m, 2)q = Im Ogee + 2ame CPR + 3ains Be seer =(-P kM (lth kth bth, +...) (di — 2am k + 3am — ... ); but Qmp — Am, kk + Ame kh? — ... = = L+hk+hjk?+...)-™, so that by differentiation Can, — 2a + Bing A? — oo = — = Oy (1 + Fy e+ Bg P+ and = (l+hk+h,k? +...) (Gm — 22k + Bains A — ...) = 5 = mot (l+hk+h,k?+...)-™ = hom, =i Zho—m,2 k+ Bho, a and we now gather that (0,1; m, n)g = (—)"(0, 1; 2—m, na; and, since (mw, v3; m,N)g=pe(1, 0; m, n)at+v(0, 1; m, ra, (u,v; mM, N)g =(-)” \f (m— 2), v; 2—m, n| : m h it being understood that the value m=2 is excluded. THE HALPHENIAN HOMOGRAPHIC SUBSTITUTION. 23. The particular case m=2 requires a separate examination. Reference to foregoing investigation shews that (Oy 2510) a is symbolically (—y dhe, so that (1, 0; 2, n), =(—)"* £4, whilst (0,1; 2, n),=(—)" hk" 0, log l+hk+hk+...) = (—)" k"+4 (hy + (Qhy — hy2) b+ (Bhg — Bhyha + hy’) 2+ ...} ) =(-)" {h, Gee + (2h, — h,?) Oh set (3h, — 3h, hy + h,*) Ono +...}. Hence (i, HS PR YOYr 1 = (=) ue = Ohy + Vv (—) {ha On.) + (2h2 — Iy®) On, 9 + (3h; — Bh, hy + hy’) On, ., + ++} Particular Cases of the h transformation. 24. The seminvariant annihilator (Gtaleriten bys becomes (He Sg ya or On, — h, On, = 2h; On, = Bh, On, —Jelstate Hence any seminvariant gud the elements les Mss 3 Pies is a seminvariant qua the elements Cigemi Cit, hs; cet Ex. gr. h,h,—h? is a seminvariant. Moreover, writing ada sy) 45, 2) = Das if the general solution of On, = hs On. = 2h, On, = 3h, On, SS ppc) be written hy bs + Gi hs dat (3) Di? QAP coor we have she bs + 8(s— 1) hy? d+ $8 (8s —1)(s—2)h SF Got..., = hy Hp, + (7) bi Hoes + (5) lot Hb.a to where H = hy 0p, + hs On, + Bhs On; + «+5 and now equating coefficients of like powers of /h,, Wiett, 2OKING ING, WHk 17 125 the 126 Mason MACMAHON, THE INVARIANTS OF indicating that, regarding the general solution as a binary s‘° in /,, 1, all of its seminvariants are seminyariants in the elements hs, hs, hy, sinlay, and therefore also in the elements a, G4, Qo, .... 25. As another interesting particular case we find that the pure reciprocant annihilator (4,1; 2, 1)a = 20p, — My On, — (2h2— hy) On, — (Bhs — Bhyhz + My°) On, — ---- Moreover it will be shewn that (4, 1; 2, 1), can be transformed into (1, 0; 2, 1),, and this becomes 40h,; of which the fundamental solutions (it beimg equated to zero) are loam Itksep [Why L053 S000 The fact is that by means of the b transformation if b b, 2 U4 aa Re: ie a 0 0 (4, 1; 2,1), is transformed to — (2,1; 1, 1), and if we write b,=(s+1)c,, it is further transformed to (i, 3 tL, De We now again employ the b transformation G2) ee thi. 9s 0 d, 2 1 dj ’ LSS d,' ° else, when it becomes ZO 2ee) 5. The h transformation do=@, h=G, d,=e2— G6, ds=e,> — 2e,e,e + &, --- finally converts it into €00¢,) and we have the complete set of solutions Qo, Co, Og, Ogy oes In detail, 1 calaaey; = = Gy 0 0 b 2c be ba = ¢ 5 = 2d, = 2¢,, 0 0 be b,° Cy c d= — 7 +27=-3824+8- by by Co 0 PAE. Ar. = 3d, + 2 — =— (5e,?— 3e,6), dy @ THE HALPHENIAN HOMOGRAPHIC SUBSTITUTION. [Py — a lists 2 Cs athe ; CC, (oS .=— = = 3 = — 4() ds bs +5 by 5 By 4 a + 30 cit ot Chih ees = 4d, + 10 4— = — (Te, — 9e,€,e, + 2e,2e), dy eo b, bb b.? b2b b4 i 6 1“%3 2 2] 1 2 1 e500 a a Sa = — 5% 4.499% 497% _ 959% 4 994% Co Co’ Cy Co° Cy 2 2 4 = sadees dhds , 12 d, iy gt us m 9th d, d, ie fe 1 Es ee == (A2e,4 — 84e,¢,°¢, + 17¢,7¢.2 + 28¢,2e,e; — 5e°es), 0 &e. 127 This then is the transformation from the elements a to the elements e which transforms the operator. As above shewn, €o, C2, @3, C4, +++ are pure reciprocants, and we find by calculation their expressions in terms of the elements «. Viz. y= Chp 1 e = ——— ST er (ba — 4a) a2), i €s = =— (a2 a3 — 3a) a, a, + 24,°), 4a, 1 és=s5~—5 720a, 3 &e. The first three of these will be immediately recognized. (551a,; — 1184a,a,2a,. + 272a,a,? + 504a,a, a, — 144a,°a,), The last one, in the bracket, is expressed in terms of Sylvester's ground forms by the formula 5B1 (5a,? — day)? — $2 (105a,a,2a, + 28a,24,? — 175a,?a,a; + 50a,°As). The forms obtained in the above manner form a complete set from which all pure recipro- cants can be obtained, but they do not constitute the simplest set of ground forms obtainable. The possibility of such a transformation depends upon two circumstances. place the operator (4, 1; 2, 1) is a particular case of the operator such that n—m+2=1; consequently the b transformation results in an operator, viz. (KH, vy, m, n) (2,1; 1,0), In the first 128 Mason MACMAHON, THE INVARIANTS OF which is linear in the elements which are coefficients of the differential inverses. in the second place any such linear operator (u, v; 1, x) can be transformed by mere numerical multiplication of the elements into any other form (y’, v’; 1, x). It follows that (2, 1; 1, 1) is transformable into the seminvariant operator (1,1; 1,1). An immediate consequence of this, of course, is that all pure reciprocants are transformable into seminvariants. The 0 transformation now transforms (1, 1; 1, 1) into an operator, viz. (1,0; 2, 1), in which the element y is zero, and this being so the h transformation produces the final simple form of operator C006, - A transformation of the Seminvariant Operator. 26. It will be observed that the seminvariant operator (1,1; 1,1) is by the successive b and h substitutions brought to the required simple form. This in fact takes place during the transformation of the pure reciprocant operator. The transformation which effects this is a) = — G, Gy = Cy" + Co, d; = — C — 30,C2 — C5, (ly = Cy + Gees + 2.7 + 40,6, + Cy, &e. 4 n! In general a, =(-)"> CoC) C2 2... = Oy a Geek. The operator then becomes Gian and ¢y, Cs, C3, Cy, -.. are Seminvariants. In fact from the above relations Co = — A," + Ae, C3 = — 20,° + 3a, a, — As, C= — 5 (— aft a) + 3a" — 4a, a, + a4, : s 2-SA+1 n4-=r — 2)! and in generai — ( In the expressions of ¢,, ¢,, ¢,, Seminvariants are at once recognizable. The s transformation. 27. If s,, s,, s,,... denote the sums of powers of the roots of the equation a” — aa" + av" — ...=0, where n= ’ it will be shewn that Vig = (Anal ec (0s, — Bay0., + 6a,0,, — 100505, — «..), Oy = (1, 15 Ly Ya = 2), = 26:80, — 8810», — 48,0, — ---. THE HALPHENIAN HOMOGRAPHIC SUBSTITUTION. To prove the first of these relations, it is easy to shew that ee oe) {pes, +(pt+ 1)h0s, 4 +(pt+ 2)hs,.» +...}, 2V gq = Atty (05, + 2hyOs, + Bh.0,, + 4hs0s, +...) — 5dy (205, + 3,05, + 4heds, + «.-) + 6d» (30,, + 4/4 0,, + Sheds, + ...) — Tats, (405, + 5,05, +...) aR po0e Herein the coefficient of Os, is P [Saghpa — dy hy» + 6doshy_,— ... +(—)”* (p + 3) Til 1 : 12 ue = = 1 a hu + hou + ees Now if which we may write in the form Ae eed a 44.7 + U0, Ai? = 4d — 5d + Gdou? — ..., and Ayo H=1thu+ hou? + ...; therefore by multiplication Ady + (Adah, — 5a) U + (Adan hz — S5dyh, + Gan) — ... + {4dghp — Sd hyo t ... +(—)?4 (p + 3)aa,} w+ ..., =4A,+2u0,Au, =4 (1 —ayu+ au? — au +...) + (— 2q,u 4+ 4a. — Ga,v + ...), =4— 6a,u + 8a.1?— 10a,u®+.... In this series the coefficient of w”~ is (4 (2p +2) apa, proving that the coefticient of Os, is (—)?* p (2p +2) ap. Therefore 2V,==(—)?" p (2p + 2) aya 0s, = 9S (jon (PED or Va= 22 (—)jP" ( 2 ) dys,» or Va=2 (G,, —3a,0,, + 6a,0,, — 10030, + ...). For the second relation Og =0,, + 2 (In — 2) 0,, + 3 (he — 2a, + 3az) 0s, + 4 (hs — 2h, + 3a.h, — 403) 05, +..., and by easy algebra this reduces to Oq = O¢, — 2810, — 38203, — 45505, — .-.. ‘i \ If, however, we write 8 ==]he ee sd) 35 Ops — AO, 130 Mason MACMAHON, THE INVARIANTS OF indicating that every seminvariant in the elements he) 4 OS Boe is also a seminvariant in the elements — $1, —S2, — Sz, +0 Thus for example from the seminvariant (yy — 44,03 + 3a we at once derive a new seminvariant —s,—4s,s,+ 3s, which is found to be equal to — 2 (a,* — 2a°a, — 54° + Saya; — 2a). It should be noted that the important operator DiS (2, 13 22) is equivalent to — 2 (0;,— 30,05, + 6a,05, — 10a;05, + ..-). The combined b and h transformations. 28. Both the b and the A substitutions being of period 2, we can combine the substitutions alternately. Thus if we first employ the b substitution and then the A and b substitutions alternately, we obtain substitutions which we may denote by (hb)? and b (hb)?7.* Tt will be found that { (hb)? (wu, v3; m,n) =(—)” ike ~ (pn— 1) pn, nt, m nm b (hb)? (u,v; m, n)=(-)r?™ {= = (pn — m+ 2), Also ley (ae my + Pe. (bh)? (uw, v3; m,n) = (—)”” la (pn +m), = 5 Sf (a) {ue g h(bh)? (p, v3; m, n)=(—)?'?™ mn te + m — 2), = my — Pe my + PP. 95 ; pun—m-+ 2, nt : > m+ pn, at, —m— pn, nf. In the first, third and fourth of these formule p may be zero or any positive integer; in the second p may be any positive integer, zero excluded. Observing that the third and fourth operators become respectively equal to the first and second when — p is written for p, we gather that we only require the third and fourth operators in which p may be supposed to be zero or any positive or negative integer. As a particular case we find that the seminvariant operator (lols 7a; h) may be transformed into (—)?(p+1, p+1; p+l1, 1), and into (—)?# (p—1, p+1; 1—-p, 1); * The substitutions are from right to left successively. THE HALPHENIAN HOMOGRAPHIC SUBSTITUTION. 131 and the pure reciprocant operator (4, 1 ; 2, 1) into (—)? 2p+ 4, 29+1; p+2, 1), a (—)?*1 (2p, 29 +1; —p, 1). Transformation by suffia diminution. 29. The operator (u, v; m,n) admits of one very simple transformation which may be repeated indefinitely. If therein we put y= 05 GhSGh a for all values of s, or in other words if we diminish each suffix by unity, the operator becomes (w+ mv, v3; m, m+n—1), and the solutions of this operator are obtained from those of (u,v; m,n) by subjecting the solutions to a unit diminution of suffix. If we employ this transformation p times we reach the operator (w+pmy, v; m, pm+n—p). This operator is effectively the same as the untransformed operator when i=, DS: VILL. Vector Integral Equations and Gibbs’ Dyadics. By C. E. WeatHERBURN (Ormond College, Melbourne). Communicated by Mr G. H. Hardy. Yi; J [Received 4 October 1915—Kead 25 October 1915.] CONTENTS. PAGE Introduction : : : : é c 5 : : ‘ : ; : ss I. The linear vector integral equation of the second kind. : : ; : “1, se: If. The iterated dyadic kernels, and solution by successive substitutions < : 5 lait III. The resolvent dyadic . : : é : : ; : ‘ : : : > ge TV. Dyadic determinants . : : : : : ‘ : ; 5 : i . ‘141 VY. The series D(A) . 3 F : é : : : F F : : ; . 143 VI. Determination of the adjoint and the resolvent : c : : : 5 ED VII. The homogeneous integral equations, and singular parameter values. : . 148 VIII. The conjugo-symmetric dyadic kernel . : 3 : : : - c los INTRODUCTION. THE integral equations that have hitherto been considered by mathematicians are scalar equations in which the functions involved are not related to any particular direction in space. In the problems of mathematical physics we are frequently led by Cartesian analysis to a system of three integral equations with the same number of unknown functions, while if our methods are those of vector analysis we find instead a single integral equation in which the unknown function is a vector quantity, and the kernel of the equation no longer a scalar but either a vector used in cross multiplication or an operator involving vectors. The vector integral equations that most commonly appear are of the form HO) I SCS) GI @)C SEO) sdcoccocescotonconodaccncaconcone (1), where u(¢) is the unknown vector function of the position of the point f, X an arbitrary parameter, f(f) a known vector function, and K(ts) a dyadic*, ie. an operator which acting on the vector u gives a linear vector function of u. It will be shewn in the following that (1) is the most general type of linear vector integral equation of the * Cf. Gibbs-Wilson, Vector Analysis, New York, 1901, Chapter v. WMorsexoxT, No; WIE. 18 134 Mr WEATHERBURN, VECTOR INTEGRAL EQUATIONS second kind, and it is to equations of this sort, analogous to Fredholm’s equation, that we shall confine our attention in the present paper. The scalar kernel of the ordinary integral equation is thus replaced by the dyadic operator K (ts) which involves vector functions of the positions of the two points ¢ ands. In developing the theory of the equation (1) along lines suggested by the theory of Fred- holm’s equation some new ideas will be necessary in view of certain important differences between the algebra of dyadics and that of ordinary scalar multipliers. For instance, the fact that the commutative law does not in general hold for the factors of a dyadic product makes it essential, if we wish to introduce determinants of dyadic elements, to formulate rules for the expansion of such determinants according to which the elements in each term will occur in some definite order. This the author believes is done with complete success by the introduction of two kinds of dyadic determinants, called respec- tively row and column determinants, by means of which two series are formed whose quotient is the resolvent dyadic for the integral equation (1). A special class of kernel will be considered (for which the author suggests the term conjugo-symmetric) which need not be either self-conjugate or symmetric but which suggests a blending of these two kinds. Such a kernel makes the vector integral equation identical with its associated equation. A set of theorems will be established for the conjugo-symmetric kernel analogous to those that hold for the symmetric kernel of © the scalar integral equation. I. THe Liyear Vector INTEGRAL EQUATION OF THE SECOND KIND. $1. We are familiar with a system of three linear scalar integral equations of the second kind in the following form to which the system may always be reduced u, (t) —d J [Ru (ts) wm (8) + Kye (ts) Us (8) + Ks (ts) us (s)] ds =f, () Us (t) — XJ [Km (ts) w (8) + Ke (ts) Us (8) + Koy (ts) us (8)] ds = fo (B) [ -.seeee eee ee ene (2), u; (t)— | [Ka (ts) wm (s) + Kee (ts) Ue (Ss) + Key (ts) us (s)] ds =f; (6) where all the functions are scalar functions of the positions of the points indicated, u;(t) (i=1, 2,3) are the unknowns, and the integration is to be extended over a de- finite fixed region S which may be a line, surface, or volume, ds being the element of that region surrounding the point s. We shall assume that the functions f(t), Knm (ts) are finite within the region considered. Let i, j, k be the unit vectors of a rectangular system. The scalar functions of (2) grouped in trios may be regarded as the tensors of the components in these directions of vectors defined by the relations u(t)=iw, (t)+jw(t)+kK u, (), FO=1f(OtiAO+kA, K,. (ts) =i K,, (ts) + j Kx (ts) +k K,, (ts), r=l1, 2, 3. AND GIBBS’ DYADICS. 135 The integrands in (2) are then equal to the scalar products K, (ts)*u(s), K.(ts)*u(s) and K, (ts)*u(s) respectively. On multiplying the equations (2) by i, j, k respectively and adding we have the single integral equation u(t) —2 | [i K, (ts) +j Ky (ts) + K, (ts)] © u(s)ds=£()eeccccccececeeees a; which may be written Wu) = I) ES(UNOUN GCS 53) eh caccencecdoasdoeossouabooaebee (1) where the function appearing as the kernel of the vector integral equation is the dyadic K (ts) =i K, (ts) +j K, (ts) +k K, (ts). This is an operator each term of which is the indeterminate product of two vectors known respectively as the antecedent and the consequent of that dyad. The dyadic in (1) occurs as a prefactor to the vector u(s), and the result of its operation is the sum of the products of each antecedent by the scalar product of its consequent and the vector u(s). When expressed in nonion form* the kernel K(ts) becomes Ky, (ts)ii + Ky. (ts) ij + K,, (ts) ik + Ky (ts) ji + Ky (ts) ij + Ky; (ts) jk + Ky, (ts) ki + Ky (ts) kj + Koy (ts) kk, the “determinant” of which} is identical with the determinant of the coefficients under the integral sign in the system (2). §2. The system (2) of scalar integral equations is then equivalent to the single vector equation (1). Conversely (1) may be replaced by the system (2). We shall need frequently to refer to another integral equation intimately related to (1), viz. the vector equation TCE Nie (G)o IK (at) ds =f (O)ieicesc ees g-ohecaceccacecios ees (3) which will be called the associated equation. In this the dyadic kernel K(st) occurs as a postfactor to the vector v(s), so that the result of its direct operation is the sum of the products of each consequent by the scalar product of its antecedent and the vector v(s). When expanded the integrand in this equation becomes i[v, (s) Ky, (st) + v2 (8) Ka (st) + v5 (8) Ka (st)] + j[v.(s) Ky. (st) + v, (8) Ko (st) + v5 (8) Kop (st)] +k [v, (s) Ky; (st) + v2 (8) Ko, (st) + v3 (8) Ko, (st)], so that the vector equation (3) is equivalent to the following system of three scalar equations v, (t) —A J [% (s) Ky (st) + v2 (8) Ka (st) + 05 (8) Ku (st)] ds = f, (0) U2 (t) — AJ [1 (8) Kos (st) + v2 (8) Koy (St) + 03 (8) Ky, (st)] ds = f(t) | 0; (t) — rf [0 (8) Ky, (st) + v. (8) Ko, (st) + v5 (s) Ki; (st) | ds =f,(t)} This system, it should be observed, is not identical with (2), The rows of the coefficients in (4) agree with the columns of the coefficients in (2) with the variables s and ¢ interchanged; and vice versa. It is a common mistake in discussing the system * Gibbs-Wilson, loc. cit. p. 269. + Ibid, p. 317. 136 Mr WEATHERBURN, VECTOR INTEGRAL EQUATIONS (2) of integral equations to assume that the associated system differs only in the inter- change of variables. This assumption is quite wrong except in the special case in which K,; (ts)= K;, (ts): that is when the dyadic K(ts) is self-conjugate. In this particular case it is immaterial whether the dyadic is placed as a prefactor or a postfactor. But in general the relative position of the kernel and the unknown vector cannot be varied at pleasure. §3. Either of the equations (1) and (3) is the most general form of linear vector integral equation of the second kind. For it is the fundamental property of a dyadic that when operating on a vector u it gives a linear vector function of u; while every linear vector function may be represented by a dyadic* to be used as a prefactor, or by the conjugate of that dyadic used as a postfactor. Hence the most general form of the integrand is the direct product of a dyadic and the unknown vector. A form that might suggest itself is axu, where a is a vector independent of u. This form is included in the above, for a vector a used in cross multiplication is equivalent? to the dyadic Ixa or axI used in direct (scalar) multiplication, I being the idemfactor, that is the dyadic whose operation leaves a vector unchanged. The case in which u is multiplied by a scalar function m is equivalent to that in which the dyadic is mI. If the unknown u outside the integral sign has a dyadic either as a prefactor or as a postfactor, the equation may be multiplied throughout by its reciprocal} dyadic and thus reduced to the form (1) or (3). Moreover the dyadic kernel K (ts) =iK, (ts) +j K. (ts) +k K, (ts) is the most general form of dyadic. For every dyadic may be reduced to the sum of three dyads, of which either the antecedents or the consequents may be arbitrarily chosen§ provided they are not coplanar. In the present form our arbitrarily chosen antecedents are the rectangular unit vectors i, j, kK We have shewn then that (1) is the most general form of linear vector integral equation of the second kind. § 4. In his classical memoir Fredholm || has shewn how a system of integral equations such as (2), in the case where the region of integration is linear, may be reduced theoretically to a single scalar integral equation whose kernel and unknown each represent different functions in various sections of the domain of integration. The possibilities of this method, such as they are, may be extended to the general case in which the region of integration, S, is a surface or a volume. If this be replaced by another, S’, consisting of the original region S traversed three times the system (2) is equivalent to the single scalar integral equation u(@)= el EE (te) ule) de=/(0)....:..0 ee (5), JS’ where, if ¢ and s are points of the region being traversed for the nth and mth times respectively (n, m=1, 2, 3), K (ts) = Knm (ts), w(t)=un,(t), u(s)=%Un(s) and f(t)=fh (t). * Gibbs-Wilson, loc. cit. p. 267. + Ibid. p, 299. t Ibid. p. 290. § Ibid. p. 271. “Sur une classe d’équations fonctionnelles,”’ Acta Math. Bd. xxvii. (1903), pp. 378, 379. AND GIBBS’ DYADICS. 137 From the solution w(¢) of this equation the unknown functions w(t), w(t) and u,(t) of (2) are found by the above relations. But though this procedure reduces the system (2) theoretically to the case of a single integral equation the method is rather cumbrous in practice as all the functions involved change abruptly and frequently within the region of integration. The enquiry therefore suggests itself whether we can work with the single vector integral equation (1) to which the system has been reduced, and develop if possible the theory of vector integral equations as an important and useful branch of vector analysis. The enquiry is all the more essential to one who works habitually with vector methods, for it is in the form (1) that the integral equation presents itself to him: and it would be a doubtful gain to give up a single vector equation for a system of scalars, even if that system be reducible to a single scalar such as (5). Il. Tue Ireratep Dyapic KERNELS, AND SOLUTION BY SUCCESSIVE SUBSTITUTIONS. §5. Before proceeding with the solution of our equation (1) we shall introduce the idea of an iterated dyadic. The algebra of dyadics makes us familiar with the direct product of two or more dyadics. The direct (scalar) product of the dyadics K(¢S%) and K(Ss) is written K(¢S)+K(3s), and is the formal expansion of the product, according to the distributive law, as a sum of products of dyads. The product is itself a dyadic and the sum of any number of dyadics is a dyadic. Hence multiplying the product by dS and summing for all the elements of the region S, we have in the limit that the integral /K(tS) °K (Ss) d3 is a dyadic. Further we may have the product of three or more dyadics, and the factors of such a product are known to be associative though not in general commutative. The products K(t3)*(K(Sc)*K(cs)] and [K(tS)e«K(Soc)|°K(as) are identical. Multiplying by the scalar product dSdo and summing over the whole region of integration for each of the variables S and oc, we have in the limit {K(tS)-[[/K(Qc)eK(os)do]d3 =/[/K(tS)+K(Se0)d3]+K(oes)do ...... (6). The process may be continued for any number of factors so that the order of integration may be changed at pleasure and the associative property used for any grouping of con- secutive factors. The factors however are not commutative. The dyadics formed in this way by successive iterations of the dyadic kernel K (ts) will be called the iterated dyadic kernels, or briefly the iterated kernels. The dyadic K, (ts) = {| K(t3) eK (Ss) dS will be referred to as the first iterated kernel : K, (ts) = { K, (tS) + K (Ss) dS =| K(tS)-K, (Ss) d8 as the second, and in general K, (ts)=/K,4 (tS) «K (Ss) dS = /K,_, (tS) «K, (9s) d3 as the pth iterated kernel. 138 Mr WEATHERBURN, VECTOR INTEGRAL EQUATIONS The operation of the dyadic K(ts) on a vector a yields another vector b. The ratio b|:|a| depends of course on the direction of the vector a; but if the dyadic K(¢s) is finite, that is, every element of its determinant finite, there will be a scalar function M(ts) such that this modular magnification ra S25) 1s (S282 NEN (25 /RSy 6 Se n = Gea El, ee) OS{ OS RRinty WSstossesbeasnc tute (18), where in the general term we have adopted the notation (17). The integrals obtained by expanding these determinants are to be interpreted as explained in the previous section. Tt is clear upon examination that the determinants may be expanded as either first row or first column determinants, the same result being obtained by either method. For the factors in any term of the expansion fall into closed groups in just the same way, the only difference being that the order of the closed groups in any term will be different. But as the integral of the closed group represents a scalar quantity, the change of order is immaterial. The series D(A) is absolutely convergent. An upper limit may be assigned to the value of the integral which is the coefficient of (—)"/n!. In accordance with the meaning of the expression “greatest modular magnification,” M, of §5, it is clear that K(ts) < MI, meaning that the modulus of the vector after operation with K(ts) is < that after operation with MI. Each element of the determinant regarded as a dyadic cannot exceed MY. Further, the scalar of any dyadic which : 2) and .K(* uae (ailmona tes see denote the dyadic determinant of the form (17) expanded according to the first row and the first column respectively. Introducing series analogous to Fredholm’s minors we shall speak of D, G doo Gp n) a ik G Seen = ay [x & Soc btn i) ae G te «0e tn t ty ... tn By) th ... tn T a | f KH ooo On ooo I ) noo coo Ipalis ihe HN Gher ane @ Sut Soobosabine 8 patos m! ] d ls erbogra ene. Chace Otis Se 29) as the nth row-minor of the series D(X): and of By Se 0-0 Sy aay aie: : -D, G a ‘ ) =a similar expression in column determinants ............ (29) Zeee Un Vou. XXII. No. VIII. 20 150 Mr WEATHERBURN, VECTOR INTEGRAL EQUATIONS as the nth column-minor of D(X). From these definitions it is clear that d” x ra caeuon Gs Umno Ueseree sas ie! K Chet mcathem § eosoac 31). i= -2 | (sits) ° +f]. es 500 AR ce SEE tRN EL Bp tir 2 ee Ge The last summation may clearly be written i -S —1\ = m | K (s; T)e ie al »K Ge nT +++ Tm "| dr, oe dt ae. is Br Gr One Mpheay Multiplying then both sides of (31), thus ee by (—2)™/m! and summing for all values of m from 0 to # as required by (28) we derive Sy Soleeeion Sp reratet Si Dn eat a) ECs A at x) - 2K (s,t:) + Dy ee ee) QZ ee° Y—1%+1 os" Yn i=2 = r| K(7) ..D ‘eet ree x) din? Tene ee (32). Similarly by expanding the column determinants in (29) we find D, ce sei ) sap, bs 30 x) °K (st) fies are a 81 So ... Sp} Sp) -.. § : = > D 1°2 —1°7+1 rn ) K st Sas leag hales bee) oe sey [., (a ast ‘ ) « K (nid) dee tee aoe ee (33). Now since D(X) is an integral function of X not vanishing identically it follows that D(a), - Doge DO) 7 a are not all zero when X=2,; and therefore in virtue of (30) that the dyadics 8, S18. $1805 D, (7 re) D, (7 Me) Bi rv), 5 ee (34) are not all identically zero with respect to the variables. Let qg be the index of the first of these which is not identically zero. This number qg will be the same whether the AND GIBBS’ DYADICS. 151 determinants are row or column determinants. For these differ only by an interchange of variables, and the identical vanishing of ,D, (...) involves that of .D, (...) and vice versa. It follows then from (32) and (33), since all the minors of index 1. Hence the pole of the resolvent is simple. § 21. Principal system of fundamental functions. The q linearly independent solutions of the homogeneous equation (14) corresponding to a singular value 4, may be replaced by a system of g normalised orthogonal} functions, that is, functions satisfying the relations - Ltrs u,(t)eu dt= iA Ce ily Heathen stewie Gbiremitatn es adele einai cata J (t) « (t) t 0 if nes * Gibbs- Wilson, loc. cit. pp. 426-436. + Cf. Schmidt, Wath. Ann. Bd. txt. (1907), 8. 455-7. t Ibid. S. 442-4, AND GIBBS’ DYADICS. 155 Then since the fundamental functions* corresponding to different characteristic numbers are in the case of the conjugo-symmetric kernel orthogonal to each other, we may, when the kernel is of this nature, replace all the linearly independent solutions of the homogeneous equation corresponding to all the characteristic numbers by a series of normalised orthogonal fundamental functions satisfying the above relations. Such a system, ULL CE) Ss ((Z:) saatetsarse Ll (Eng ain ctaerarss terete clo tists vtec die meres (46), may be called the principal system of fundamental functions. We may assume that these functions are placed in the order of the increasing magnitude of the characteristic numbers to which they belong. Any solution of the homogeneous equation (14) is linearly expressible in terms of a finite number of functidns of this series belonging to the same characteristic number. § 22. It may be possible to express the dyadic kernel K(¢ts) as the sum of a series of dyads whose antecedents and consequents are the vectors u,(t), u;(s) of the system (46). Suppose it is possible to do this in such a way that the antecedents of the dyads are the successive functions (46), giving BE ((65) NU (ONC nes (EC) | Cy toi Ul (O) Grit iece | saeeces cece selese vcs se (47), where the consequents ¢, are to be determined. If this relation holds, so that the second member is absolutely and uniformly convergent? when the number of terms is infinite, we may act on any vector a with each side as a prefactor, obtaining K (ts)ea=u,(t)C,ea+u,(t)C,eat..., where all the terms are vectors. Multiplying each side by u,(t)+* and integrating, we have, in virtue of the orthogonal relations (45), [ fu, (t)+K(ts)dt]}ea=c, ea. This holds for any vector a. It follows then that w, (5) nr c, = fu, (t)*K (ts) dt = When, therefore; the representation (47) is possible, it becomes u, (t)u,(s) , u.(f) u,(s) 7 + X. +... K (ts) = the functions u,(¢) forming the antecedents, and the functions u,(s) the consequents of the dyads in the second member. This series corresponds to the bilinear series for the symmetric kernel of the scalar integral equation. Conversely it may be shewn that if the series (48) is absolutely and uniformly convergent its sum is equal to the kernel K (ts). § 23. The following theorems for the conjugo-symmetric kernel may also be established without difficulty : * The solutions of the homogeneous equation (14). + That is to say, the series obtained by operating term by term on a finite vector is absolutely and uniformly convergent. (1) The necessary and sufficient condition that a continuous vector function h(t) may satisfy the identity (feels) ons Oe Perera (49) is that for all values of n 156 Mr WEATHERBURN, VECTOR INTEGRAL EQUATIONS Ju, (s)*h(s) ds = 0. (2) Any vector function g(t) expressible in the form g (t) = | K (ts) +1(s) ds, where 1(s) is a continuous vector function of the position of s, can be expanded in terms of the fundamental functions (46) according to the Fourier rule, B()= = uy (t) J 8 (t) © Un (¢) dt = ~ a, (t) [1 (£) «Up (t) dt n n We shall say that a conjugo-symmetric kernel is closed when there does not exist any continuous function h(t) satisfying the relation (49). It may be proved that (3) Every closed conjugo-symmetric kernel has an infinite number of singular parameter values. In conclusion, Schmidt’s formula* for the solution of the non-homogeneous symmetric integral equation may be extended to the vector integral equation (1) where the kernel is conjugo-symmetric. Thus (4) If X is not equal to a characteristic number the equation (1) with conjugo-symmetric kernel has a unique continuous solution given by u(t)=f(t) —AZ Un (2) Bat (OLY (OC Lille cenrmannenesneseonescnocc: (50). n x— An If however X is equal to a characteristic number A,, to which correspond g fundamental functions of the principal system, for the above solution to remain finite it is necessary and sufficient that f(t) be orthogonal to each of these qg functions. If this condition is satisfied the solution of the integral equation becomes n(f)=8@) oa OS. Le Gem eee ee (51), 1 n Xn — Xn where the quantities a; are arbitrary constants, the first summation including the funda- mental functions belonging to 2X», and the second the remaining ones of the principal system (46). That u(t) given by (51) is a solution of (1) for the singular parameter value A,, is easily verified by direct substitution. §24. From the results established above for the conjugo-symmetric kernel a proof may be deduced+ of the sufficiency of the conditions found in $17 for the existence of a solution to the non-homogeneous equation (é)— Ay f esh om (ade =F (6). sc ccvesnpeneewtvonversexeya (39), * Cf. Schmidt, loc. cit. 8. 454. + Ibid. § 13. AND GIBBS’ DYADICS. Wor where A, is a characteristic number of the kernel, not supposed symmetric or conjugo- symmetric. From this equation and its associated equation Vi) Na few (8) 0 EK (G6). ls" ei(E) ULI. seated tu ato deans: (52), we readily deduce g& (t)— Ay | K (ts) + 8 (s)ds=V(t)—A.[Q(ts)eV(s)\ds ...ceecececeeecees (53), and Se@®-s(t)dt=(v(+[v(t)—A,{ Q(ts)ev(s)ds]dt ..... ee. (54), where Q (st) = K (st) + K, (ts) —\, | K (sc) « K, (tc) do; K,(ts) denoting the conjugate dyadic of K(ts), that is the dyadic which acting as a prefactor is equivalent to K(ts) used as a postfactor, and conversely. The kernel Q(st) is conjugo-symmetric. For if r is any finite vector Q (st)+r=K(st)er+ K.(ts)+r—2, | K (sc) K, (tc) erde =reK,(st)+reK/(ts)—X,{/reK(tc)+K, (sc) do. The transformation of the integral is a consequence of the theorem that the conjugate of the product of two dyadics is equal to the product of their conjugates taken in the reverse order*. The last equation may be written Q(st)er=re Q(t), proving that Q(st) is conjugo-symmetric. Now if v(¢) is a solution of the associated homogeneous equation TD) = Ral PA QYOLS (CSS saoudsasoctoccsepepesppecbeodseuc (40) obtained from (52) by putting g(¢) zero, it follows from (53) that it is also a fundamental function of Q(st) for the characteristic number X,. Conversely, if v(t) is a fundamental function of Q(st) for this parameter value, it follows from (54) and (52) that it is also a solution of (40). The linearly independent solutions of (40) are therefore identical with the fundamental functions of Q/(st). If now we transform (39) by the substitution (Sar (al Ve CRES(GONGES® Stace soocucus cocudecseeuoonece (55) it becomes w(t)—2r,/ Q(ts) + w(s)ds=f(t). But by the preceding section the necessary and sufficient condition that this equation may admit a finite solution is the orthogonality of f(f) to all the fundamental functions of Q (st) corresponding to A,, that is to all the linearly independent solutions of (40). Therefore in virtne of the relation (55) this is the necessary and sufficient condition that (39) may admit a solution for the singular parameter value X,. § 25. Singular Kernel. We have up to the present assumed that the dyadic K (ts) remains finite, ie. that all the coefticients K;,(ts) [?, r=1, 2,3] in its nonion form are finite for every poimt t, s of the region S. In many physical problems when the points t and s coalesce the kernel K(ts) becomes infinite like 1/r*, r being the distance between the two points. But as shewn in $6 the integral equation (1) may be replaced by (8) in which the kernel is the nth iterated kernel of K(ts). When a is not too large, this iterated kernel will be everywhere finite if m is sufficiently great, and the methods of the * Gibbs- Wilson, loc. cit. p. 294. 158 Mr WEATHERBURN, VECTOR INTEGRAL EQUATIONS, Etc. preceding pages will be applicable. It may be shewn, as in the theory of the scalar integral equation*, that if the region S of integration is ‘a given surface this method holds for a< 2, and if a given volume for 2<3. The theorems of {19-23 on the conjugo-symmetric kernel are true when K(ts) is singular, provided its discontinuities are regularly distributed and the integrals {[r-K(st)P ds, and /[K(ts)erPdst are finite and continuous functions of the position of ¢, r being any finite and continuous vector. In the foregoing pages the author has dealt only with the theory of the vector integral equation. Applications of this theory to various problems of mathematical physics will be discussed elsewhere. * Cf. Fredholm, loc. cit. pp. 384-390; Heywood and + The square of a vector denotes as usual the scalar Fréchet, L’équation de Fredholm dc., Paris (1912), pp. 141- _ product of the vector by itself. This is equal to the square 145. of its tensor. IX. On certain Arithmetical Functions. By S. RaMANUJAN. [Communicated by G. H. Hardy*.} [Received and Read 25 October 1915.] 1. Let o,(n) denote the sum of the sth powers of the divisors of n (including 1 and »), and let o;(0)=$E(—5), where €(s) is the Riemann Zeta-function. Further let Dre (2) = o, (0) os (m) + a (1) os (n—1) +... +0, (M) G5 (0) ......eeeee eens (1). In this paper I prove that T@+H)PC4) C74) 6+) Sr, (”) = Tr+sii (M) D(r+s+2) E(r+s+2) + aatnae tO > ao od =a) NOr+s(n) + O {nt @+8+1) BS fo (2), “+8 whenever 7 and s are positive odd integers. I also prove that there is no error term on the right-hand side of (2) in the following nine cases: r=1,s=1; r=1,s=3; r=1,s=5; E—we—1; P=1,s=115 r=3,5=3; r=3,s=5; r=3,s=9; r=5,s=7. That is to say >,,s(”) has a finite expression in terms of o+5,,(”) and o,1;,(n) in these nine cases; but for other values of 7 and s it involves other arithmetical functions as well. It appears probable, from the empirical results I obtain in §§ 18—23, that the error term on the right-hand side of (2) is of the form Ore asad HONEY Ce eet te netaaes meee oen Mee caes eae (3), where € is any positive number, and not of the form ON as | eas cer REPRE cocce oor or Cee PERE (4). But all I can prove rigorously is (i) that the error is of the form O {nt +541) in all cases, (ii) that it is of the form OG AV Ga aa Ul aren Beret 2” 9. GAC een aciae eee (5) if r+s is of the form 6m, (iii) that it is of the form ONS At ts) A Bee re ernsee cer oc os aoe cs ae eee (6) if r+s is of the form 6m+4, and (iv) that it is not of the form ORO UAE SNE, se Meee Mere ns ss cag eeoass (7). * IT am indebted to Mr Hardy for his kind assistance and advice. WO SOSH SOR D- 21 160 Mr RAMANUJAN, ON CERTAIN ARITHMETICAL FUNCTIONS. It follows from (2) that, if * and s are positive odd integers, then T(r+1)P(s4+1) (r+ 1) &(s +1) =,s(n) ~ EG ees) AGI aa) pT (Dea eae oc been (8). It seems very likely that (8) is true for all positive values of r and s, but this I am at present unable to prove. 2. If S,..(m)/o,1¢:,(n) tends to a limit, then the limit must be Re ORGY Set) Cer) T(r+s+42) E(r+ts+2) — For then s. >, m1) : Dr,2(1) +2 Sa (Oyabes Ge C2) Lim = Lim me) ae wo Ortst1 (n) n> Or+s11 (1) + Gress (2) + --- + risus (n) ae ee re (0) + 5,4 (1) + Sy (2)a8-+ 21 Or+sq1 (0) + Orisyr (1) @ + Foe oe SS: = Lim g ew >1 Pris ? Iz A BS hiktnd (9). where Sra OHO) eee pa at ye pace shad statins ase steeeeeees Now it is known that, if »>0, then » Pet) e(r+)) Se 2 ee (10), as «2—1*. Hence we obtain the result stated. 3. It is easy to see that o,(1) + o,(2)+¢,(3) +... + o,(n) = Uy + Us + Ug + Uy + ee F Un, where warerg srg + |), From this it is easy to deduce that ox(l) 032) sb or(n) bob yee ae ee (11)+ and T(r+ DT (s+1) CoE Telseone E(r+1)n o,(1)(n—1)§ + (2) (n — 2) +... +6,(n-1) 1 ~ provided r>0, s>0. Now a, (n) > n', and o,(n) < n§ (1% +2 +37? +...) =n8F(s). From these inequalities and (1) it follows that Pe 1) ee Ok Cate a ON ihe a) Lim rie > P@+s+2) 4 Cae oil O eRe nritatgoents spesucnacondc (12), if r>0O and s>0; and = y 2 Ta ee) ee i Cts (13), 1 nrteti S (r+ s+ 2) * Knopp, Dissertation (Berlin, 1907), p. 34. + (10) follows from this as an immediate corollary. Mr RAMANUJAN, ON CERTAIN ARITHMETICAL FUNCTIONS. 161 if r>0 and s>1. Thus n~’*~Y,.,(n) oscillates between limits included in the interval T(r+1)T(s4+1) T(r+s+2) T(r +1)P(s+1) g(r+1), LD (r+s+2) E(r+1) f(s). On the other hand n~*“a,,,,,(”) oscillates between 1 and {(r+s+1), assuming values as near as we please to either of these limits. The formula (8) shows that the actual limits of indetermination of n~7~**,.,(n) are P@r+)0(s+1) €@ +) E(s4+1) P(r+ 1) (s+]) (r+ 1) &(s +1) E(r+s+1) T(r+s+ 2) C(r+s+2) ” T(r+s+2) E(r+s+2) Naturally (r+1)&(s+1) _ E(r +1) €(s +1) €(r+84+1) a FEES ED) < (r+ 1) £(s)*. What is remarkable about the formula (8) is that it shows the asymptotic equality of two functions neither of which itself increases in a regular manner. 4. It is easy to see that, if m is a positive integer, then cot 14 sinn@ = 1+2cos@ + 2 cos 20+... +2cos(n — 1) 0+ cosné. Suppose now that zsin@ «sin 26 asin 30 | ) l-z 1-2“ 1-# = (4 cot 40)? + C, + C, cos 0 + C, cos 20 + C; cos 30+ ..., 1 1 (cot 56+ where C;, is independent of @. Then we have C= 5( z oe F i fs ) SF Ni Se> Tae eS ee eu, 2 |\l—2, 1-2 1-2) °°" a 2 i a om ae a 72 (G—a* d—e#y* day eal z a 22:7 328 ) 1 alee a = oo Ba50 Coad aGapos saanoRmeace coceuce (15) Again 1 zr gra gn? nts Vo aes ae ee eres £L grr ge grt? xe gress 7 Pe la To Toa? Toe Toor il L gr ra ar gr £ 1 2 i eee TE ete Ser Pal _ * For example when r=1 and s=9 this inequality becomes 164493... <1-64616.,.<1:64697.., <1°64823.... 162 Mr RAMANUJAN, ON CERTAIN ARITHMETICAL FUNCTIONS. Hence (O- 4 1 x“ gn a gine a —£")= 2 AF c= ox) hy (- coat ae 1 z Gp Tee ie -5{(1 + | +(1 = eee a ( 1-2" 1-2 #3 iL n a a” nx” i 7 — eRe Arise caciaraneenanenooaeon Ids 16). That is to say Cn day 20a) (16) It follows that wsin@ #sin2@ «sin30 i y pies + af (Zoot 5 ren ee ee = zoc0t56) + weosd | scos 20 , a cosd0 -(5 2 (1-2) (-2#) (1—-#) wee 3 s(t os 8) + ae: Ta ey ee (1 — cos 34) + } (17) 2 lz © =a 1-2 SOT) TH wee Pereececes le Similarly, using the equation cot? $6 (1 — cos n@) = (2n — 1) + 4(n —1) cos 6 + 4(n — 2) cos 20+... + 4 cos(n — 1) 6 + cosné, we can show that eae il 1 x 2a° 2 3x? = {oot 3 asia eee C — cos 6) + I al — cos 20) + iat — 00830) +.) PH 1-# 1 l il 2 1 La =f — aS as = 5 2 9 (Scot? 58 +75) +43 7 + 008 8) + (5 + cos 26) 3. 7 = (5 + cos 38) + a eee. (18). For example, putting @=2m and @=47 in (17), we obtain 1 az Te Yi xz = (5+ = a +...) i=. 1-2¢ Io eee it al @ 2a? 4art 5ad =a Se See eee cate 19), sets(i-eti-e tiie ) (19) where 1, 2, 4, 5,... are the natural numbers without the multiples of 3; and G amps ax a Fi 2 4°1l-@ eee | 2a" 3a8 5x? il iL aL r =igta(restiostretist) see (20), where 1, 2, 3, 5,... are the natural numbers without the multiples of 4 Mr RAMANUJAN, ON CERTAIN ARITHMETICAL FUNCTIONS. 163 5. It follows from (18) that (alg 2 +558 — Fst Ss ) lies ig? = apt 3 %s— ig (a where S, is the same as in (9). Equating the coefficients of 6” in both sides in (21), we obtain (ats 64 : 68 ‘ TiS + FS —--) acisieleresterecere (21), —2 5 , ant Wn 35 Ne 55 Sn4s =" Cy 3Sn—1 a HSH Sines IF ORS OS ase SF oop ar Ores Sr 1 S;. ..(22 ), n! nQ — pnere "rin =r)!’ if n is an even integer greater than 2. Let us now suppose that mMm=CO N= ®, ;(“) = & 2 DUES Ee WR ee sectatstas oecceeene eens (23), so that OF (a)i= ©,,, (a); 1l’a QDs y? 38a? ’ \ 2 wa M4 2 Bo ARE SR a (24). w()= aap t Gat ah Further let x 22? 32° i So) ee (5+ a 1l-# os =) = Seen Be ) i ee (25) = ae ad hee oes laa yo a ba 25 4? 3573 ) a doo =—5 Ss= —5 R=—5048,=1 504 (+ =i -Then putting n=4, 6,8,... in (22) we obtain the results contained in the following table. TABLE I. 1 — 240, , (a) =P. 1 +2408, , (x) = Q. 1-504, ,(x)=R 1+ 4804, , (x) = @. 1 = 2644, , (x) = QR. 2 oF ee * If x=q?, then in the notation of elliptic functions 2 2K\2 7 pr = (=) (G+e-3), 1 T K q= Sa (=) (1- #2434), at 7S Feeeeaaee = = (2) +99) a -24%) 1-98. 164 Mr RAMANUJAN, ON CERTAIN ARITHMETICAL FUNCTIONS. TABLE I (continued). 691 + Hak (a) = 441Q3 + 250 R2. 1 — 244, ,, (2 = (Gre 6 it 8. 36174 ea (x) = 1617 Q@ + 2000QR. 9 43867 — 287284, ,; (x) = 38367 GR + 5500 Re. 10. 174611 + 132004, ,, (x) = 53361 @° + 121250 (7A. 11. 77683 — 552, » (x) = 57183 QR + 20500 QR*. 12. 236364091 + 1310404, ., () = 49679091 Q* + 176400000 @*A* + 10285000 F*. 13. 657931 — 249, .. (x) = 392931 Q@°R + 265000 Q°R*. 14. 3392780147 + 69604, ,, (x) = 489693897 Q? + 2507636250 YR? + 395450000QR+. 15. 1723168255201 — 1718646, .. (x) = 815806500201 @°R + 881340705000 Q°RS + 26021050000 R°. 16. 7709321041217+ 326404, ., (x) = 7644121732175 + 5323905468000 Q@°R* + 1621003400000 @?R* In general L(G) E> : 720®, ,(x) = PQ—K. 10084, ,() = Q@— PR. 7200, 5 (x) = Q (PQ- R). 15844, ,,(«) =3Q + 2R?— 5 PQR. 655204, ,.(«) = P (441 Q* + 250 R*) — 691 QR. 144, ,,(x) = Q (3Q° + 48? -—7 PQR). &, (a= Se POR i sees tere (29), where 1<2 and 21+4m+6n=s+2. This is easily proved by induction, using (28). Mr RAMANUJAN, ON CERTAIN ARITHMETICAL FUNCTIONS. 165 7. We have ea dP te La 24@, . (x)= 12 AiO) ome we tee 30 Ae te 240@, , (x) Sf arterteteetessrertaneceees (30). dk PR-=@ @, = — 504D,, 5 (2) = 5) Suppose now that r + 4h. S In general CoD (C2) ad Gi iene GUO ET TEEN Goo Gedoce Gon oncioonOesaeeGOneeOee (32), where J—1 does not exceed the smaller of 7 and s and 21+4m+6n=r+s4+1. The results contained in these three tables are of course really results in the theory of elliptic functions. For example Q and R are substantially the invariants 92 and gs, and the formulae of Table I are equivalent to the formulae which express the coefticients in the series gst gus 2 ga © Sosgsus PC) ah 90 + 98 * 72007 6160 in terms of g, and g;. The elementary proof of these formulae given in the preceding sections seems to be of some interest in itself, 166 Mr RAMANUJAN, ON CERTAIN ARITHMETICAL FUNCTIONS. 8. In what follows we shall require to know the form of ®,,(#) more precisely than is shown by the formula (29). We have LE Ge) Our) She aQth” it cee ee (33), where s is an odd integer greater than 1 and 4m+6n=s+1. Also d m 7) — ‘m ne m n m m— n+l m m+2 7 2 5 (QR) = (F+5) POWR -(F@ RM 4 Que ’) ena (34). Differentiating (33) and using (34) we obtain ®, 5.1 (2) = ye (8 +1) P{EE(—8) + Dos (2)} + TK n QR” ....-. cere (35), where s is an odd integer greater than 1 and 4m+6n=s+3. But when s=1 we have ae ioe ®, » (a) CVI RIS TeT Waa wy een cage Ye ewer (36). 9. Suppose now that F(a) = {4€ (—r) + ®,,-(a)} (FE 8) + ®,,<(x)} ACS ase f(1—8) 4 (2) VEC ICGE Dee DieED r+s arts (7) T'(r+s+2) €(r+s+2) x fh f(—r—s—1) + Do risis(Z)} ..-- (37). Then it follows from (33), (35) and (36) that, if r and s are positive odd integers, Tia C9) LCR a a Meer ee ces ese Sereno nS (38), where 4m + 6n =r+s4+2. But it is easy to see, from the functional equation satisfied by €(s), viz. (27) TG) GG ies dors = 3,6 (ls) seeeees ener tee ccmceneearena (39), that T+ (0) ) al Re RRB ence ses conock coder coos osocuaaenn (40). Hence Q*— R? is a factor of the right-hand side in (38), that is to say Peg (C—O) OE wen Oren anita cane penton eee (41), where 4m + 6n =r+s—10. 10. It is easy to deduce from (30) that L = log Ge) = Pw. aceon ncesanas unseen cee (42). But it is obvious that d PSe7- log: fin {ek — te") (1 a?) PA ie enctenen ccteneensene (43) ; and the coefficient of z in Q@— R® is 1728. Hence O— R= 1728 {1 — a) (1 — 2") 0 — a?) ne ena sccas ens tareene (44). But it is known that {(1—@) (1— 2) (1— a) (1— a)... }8=1— 3a + 5a%— Ta®+ 9a —... oo. eee (45). Hence QO =F =17280'— see bat — Tahitian) Giwenths edenaeesneoceve. (46). Mr RAMANUJAN, ON CERTAIN ARITHMETICAL FUNCTIONS. 167 The coefficient of 2 in 1—32+5a°—... is numerically iess than (8v), and the coefficient of 2” in @*— R? is therefore numerically less than that of 2 in 17282 {/(8v) 1+ +2 + 2°+...)}§. , 18a 252° Bia > But al+e+ebat+...8= + +o 4... Sydataretatenstelersie sce (47), and the coefficient of ~” in the right-hand side is positive and less than W(ptptat): Hence the coefficient of 2 in Q°— R? is of the form v0 (7) = O(0’). That is to say P= IPSS OOO aaasuss enue goon wood Been eeeanens: (48). Differentiating (48) and using (42) we obtain Sed (CBE ele) — 8 Os (W2) ai sisi eassc sect sce oseuecesees (49). Differentiating this again with respect to « we have A (P?— Q)(Q°— R’) + BQ (Q@— RB’) = = 0(v")2", where A and B are constants. But P?—Q=— 288 ®,, (a =—288 |=" Baa, a ene eo? ay Geer ii a: and the coefficient of z” in the right-hand side is a constant multiple of vo,(v). Hence (P?— Q) (@ — BR) = 2 Ove, (v) 2 = O(v") ‘ = = O(v°) fo, (1) +o, (2) +... +o (v)} = TOO") @, and so ESSE ee eee (50). Differentiating this again with respect to x and using arguments similar to those used above, we deduce 13: (GT PeSO C) (cmee ee ten eee (51). Suppose now that m and n are any two positive integers including zero, and that m+n is not zero. Then QR (Q = R?) = Q (Q ae R?) (Qa Re => OW) a {> O@*) #}"7 {> Oe") a”}" = S(O) (vy) a>O (yim) 2>d0 (vy) ay = > O (GTR tas) 2’, if m is not zero. Similarly we can show that Qn Rr (Q° —s R?) = R (Q§ = R?) (Quy RL —i>y O (GAS) x”, if m is not zero. Therefore in any case (GPa RP) OEE (OGURA) ire © cans okdpccocadacisacoCeoonARGOAE Rea eCaBeeeG (52). Norse xox: No. DX. 22, 168 Mr RAMANUJAN, ON CERTAIN ARITHMETICAL FUNCTIONS. 11. Now let r and s be any two positive integers including zero. Then, when r+s is equal to 2, 4, 6, 8 or 12, there are no values of m and n satisfying the relation 4m+6n=r+s—10 in (41); consequently in these cases When r+s=10, m and n must both be zero, and this result does not apply; but it follows from (41) and (48) that : y PR CeO i) Ie eRe eee cee eco se acecee: (54). And when r+s>14 it follows from (52) that IHS =>) OC eg eeaepensoocosonoccscscaactocednsbed: (55). Equating the coefficients of 2” in both sides m (53), (54) and (55) we obtain EP (rt DP (s4)) E(r+) E(4) SS = ; >,,s(”) T(ir+ s ae 2) C(r +s ma 2) Or+s+i (n) 7S i Re + sitet) HOt =s) NOrre (1) Hyg (1) . one: (56), r+s where Hen (0) =O Gees — 2s 46,08, Li E,..(n)=O@), r+-s=10; E,,,(n) = O(w**), r+s>14. Since o,+54:(n) is of order n’+** it follows that in all cases Rir+ D641) Sr+VE(+)) 2,3 (n)~ Tir+s+2) Erts+2) Cparrnn (0) oacongaas saeacecet (57). The following table gives the values of =,,(n) when r+s=2, 4, 6, 8, 12. TaBLE LV. 1 Syalo = B00) = m0) 2. 3,,(n)= 105 (n) lee (a), Baal) = 39 4, 3y5(r)= aeorie See (n) ee sae 6. S77) = ee (n)_ 7. Berl) = Fogg 8 alm) = Sh 691043 (n) — 2730n0,, (nr) 9 Su(n)= 6559 Mr RAMANUJAN, ON CERTAIN ARITHMETICAL FUNCTIONS. 169 12. In this connection it may be interesting to note that a, (1) a(n) + 4, (3) os (m — 1) + 0, (5) oy (m — 2) +... +o, (2n + 1) o (0) = 345 o; (20 + 1) soe, (58). This formula may be deduced from the identity Lx 3x? 5°a8 x 3a? 5a \ — KQ) Taner ee QS tpoetist:) Seana! (59), which can be proved by means of the theory of elliptic functions or by elementary methods. 13. More precise results concerning the order of H,,(n) can be deduced from the theory of elliptic functions. Let Then we have = Q = $°(q) (1 — (hk'y} | R= Gag) (ee bak eR OE Wl neled..cohsiel (60), = $2 (q) {1+ 3 (kk) Vfl cu’) where $(q) = 1+ 2¢ + 2¢' + 2q°+.... But if FM=F1-Q-gyd=@)- then we know that at f(g) =mEKt *o) 2b f(— q) = (RKY $6 (Q) ah (q?) = (kh’)* $(q) BiG) =KEF $(q) It follows from (41), (60) and (61) that, if r+s is of the form 4m+2, but not equal to 2 or to 6, then fi (r+s—4) (— q) MG) K Cp) (62) Tine (q°) = ro =) @) = n pon Ee @) ryalotaxel siefernvevele/alatarclete)steislovern and if r+s is of the form 4m, but not equal to 4, 8 or 12, then 5 tps (r+s—6) (- q) “" : ) Oa jee (Ci ) F,.3(q ) =" rin (pr) Ca { f8(q) — 16f° (q°)| - (ee pag (63), where , depends on r and s only. Hence it is easy to see that in all cases F,,(q°) can be expressed as Shea AGA Oya sn ae) G7 YS &Ka,b,0,d,e,h, LEG q)} ae | Foe (aoe } a aac 7G DSI") where a, b, c, d, e, h, k are zero or positive integers such that a+b+c+2(d+e)=[3 (r+s+2)], h+k=2(r+s+2)—33(r+s4+2)], 170 Mr RAMANUJAN, ON CERTAIN ARITHMETICAL FUNCTIONS. and [a] denotes as usual the greatest integer in 2 But ee OY, Un wee 24 OA A, BAC USCS = VE aa ieee 12 32 52 P P@M=c = 3q° + 5q° _ 79° ates he . 12 be 7 LIP 22S aensce wieoscee cance (65), 5G) ae 24 24 24 —“=g—5q +7q —llq +... thee) : s 4 AGP) 5 3 3 3 TONG) osteo Bes PED q —2q° +4q° —5q° +... where 1, 2, 4, 5,... are the natural numbers without the multiples of 3, and 1, 5, 7, 11,... are the natural odd numbers without the multiples of 3. Hence it is easy to see that n-3 (a+b+e)-d-e Bee (n) is not of higher order than the coefficient of g” in 1 s al 2 at $" (q*) b? (q2*) ° (q3) {6 (G24) & (G)}? {b (G2) $ (G2D}* 6 (G24) $F (GQ), or the coefficient of g*” in gute (q°) pai (q) p° (q°) $° (q"°) ° (q?) go (q°). But the coefficient of g” in ¢*(q?) cannot exceed that of q” in ¢?(q), since Gi (Qe) = 20g"); Sek dan, ceaee seen eer (66); and it is evident that the coefficient of g” im @(g*) cannot exceed that of g” in $(q*). Hence it follows that nN -4 (r+s+2)] Ides (n) is not of higher order than the coefficient of g*” in p* (q) $° (g*) $° (@), where A, B, C are zero or positive integers such that A+ B+ C=2(r+s+2)—2[3(7 +54 2)], , and C is O or 1. Now, if r+s>14, we have A+B+C212, and so A+B211. Therefore one at least of A and B is greater than 5. But PQ OO) 0° cern aie ae (67). Hence it is easily deduced that b4 (q) b2 (PYG (GDH TO fd 4 HF EO) — Ty oe ee gwdome nace adebebe (68). It follows that Re (= Onm ae Rs 2) on eae einer cee (69), if r+s>14. We have already shown in § 11 that, if 7+s=10, then PRS ON a soicsavisinsesucacigeoecnanmesse empeaaee et (70). * See §§ 24—25. Mr RAMANUJAN, ON CERTAIN ARITHMETICAL FUNCTIONS. al This agrees with (69). Thus we see that in all cases E,..(n) =O {nt r+s+1) NGO RGRORO CECE MACE CES enna ERTS (A) and that, if 7+s is of the form 6m, then 1B COC AUG) SERRE ee oe (72), and if of the form 6m+4, then Fh, {CERO NG SL as 9) RO RE 0S a (a3): 14. I shall now prove that the order of £,,(n) is not less than that of n?(+), In order to prove this result I shall follow the method used by Messrs Hardy and Littlewood in their paper ‘Some problems of Diophantine approximation’ (IT)*. Let q= er, q = eT h _etdr where = a+ br ’ and ad — be = 1. Also let Vie at+br Then we have BOG PUG AAS ACS, 7) Soden aceon oneennonendocesecrer (74), where » is an eighth root of unity and %, (v, 7) =2sin wv. g? I (1 — g™) (1 — 2g cos 2arv + 9!) ccccceececesevee (75). From (75) we have i log 3; (v, T) = log (2 sin mv) + 4 log g — » So ag (76). n(1—@*) It follows from (74) and (76) that 2n 2 } log sin wv + 4 log v + $ log g + log w — Sa ae a : x 37" = +2 cos 2n7rV) ais n=) ( Equating the coefficients of v**! on the two sides of (77), we obtain = log sin rV + 4 log V+ } log gq’ — ribvV — 1s¢ 28 ¢4 38 qh J sti JL Bes (a +br) ad eae ee 18 f° 28¢q/4 BFq’° = Ee e £E( s)+7 =e ae provided that s is an odd integer greater than 1. If, in particular, we put s=3 and s=5 in (78) we obtain ‘ 1l¢ 28 ¢4 33° (a + br) {I+ 240 € =F eer I au ae ck 7 = and ge a (a+bry {1 — 504 7 + a a oe, + . | Le gia Zid 3q* 1— 504 (5 eke | ee 80 7 =G I= 1-¢ 0) * Acta Mathematica, Vol. xxxvu. pp. 193—238. 172 Mr RAMANUJAN, ON CERTAIN ARITHMETICAL FUNCTIONS. It follows from (38), (79) and (80) that (Gb ber en) lige (9): aan meee eee (81). It can easily be seen from (56) and (37) that Te SEA AGING 0s cag5coeceanee ee ee (82). 1 Hence (a+bry 3? s E,..(n) @" = > VIEAGONGE® ceseescsosocosconnsacoéc (83). 1 1 It is important to observe that E Mae = r)t+€(—s) €A-r)+EQ-s) P(r+1) (s+) E(r+1) €(s+1) ie 2 r+s T(r+s+2) E(r+s+2) FEO) sis capelcs sat ven SMC b aes Me ae Pee eRee eine foie tv ole sete enee staeelacMeeaeia oreemeanes (84), if r+s is not equal to 2, 4, 6,8 or 12. This is easily proved by the help of the equation (39). 15. Now let r=uty, t=e™% (u>0, y>0, 0 (AF) {| B,,,(1) | e="*—| By, .(2) | e™ — | B,,4(8) | e-** — ...}...(87). Mr RAMANUJAN, ON CERTAIN ARITHMETICAL FUNCTIONS. 173 We can choose a number )y, depending only on 7 and s, such that | EZ, 2(1)| e?™ > 2 {| £,,.(2)| e2™ +| #,,5(3) | eo™+...} for N>A,. Let us suppose X%»>10. Let us also suppose that the continued fraction for w satisfies the condition LUNA ilin Sa (fst B> PINON aopopaootaRccco bob eee nents (88) for an infinity of values of n. Then / if r+8+2 > 4] E,,.(1)| (Ge) eH > K (dings ec eeeeveees (89), S E,, s(n) g®™ il Where K depends on r and s only. Also Qn Uni = 1/y, 3 ea (ke } is K dma Vy NM (log (1/8) ~ v= 8)" Tt follows that, if wis an irrational number such that the condition (88) is satisfied for an infinity of values of n, then bs BEN Ge ALG BEN EEE ANNE EE edcctessbsee ene (90) 1 for an infinity of values of ¢ tending to unity. But if we had E,,,(n) =0 {nt +9), then we should have SE. (n) | =0 {((1—#)~ 3 Weasaec) I. which contradicts (90). It follows that the error term in &,.,(n) is not of the form DR TRE)) So Ange erates cet ae yee ontaer a aatecsicwonde (91). The arithmetical function tr (n). 16. We have seen that E,, ,(n) = 0, if r+s is equal to 2, 4, 6,8 or 12. In these cases =,,(n) has a finite expression in terms of Grist (m) and oyss4(n). In other cases ¥,,(n) involves other arithmetical functions as well. The simplest of these is the function t(m) defined by Sie (yee es a) = a8) ed cet cee sensi: (92). These cases arise aren r+s has one of the values 10, 14, 16, 18, 20 or 24 Suppose that +s has one of these values. Then 1728 S E,, s(n) a” (P= RB) ED) is, by (41) and (82), equal to the corresponding one of the functions POM ery (3) (OR Ozh In other words > E,,5(n) a" = E,,.(1) 27 (n) a” {1 i 1 1 ao a 0 i poneonuee (93). 1 Ga €(l—r-—-s) 1- 174 Mr RAMANUJAN, ON CERTAIN ARITHMETICAL FUNCTIONS. We thus deduce the formulae D(C = OF aN) hax Go) Wapaenpicaace ccc ceuscaaddecshacoocdseed (94), if 7+s=10; and Orts—1 (0) Ey, (n) = /95- (1) {Gaon (0) 7 (n) + Or+s—1 (1) T(n —1)+...4+ 6745-1 (n re 1) 1 (1)} cca oa (95), if r+s is equal to 14, 16,18, 20 or 24 It follows from (94) and (95) that, if r+s=7/ +5’, then JI CQ) 3-2 (y= 13 1 WIE AGO) Saecanccanacan-cocaccopapdnase (96), and in general Dis MOD) 82 Kl ACO ERGO) caoassenotearcoasaoccsenese (97), when r+s has one of the values in question. The different cases in which r+s has the same value are therefore not fundamentally distinct. 17. The values of t(n) may be calculated as follows: differentiating (92) logarithmically with respect to 2, we obtain [Deel (0) BOI 3 TM) a. Fs snenseies ues aesenesectaeetttce (98). Equating the coefficients of x” in both sides in (98), we have T(n)= as jo, (1) tT(n—1) +0, (2) T(m— 2) +...4+0,(n—1) T(1)} ...., (99). If, instead of starting with (92), we start with S T(n) a” = a2 (1—3e@ + 5a°— Tae +...)8, we can show that (n—1)7(n)—3(n—10) tT (mn — 1) +5 (m — 28) t (n— 3) —7 (n — 55) tr (nm —6) ereontOn Pe uictra(Sr— 0) }] terms'=(0; Sees. e.-eaeeeeeisee reset (100), where the rth term of the sequence 0, 1, 3, 6, ... is }r(r—1), and the rth term of the sequence 1, 10, 28, 55, ... is 14+$r(r—1). We thus obtain the values of +(m) in the following table. TABLE V. | n | T (n) n T (n) Hi l +1 16 + 987136 a — 24 || 17 — 6905934 3 | + 252 | 18 + 2727432 | 4 | — 1472 19 + 10661420 5 + 4830 | 20 — 7109760 6 — 6048 21 — 4219488 7 — 16744 22, | —12830688 8 + 84480 23 + 18643272 | 9 — 113643 24 + 21288960 10 — 115920 25 — 25499225 ll + 534612 26 + 13865712 12 — 370944 27 — 73279080 13 — 577738 28 + 24647168 14 + 401856 29 + 128406630 15 + 1217160 30 — 29211840 Mr RAMANUJAN, ON CERTAIN ARITHMETICAL FUNCTIONS. 175 18. Let us consider more particularly the case in which r+s=10. The order of E,,,(n) is then the same as that of t(n). The determination of this order is a problem interesting in itself. We have proved that E£,,,(n), and therefore +(n), is of the form O (n’) and not of the form o(n°). There is reason for supposing that 7(n) is of the form O(n? **) and not of the form o (n®). For it appears that 5 = = Grete Ge eee (101). This assertion is equivalent to the assertion that, if =", 1 Dx 2s 8... Dp > where p;, Pp, --. py are the prime divisors of n, then 1 n for an infinity of values of x. . 23. In the case in which a=4 I find that, if ma (5M, 10 TT Te akin es LO Oa * y,(n) is Dr Glaisher’s (n). Mr RAMANUJAN, ON CERTAIN ARITHMETICAL FUNCTIONS. 179 where 5, 11, 17,... are the primes of the form 6m —1 and 7, 13, 19,... are those of the form 6m-+1, then Wu) _ (yy stantar+.. ee Care aC eC eee (127), Jn sin 6, sin 6, u/s where tan 0, = 1 +30 (0< 0,<~7), and w and v are the unique pair of positive integers for which p=3u*°+(1 + 3v)?; and that vv,(n)=0 for other values. In the case in which a=8 I find that, if n= (2775170 2. 7%, 1373, 19%. , where 2, 5, 11,... are the primes of the form 3m—1 and 7, 13, 19, ... are those of the form 6m-+1, then Yul) __yyetaten tin BCL +a)@ sin 30 +a,) (128) n/n sin 36, : sin 30,, tee _isielsjelele sia ° where @, is the same as in (127); and that W;(n)=0 for other values. The case in which a=12 will be considered in § 28. In short, such evidence as I have been able to find, while not conclusive, points to the truth of the results conjectured in § 18. 24. Analysis similar to that of the preceding sections may be applied to some interesting arithmetical functions of a different kind. Let ¢'(q)=1+2 2s FiO) tes oh taciescbtsiesenetoe astectere ise law aeerasle taro (129), where $ (q) =1 +4 2q 4+ 2q* + 2¢°H+ ..., so that r;(n) is the number of representations of n as the sum of s squares. Further let Re (ea oe tea Ga NG ea q aor g S8.(n)4q (= nee oem eon een) s5d0Re (130); na [es qd 9s-1 g¢ 381 g \ 2-1) B38, (n)q"=s (Tt + Et) ee 31), ( ) : os (n) gq" = s ee eGaeeTs ] (131) when s is a multiple of 4; x fs q Qs @ Qs41 ~ 9s _ , nr — ) (2 1) By 281s (n) q es adhucieencnnees (182), when s+2 is a multiple of 4; 1 it qd 98-1 g 3e41 ¢ 1s q 3s Of: js q° Nn — 98 == = — B,38(n)4q We a ee ee = ser =) 180 Mr RAMANUJAN, ON CERTAIN ARITHMETICAL FUNCTIONS. when s—1 is a multiple of 4; ao 1s qd 9s-1 ge 331 ¢ ) 1s" qd 3s g 581 g ; 2 Sag (2) Q™ = 28 -+...})—2 = one Hi, =535(n)q Gee l+q 14+¢ 2(7=4 l-g ' 1-¢ when s+1 is a multiple of 4 In these formulae B=, B= Sees, Ba ee are Bernoullis numbers, and E.=1, £,=1, #;=5, H,=61, H,=1385, ... are Euler's numbers. Then &,(n) is in all cases an arithmetical function depending on the real divisors of n; thus, for example, when s+2 is a multiple of 4, we have (2! = 1) B,Ss @) Se logan) — Qos, Giff éesnnr beets (135), ; where o;(a) should be considered as equal to zero if # is not an integer. Now let TEGO SO (GO) CaCO) mmpnercedoedeoceedccus5 1c s58dean606 (136). Then I can prove (see § 26) that CaCO} cesnwes 2d Sag hs akb -eae< GOO ae eee (137) if s=1, 2,3,4; and that etn O near TITS) 1.5. 2k ane eee oe doers (138) for all positive integral values of s. But it is easy to see that, if s>3, then ET <—Oag (10) Kos. 225. oddsene score so nacaeeron tenn (139), where H and K are positive constants. It follows that 755 (0) CO) ee Borean ne Sonos aneeenobocooneppbacsees (140) for all positive integral values of s. It appears probable, from the empirical results I obtain at the end of this paper, that CENT 5) | 2 Ae eae ay irda at oak (141) for all positive integral values of s; and that ec(MebomeOS)) 2 /e:oaie ek oe eee (142) if s>5. But all that I can actually prove is that bey a Or SUE) oS pea ce agen Sipatenscc ree (143) if s>9; and that es; (n) + 0 (n?8- DY dadveevd oe eee a eae (144) if s>5. 25. Let LAD= Seu (1) qr = = Aitega' (22) — "Gag (IU) fi a netaeeien ncaa cant iesiet ae (145). Then it can be shown by the theory of elliptic functions that fos (Q) = O* (Q) > OOD he eres cE BOA th StU (146), 1 : Noo oor (I) Dierolli sa = 616/18 (— g) f4 (92) —128 = 1 0 (9) Ban G7, (92) Fue) a. 6. 50521 fy (g) = 1103272 (— ¢) f? (¢?) — 821888 we. 7. 691f,, (q) = 16576f% (— q) — 32768/* (q?). It follows from the last formula of Table VI that Sot eM) —=\(— 1) A259 T(r) — SAT GEN) oreeke reese sersee ns (148), where t(n) is the same as in § 16, and r(w) should be considered as equal to zero if « is not an integer. Results equivalent to 1, 2, 3, 4 of Table VI were given by Dr Glaisher in the Quarterly Journal, Vol. xxxvut. The arithmetical functions called by him x1(n), Qn), Wn), O(n), U(n) are the coefficients of g” in fe (¢) (KO He gave reduction formulae for these functions and observed how the functions which I call TAG) SOC) Cae ONC) SAC) KC) (7), @2(n) and e@,(n) can be defined by means of the complex divisors of n. It is very likely that 7(n) is also capable of such a definition. 26. Now let us consider the order of e;(n). It is easy to see from (147) that f2;(q) can be expressed in the form > Karen if? (= O}" eae ; Aes SQ iF (GA) deremeas crest (149), where a, b, c, h, k are zero or positive integers, such that atb+c=[3s] h+hk=2s—3 [2s]. 182 Mr RAMANUJAN, ON CERTAIN ARITHMETICAL FUNCTIONS. Proceeding as in $13 we can easily show that n~ 2185] ¢,, (n) cannot be of higher order than the coefficient of g” in AG) DENG DOG) <= se « sos sche enaea eee erecta (150), where C is 0 or 1 and A+B+C=2s— 2[3s]. Now, if s>5, 4+ B+C>4; and so A+B>3. Hence one at least of A and B is greater than 1. But we know that g(M=LOW)¢" It follows that the coefficient of qg*” in (150) is of order not exceeding ne (A+ B+C)-lte Thus Ba (= Oe me a Bel he) ya ee eee eee (151) for all positive integral values of s. 27. When s>9 we can obtain a slightly more precise result. If s>16 we have d+ B+C212; and so 4d+B>11. Hence one at least of A and B is greater than 5. But g° (q) = S30) (v*) q’. It follows that the coefficient of g*” im (150) is of order not exceeding nb (At+B+C)-1 ’ or that 3 (n) = O Cab SUAS). Sates an eee ee (152), if s>16. We can easily show that (152) is true when 9 5. I find, for example, that = AD dS a eee (155), where ut ma) (ST) a)” 3, 7, 11, ... being the primes of the form 4n—1, and 1 i, = = = 5) * (1=2e,. 5 + 5) (1 — 2¢,,. 138-* + 13*) ... 5, 13, 17, ... being the primes of the form 4n+1, and Cp = U? — (40), where w and v are the unique pair of positive integers satisfying the equation u?+ (4v)?=p’. The equation (155) is equivalent to the assertion that, if (Suse item liale 52)? 3, 55). Use ite. where a, is zero or a positive integer, then Co (nr) a, Sin 4(1 + a5) 6, sin4(1 + as) A; as aaa 1) sin 4 6, = sin 4 0, Sey we eevvcecescnce (156), where tan @,= ~ (0<6,< 47), u and v being integers satisfying the equation u?+»?=p; and e,.(n)=0 otherwise. If this is true then we should have | an (n) lea << te KC) Nest qoapanbocosseeodoadeg coop Ineas TIeECee (157) for all values of n and Ey) (n) a =| |SSS0/ PRS RES AO EC Oncor osdoa node boo sneunereanBene 158 € (1) 8) for an infinity of values of n. In this case we can prove that, if n is the square of a prime of the form 4m—1, then Eo (N) — =n. Gin CL) Similarly I find that S A. (n) % ( 1 ) aie ey (1) i rea ee p being an odd prime and ¢,?n™ for an infinity of values of n. X. On the Fifth Book of Euclid’s Elements (Fourth Paper)*. By M. Jove torr MA, 1.) ScD EF RS., Astor Professor of Mathematics in the University of London. [Received 11 December 1916. Read 5 February 1917.] 1. THE object of this brief paper is to continue the discussion of the famous fifth definition in Arts, 4—8 of my third paper *. In Art. 5 of that paper I stated that I believed that Stolz in his Vorleswngen wéber allgemeine Arithmetik, part I. p. 87, published in 1885, was the first to reduce to two the number of independent sets of conditions comprised in the fifth definition. 2. My attention has been recently called by my friend Mr Rose-Innes to a passage in De Morgan’s article on Proportion in vol. xtx. of the Penny Cyclopaedia, published in 1841, from which it appears that the possibility of this reduction was known to him. It will be seen from the foot-note he appends to his demonstration that he was aware that his demonstration was not exact in form. The words “of given nearness” which he uses are difficult to interpret. I have however endeavoured to complete the proof on what I suppose were his lines of reasoning, or in the event of my having misinterpreted his words, then on the lines which his argument has suggested to me. He says (lc. p. 52, column 2): “Tt is however perfectly allowable to leave out of sight the possible case in which a multiple of A is exactly equal to a multiple of B; since if the test be true in all other cases, it is therefore true in this. For, if possible, let 44 = 7B, and 4C be (say) greater than 7D. Then m(4C’) exceeds m(7D) by m times this difference, which may be made as great as we please, or 4mC and multiples succeeding it, may be made to fall in an interval as many intervals removed from that of 7mD and (7m+1)D as we please. But 4mA is equal to 7mB, whence (4m+1)4A, We. must fall among the multiples of B in intervals of given nearness+ to the interval ot 7mB and (7m+1)B. Consequently the multiples of A followmg 4mdA_ cannot always fall among the multiples of B in the same intervals as the same multiples of C among those of D; and the rest of the test cannot be trne unless 4C=7D; that is, if the rest of the test be true, then 4¢0=7D.” * The first paper will be found in vol. xvi, part rv, the + (De Morgan’s foot-note.) We leave the reader to put second in yol. x1x, part u, and the third in vol. xxu. this demonstration into a more exact form. Vion xexcl,, Noo: IX: 24 186 Dr HILL, ON THE FIFTH BOOK OF EUCLID’S ELEMENTS. 3. To prove that 4C cannot exceed 7D, it is sufticient to show that if 40 do exceed 7D, then a single pair of integers p,q must exist such that whilst pA qD. But De Morgan’s idea seems* to be to find a number » such that if ¢ be any positive integer whatever or zero then another positive integer w exists such that whilst (n+t)A < wB, (n+t)C >wD, 4. I proceed to complete the demonstration on these lines, It is supposed that, if 7, s be any two relatively prime integers except the pair 7, S, It is given that if rA>sB, then rC>sD; but if rAs,D. Then I imagine De Morgan’s first step was to take an integer m such that m(7,C —s, D) > pD, where p is any selected integer however large. In order to reach the result set out in Art. 3 it is enough to take p=1. Let us take for m the smallest integer which makes m (7,0 —s,D) > D. Next since 7,A =s,8, . (mr, +t) A= (ms, + =) B. ay, : St Let w be the integer next greater than ms, + = 1 St 2 Syt o. MS, 5 we=ms,+ ms +1, 1 1 and (mr, +t) A qs + 2") D+m(7,0—s,D) 1 > (ims; + =) D+D,; 1 . (mr, +t) C> wD, but (mr, +t) A < wB. This is true when ¢ is any positive integer whatever or zero. Hence the multiples of A, from and after mr,A, are not distributed amongst the multiples of B in the same way as the multiples of C, from and after mr,C, are dis- tributed amongst the multiples of D. - 4 St / St eal = aU 1) [Since (ms, ate = ) D; Oo esl Sey “. (mry+t)A= (ms; + =) B. 1 ‘ Sit Let w be the integer next below ms, + ee 1 st Sit “. ms, +2 —-1lsw wB. Now r,C + (s,D —7,C)=5,D, st (mr t+t)CO+ (m af *) (3, D—7,C) = (ms, + =) D; sit ve (mr, +t) C+ m(s,D—™C)< (ms; a =) D; ? t “. (mryt+t)C+D< (ms, + ~) D; 1 . (mr, +t) C < wD, but (mr, +t) A > wB. This is.true when ¢ is any positive integer whatever or zero. 24—2 188 Dr HILL, ON THE FIFTH BOOK OF EUCLID’S ELEMENTS. Hence the multiples of A, from and after mr,A, are not distributed amongst the multiples of B in the same way as the multiples of C, from and after mr,C, are dis- tributed amongst the multiples of D. 1 1 ’ Sit Sit | Since NUS, + = 1lsawB, then mA cannot be a multiple of B. Consequently an integer n exists such that nB = D. i Then since $e and C= wpa dD, mr mM m By ee a y m m A OE IN By T, m ; iD) *But this cannot be, because m can be taken so large that ee less than any : 8 quantity of the same kind as D, and therefore less than C-—D; 1 8 0 + =D: vr * This is the starting point of the proof given in the next article, where however m has a different meaning. Dr HILL, ON THE FIFTH BOOK OF EUCLID’S ELEMENTS. 189 (ii) Next, if C<7D 1 2 eae | then since aie a and (>— D, YY; m -m™ Ss, n+1 n eG ae i m m s ape as m which is impossible, as_ before. ESA r; Consequently, Cp: r , AO=SBID 6. The following proof, which I gave in my Second Paper, Art. 67, was sent me by both Prof. A. C. Dixon and Mr E. Budden. It is, in fact, Stolz’s proof reversed, but it seems simpler and more direct. The enunciation is of course the same as Stolz’s. (i) Suppose 7,4 =s,B, but 7,C >s,D. Let m be the smallest integer which makes m(7,C —s,D) > D. * mrC > (ms, +1) D, but mr, A =ms,B; *, mr, A < (ms, +1) B, whilst mr, C > (ms, + 1) D, which is inconsistent with the condition that all values of 7, s which make rA < sB must also make rUC, J. (mr, +1) C < ms, D, but mr, A =ms,B, . (m7, +1) A > ms,B, but (mr, +1)C sB must also make rC>sD. Therefore 7,C <5, D. Consequently, 70 =s,D. 7. It appears that the possibility of the reduction to two of the number of sets of conditions in the fifth definition was first enunciated by De Morgan, that the first clear and unambiguous proof is due to Stolz, but the simplest and most direct proof is that due to Prof. Dixon and Mr Budden. XI. The Character of the Kinetic Potential in Electromagnetics. By R. Harcreaves, M.A. [Received 14 November 1916. Read 5 February 1917.] THERE are three important volume integrals over an infinite electromagnetic field derivable from Maxwell’s equations as modified by FitzGerald and Lorentz. One deals with flux of energy, another with flux of momentum, while the third gives an expression for the difference between electrical and magnetic energies. This last quantity has been called the kinetic potential, and the term carries with it the suggestion of an advance from the electromagnetic stage in which an infinite field is considered, to a dynamical problem in which the activities of the field are summed up in a single expression dependent on the coordinates of charges, and on their derivatives with respect to time. A normal kinetic potential constitutes in fact the complete statement of a dynamical problem. There is one primd facie reason for doubting the normal character of this function in Electromagnetics, viz. the necessity of accounting for dissipation of energy and momentum. It is the first object of this paper to establish a departure from normal character, and to fix its precise nature. As dissipation is due to flux at infinity, which is a feature of the other integrals cited above, it is plain that if the kinetic potential fails in any of its normal functions we should look to these integrals to fill the lacuna. It is our second object to shew that they are adequate for the purpose, and to deal with the method of applying them. § 1. Of the two main purposes which the K.P. serves, the derivation of expressions for energy and force, the former is that in which help is specially needed. Sommerfeld in his memoir on potentials calls attention to the difticulty (no doubt experienced by others) of dealing with the energy, on the ground that a quadratic function of electric and magnetic vectors is to be integrated. But the K.P., which is also such a function, is integrable so far as its effective part is concerned, when the potentials are known. If the method of k.p. had complete validity it would be possible to turn the difficulty as regards energy by deriving it from the K.p. The form obtained for the K.P. is >: ay (Fe + Gv+ Hw — VV) re a in which (uwvw) is the velocity of an element e of charge, V that of propagation, w, F, G, H are potentials, and y is an integral not evaluated. For use as a K.P. a function which is a complete time-rate is entirely nugatory. The sum > 5 (fu + Gv + Hw-—wV) Vor. XXII. No. XI. bo OU 192 Mr HARGREAVES, THE CHARACTER OF in the case of a finite number of point-charges is a completely integrated form when the values of the potentials are known, and the way is therefore clear for the application of the method to point-charges. This then is a case in which we can make the enquiry as to the character of the K.P., and the proper use of the other integrals. Point-action has always been the basis of mathematical treatment for continuous distributions, and evidently has a more immediate application in an atomic electrical theory. With respect to the other function of a K.P., the derivation of force, there is no such disability in the direct application of electromagnetic methods. The connexion between the methods may be briefly stated. If Ww, ... are values due to the action of ¢,, estimated at a place (x:y22.), where at time ¢ there is a charge e, moving with velocity (u.v,W»), the force on e due to e, has the « component 2 {X, + (VC, — web;)/V} =—6& ( por, - ane 4 fala & ge) _ &Ws = V ot Oar, / V \dr ye V \0z, 0a » (OF F P =— = + Uy = + U2 = a i) a 3 = (Pyity + Gyr + yw, — yp i); ; : OUr ‘ = ibe i i ewe . eT , 2 2 since for a point-charge AE do not exist. Now ay includes the whole dependence of F; on time through the motion of e,, and therefore OF, es oF, ae OF, . & oF, of Gm By, Oe, q : dF, ; f is a complete time-rate aE Thus if we write L= > (Fit + Gv. + Hyws— WV); ane ae and the above expression has the form ab_d a Ca, dt OUy We observe at once that this is a derivation, not from the kinetic potential sph Us + Gv. + Hyw.— Ww V)+ a (Fru, + Gov, + How, — 2 V), but from the first section doubled. This is a first hint of departure from normal conditions. In the statical problem there is equality of the two sections; in the general problem dynamical equivalence is consistent with inequality if the difference is a complete differential df coefficient with respect to the time, or more briefly if the difference takes the form ad °° that there is no immediate certainty of breach of normal conditions. § 2. To make a further step the formula for the potential of a point-charge is expanded in a series proceeding by inverse powers of V, the velocity of propagation. ‘This value is required only at the point where the second charge is placed, and is used to give a corresponding expansion for the K.P. The groups containing odd and even powers of V are considered THE KINETIC POTENTIAL IN ELECTROMAGNETICS. 193 separately. It then appears that the terms of even order in the two parts named above é : : d q do in fact differ by a quantity of the form 2. The even sections of the two parts are therefore equivalent, and their sum may be accounted a true kinetic potential, which if it stood alone would represent a system conservative in respect to energy and momentum linear or angular. The terms of odd order possess a property the antithesis of that of the conservative : : d ; : group, viz. the sum of these terms in the two parts has the form af Thus, if we write dt Ly and L’,, for even and odd sections of one part, we find that Z,,—Z., has the form = Z while Z’,.+ L’ has that form. The force of even order on e, given by 2L,. is also given by 2£., or by Ly+L,; but the force of odd order given by 22’, is also given by — 2L’x, while Z’,.+ LZ’, which we expect to be the K.P. in fact yields no force. There is then a distinct breach of normal conditions in the group of terms of odd order specially associated with radiation, which we may call the dissipative group. § 3. The volume integral involving rate of change of energy has the form j= = > (e+ ny + €2) + tlux of energy (or total radiation), if H is the total electromagnetic energy, & the # component of electromagnetic force on a point-charge with « for component of velocity. But if we use # and & for the sections of even order, #’ and & for those of odd order, we have dE 3 0 = +5 (Ee + ny + &) in virtue of derivation from a regular K.p.; and WEG rs gr 0= aes + 2(&e+ 77+ 62) + flux of energy ; or in effect the flux of energy is a quantity of odd order in V. This flux is found by use of simplified values in an integral over an infinite sphere. When this is evaluated it should appear that the sum of radiation and a rate of working is a complete time-rate, and then an expression for #’ can be found. The equation for loss of energy is got by writing = (E+ E)E+( bn) p+ (E+) 4) = -3 Get tut Po where 7’ is the material kinetic energy, and f, a component of mechanical force if any such exists. That equation is a dt We have first the use of a formal relation between different electromagnetic quantities (1 +E+#’)+ Radiation (or flux of energy) => (af, + f, + Z.). to give an expression for H’, and later, when we associate electromagnetic with extraneous mechanical forces, or definitely postulate their absence, we have an equation for rate of loss | of total energy. 25—2 194 Mr HARGREAVES, THE CHARACTER OF The flux of momentum is also of odd order, and the second volume integral gives a formal relation of the type O=&+ = +fux of momentum, which is connected with mechanical forces by writing & + & = Xm#—/,. Here again two quantities, now a flux and a force, are to give on summation a com- plete time-rate, and we infer the expression of a momentum p’ belonging to the dissipative group of terms. In the case of each integral a condition is to be satisfied which will give us the assurance that the flux at infinity is correctly treated. Also it will be found to involve the localizing of the parts of these mtegrals, a problem solved for the conservative section by the use of K.P. § 4. It is understood in Dynamics that all coordinates are stated with reference to one time. In Electromagnetics the. primary position is that coordinates of a source are referred to the time of departure of a wave, and those of a point affected to a time of arrival. As a preliminary to dynamical treatment we require that coordinates and velocities of the source should be referred to the time of arrival. If (x.y,2,t) denote place and time of arrival of a disturbance originating from e, a point-charge at place and time (a'y/2,'t’), a... being functions of ¢’, this reference is made by the use of ro —= (Ge — ar) (Yael ee), aNd oi t—a7e Ve eee eee (1). The values of potentials at (ay.2.t) are = 6) (ray ie Vy 3 = eh Vee re eee (2); and these values are to be expressed through (1) in terms of 2,—a,, uw, %..., where msi ()), eel =a OReose We use r for N (a — 2)? + (Yo — i)? + (@— A)’, and (#yz) or (lr, mr, nr) for (@—21, Yor-Yi, 2—-%). The values of potentials and of the resulting K.p., which would follow from a treatment of (2) by successive approximations, may be obtained more rapidly as follows. The expansion of yf, is fal De r Ds r2 Ds rs thea int rapt Phan + ene (3), where D, or = denotes differentiation through «,y,2, and their derivatives mw, .... If we operate with V? on y,, where 24.2, are the space coordinates in V*, and use the equation V?r"=n(n+1)r", then noting that V? and D, are commutative, we get - : a 1 De r Dr? Div VF es i+ (7) eel oe) a a = the fundamental equation for y,. For the expansion of F, we have = 0 D Der Ds a2 Pa fp tes iat + } THE KINETIC POTENTIAL IN ELECTROMAGNETICS. 195 It follows that the expansion of LZ,., the part of the K.P. due to the potentials of €, at (#24222) the place of e,, is given by V2— Su, us, c= D, (V2 =U) fe D2r( V2 — Suu) Ds (V2 — Lu, tr) ” ) 2 V V1 ae ie pees 21,.= = a | Consider separately the terms of even order in V with a view to obtaining an equivalent : Brae df : ee form, i.e. one differmg by a term = of symmetrical character. Now in (5) where a K.P. wae - : : F d, Br is in question «, as well as a, is a function of t, and we may use D, or FF for dif- aL: ; ae d ferentiation through w, and its derivatives, so that —=D,+D,. A term Def may be dt — replaced by — D,D.f, a term D,f by DD 2f, and so on. Hence an equivalent of the even group in 2Z,, is Z where |= = 2auo° DDyr (V2? = me) _ D?2D?2Z7?(V?— uu) | _ = a¢ == THAI i aia0 €1 2 a r V2) V2 a form symmetrical with respect to the two points, and therefore replacing even terms of 22. for e, and of 22., for e,. For the group of odd terms, D, may be put outside the bracket, and even powers of D, within it replaced by —D,D,, D?D,,..., giving for the equivalent K.P. 6:6, | V2?— Suu, D,D,72(V2— Sau) D2D27*( V2 = duu) S —— = Be ae 7). 21’, = D, ie 7 V23! VAN An equivalent of the odd terms in 2Z,, has the same form with D, outside the bracket, and the sum has therefore the form = The points stated in § 2 are therefore established. § 5. As a formal example of these results we may apply them to the case of charges e, and e, distributed uniformly over the surfaces of concentric spheres of expanding radi 7, and 7. If we write e,d@,/4a for e, and e,dw./4m for e the integrand for joint terms (containing e, and e,) contains direction cosines only in the combination %J,/,, and an element of integration dw,dw,/(47)? may be replaced by dn/2, where n= I,/, and has limits —1 and +1. Whether in (6) or in the bracket of (7), terms of like order in V give an operator (which is a power of D,D,) acting on D,D, (re + 72 —2nryre) 2 + (im +1) (m+ 2) nts (2+ 72 —2nryr2) 2. m-1 This quantity is equal to —7,7,(m+1 a L—n?) (r2 +1? — 2nryre) 2 , y 4 ( dn ( and the integral between limits —1 and +1 for n vanishes. Another form of the relation is r+ | {(m + 1) ryr.(1 — 7?) R™ + 2nk”™™} dn =0 - -1 with R?=r?2+r2—2nryr,. Thus in (6) and in (7) the total is reduced to the term of 196 Mr HARGREAVES, THE CHARACTER OF lowest order in V~, which in (7) is a constant; Le. there are no odd terms, and radiation 2 : : *ldn 2. is absent. The conservative K.P. is reduced to =a R oF to as if r,>7,. From = 2 22 z this we infer at once the terms —S a and — due to the actions of elements of one a7) 21» sphere on other elements of the same sem and the K.P. is that due to a statical system. This simplicity however cannot attach to the potentials, for the vector potentials do not vanish and so the scalar potential cannot be independent of the motion. We may examine these potentials for a single sphere, radius 7,, charge e,. Symmetry justifies us in writing F, G,H=(%, y, <) x8, 0G OF : : : which through Eee makes the magnetic vector vanish and so involves absence of radiation. The equation a Ghee =" dx Oy V ot is then represented by aS 28. Loy or -- —— = ann => 0 wc le’a.onlelelslatelegnteialtiatelalelsielniaielelelatetatulstalaberetetats (8 a) eee lees Oy _ 10S dw Be es iat oe ae electrical force is reduced to the statical value if 10S Oy _ V of tor RS We SPR o5 Soocbbananscosraccusoste xc (8b), according as r2 7. Now take (0, 0, 7) for the point at which we are seeking potentials, so that F=0, G=0, and H=S. By (8) and (4), with R?=7*+7%—2nr,r, we have & fa (al Dee Re DORs ge fe lat V2) V33i Wear | ie esi +L) Dede DSi). D2 (Ren) a oe (9). Q OV a Ra Te W229! = V3! ome ndn The properties (8a) and (8b) then hold, if Qo ft +1 n= | r? R™ndn+ D,| rR"™dn=0 or =| =| a +1 Re*dn eae ee and Ae dD, ih (m+ 3) (m + 3) + is ie R™ndn =0. The latter is obviously the result given above, and the former is the second form given to the same result above. These properties enable us to write THE KINETIC POTENTIAL IN ELECTROMAGNETICS. 197 according as r27,: the quantity ¢@ is equal to : Die Dee DE wom =. voi vesit pear | Ops: ery , and V?¢— a He 0 or +2 according as r=7;. We have thus a satisfactory account of the properties of y and S, in virtue of which complicated values of potentials due to the expanding motion are consistent with the system as a whole having the potential energy of a statical system. § 6. With a view to shewing that the method of partition into conservative and dissipative groups leads to reasonable results, we examine briefly the opening terms before proceeding to a general method of realizing the differentiations involved in the series. For the conservative group the efficient terms to the second order are Ces Duy Us + Dlu, Sluz : L=% (- Dp Et) ceeetterseeteeesesseeeen (10a), and the corresponding terms in the energy are p= (1 peat Sas, Foe, ACen Beco (10). The relation of (10a) to the formula of Clausius may be noted. His first simplification of an original more general formula gave a value 18 > s SIo,. SIn/_)) U= =F {Rk 24 Us + hy (Luu. — Tu, Tlue)}. With respect to 4, he suggested that simplification in the force may be obtained either by writing k,=0 or k,=—k. The first value is that which he preferred, and with that value the formula is known as Clausius’s potential. The present form is obtained by taking k,=—k/2 and k=1/V*. It is the form properly belonging to Maxwell’s theory, and here it is understood that in the application to moving charges it gives first and second order terms im a series of terms of even order, constituting the K.P. of the conservative section of the forces. A component of force is given by _ OL d oL = if TH = inte —— (36,7 — Su? + 2, Ue + re ——— aN 1 1 wt + V2 2V2 in which o,= >lu, ae o,=lu,. The conditions for the existence of integrals of linear and angular momentum are satisfied, viz., OL ol ——— + a 0, 0%, OX ae Bi 1), a ee ~ Oy Y oa BOUTS Gly em vOUs Naktats ; the terms contributed to angular momentum being ne and oe aL 13y, 99 “EG ; Th 198 Mr HARGREAVES, THE CHARACTER OF . ae 3 7 a 2a .. For the group of odd order the initial term in 204 18 ays wales yielding a force component , 2e, ey =. a= Qa tlh ceeees trees tee eetee tee tersceteeneenens (1la) The sum £ 26, es 2@, 3 te, = (Eu, + &'uz) = a Y(hut w rt.) = 5 32 Dh, vs + Tw 2.) — 3\ OS ie and a comparison with a ¥ (&/u,+ &/u,) + Radiation = 0 suggests the inference of a term in the expression of the energy H’, equal to 2e, es = (one! Dini)” ..... Sees cue eee (11b) d : diati apa Aes. . VW and a term 10 radiation equal to Bys faietetaletsfeistoleiel®.«.e)ais erelelelotetetetetetstecstatateeloislslelateteietetestar (11c). Integration over an infinite sphere (v. inf) confirms the inference which is sufficiently obvious in this simple term. The self-terms which may be inferred from these terms in the mutual action are: erie Se aa : 2e? in force 3ps> 2 radiation 3 and in energy — Ps ee (12) Taking these in conjunction with the ae terms, we have to this order: the « component of force on @, is = (GPS SSC) Rene eee coecandopcecdscosaocssenact (13 a), the term in radiation of pair is sya (eat + ev caled Some tees RT eee (13 b), and the term in expression for ag is | SGN F- C5tte)' (Ext + Osa) saradaneweasinwe tine eee (13). == Thus we find that the main term in the odd group gives Larmor’s expression for radiation. The sum of the « components of force on the pair is 2 (e,+ @.) (Qi, + estiz)/3V*, which vanishes for a neutral pair, so that only the relative motion is affected. Again, if we treat as approximately true the equations m,%,+m.t%.=0, which are exact when only the dominant term of zero order is taken, the supposition- e, : e¢ = m,:m, will make the expressions in (13a, ),c) vanish, ie. the main terms in radiation and the force associated with it vanish to this order. In effect then we have two types of pair physically differentiated from others, (i) the neutral pair, (ii) the like pair for which e,/m, = e/m. §7. ‘To carry out the ditferentiations in (5), (6) and (7) it is useful to write a, ds, dy, ... for (D,, DY, Ds...) 77/2, with b,, b., bs, ... for the operator D,. The opening terms are Q=—X(%—2,)%, d= Zuz—Z(a,—%)%, G=32my%— = (e—-M,) ii, a, = 382u? + 42 i, —S(a,—m) ti, as = 1lODUti, + SE yii; — TZ (qy— ay), wb cere (14a) b, = (@— 2%) Ue, b,= Due + E (a, —a) ty, ... i THE KINETIC POTENTIAL IN ELECTROMAGNETICS. 199 and we note Dia, = — Vy, Dydg=— Dh, Dib,=—Sujw, D,b,=— Suto, =| da, da, | eter (145). . : =+6,, ===O.— Dj, S- = A3— Dit, ... dt dt dt The numerical coefficients in (14a) are those of the binomial theorem, with the first coefficient halved for letters of even index ay, d,.... making r In this notation the opening terms of (5) are ON bes 1 1 1 ae ; = ae {as — a?/r? — Zuo} + 31 (as — BX tute) 12 . 1 2 So . = As! 2/2 5 9 /y.2\2) Mee (7? (dy — 427i, Wy) + 40, (As — 2Ett, Uo) — 42, Ue (Gg — a 2/2*) + 3 (Ag — a2/7°)?} Cc 1 (o.2 KN Qy5 sy © 5 ) ~ + 307" {72 (a5 — SE, Uo) + 10, (4 — BVI, Uy) + 10a, (2a, — YW) — BasE%w, uy} +... ... (15). The opening terms of the symmetrical form (6) are L Sr. — ay),/7" ona SS ee exes Fa 2Ver 1 , : : + eWay. {2 [( Sa)? — 04°b,2/7*] — (ae — a2/7°) (b, — b2/7?) — 2 (ay Day te, +B, St, wz) — B72 Sty, ih ate eee liG)s 1 5 5 oe 2s oO, 05 r QV 1 a] 9 2 2 Sv 2 9 . SS 2 ° . SS . . a . oll sa 12 Ue P= oon?) — (Lu? — oP — 71764) (LuUF = CL +TS2) + WW ( Gy LU ty — Gy DyUy) — B72 Vy, Nig} +2775, 2[ (Saw) ew ré,)(Su 41r6y)+ 2r(a, Su, % Dus) —3r2S in the other notation used above. The opening terms of (7) are 21 =— Eth, us + gy @e@ i 3V2 15V% (Asbo + 24g Ey ty + 21? Vii, thy + Tay Diy ty + 3, Sti, Uy — OZ, Uy Tru). ..(17). §8. In (15) we have terms of types ay, a,a;, @,°, a1 associated with a denominator V+ Thus the expansion (15) will represent a succession of terms of diminishing importance if, u,/V being of the first order, 7,/V? is of the second order, r?ii,/V® of the third order, and so op, On what does the fulfilment of these conditions turn? If we admit the dominance of the first term —ee,/7 in the K.P. the equations of motion are in the first approxi- Mation mst, = e,@l/7? = — m,%,.... If there is only electromagnetic mass, and u,/V is not hear unity, that mass is a finite multiple of e°/V°c,, where c, is the radius of the electron, and then rw,/V? will be small if ¢,/r is small. If the whole mass is greater than this, the smallness of ri,/V® can be secured with a less magnitude of orbit. Differentiating the last equation written ¢,¢,(@— 3121e)/7 =— m,ii,, and r°%i,/V* is of order > x —- of a pair is considered the successive orders of acceleration will conform to the statement above, if the ratio of the relative velocity to V is small, and the orbit sufficiently great. In Vou. XXII. No. XI. 26 Thus when the motion 200 Mr HARGREAVES, THE CHARACTER OF a conical orbit it is generally true that v° and fr (v and ff relative velocity and accele- ration) are of the same order, ie. if v/V is of the first order, fr/V* is of the second order. But there is a case of exception for a hyperbolic orbit near an apse where 2: fr=e+1 for an attractive, e—1 for a repulsive orbit; and if e is very great the smallness of fr/V? would not secure v?/V? being small. It will appear later that the case of large «absolute velocities can be dealt with if accelerations are small. §9. We consider now the method of dealing with a kinetic potential in which accelerations of any order appear, and in particular with the two-point case of the con- (n) servative system. Let Z be a function of ##4#...2, and write (n) B= X,24+X,¢+...4X,2¢-L, where X,... X, are to be defined. Then GE (ot CI eee te.) ee dt Ox 0a: meee) \ (n) 0 & On Pe ey Hues 2 dt dt Sea = e (n) (n+1) + X, €+...4 Xn, 2+Xyq &. : : dE We now define X,,, X,_,--. in succession so as to reduce the value of — to the first ; dt column. Thus . ee oL vee OL dX, ob d ob Geom yp Ui “(=| a Cen (n) ? 02 x dt 0 « dt 02 and if we also introduce possenseaeeente (18), Xx _oL dX, ob d OL d? OL _ On «Co dt Ssiéaet:C‘éi we have = +X,¢=0. If there are several variables we write ({n) E= (NX, Gt pXok +... ++Xn 2) — L, then = =U). Ge MEA Vp eapepene concnccobosaanboncgctodce ic (19). Admitting that the quantity X,, occurring in the variation of Z, represents force in the extended scheme, then Z, in virtue of satisfying identically the equation (19), is the energy function attached to Z. If there is force 7, external to the scheme of k.P., then we have X,+f2=9, and oo ne dE : : i 2 > af,, or more generally aE 2 fy fe,; with no such force # is constant. r $10. In the two-point system (#42) (@Yyo2,) the total linear momentum P in direction « and the total angular momentum WV about the axis of 2 are given by P=,X,+2%,, and N= &,¥,— UX. + Vo — yy Mot BV 5 — NXg t coe Pocseeeee esses (20). + 22 V\— Yo oXy + Ho 2Va — Yo Not ss. i THE KINETIC POTENTIAL IN ELECTROMAGNETICS. 201 : ; a, SDDS df ae 3 We have made use of equivalent K. P.’s differmg by zs ; are the quantities P and V given . . . . (n-1) uniquely? Let f be a function of the type of Z above, but proceeding only to x, ,...; then a K.P. Be eet Of. oF 5 Gf of 5 Of Tt aa -o + ay n= + Y, ae rs +), reg at | will shew for each w or y Oba of ae oF df | BL ah | _(@) ~ =i)’ ey Py A@=2) alae Ae 1)?" Oa dt Ow : ae Si) Ra NG Nee Re eS oan (21) af 3 of of and therefore Xn= aay? Ca cee ~—-- | Che iB @) & result from the comparison of columns above. From the last follows X,=0, ie. there ; ‘ x df is no force, and from = (#,,X,+ 4,,Xo+.. j=9 or L it appears that there is no contribution to * energy. The contribution to linear momentum is Ope OF Ay ted, = ees ort On which vanishes if 2, and «, appear in the combination (z,—,) as they do in our case. The value of NV obtained by substituting in (20) the values of X,... given by (21), will also vanish if () SOR a OE Gi = 4 GF of of 0=a, iv + ty 5 + #, = tenn (me! +H, Pet 1+) bape to =(tege +...) (22), 1 1 1 pal 1 2 Lo a less simple condition than that for linear momentum, but one satisfied by all the elements used in constructing the formulae for ZL such as a,, tju.,.... As an example of the type of proof applicable, suppose the property to hold for a;, and operate with D, on (22) written with a, in place of f Now D, as applied to az, is (me + ip ea oe * Buy Sits Ga The difference Gai. <2) “mee nate) 0 Baty _ = Wi Dy (« ay, + Uy, av, +...4+ u, am) As (« ay, ap ese ste Uy =) D,a;= =U, —— ae i, uy oii, 5 _ may 0 oe or D,( a ou + =) A3\= (« yt we tH sq) an operator of the same type extended to cover the new form i in a. The same is true 3 : : a for the section y, ods ..., While for the other two sections 7, ..., Yo 2 ... the operators ¥ y: i 0a OY> Ox, are reversible. In order to make it clear that the value given to WV is correct we form EN. d Ried yo ie +i (M+, Y.) + & (i+ GpsF) to fod tf = (eh ile etl =— %1Y, + lar Fie ty 5 ae Rogier 3) Ieee ocr =—(#,.%-y1 X) — (#204 7 — Y22X 9), 202 Mr HARGREAVES, THE CHARACTER OF in virtue of J satisfying (22). Hence, with ,X,+/,,=0, we have dN : : q : de Dd — YiJx, + By, — Yoda, Thus the condition (22) plays an important part; as applied to / it ensures K.P.’s : : d which differ by J dt to LI it gives the proper dependence of rate of change of angular momentum on external force, i.e. it makes the K.P. system alone conservative in respect to angular momentum. angular momentum, while as applied As an example of derivation, by using (15) or (16) the fourth order term in the value of # is found to be ee [2 {(22y ws)? — a2b2/74} — (ag — 2/7?) (by — b2/7*) + (Sy Ue — Bad, /7?)( da — a2/r? + b, — 62/7?) + (ash, + t)b5) + (5a, — 2b.) Sty a + (5B, — 2a) Vey tty + 37? E (ti, Us + Wy tin — Hy it.)]...(22 a), while the z-component of force on e, of the same order is €,€. : Sn Ae as ba 7 . 5 a 3 Vie E {3 (dy — a°/r°) (dg — 5ay?/7*) — 420 Us (dg — Ba,2/7*) + 2a, (dg — ZZ, Us) — 7° (ay — 4EHi, Ws)} + 4u, {7 (ad, — 38a2/r°) — Sinus} — 2rit, {3 (dz —a2/7°) — 2m vs — 4a, b,/ 77} — 4rii, (2a, + b,) — 37H, § 11. I now propose to state what I have found possible in the way of emancipation from conditions as to smallness of velocity in the case of the conservative group of terms, acceleration being still accounted small. In (6) write #,—#,= and perform the operation V? with wzyz as space variables. We get v D,D, , _ i eg VL = a ee or — Wi dt dee’ Pee cecceteen eens cveneecescce (23) as a differential equation satisfied by the symmetrical form of K.p. If here we write 1, SRNL 2 an 2 d: ee PoE Lies di ot du i ‘dys “ae di ot | "Oe oy “0B” where a operates on w,%,...,and then suppress cae we obtain at a ae PP ot’ ot y= Ge 0 ) a ) é VeL= { ( 2 No Ws ) eee ececcesceces oF, ip (% aa + 2, e + W; =a) (w an + Us By + Wa L (24) for the quasi-stationary term. A solution is the reciprocal of the square root of a quadric whose coefficients are the minors of the discriminant of V*2a* — Xww=u«; this quadric is r? {(2V? — Sry ue)? — Su? Tus} + 2(2V2 — Vayu) Sau, Seu, + Vu? (Lau)? + Tu? (Zew)...(25), and the term* in the kinetic potential is DL = = 2¢,¢,( V2 — Sy u)/7 V(2V2—Suy wy + 0,02)? — (Su? — o7)(SuzZ—o.2) ...... (26). The positive character of the quadric is assured if 1>«*cos?a where V4«!= du2Due and 2a is the angle between the directions of the velocities. This is readily seen by * If in (6) the differentiations are made, and a,..., Dit, uw... omitted, the result agrees with (26) when dg, b, are written for Du,*, Du,". THE KINETIC POTENTIAL IN ELECTROMAGNETICS. 203 referrmg the primary quadric to axes two of which are the internal and external bisectors of the directions*. If the formula (26) is applied to the example of two spheres we get z 6 [2 (V2 = nite) dn _ 102 lay +1 Simo, mA Ve (re + 72 —2n7r,1s) —TyPotis (1 — 72) 2Vryr2 | => sr [Vv V2 (r2— 1) —VV? (rz ny ]= -2%, Ve 1s agreeing with the previous result. If we seek in (26) a suggestion for the form of the self-term, one course is to-make €:=@, Ue=%, halve the value and take the mean for all angular positions. This gives iT eee = oa 5 doe (V?— gig ag 2Vr VV2— Sue + (Slay? 47 4Vq1 YS Gh 27) Heal Oe | Vin oy 8 Va, V-q with ¢,?= Zu? and m, V? = a m, being chosen so that the term involving 21? is o zu, The Ly Ides inertia in the direction of motion is lq?’ , the transverse is — 1 qh aq” If we write WU =(m, + m) U= mm + ms, Uy—% =u and suppose uw small in com- parison with U, then with 3MV2? V?— Q, V+Q — Py Ae tT 3 vq 8 7—Q: Q@?= > U2, : na mm, jl dl, s , 1 dL,\ (& uy? a Int a= ln Sars gag = * (e990) as representing terms in (wvw) as far as the second order. Under the same circumstances the main term in (26) is — ¢,e,(V2— XU?) V V(V2 — 0?) r+ (S2U? : §12. We now consider the information given in respect of the non-conservative group by the integrals over an infinite sphere, and at first deal with a single source. The section of a space derivative of potentials which yields finite values is oY == ae where (c—,)/r,=h. Consequently at a great distance 7 Scans lb 6G oH X=75(a¢-m)> aay \ Mh ap a =) OF 6G oH Low _ 5 and as A ae ay + = 2. Ties 0 eae estes ae (28), : OY _, OF 0G oH | yields p= l, a +m, BE +n at * Or by use of velocities 0, 0, w, and us, 0, w,, when the condition is 4V+—4V*w,w.—u,?w,°>0 or in general terms (2V?—Zuju.)? — Du? Du," >0. 204 Mr HARGREAVES, THE CHARACTER OF we have X=nb-me, a=mZ-nY, WX=0, Zha=0 as for a plane wave. We have also Sa?= =X? =@ say, and Yo—Zb=16, %h(Ye—Zb)=4 4X, +mXy+ mX,=hé= Vo—B ...(29); viz. since Xe =4(¥2?4+ 77—X2)+4(F +e — a’) or 6—X?—-@", and X,=—-XY-ab, X,=—-XZ-«a, we have LX,+mXy+uX%,=h6-XE1,X —alha=he. Thus at the surface of the great sphere a component of Maxwell's stress is also V x com- ponent of momentum-content, or it is V~ x Poynting’s vector. The sphere is taken with (a,7,2:) as centre. If it is supposed fixed, a factor 1 —o,/V is needed for the Doppler effect and the radiation integral is 1 (V—-a, ml 7 Cre: if it is treated as moving there is no Doppler effect, and the integral is 1 = | (VEL, (Ye — Zb) — v,6} r2do, which on identifying v, with o, again gives ij Ep2 Az | ( _ o,) Ered. There is the same feature in connexion with the momentum integral which is 1 sV-—a z!— ie neta for a moving sphere the form is = \z1X. anaes ~2b); rida, reduced to the above by (29). Now y being e,/{7,-=(x—2,)u,/V} or Ve,/r,(V—o,), and F being u,e,/r(V—o,), we have at a great distance Oy Va, oF 4 f ty, fe MG, et = 74(V—a,)’ Of (V-o (Vv (Gq) while e = mel — : j Su,? 26, Suu V? — Su?) 67 t re ca e Uy on hu ( i) aa oe ri(V —o,) { V—oa,? 24 (Vi=o;); (V—a,) and the radiation isan is — 4.26 >>) Yi V2 — du Ve" dw a >) 2G, 2th _ ( Uy "| —— = ah {3 1° V—o, (V—c," -) Wen = 2e/ “ie f = (u ui | tt mec ceceeeeeseee (30). => S532 bam rth 3(V2—Zu7) | [= ts — Du?) | THE KINETIC POTENTIAL IN ELECTROMAGNETICS. 205 The corresponding integral in the equation of momentum has the extra factor J,/V in the integrand, and its value is u,R/V2. § 13. When two sources are in question the joint terms in energy-content are > (X,X. + ads), and as the difference between (#—4«)/r; and (x—~«,)/r, would involve 7, we may take l,=1,, and the last relation of (28) makes Lads — A Xe — Sl, X, 3X, = DX, Xs, so that the terms in energy-content are 2}.X,X,. The value of LY.X, is € > re (V = 0,7 (V— 0.) The Doppler effect, or effect of moving surface of integration, will be dealt with as SS * oe SS Vz Gy QUyUyg To QUy Ur = Lys) 6,6: {Stns + — ( nts) O16 V—a, a Vi2ca a (V—«a)(V—a,) follows. For the flux of momentum localized at ¢,, and for a corresponding flux of energy, integration is taken over a sphere with centre at ¢, with the Doppler factor for that centre and the integrand {X,X,. As X, depends on w a function of 4, the phase-difference between #, and ¢, will be calculated for points on the sphere, and thereby the expression will be brought to one time. Corresponding parts due to a sphere with e, as centre will give the flux of momentum localized at’ e,, and a corresponding flux of energy. These fluxes each using half the energy-content will constitute the total. It is proposed to give evidence that this method ensures the adaptation to the forces stated to be essential. The times are connected by V (to —t) = 1 — My = & (a, — 2) (a4 — 2)/7, = Sh (a, — Xi); im which #, 1S #(é). -If now we use #2, %.,-.. for a(t), #2.(&),.--, Ds (to) = y+ (to —b) Ue oe Uy +. and VAC a) = th fr. + Us (te—t,) + = (4-4)? + aa : or, with z now for relative coordinate #,—a,, and o,=Sl,z, (—t) _ Co 22 — t,) -= (t2— e = 4) | , ..(32). ie a fa com adi oy on org \Os Fx awe | a 2 2 Vo, 2(V—a,) oma 13 apy = -o»){ With this difference we have U2To Usoy” Usoy Wn Oy Fp Vise on aos aay 2(V —o.) and similar forms, and in particular Uy (to) = Uo + gives — 206 Mr HARGREAVES, THE CHARACTER OF §14. Thus denoting by R(e,) this first section of the joint term in radiation, we have on introducing the Doppler factor = dw ss tty | FeBtyty (V2 — Trt) G1de ae) = vlor-en—en | {B sits as V—o, (V—a,)(V—c,) o> 6, DU ti te i God tly — TU Ue, G,—(V? — Duy Us) G52) aye a tai Wen (Kano. | + ce . {Sty tin + ..-} 20,65 6, DU, te) Boog, (FZ (V2—-Tuw) Gey : ay Binet 5 s a Ca) im=a: aes 2 | ee in which the lines after the first are due to phase-differences. The z-component of flux of momentum is the same integral with the extra factor /,/V. For the moment we require these evaluated to the order V-> for use in conjunction with force on e, and work done on @. The evaluation is IML(E)\= 151 Vs [4adw, ti, + (8x, + Tus) Sty tty — ty [bs — BU te} + WL Vw (us —U,) — ti, Zar ]...(34a), R(e)= a ye Sit Uo + = AP [(22u? + 9S we + 9b.) Tri, tty + 27? Lr Wy + (11b, — 3a,) Vr itin + (dy — Day") (bs — BE uy tig) + Vt Us (b; + YE %e) — Zr ty (b; — Zw ty)]...(B48). Now, working from the formula 2Z,, in (17), a ee i +inyl- a (b; — 5Sryiin) + 10%, Ery tip + 10d, + 10%, (2b, — Su, U2) + 5g (3d, + ay) + 27°] ...(35.a), and Sé&'4,= a = Dt tin + ah [a, (bs — 5E ws) + 10b; Er, ti, + 206, Er, ti, +5 (3b, + ) Sutig + 27 Zu, We] ...(85b), Thus the sum | | 2e,e, K F,,(@,) + & = = Ea tly + te }— a (by — 4204 tin) — UW, (b; — BE Ute) + Ue (b, + TEU Me) + thy (9b. + 22a, Uy — Tuy?) + tip (118, + G) + 277%, 3 ...(36) and the sum F d | 2e,e, ers: 2 R(e,) + LE/u, = ai see. Sth ts +7 = {27° Du, tio + (11d, + a) Dati + a, (0, — 42H tin) + bw (te — Uy) + Tuy ty (QE? + OE, to + 00.) soled) dE’ (e,) rk Next we observe that with these values of #’ and p,’ we have the relation H’ (e,)==p,'u. We now form the quantity Sw F,(e,) and subtract it from R(e,), ie. we take from the total rate of loss of energy that which is of mechanical order, due to flux of momentum, and denote by S(e,) the pure radiation which is left. We find @) 2 Lu, Fz (¢;) ~Tisl ai [— 4a, See tin + (BBUF + TE ly) Dry ti, — Sey ry (b; — Tq %) + QP ty Dt Wy — Tayi, Tai], THE KINETIC POTENTIAL IN ELECTROMAGNETICS. bo =) “I and thence S(e, a= 5 Vs LB ty, + = a [= La, (by — 420 ti) + 27? Diy, + (110, + ay) Tri. + Lith tly (Db, + uy Ue — Vu") — Tuy ty (b; — BZ ay Weg) + Vt Uy (bs + TE ry te)]-..(38). S is linear in respect to (v,%,w,) and we find that S(e,)=—p,/%. We have then a scheme of relations & +f, Ore =0,. 28% Rie) ee eG | dt EF’ (e,) = Spy, S(e)=— =p th, R(ej=8@) Eu Fe)| Thus the conditions laid down that the sum of force and flux of momentum, and the sum eters (39). of rate of working of force and rate of radiation, should give time-rates of momentum and energy, are satisfied to the fifth order by our method of treating the integrals over infinite spheres. The position is confirmed by the simple relations which are then found to obtain between energy and momenta, and also between pure radiation and momenta. I have also evaluated the flux of momentum to the seventh order, where the flux contains upwards of 50 terms (in the compressed notation with a, b), and this condition is again satisfied. The radiation condition of this order was not examined, but I feel little doubt that the whole scheme of (39) is exact. But if this scheme is of general validity, it is evidently possible to proceed in a different order, viz. to find integrals R and F, infer S and thence p and so & and £, that is to construct the whole scheme of forces of non-conservative order with expressions for momentum and energy, from the integrals at infinity. The advantage of this is that -these integrals can be evaluated without reference to the magnitude of w,/V, w./V; and there remains only the condition that the phase-differences should be small enough to admit of treatment by expansion. The application of this method to the self-terms gives the result g= 2e,° (ws ae i, )* t 3V (V2—Su ay ew rol 2e, 3 p=- ee& 2 dai, + Uy Uy ty ) 3V (V2— Su?) (ea V2— Su)’ ite 3 (V2= Su?) ; 26° Uy DA, + Bt, Day . 3, (Lryw)!? — Ta = 1 5. TEP O a a 9 Sy, 2\2 3V (V2— Su?) V?— su? (V?— Su??? with R and F as given above in (30). This agrees with the value of & given by Abraham. §15. The integration for the joit terms can be carried out by exact methods, and We propose to give this integration for the main terms, i.e. omitting those dependent on phase-differences. The integrals in we dw 4a | (V—o,)(V —oF Vor. XXII. No. XI. Say Cc ou => - a 6 DUy Us Dit Us _ ( Uy Uy) Oy Fo R(e)= [Ete + ee ee aa Pad bo ~T 208 - Mr HARGREAVES, THE CHARACTER OF are derivable by differentiation with respect to w, or wu from a fundamental integral* over a sphere, viz. = (V2 — mw) do Fe = pe aml 1 fen ea in (V—«,)\V—o:) 2VG@2—AB °C-V@?—AB 2 °1- | a (41) with the notation | A=V?—Su2, B=V?—-SuZ, C=V2-—Smw, 22=(C2?—AB)/C | In differentiating it is also convenient to use y?=Ab/C*=1-—a*, and then from loy % , Ue Loy _& uy you, a Be? Y OUy BG we form = sae (-“ mi +2), where fi=y zr : ae, df, Extending the notation, Le. putting fe=Y Gy? .-, we have 14+fA=H¥(F+A) L=PA+3A+H) H=~PAF+8At 5fathh leading to = 2+2A—f=—y"*(Atf) and 2%— f= QA+BAth, which with y°= AB/C? can be used to modify forms of the resultt. The integration yields: ae Lihte (foa—fi _ fath\ — Vth Vtetle (fot fs , 2a—Js ( Be (Aaa) R(e)= CO \ Be ABET G i Lt Uy Vthtly (cae WE = ie Let, Tuythe (4 +s rae — fs) BC A C AC C B 4 atl Sith é eastcfs 2fs~ Js) (42a) =. s Fn) | eager estan es 2a), * Using velocities (0, 0, w,), (v2, 0, wo) and putting V=1, we require =f. Be phi dnd Be 1 i } _dn 4m 1l-w 1n) (1—w, n— Uy V1 — 22 cos p) 2} -1(1—wyn) Jiu — Bn, +n? (us? 24 w,2)’ for which we quote | dn a jog bn+-y (en—b) + N(a—2by + ey") (a—2bn+en*) | (y-n) Na- 2dn + en? "Va — 2by+cy? wan : and the integral is then ca ’ 1 log + t1) { (aw, ~ b) + (c = bw,) + (1 —w)} von = 2bw, +e 2 Vaw,2—2bw,+e — (1—2w,) { (aw, — b) — (c - bw,) +(1+w,)} Naw? — 2bw, +e 3 me, 1 = wy Wy + Naw? - bw, +¢ 2 J/aw,? — 2bw, +¢ 1 = ww, We - Va Ww)? — 2bw, +e f To get the last form cross-multiply the fractions and use aw, — b+ w, (¢ — bw,) = (w, — we) (1 — wy wg). Then note that aw,? — 2Inw, + ¢=(1 — wy wy)? — (1 — wy) (1 — u.? - w,*), write V where 1 appears, and the general character of the result is evident. + Opening terms of expansions: fx are fant oe pee Skt iat pate 82? 8x4 Peek Heskjar i alsin lS raaamiapencss THE KINETIC POTENTIAL IN ELECTROMAGNETICS. 209 Palen) = Sop | 4 FF ta ep, — fy) — Seat 4 fy Sut, Sant Dtyus Vustis + tae BES (ft fi) + a f+ 2. Lue Duh eae LUs the BC? (Js ; — fi) -—-—> (S37 . + Ber (f —fi)+ (2f.—f:) Lint, Vuyis ABC ty (Tuy te zak the Siu, Duit,\ - ee +a B G2) ith) + B(= C = 7 ) A] alee giesie eee (42 b): Li, Un Sis Us ean Gia= ts) + BC Mei) thence S(e)= €:€> Suzi, Ducts Sis Duty My Uo 5 oe (=3 (23 fh) ape (Atf) + = HE Va) Lath Teyily C? Diy zs ths aay — Sapaue (42 c), and Dus ta 2Us the Uo Tuyo > lo E’ (@)= If here we write F,(e,) =a, +. Bus + ai, + Byis, then R(e)— 8 (e) = Sm F(a) = V2(a + 8B) — (Aa t+ CB — a Sui — B, Emin), if for Su,2 and Sw, we introduce V?— A, V?—C to compare with the forms in (42). It will be found that R(e,)= V*(a+ 8). Thus we can go a step further, and say that the evaluation of F is the only integration needed. This does not appear to be affected when the phase ‘terms are introduced, though the form of F is thereby extended to F’,(€)) = yu + (au, + O%,) + (Bro + Bytig + Boiiz + ...). An example of this is seen if we look back to the fifth order value of F,(e,), when we find that V?(a+ 8) gives correctly the third order term in R. Also in obtaining the value of F,(e,) to the seventh order I found that these terms give correctly the value of the fifth order term in R, a further addition to the evidence. § 16. A case in which the formulae are much simplified is got by writing U = (mh, + Motz)/(mM, + Me), UW= Ue — and then assuming that wu is negligible in comparison with U, while U is negligible in comparison with zw. Thus in effect we put U for wu, or uw, and U=0 makes w, =— mu/M, %, = mu/M, where M=m,+m,. As a first poeees we then get stay a nt Re)=- ie 2 ss eo (8) Uke) “oo = ae ons 7 27—2 210 Mr HARGREAVES, THE CHARACTER OF with the same values for S(e), R(e) and F,(e). Hence with the self-terms the totals of radiation, and the sum F,, concerned in total linear momentum, are 2 (€ ms — &m)? V { Sw ee (2Uu)? | g k= >U?) ae UR 3uP (V2— SUP” (V2 = SU2yJ ’ "i V2 2 ee ~ These quantities vanish for a like pair with the property e,:@ =m: m.; while for a neutral R= (44). pair (€,m.— e,m,)?/M* =e, or e,* The sum of the self and joint terms in the momentum of e, is 7.) 20: (Me —esm,) ( . U>Uu zs Pp (@) = 3 MW Vivi=s SU?) V2— Sl? ele wins aie vinci s)u elejelsle vive cia’ (45 a), and therefore the total linear momentum is 2 (G+ @) (Gm. — em) (. — UsUu 3 3MV (V2— S02) V2 SOR ces (45 D), which vanishes in both special cases. § 17. It is proposed to examine in detail how the primary motion is modified by the main dissipative terms, those of the third order, which we shall treat as small quantities while gnoring the conservative terms of the second order. The equations of motion are Cie ees eel ea, a + 3 ya (erth + @ils), Mwy =— a 3V8 (Githy -F @ntin)! J.ceeeees (46), Mz Uy = and the integral of linear momentum is (Aaa VAR AO V2) oneacosnsooussonoccese (47 a). 2 TU, + Ms Us — 373 For a neutral pair this reduces to the primary form, and the relative motion only is affected. For a like pair with the ees @, 2G = ™M,:Ms, (47 2) viele (e+ e,)? t/3MV* MyUy + MzUy — P = {(m Wh + Meoto)¢-o — P} e i neg ed) . which requires my, + mst. =P a and therefore always, unless we are to suppose that linear momentum can increase indefinitely. In this case the problem is reduced to that given by the primary terms. But if we take the problem of (46) with general values of e, and e,, and introduce in the small terms of (47a) values resulting from the primary solution, we get ? 5 Nae My, # My — = eet ee 2) ( (= = “ ) = IPR tere hos bien 8 MEER (47 6), 1 and for relative acceleration C621 / 1 1 ) 2, Msii / € e; ) i= . 3MVs (= m, (pe Ga Mo 2 9. Ne With the notation j= -syr ae = (1 . 1\. Qi we have li + mi + nw = ail oa ) + ai Rao SUCRE RCODO SOT COO (48), v2 \m, Me r and : a —yl=— g (wv — yu), ... | shewing central acceleration and areal change in convenient forms, THE KINETIC POTENTIAL IN ELECTROMAGNETICS. 211 With multipliers w, 1... m (46) the equation of energy is d [im Ms C4 92 , , 2 ; ae 5, 2 g Stat = o Suse + = ane & (G1 + Cote) (yt, + Cotto) | = ap S (ety + Grte)?...(49 a2). : ? :

) SS ORONO KH HER SOTORCRCOE UAE 49 au + 2M ~ Mr? (29.2): : eee 2 ; ae Peres eee The rate of radiation 5 73 = (Cats + Cetin)” has the value — oF in fact dz ees “3 a =P = sade sce eee ease ince eee eee ae niso ans canal (50). The loss of energy by radiation in one revolution is dt pae fr dO _ au - _ 4&PT (5 = peves|, ay iE = ie "(=e cos 6): dé sarge + e)...(51 a), and the mean rate of loss, on division by the period, is _ peres (2 +e’) 2a‘ (1 — e)2 where in the small terms we have assumed the results of an elliptic orbit, & being the constant of areal description, and e, a, and 7 having the usual meanings in Conics. S552 ae Sele sonst (515), §18. So far it has been possible to dispense with an account of the actual deformation of orbit: this we proceed to consider in the simpler case where the motion is plane, and the position in polar coordinates is defined by - dh__?p 2 le ee a 3p d 1 dé Po. Pp CPE h? Mm) hr dr €,€/ 1 1 = a ) k? \m, Mm Pear oscars (52). ™m, , DEM cos 0) The solution based on the fundamental solution h =k, = when radiation is neglected, is h=k(1+«)—7 (0 +esin 8) Gale So 7° Te : =f: F (20 —esin 6 — 3e’sin @ cos @) \..-.(53 @). z 7 -=1+ecos@—2«+acos@+ PB sin @ +F (20 2a cos @ + ée* sin d cos 8) dr @r ’ dd’ d& have h=k at 0=0, by taking a=x=0, B= =% (2 2 is + e). It appeared however in The orbit can be made to agree with /=7r(1+e cos @) in respect of r , and to dealing with the period and some mean values, that oe formulae are obtained by making this agreement correspond to 6=7 in the middle of the circuit. We make use of this deter- mination, which gives Ur — We SP (5 ee 2 2 a2 Ohi” B=7) (2 até SOc ORC CORO ROBO E DEE (53 b) 212 Mr HARGREAVES, THE CHARACTER OF In (53a) we distinguish between a cyclic and non-cyclic perturbation, Le. we write = 1+ecos@+ U, in the first revolution, and T= 1+ecos 0+ Uy + Ue or L+ecosd+U, +47 7 (4 +e cos 8) in the second revolution. Thus we suppose that the @ which occurs in U, is always between 0 and 27. , If then the constants of the modified ellipse in the second revolution are I’, k’, e ray (it er), e (1 +47), é =e 4 Ea kl a a es a : areal Ree NO (54). Toe j —_ 2p (2 aes L, Perit — Bee oS , kl —é) oi ) The period in the orbit is Ao ae males 3p (a — 0) dé 2 le 7 ee 1+ ee + | (1 + ecos 0)?’ where terms containing sin@ are not written. This gives a normal value to the period, viz. Qari? /k (1 —¢) ~ Qara?|V — e,e.M/mymy. The term to be added for the next revolution is P fie (— 6p7r) dé ee: _ Sprtl(2+e) Jo kl +ecos 6) (1 —e&)? k ie. the next period is ri ss faie (> 2) Sa = i ora"? |v Selene k(1 ey | kl(1—e*) | 1 [ which conforms to the modification of the fundamental ellipse. In the same way the mean potential energy taken for a revolution is » [dt ee [rdée mi fen) 3p (a — @) a dé NT Lh I |p Qkl = «Bkl(1+ecos@) ~~} 1+ecosé Q7re, e2l _ 122 : a ES ee wi (55). yi = i : a rolath - — mm ade is The mean kinetic energy of relative motion or OMT Dwdt 1s gl eee 2 ae oa} +..| Ber IMT Jo ~ 1+4ecos6 kl ~ T+ecos6 * 1 +ecos6) 1+ecos@ mya ke my mz ke _ _ es b = iia wal ie ee ee (56). These values are normal, and it is readily shewn that in the next revolution the means retain the same form with the altered values of a. These values give as the loss 9+ 9 ) oe Pee Te) in agreement with (51 a). cis of energy in a period, == 2 a a THE KINETIC POTENTIAL IN ELECTROMAGNETICS, 213 § 19. The loss of energy is associated with a contraction of the major axis, and a diminution of eccentricity. The mean kinetic energy of relative motion is inercased by an amount equal to the loss of energy by radiation, and the mean potential energy bears the double loss. This is an immediate consequence of the maintenance of the relations eR) and, 1: Uh pee ene cee sews: (57), where JZ’, and U, are mean values of relative kinetic energy and of potential energy. A brief statement may be made in respect to hyperbolic orbits, where the total radiation may be calculated, a quantity which in the theory of point-charges represents the radiation due to a collision. For the attractive case make the comparison with the fundamental orbit J=7r(ecos@?+1) at O=0, ie. take a=0=« and 8 =f (2 + Sa e). The radiation in the complete orbit is 2pe,é, [7% és 1 Go : = i ih (1 + ecos 6) dé = — = (2 + e*) (7 — &) + 3e sin O,} ......(58 a), 7 where @,< 5 and ecos@,=1. For the repulsive case we get Pers . eee ETE (2+ 2) Oy — Be sin Oa} eee cevc scons eeconneeee (58 b). : Dee The value of the constant p when e, = —e,=e is P= 30% mime? and if we suppose : : : . : 22 /e€\? that e, is a negative electron, so that the ratio m, : m, is small, then P=aplan) . The 2 Ms A ae 5 266 - = = a = s coefficient of the bracket in (58) is then ome — yar if J is eliminated, it is 2 e2\5 3m,' V3 7) - For the elliptic orbit formula, (51 5) is 2+ ¢ ; (2) e\' 3V2 (1 —&): \M _2/ 2 \ and the number £ occurring in the orbital changes is 3 (ackV) ; It may be noticed that if @ is taken inversely proportional to temperature 6, then the kinetic energy of relative motion is proportional to @, and the rate of radiation to This is no doubt a significant point; but application to the thermodynamics of radiation probably demands a statistical treatment of a large number of elements and the groups which they can form. § 20. I have also solved the problem of the primary motion as modified by the terms of second order, which when e is finite gives more trouble in the integrations. The results are of the type found in discussing the question of a modification of gravity as applied to explain secular changes in the orbit. For the present purpose their importance seems hardly commensurate with the space needed to prove them. 214 Mr HARGREAVES, CHARACTER OF THE KINETIC POTENTIAL. In conclusion a brief review of some points in the paper is added. The fact that a form 2Z,, is used to give force on e while 2Z., gives force on &, raises a presumption that the kinetic potential has not a normal character. The departure from normal type is not easy to locate exactly, without the use of an expansion proceeding by powers of V—, an expansion certainly valid for a wide range of motions. It is then definitely located in the section of terms of odd order, and these terms only are concerned in radiation. The even groups in Ly. and £., are shewn to have dynamical equivalence (§ 4), and to form an entirely conservative system if treated alone (§10). This conservative section, when acceleration is negligible, admits of a quasi-stationary kinetic potential without assumption as to smallness of velocities. Electromagnetic force is known in respect to odd or even sections: in the conservative section an expression for energy follows at once, in the dissipative section not until radiation is evaluated. Closely connected with this is the question of localizing momentum and energy, i.e. distinguishing the parts attached to the two charges, a problem solved for the conservative section by the use of the kinetic potential. For the dissipative section it is necessary to call in the aid of fluxes at infinity. In view of the fact that two centres are concerned it is not immediately evident how this flux is to be treated. But the fact that we are using information furnished by two methods implies that a correspondence is to be found which will be a criterion of correct treatment of the flux. The agree- je—-ment of two methods of reduction to a one-time system is involved. This adjustment is in fact attained as far as the approximation extends, and it is presumed that the scheme of relations (39) so deduced has general validity. This carries with it the possibility of presenting the radiation from two sources in a form free from limitation as to the magnitude of velocities; and also of deducing expressions for the terms of odd order in energy, momentum, and force directly from the fluxes at infinity. The integrals concerned are all derivable from one fundamental integral (41) involving the sources in a symmetrical way. It will be noted that the argument in general deals with joint or product terms in the action of two point-charges. The transition to self-terms for the dissipative section presents no difficulties; in the conservative section infinite values would appear. It is only in this connexion that the necessity of giving finite though small dimensions to the electron arises. The method used in the text does not postulate definite structure, but I think the decision in the matter must be left to experimental evidence as to the ratio of two inertias in the case of rapid motion. a (Recetved 1 December 1915; read 21 February 1916.) XII. The Field and the Cordon of a Plane Set of Povuts. An Essay in Proving the Obvious. By Eric H. Nevitte. CONTENTS. SECTION PAGE 1. Introduction and summary 2. Explanation of logical symbols need in this paper ond nee desaribed in Guts I of the Introduction to Principia Mathematica bo = oo 2 3. Definitions adopted from the general theory of sets of Sofas 219 4. Straight lines and rays 222 5. Chords 224 6. Triangular soos 224 7. Parallel lines, and rays Gemeeedl in parallel aes 225 8. Leaves and clipped leaves 225 9. Sectors 227 10. Circular domains 230 11. Theorems relating to chow fearon pectors: ane aronler Gini ; 231 i] bo 12. Primary chords of a set 13. Secondary chords of a set 14. Cross points of a set 15. Convex sets . 16. Excluding sectors and the Sxiidiie eee of a aan 17. The classification by means of excluding angles 18. Points for which the excluding angle is equal to z 19. Points for which the excluding angle is less than x . bo bo bo bo ww ww O=sI o oO ree any 5b bw bw BR w? LCi) 20. Excluding leaves and the points outside the field of a set 244 21. The field and the cordon of a set : 245 22. Elementary properties of the field and the contin 246 23. The domains inside and outside a cordon 248 24. Analysis of the cordon of a set 250 25. The nature of a cordon and of a convex are 251 26. The fundamental properties of the field of a set 252 27. Fields, cordons, and convexity with respect to relations in peneral : 254 28. Fields in Euclidean space of more than two dimensions . : ; : : : 255 29. Geodesic fields on a sphere . F : : : : : F : ; 256 30. Zeroes of functions of a complex eamteblels a suggested line of research . , : 256 1. Introduction and summary. The objects of this paper do not include the introduction to its readers of the sets of points with which principally the paper is concerned, for with these sets every mathematician is well acquainted; the set which is here called the field of a set I is the least convex set containing I’, and I have given the name of cordon to the boundary of this set. Vor. XXII. No. XII. 28 216 Mr NEVILLE, THE FIELD AND THE CORDON In a number of simple cases the field of a set I’ may be defined in terms of centres of mass: thus if T is a curve of finite length, and line density is given to I, the position of the centre of mass of IT’ depends on the distribution of density, but there is a certain region F‘T such that however the density may be arranged, continuously or discontinuously, provided only that the density is nowhere negative, the centre of mass lies in FT, and such moreover that whatever point « of F‘I’ is chosen some arrangement of density can be contrived which brings the centre of mass to #; it is this region F‘T which is in this case the field of T. Again, if TI consists of a finite number of points in a plane, there is only one polygon, in the Euclidean sense of the word, which has all its vertices at points of T, has no reentrant angles, and has all the points of [ in its interior or upon its sides; in this case the sides of this polygon compose the cordon, and the field consists of the cordon together with its interior. From these two examples the importance both of the field and of the cordon will be recognised; the field is involved in almost all mean-value theorems in multiple integration, and the cordon, apart from its relation to the field, is to be found in use in the Newton parallelogram for dealing with branches of a curve and in the Puiseux diagram in connection with linear differential equations. The bulk of this paper is concerned solely with plane sets of points, and what is offered is a definition of the field of a set [ in a form at once applicable to plane sets of all kinds, limited and unlimited, open and closed, and convenient for the development of the properties of a field. The essence of the definition consists in the use of a geometrical element of which, as far as I know, the introduction is a novelty ; this element, which I call a leaf, consists of a point together with all the points of some line through it which lie on one side of it and all the points of the plane which lie on one side of this lime; no set of a quite simple character can reasonably be called a half-plane, and there is no set simpler than a leaf which more closely resembles its complement. A point « is said to be outside the field of T if there is a leaf which includes « and includes no point belonging to I’, the field is the complement of the set composed of points outside the field, and the cordon is the common boundary of the field and its complement. The cordon in general les partly inside and partly outside the field, and an account is given of properties distinguishing the two portions of the cordon. After such explanation as seems necessary of the notation adopted, the paper proceeds to exact definitions of the sets of special kinds which are used, it being stated in each case what part, if any, of the boundary is included; the theorems of which use is to be made are enunciated, proofs being omitted, but in two cases where existence theorems (11°37, 11-42) are required constructions are given. With the definitions of the primary (12:11) and secondary (13:12) chords and of the cross points (1411) of a set we come nearer to our main topic, and the digression to the definition of a convex set (15°11) is not irrelevant. Two ideas of which much use is subsequently made are next explained; roughly, the ex- eluding angle (1612) of a set for a point is the angle of the biggest sectors which have the point for vertex and have no points of the set within them, and a sector = is called a limiting excluding sector (16°13) for a set [if = itself contains no points of I’ but every sector with the same vertex as } of which = is a proper part does contain at least one OF A PLANE SET OF POINTS. 217 point of [: in this connection it must be noticed that in anticipation of this stage our formal definitions (9:11, 9°12) of a sector are such that no existent sector contains any part of its boundary. Each point of a plane falls with respect to a set I into one of three classes, denoted by U‘T, V‘T, and W‘T, according as the excluding angle of I’ for it is greater than, equal to, or less than, 7, and the only propositions in the paper of which the proofs are tiresome are those (1835, 19°45) which describe characteristics of the sets V‘T and W‘I. These propositions established we define the field (20°21, 21:11) and the cordon (2114) of a set, and we have no difficulty in demonstrating so many properties of these sets as to render it evident both that our definitions are well adapted for the development of formal proofs, and that the subject is one in which what is obvious is true. Something is said of the nature of the field of a set with respect to any relation in space of any kind, and of the properties of the field in Euclidean space of any finite number of dimensions, and the paper concludes with suggesting a line of research. The principal theorems regarding plane sets of which the paper contains proofs may be summarised as follows: I. The field of a plane set I’ is composed of the points of [ and the points lying on primary and secondary chords of [; if the set I’ is connected or is the sum of two connected parts every point of the field belongs to I’ or to a primary chord of [: (26°15, 26°31, 12°14). II. The field of a plane set T is itself a convex set containing I, and is the set formed of all the points common to all convex sets containing [: (22°34, 21°22, 26°59). Ill. If a set is enlarged by the addition of any part of its boundary, the cordon is unaltered : (24°23). IV. The points inside the cordon of I are the cross points of T° and the points lying on secondary chords of IT, and they are the points for which the excluding angle is less than 7: (23°36, 19°45). V. A point. lies outside the cordon of T' if there is a straight line parting it from I’: (23°59). VI. If the cordon consists of the whole of one straight line or of two parallel straight lines, the excluding angle for every point outside the cordon is equal to z, but in all other cases in which the cordon exists the excluding angle for every point outside the cordon is greater than 7: (23°65). VII. A point of the cordon of I’ which belongs to the field of [ belongs either to T or to a primary chord of [: (2414, 24°16). VII. A point of the cordon of [ for which I’ has an excluding angle greater than 7 either belongs to [ or is a limiting point of [: (2414, 2415). IX. Unless I’ consists of only one point, every point of the cordon of T is a limiting point of points on primary chords of [: (24°13). 2. Haplanation of logical symbols used in this paper and not described in Chapter I of the Introduction to “Principia Mathematica.” Except in the use of the letters B, C, D, F and in the absence of any sign of assertion, the logical notation of this paper is that of Principia Mathematica. There are only a few 28—2 218 Mr NEVILLE, THE FIELD AND THE CORDON symbols used here which are not among those explained in Chapter I of the Introduction to that work, and by describing the use of these I hope to render the paper intelligible to anyone familiar with that Chapter. If a symbol is introduced to give brevity to a few proofs without any implication that the idea associated with it has permanent value, the definition introducing the symbol is called a temporary definition, and the letters Df which distinguish a permanent definition are replaced by Dft followed by an indication in square brackets of the extent to which the new symbol is to be used. If « is a class whose members are classes, that class whose members are all the terms which belong to every member of « is called the product of « and denoted by p‘«, and that class is called the sum of « and denoted by s‘« which is such that a term belongs to s‘x if and only if there is at least one member of « to which it belongs: 2:21 pie=FZ{(a).aexDxea} Df, 22 se =@{(qa):aex.xcea} Df The number of members which a class a contains is denoted by Ne‘a; thus if Tis a set of points, Ne‘I’ denotes the number of points in the set, while if « is a class of sets Ne‘« is the number of sets belonging to x, but the number of points concerned in the con- stitution of « is Ne‘s‘k. The authors of Principia Mathematica have occasion to use two different pieces of symbolism for the one idea of the class formed of those terms which have a given relation R to a single term; in our applications, the terms in question are in fact always sets of points. If R is a relation which is not in all cases one-many which holds between one set of points and another, the class of sets whose members are all the sets which have the relation R to T can be denoted by either RT or (sg‘R)‘T. The relations which we denote by single letters are all as a matter of fact one-many relations: if R denotes one of these relations, there is only one set which has the relation R to T, this set can be denoted by R‘T, and the class of which this set is the only member is u‘R‘T. Thus almost all of the cases in which we require a symbol for a class of referents are cases in which the use of an arrow is inconvenient or its appearance unsightly: for example, to print an arrow above the group of letters exlf, which occurs in one of our most important definitions, would not only be inelegant in formulae but also affect the spacing of lines if the resulting combination occurred in the body of the text. To limit the groups of letters used in expressing relations to groups with which we do not object to printing an arrow would in many cases prevent such a choice of letters as assists the memory, and uniformly we adopt the alternative notation; for example, the group of symbols that occurs in 20:22 is (sg‘exlf)‘I’, and in 43 we use (gs‘e)‘x + rather than ef” for the class formed of the sets to which « belongs. There is one feature of the notation of Principia Mathematica to which attention is not called in the first Chapter, although it is recognised later as a natural outcome of the theory of classes there explained. Possession of any property which objects of any kind, individuals, classes, relations, may have, can be treated symbolically as membership of the OF A PLANE SET OF POINTS. 219 class of objects possessing that property. For example, we have presently to define what is meant by the assertion that a set of points is united, and the symbol which is intro- duced to correspond to the idea of a united set is a symbol not for an adjective but for the whole class of united sets: we write ['e Ud just as we write wea, but while in the one case we read “# is an «”, in the other we may read simply “I is united”; the dis- tinction between possession of a property and membership of a class is one of language alone, and needs no embodiment in logical symbols. An immediate consequence of this result is that, if we wish to denote that a set has one or other of a number of properties or has several properties simultaneously, we can use the ordinary notation for the logical sum or for the logical product of classes: for example, the numbers 0 and 1 are themselves regarded as classes, and the condition 'e0u 1 is equivalent to the condition Te0.v.Te1; similarly important properties of sets are expressed by the terms complete set and congre- gate, Cp is used for the class of complete sets and Cg for the class of congregates, and to write ['eCp a Cg is to assert that TF is a complete congregate. It is chiefly as a form of shorthand that the notation of Principia Mathematica is required in this paper. Nevertheless to frame definitions in the form which this notation is best adapted to express is the surest guarantee that the ideas involved are logically precise, and in this connection I owe thanks to Prof. Whitehead himself for criticism of my manu- seript which has led to considerable modification in the formal definitions contained in the earlier sections. 3. Definitions adopted from the general theory of sets of points. The explanations yet given are virtually extracts from Principia Mathematica, accounts of general logical symbolism. Next must be described the notation used to express certain ideas peculiar to the theory of sets of points but common to all parts of this theory, and this is done quite briefly, the reader being referred for a fuller discussion of the ideas in- volved to a paper shortly to appear in the Acta Mathematica. Throughout I use ¢, u, », w, 2, YZ for individual points, zy for the distance between x and y, I, A, ©, ®, WV for sets of any kind, V for the universe of points, that is, except in sections 27—29, for the set composed of all the points of the plane in which our sets are supposed to lie. A denotes the null set of points: that is to say, to write I'=A is to assert that there are no points satisfying the conditions that define membership of I’, so that for example the formula T'AA=A expresses that [ and A have no common point; q!I denotes that I is not null, and is the contra- -dictory of [= A. If I is any set, I denote by C‘T the complement of T, the set formed of all points which do not belong to I’, by D‘T the derivative of I’, the set formed of all the limiting points of I’, and by G‘T the set obtained by completing I’, that is, the set Mu D‘T obtained by adding to VT all those of its limiting points which do not belong to it; also I denote by Y‘T the edge of I, that is, the set M'a D‘C‘T formed of the members of [ which are limiting points of the complement C‘T’, and by B‘Y the set known as the boundary of TP, that is, the sum of the edges of I’ and its complement, and I describe the set [T— YT, 220 Mr NEVILLE, THE FIELD AND THE CORDON which is the same as [— D‘C‘T, as the set obtained by clipping I’, and denote this set by HT: 321 CA Vi fe 22 DV=Z{p >05,(qy)-yel.0 22y} . D:O0CA. BECCA. (Ov) n BA=A, shewing that if a set is not connected there is a boundary which it does vault. Thus we have 74 Cd=PiqiPadA.qilaCA.d.q!0n BA, the fundamental theorem that expresses the precise degree of continuity belonging to a connected set. From ‘74 it follows that every united set is connected, a result of which the symbolical expression is Td Ud C Cd; the converse of °75 is proved false by the actual construction of connected sets that are not united. 4. Straight lines and rays. In the following discussion of certain parts of the theory of sets of points in a plane considerable use is made of sets of several particular kinds, which we commence by describing, and we reserve particular symbols for sets of these kinds. To denote that a set T° consists of all the points composing a straight line we write TeStl, and we use g, h, and & only for straight lines: 411 Stl=straight line Df. If / is any line through a point a, the set h—t‘x consists of two similar cells, one on each side of «; each of these cells is called in this paper* a ray, and of these rays @ is called the source and / the line. For formal definition we take "12 Ray = e (qa, h) heStl.weh.Tex(h—w)} Df, * The most useful sense of the word ray in pure Hlements of Quaternions, § 132, ex. 4 (p. 119 of the first mathematics is to denote a directed straight line, but (1866) edition; pp. 121, 122 of the first volume of Joly’s directed lines are not required in the study of fields and (1899) edition), uses ray in precisely the sense adopted cordons, and the word ray is convenient; Hamilton, here. OF A PLANE SET OF POINTS. 223 and for rays we use a and ¢; to denote that # is the source of the ray a we write aefa« or zfea, and to denote that h is the line containing the ray a we write hlea or aclh, the use of any of these expressions being taken to imply that a is a ray and also that « or h as the case may be is a poimt or line; a ray a has only one source and only one line, and these may properly be denoted by fe‘a and lc‘a, but the rays issuing from a common source « form a class (sg‘ef)‘# and the rays situated in a line A form a class (sg‘cl)*h and both these classes have infinitely many members : 13 lo=h@ {ae Ray. heStl.aCh} Df, 14 el=Cnv‘le Df, a5 le e l> Cls, 16 ef =@2 {ae Ray.aex‘(lefa—u'x)} Df, “aly fe=Cnv‘ef Df, 18 feel Cls. If a is a ray, le‘a —e‘fe‘a consists of two cells, each of which is a ray; one of these rays is the ray a itself, the other is called the reflex of a and we denote it by rfl‘a: Al rfl = @ {a e Ray .c=Ie‘a — tfe‘a—a} Df, ‘22 ae Ray D: E!rflfa. rflfae Ray. If x, y are any two distinct points there is one and only one ray issuing from « which contains y, and we denote this ray by «— y; the reflex ray, which issues from the same source in the direction away from y, we denote by ay=st\jaefx.yea} Df, not as (@ jaefa.yea}, for although By ertyDE!%G lacie. yea, and for any class of sets y 33 Blifyd.iy=s‘y, the class @{aefx.xea} is the null class of sets, and (@{aefx.xea} does not denote A but is meaningless; on the other hand s‘@{aefw.zea} denotes by definition the set 2 {(qa)aefx.x,zea}, and since the condition (qa).aefa.«,zea can in no way be satisfied, this set is the null set A. Thus ‘31 yields as we desire B4 vty d:c2—yeRay.coyefc.yexr—y, 35 t—>uv=. Considerations somewhat similar prevent us from defining #«y formally as rflfe—>y: the null set is not a ray and this definition would leave #<~z meaningless; it is sufficient to put 36 ve—-y=sl\aefic.yerfiifa} Df, Vou. XXII. No. XII. 29 224 Mr NEVILLE, THE FIELD AND THE CORDON and then we have 437 ety Ji:7e—ye=riey, 38 ce—ar=N. The set e<—~y must not be confused with the set y—»#; both are null if y coincides with a, but in general the former is a proper part of the latter : “39 etydi:2eyCyre.qlyor-“2ey. 5. Chords. If y is distinct from 2, the common part of the sets sy, y—>w is the set formed of all points between « and y on the line through them; in any case this set is called the chord xy and denoted by #—y: 511 L-y=xraynyoun Def; the set obtained by adding to this chord the poimt # is denoted by w+ y or ya, and the set obtained by adding both the end points 2, y by Hy: “12 gey=2—-—yvia Df, als eay=a2—yvity Df, ‘14 cHy=r-—yviicury Dé. Two useful elementary propositions are 21 ZELY =-LZ=H->Y, ‘22 Z€Le—Y=LEY —Z, of which the second is equivalent to "23 2e—y=2 {vey—z}; and we use also “24 y+u.=xeD(x—y), 25, y+a.=reD(aHy). If y coincides with «, the chord —y is null, but the completed chord «+ y has the one member #; the case of coincidence is the only case in which the derivative of the chord is contained in the chord, and also the only case in which the finished chord is not contained in its derivative : 26 Y=e.=.2—y=A, Hf Y=HR.=.L£Hy=o. 6. Triangular domains. If three points , y, z are not collinear and w is any point in the interior of the triangle of which they are the vertices, there is a length p, namely the length of the shortest, or of one of the shortest, of the perpendiculars from wu on the sides of the triangle, such that every point v whose distance from wu is less than p also lies in the interior of the triangle; in other words, the interior of the triangle is a domain. This domain is called the triangular domain OF A PLANE SET OF POINTS. 225 ayz, and we denote it by tridom“(z, y, z); it can be defined formally in terms of chords, a simple though unsymmetrical definition being 611 tridom‘(z, y, z)=% {(qv).vey—z.uex—v}—(a@—y)—(x—2z) Df, where the chords «—y,«—z, which in general have no points in common with the set a {(qv).vey—z.uex—v}, are subtracted in order that we may have 12 (qh) -heStl.a, y, eh. tridom“(z, y, z)=A, an implication which can be replaced by the equivalence “1183 (qh).heStl.z, y, zeh.=. tridom“(z, y, z)=A. Since the framing of a definition more symmetrical in appearance than ‘11 finds a natural place later in our work, we content ourselves for the present with 11. Following a course which we take in a number of similar cases, we write ‘14 , Tridom =P {(q2, y, z)- TC =tridom“(z, y, z)} Df, and we must note the property implied in the name, expressed in the theorem ‘15 Tridom C Dom, which is true even if the domain is in fact null. - 7. Parallel lines, and rays contained in parallel lines. To denote that two lines h, & are parallel we write fh prlk, it being understood that the possibility of coincidence is not exciuded. Since our space is the Euclidean plane we can write 711 prl=hk{h,keStlih=k.v.hak=A} Df, but an interesting alternative rests on the fact that if / and & are not parallel they divide the plane into four pieces: 12 h, ke Stl. 3: Ne&e{C(huk)}=2.=.h=k, 13 h, ke Stl. D2. NeKe{C(huk)} =3.=:hprlk.htk. 14 h,keStl.d: Ne{C(hvuk)}=4.=~h pri k. Rays in parallel lines may have either opposite directions or a common direction. Utilising a simple criterion for two rays to have opposite directions we can take as definitions 21 opd = @¢ ‘a, ce Ray -(qh, kT, A).h, ke Stl. T, AekC(aucu fea fete). hCT.kKCA.T+A} Df, "22 cod =opd? Df. 8. Leaves and clipped leaves. If h is a line through a point z and a,c are the two rays forming h — ‘x, the sets a, ¢ are ordinally similar and so are the sets G‘a, G‘c, but since ave is not the whole of h and one point of / is contained both in G‘a and in Gc, neither @ nor Ga can properly be described as a half-line. Similarly if A is any line, Cth is formed of two similar 29—2 226 Mr NEVILLE, THE FIELD AND THE CORDON cells, but neither of these can be called a half-plane, and we call them clipped leaves, writing $11 Clilf=P {(qh).he Stl. DexCOh} Df and using YT and © for sets of this kind. If YT is a clipped leaf, the complement of its boundary is composed of two cells each of which is a clipped leaf; one of these is T itself, the other we shall call the reflex of T and denote by rflx‘T, but as we wish to postpone the formal definition we denote it for the present by O*G‘T, noting that 12 Te ChifD C6G‘T ¢ Chilf. If « is any point of the boundary of a clipped leaf T, and if @ is one of the rays with source 2 contained in BfT, the sets Tua, CfG*T urfl‘a are similar mutually exclusive sets whose sum omits from the whole plane only the one point a, and the sets Tv G‘a, GT v Gerfifa obtained by adding to each of them the poimt # are similar, their sum is the whole plane, and their only common point is 7; neither Tua nor TuG*a can be called a half-plane, but Tv G‘a is a typical set of a kind of which we have to make much use, and we call such a set, that is, the set formed of a completed ray and all the points on one side of the line containing the ray, a leaf, and the source of the ray we call the pivot of the leaf. The definition of a leaf that follows explicitly the description just given is “21 Leaf = {(qa, A). Aex*Clefa. T=cfeauvavA} Df, but an adequate definition which formally is simpler could be derived from the theorem "22 Leaf =f {(qa).ae Ray. YSC'T =a. YT =Ie'a — a}. For leaves we use M and N, and in virtue of ‘22 we may take for the definition of the pivot of a leaf M 23 pvt = 4M {Meleaf.a=fe‘Y‘CM} Dé Certain elementary properties of leaves have to be noted for use: Bil Me Leaf.ceM.3:(qN).NeLeaf.2pvtN.NCM, By Me Leaf. y,zeOS‘M.rvey—z.dxe0™M, which implies 33 MeLeaf.cey—zaM.d:yeM.v.zeM, a result that proves valuable, and 34 MeLeat.ceM.D3ac DSH, which is used in conjunction with “44 below. Clipped leaves are to our main purpose of less importance than proper leaves, but they are simpler in nature, possessing the properties expressed by 41 Chilf C Domex and by “42 TeChif.veT.D:(qh).-heStl.h part (tS, CT). But the results “43 TeChif.veT.3:(qM).MeLeaf.ceM.MCT, OF A PLANE SET OF POINTS. 227 which can be strengthened into “44 TeClilf.ceT.D:(qM).MeLeaf.cze M.MCT, and “45 Me Leaf > HM e Clilf, which is to some extent a converse of “44, enable us often to secure the advantages of ing with clipped leaves. operating wit pp 8. 9. WSectors. If a,c are two rays which have the same source x but do not coincide, C(t‘r vavuc) is the sum of two domains each of which has t“cuave for its boundary and is called a sector of a and c; if A is one of these sectors, G‘A is obtained by adding the rays and the source to A, and therefore the other sector is C‘G‘A. When c coincides with a, the set C“(u'vavuc) is a single domain, but it is convenient then to regard the null set as a sector of a and c; in this case if A is the existent sector C“(u‘x ua), the completed set G*A is the whole plane, and C*G‘A being null again represents the sector, although C‘G‘A is not A but G‘A. For definition of sectors we can take gala! Sectex =P {(qa@, a,c).a,cefa.TexnC(cuavue)} Df, defining existent sectors, followed by 12 Sect = Sectex vifA Dt; we reserve for sectors the letters =, T. The properties of sectors first to be noted are a3 Sectex C Domex, implying “14 Sect C Dom, and 15 > e Sectex D C“G‘S € Sect. It is possible to replace ‘12 or ‘15 by 16 Sect = Sectex v O**G‘Sectex, and indeed to deal directly with sectors by starting from Ie Sect = f° {(qa, a,c, A):a,cefe.AcekC(isvavuc):TH=A.v.T= CGA}, which is effectively a combination of ‘11 and ‘16. It is convenient to have symbolism expressing that > is a sector, existent or null, of a and c¢, but a direct construction is impeded by two considerations: unless a@ and c ' have a common source, the definition must not lead to the null set but is to fail altogether ; nevertheless, the definition must depend primarily on the pair of rays, not on the sector, to meet the cases of the null sector and of the clipped leaf. To this end we write A All sectex = dy {(qu, a, c):a,cefa.y=Uavuiic.LexC(uevauc)}| Df, v9) sect = 34 {(qw, a,c, A):a,cefx.y=tavic. Xen (Ue vauc):S=A.v.5=CG‘A} Df, 228 Mr NEVILLE, THE FIELD AND THE CORDON obtaining implicitly definitions of ¥ sectex(i‘a vt‘c) and Esect(t‘au ec); to avoid the use of the argument Ufa vl‘c we substitute (a,c), and we have 9:23 Y sectex (a, c) = :(qx).a, cefa.LexC(lwvave), ‘24 sect (a, c)=:(qa, A):a,cefe.AenC(rvavuc):2=A.v.2=CGA Corresponding to ‘12 is 25 Ssect (a, c) >: E sectex (a, c).v. 2=A, but to obtain useful propositions we must exhibit the conditions under which the null set enters: 26 fefa = fefe. ate: D.Esect (a, c)= & sectex (a, c), PH a=c.2D:E!sectex“(a, c) . (se‘sect) (a, c) = sectex“(a, c) v UA. It is hardly necessary to add the propositions . 28 Sectex = : {(qa, c) = sectex (a, c)}, “29 Sect = Sq a, c) > sect (a, c)}. The relation to a sector = of rays a, c by means of which it is defined is expressed by calling these rays bounding rays of the sector, and we write 31 br=@>{ {(qc). sect (a, c)} Df, “32 rb=Cnv‘br_ Df. If S is a sector of a and c, the common source of a and ¢ is called a vertex of =, and we put 33 vx=2S {(qa, c).a,c efa. > sect (a, c)} Df, “3B4 xv=Cnv‘vx Df. If x is a vertex of > and a circle is described with centre «, the ratio of the length of the part of the circumference within = to that of the whole circumference is called the angle of > and donee by ang‘; this angle is a definite one of the two angles between the bounding rays of = which issue from #. If the angle of an existent sector is not equal to 7, the vertex is unique and the bounding rays are definite, but if the angle is equal to 7, the sector is a clipped leaf, every point of the boundary is a vertex, and every ray contained in the boundary is a bounding ray, peculiarities for which allowance has been made; to justify the use of the symbol ang‘ we have to remark that if the vertex is not unique the angle is the same at every vertex. From the definition, “41 = sectex (a, c)D C6G*S sect (a, c), “42 fe‘a = fe'c . D. Ne*(sg*sect)(a, c)=2; the sum of the angles of the two different sectors derived from one pair of conterminous rays is 27, and this is true if the two rays coincide, the null sector hi aving angle 0 and a sector of the form O*(‘a, where a is a ray, having angle 277: "43 (qa, c).%, sect (a, c).2+T: 3 ang‘> + ang*T = 27, OF A PLANE SET OF POINTS. 229 The complement of a sector = is not a sector, but is a completed sector or a completed ray according as C‘G‘2 exists or is null, and in particular the universal set V is not a sector, the most comprehensive sector being the complement of a completed ray. It is hardly necessary to remark that in connection with some problems the valuable sets might be the complements of what we are callmg sectors; in that case V would be of the standard form while the null set would not. The propositions “44 > e Sectex D ang‘S v ang‘C*G*S = 2r, 45 > e Sectex . vx >. Davx CGS, “46 > «Sectex.abr>. abr O*G*> are true even if the angle of = is equal to 7, for = and C*G‘> acquire simultaneously the peculiarities consequent upon the possession of that special angle. Of value to us in relation to any sector > is the sector which we call the reflex of >, which may be described as the reflection of = in a vertex of ©; even if the angle of & is equal to 7, the reflex of = is unique, for then the reflection of = in each of its vertices is the same. We write ol rfx =PS [= e Sect .T=s@ {(qzr).cvx=.aefz.rfifaCS}] Df, to which an equivalent form is 52 rfix =f £[¥ Sect. P=3 (qa, y)-uvxd.yed.cvey—2z}], and we have 53 > € Sect D rflx‘> ¢ Sect, “54 ang} >a7=q! =a rflx‘s, 55 0< ang <7=q! CGS a rflxSCSGSS, “56 ang} > 7 = OG‘S C rflx‘S, Dil ang’) < wm =rflx‘S € (GS. From the last two formulae, “61 ang*> =7. =. rilx*> = CGS, and other distinctive properties of sectors of angle 7 already mentioned are 62 ang‘} = 7 .= Se Ciilf, and ‘63 = eSectex D : ang} = 77. =. Ne“(sg‘vx)*> +1, ‘64 ang‘> =r. Sect (x y, sz). Duc y—z, which we require in the sequel; a slight but useful modification of °56 is 65 ang’> Sa .=. > v reflx'S = 0(sg‘vx)‘S, which involves : 66 angs$= >a =. Ne‘C(S urfix‘S) =1, 67 ang’> = 7 .= C(S uv rflx‘S) € Stl. 230 Mr NEVILLE, THE FIELD AND THE CORDON 10. Circular domains. The last particular set which we have to mention is the circular domain, consisting in normal cases of all points inside a circle. Many of the properties of normal circular domains are shared by both the null set and the whole plane, and therefore we include these sets in the general definitions, which are 10711 Circedom =f {(qa, p).T=9(ay< p)} Df ‘12 circdom(«, p)=9 fay 0.2 =4 (xy Cls. The use to us of circular domains is in connection with limiting points of sets, for 3:22 is equivalent to each of the theorems 31 ceDT.=.p>05,q!T a {ciredom (a, p)— ea}, 32 ceOD'T .=:(qe).ccente .a2CCT ut. . * A different order of ideas includes the clipped leaf no use can be made in the study of limiting points. as a form of unlimited circular domain, since the straight } If R is any relation, & 8 denotes the same relation line is in one sense a form of circle; in that work however _ restricted im application to members of the class £, that is, the distinction between the inside and the outside of a denotes *7(« Ry.yef); similarlya4 R, a4 R [8 denote circle tends to lose importance, and the valuable construct 29 (« Ry.xea), £9 («Ry.cea.yeB). See Principia is the cell of the complement of the circle, a set of which Mathematica, * 35. oe OF A PLANE SET OF POINTS. 231 11. Theorems relating to chords, leaves, sectors, and circular domains. Many of the properties of the special sets we have described are useful to us chiefly in the form of existence theorems. Thus we require lll wey—z.ycentH.zcentZ.0}CT.q! >.> Ne%e(T—3)=2 gives significance to 15 LVX>.eVXT. > vsGlaeix.abr=}CT.q!>.y,zeT—L.angTH7. Di ye K_sz.vV.qly—zae. Of a different kind are Ail CVE et << ane eo (qu) sang D— 7.2 vx De hes gts — 1; 2 22 x vx > .angs> >7.2:(qM).MeLeaf.ceM.MC 3 vt‘a, 23 ang’> =7.abr>. (2 vave‘fea) e Leat, “24, ang) 7 .£ovmy. (qr). Te Chilf.ceBT.TECS, “25 Me Leaf.ceM.3:(q7T).TeChif.cze BT.TCM, the last of which we use in the form 26 Me Leaf.ceM.3:(q>). angst =7.a2vx>.2CM. By actual construction 31 T,O¢Chif. ht a BOe1l.ceTaQ.4a,cefx.acod BT —GS0.ccod BO-GT. _ D:(q=). = sect (a,c).ang*= >7->CTvO, 32 T,OcChif. BT a BOe1.ceT—-Q.aefz2.acod BT—-GO .c=rx (BST an BO). >:(q>).>= sect (a,c) angs> >a7r.>eTv0; ‘31 implies 33 T, Oe Clif. Ta BOel.ceTaAO.3:(q>d).angs> >7.avxdy.2CTuvO, ‘32 implies “34 T OcChif. Ta BWel .zeT—O.5:(q>) -ang’> > 7 .2vx >.> CT vO, and by an interchange of T and © implies also 35 T, Ne Clif. BT a BQe1l.ceQ—T.3:(qz).ang*> >7.2vx>.>CTv0; and since 36 TyQX=(TaQD)v(T-D)v(Q-T) we have from °33, 34, 35 37 T, De ClilfD:: BST pri BO. vs. ce TUOD: (qd). ang >7.evx>.2>CTv A. The last result implies 38 YeChilf.anes> >7-2eT.D:(qT)-angT >7.a2vxT.TCT v2, Vor. XXII No. XII. 30 232 Mr NEVILLE, THE FIELD AND THE CORDON for among the clipped leaves contained in a sector whose angle is greater than 7, some have boundaries not parallel to a particular line B‘T. Somewhat opposite in character is another result proved by construction. We have 1141 aneS >7.a2vx>.ccente.gqif.heStl. Bin BeCh.ceCh.T=K*(a, Oh): Dis Wel@hife arent 0 Gero = if ang‘= < 27 and = is limited, h is determined by the condition of passing through both the points common to the circumference B‘= and the pair of completed rays B‘S, # is necessarily outside h because ang‘=+7, and the cell of C‘h which contains «x is contained in S>vE because ang‘= >7; if ang‘>=27, BY n BE is a single point and h may be any line through this point except the line through « itself; we make no use of the latitude allowed, for we require ‘41 only for the sake of the existence theorem implied, namely 42 anos> >7.E! cent‘ . cent‘Evx>.5:(q7Y).TeChilf.cent*‘@eT.TEC Suz. 12. Primary chords of a set. A point is said to be on a primary chord, or simply on a chord, of a set [ if it lies between two points of [ on the line joining them, and we denote the set formed of points on the chords of [ by S‘T: px ST = {(qy,2).y, ze .vey—z} Df. If three points of [ are collinear, the middle one is a member of both I and S‘T, while if T consists of only two points neither of these belongs to S‘l: there is no general relation of inclusion between I‘ and S‘T. Of more value for technical purposes than S‘T is the set defined by a, LV =2{(qy,z).y,zeT.cveyHa} Df, which we call the set derived from I. by simple linkage, the principal advantage of this set being that from the definition 13 Gee Ty The fundamental relation between the sets S‘T’ and L‘T is ‘14 JER! = IO ISIE which may be expressed in terms of operators only, in the form "15 L=TIvS, I being the operator of identity; but the elementary relation 16 See ee is often useful. Since I’ and S‘T are not mutually exclusive we cannot express S‘I’ simply in terms of [ and Z‘I’, but we have sly ST =2 {xe LT —‘a)}. From ‘13, "21 qiloOqi LT, so that indeed "22 qireaq! LT, which is equivalent to ‘23 | De) WP 7 De Wa OF A PLANE SET OF POINTS. 233 this is a convenient point at which to note 24 Wek E‘Rie il. "25 Net SAP he As to the existence of S‘T we have 26 qi ST=:(qy,2)-y,ze0 .y+z, that is “27 NeT >1l=q! ST, so that 28 TeQul=.ST=A. From ‘11 and 5:24 31 qi sdL> DG Dest: if T is null it is contained in every set, and therefore 32 IM GW) s (7c INGAP Ss 1) IDC IIESEI while 33 Teldq!C—- DST and so “B+ Tel=q!l—-DS‘T, 35 NeT+1=FC DST. From 35 we have 36 INC APS AES DIS 7 DGS Dye and since if Tis a unit set D‘T is contained in every set we can assert without hypothesis Bi Dal EDESET so that from 14 38 DSETI— DEST Of another kind are “41 INCOMES UE Re 1 A Me “42 SeiuGDSSs that is 43 NeP+1.= LT e Ds, “44 ST ¢ Ds: both S‘T and Z‘T are dense sets except when I has but a single member, in which case /‘T also has one and only one member and is not dense, but S‘T is null and formally is dense. By combining ‘37 and °38 with -43 we have ‘51 NePF1IIT uD uv ST C DST, a result used later. As we shall see from examples, neither S‘[ nor L‘T need be complete, but from the elementary propositions ‘61 (y—z)¥(u—w) vv (z—w)eCg, 62 (yHz) ¥ (vow) v (zHw)eCl a Cg we have 63 S‘T'e Cg, We LT Ud. 30—2 234 Mr NEVILLE, THE FIELD AND THE CORDON From the preliminary propositions 11-11 and 11°13 we have immediately 12°65 SDT C DST, 66 S(l — DCT) CST — DICS'T, and ‘14, °37, 65, and -38 imply 67 EDI GepsE an without hypothesis as to Ne‘l’. We have sometimes to use 68 INTE (aN 5 Dig SORCERY “69 ID EAN 3) 5 LEAS EN but we use them as a rule without explicit reference. For a few purposes it is convenient to write “(fl S(T, A) =2 {(qy,z).yeT.zeA.xey—z} Df, ‘72 LT, A)=2 {(qy,z)-yeT.zeA.reyrz} Df. with which notation “Te SUD, IDSA ‘74 i= (eel) 2s 0 We have no need to enunciate results corresponding to all those given for S‘T and L‘T, but we note that while the use of 1111 gives information concerning the sets S*(D‘T', D‘A) and L‘(D‘T, D‘A), by using 11°12 we can draw the conclusions ‘75 S(T, D*‘A) € D‘S(T, A), ‘76 LT, D&A) C D‘L“(T, A), with the particular cases ail SHE LD) (SOLIS ie ‘78 IG (US, JOA) \S JOTI EEN An important relation between the set L“I, A) and sets of the form ZT is most simply written in the form $1 L(PvA)=LT uv ‘Av LAL, A); this is certainly redundant, for [vA is contained both in Z‘Tu L‘A and in LL, A), but it is the most useful form, and since even if we write 82 I(T vu A)=LT uv LA S(T, A) we are not secure against repetition, ‘81 if replaced should yield only to 53 [(luA)=TvAvSTv SAu S(T, A). The set Su A) cannot be expressed by any formula similar to ‘81, but “84 S(T vu A)—-(Tv A)=ST vu SAvu S(T, A) —(T vu A). From ‘81, ‘41, ‘67, and ‘78 we have ‘85 Delay eG Gori T and ‘16, °85, ‘38, and ‘28 imply s ‘86 SGT C DST, although this cannot be deduced from ‘84 without the help of 35 and °37. S| OF A PLANE SET OF POINTS. 23! 13. Secondary chords of a set. If uw, v, w are three points of a set I’ no one of which lies on the primary chord joining the other two, the chord joining any one of the three to any point on the primary chord joining the other two is called a secondary chord of I’, and we denote the set composed of points on secondary chords of T by 7*T. This description is designed to bring 7*T into relation with S‘T, but in fact a point is on a secondary chord of I if it lies in a triangular domain whose vertices belong to [T. By means of the operator LZ we can give a sym- metrical appearance to the definition of the triangular domain «yz, for 1311 tridom“(a, y, 2) = L?*(t'a uv fy v 62) — Lute v ify u Lz), and 7*T is defined formally by 12 TT =2 {(qu, v, w).u,v, wel. cetridom(u, v, w)} Df. Since oie x,y, ze D tridom“(s, y, z) C TT, we have the important theorem 22 TT « Dom, implying ‘23 q!2‘T > 7‘T « Domex. It can easily be shewn that 31 Sze Sw Lely but S‘T and 7‘ are not in general mutually exclusive, and indeed 32 Newari Lek Oy S bia Lely while on the other hand neither the set S‘(—7*T nor the set 7*'—S‘I’ plays any part in the developments we make. Corresponding to 12°68 and 12°69 we have 33 IGN LG TEA: The set L*T, the set derived from I by double linkage, is one of the most interesting of the sets connected with I’, and the value of 7‘T is owing partly to the fact that a graphic analysis of Z*T, though not an analysis into mutually exclusive ‘sets, is given by “41 PV=TuST v TT. Proposition ‘41 written in the form “ol PV Eo Te has a curious result when taken in conjunction with the hypothesis that [ is connected or is the sum of two connected parts, which can be used in the form given by 52 T=0v®.0,ReCd. =2u,v, wel Dur w(qy, 2, A).y, zeluvlvviw.y+z.-AceCd.y,zeA. ACI: of any three points of I’, two lie in a connected set contained in T. We have 5S} vetridom(t, y,z)). Ky, O(tcurezvxet) + Kz, Oe veezuret)} 236 Mr NEVILLE, THE FIELD AND THE CORDON which implies immediately 13°54 xe tridom((t, y,2).AeCd.y,zeAd qian (Uavaezuxet); also “541 qiAnt‘c=a2eA, 542 zeA.q!Anrez.dreSA, and in the notation of 127 543 qiAncet=reS{A, Ut) and therefore 544 AvitCr.qiAnzet.dIceST; from °54, 541, 542, 544 5) xe tridom(t, y,z).AeCd.y,zeA.AuitCr. Deel u ST, and °55 with 12 and ‘52 gives 56 T=0vu®.0, SPeCd.ceTT. dee LT, that is ‘Did T=6v%.0,eCd.O7TCLT. From °57 and ‘51 comes 58 T=0v®.0,8eCd .IT=LT, a result which we shall appreciate more fully when we are better acquainted with the set LT; 58 of course implies 59 P=0v%.0, BeCd.d:n219.10°T=LT. 14. Cross points of a set. We call a point « a cross point of I if there are two chords of I’ having « for their only common point, and we denote the set of cross points of [T by X‘T: 1411 XT =2 \(qt,u,v,w).4ur7,wel.te=(t—u)a(v—w)} Dt. As with S‘'T and 7‘T, so with X‘T, ‘21 MGA ACD CX 0) cy << piDyyieL No xc and so from 13:22 33 qi@TOTT v X‘Tc Domex, or since the null set is a domain ‘BA TT u XT e«Dom. 15. Convex sets. A set of points I‘ is said to be convex when if two points y, z belong to I" every point of y—z necessarily belongs to ['; we write ['eCyx to denote that T’ is convex, the formal definition being 1511 Ovx=f{sTCcr} Df Convex sets of points have many important properties, some of which we shall develop as we proceed. From 12:14 we have 12 eCysa— ee — a relation often more useful than the fundamental one on which the definition is founded. Since ‘11 with 12°68 implies 1183 IN @ Crp DISA (eS Ie we have from 13°31 14 a Gyx 2 ih G Ie and so also, using 13°22, 5 Me Cvxe eel urxe ly Gul From 12°64 and ‘12 16 Cvx C Ud, a result which has its use in connection with the nature of the boundary of a convex set. Possibilities in the relations to a set I’ of the sets S‘T, Z‘T can be illustrated by means of lines, rays, and leaves. If [ is a line or a ray, S'T and L‘T both coincide with I, and T is convex, although if T is a ray it is not a complete set. If I is a leaf, SST consists of all the points of [ except the pivot, and L‘U coincides with T'; a leaf I is convex, although there is a whole ray V‘C‘T which consists of limiting points of [ not belonging to I. We have already in 832 asserted implicitly that the complement of a leaf is a convex set. 238 Mr NEVILLE, THE FIELD AND THE CORDON 16. Excluding sectors and the excluding angle of a point. We say that a sector = excludes a set TI if no points of I lie within the sector, writing 1611 exsect = SP {Se Sect. 2a IPSN Drs we have to remember that the boundary of a sector is not contained im the sector, so that Sexsect I is not inconsistent with (qz).evx>.aeT or with q!Ia Bs, and also that the null sector has every vertex and excludes every set. From the last convention it follows that the class of numbers anes {vx > . > exsect I} contains the number 0, and from 11°21 it follows that this class is a stretch; because the bounding rays of a sector are contained in the complement of the sector, this stretch cannot have an upper limit which does not belong to it, that is to say, the stretch has a maximum, and we call this maximum the excluding angle of the point « and the set T, writing 12 ea‘(x, T) = max‘ang‘“|(se*xv) ‘x nm (sgfexsect)‘T} Df. A sector of which a is a vertex, which excludes IT, and has ea‘(x, I) for its angle, we call a limiting excluding sector of « and I’, and we write 13 = les (a, 1) =: ane‘ =ea(2,T).cvx>.Tat=A ODF; the class (sg‘les)“(«, [) certainly exists, although if ea‘(z,T) is zero the members of the class are null sectors: 14, al (sg‘les)(a, DP). From the nature of a maximum and from ‘13, 210 gil —e&.Sles(2,l).aefz.abr>.3: TeSect.aCT. pq! Pan(T—3%) so that also 22 «gqil—«a.Sles(¢,T).aefa.abr>. D:..q!Paa.vi0< p27 D,(qT).ang‘T=p.abrT.qila(t—3S); 93 qiT—es.dles(a, f).avxT.2vus(sg*br)> CT. q!>.Kex(T— 2). I qiPak has value because its bypothesis includes that of 1114, implying that T— has two cells; 24 qil—«a. >, Tles(¢,r).>+T.>.3aT=A, 25 ang‘> =ea(¢,T). 2xve.I:qilnd.v. dles(a, I). The case in which I’ has only the one point # is peculiar; in this case every sector of which # is a vertex excludes T, and 27, the greatest angle a sector can have, is the excluding angle: 31 ea‘(ax, us) = 2rr. 3ut V—c is not a sector, and the limiting excluding sectors are the complements of completed rays issuing from «# If y is any point other than «, the excluding angle OF A PLANE SET OF POINTS. 239 of y for t‘e is 27 but there is only one limiting excluding sector, the complement of “yvyre: 32 ea(y, Ua) = 27, 33 y+a.>les(y, x). r= Ch{e'y v ya}. It is convenient to note ‘BA ea‘(a, T) < 279 DA IT — esa. The case in which the excluding angle of « and T is zero also is peculiar; the ex- cluding sectors are the null sectors of which the various rays issumg from « are the bounding rays, and every existent sector with vertex « contains points of [T: 35 ea‘(a, D)=O8Ssavx>.q@!>.33q!Pa >. There may or may not be rays from # which do not contain points of TI’, the existence of such rays being from our point of view irrelevant. We have of course “41 TCA. Sexsect A. >> exsect I, which implies ‘42 [CAD ea‘(a, T) > ea‘(a, A). A particular case of ‘41 is 43 S exsect GIT D & exsect I; on the other hand, because } is a domain, D‘C*S is contained in COC‘, and therefore G‘C*> is identical with OS; hence “44 IME CSS SCANS CDs that is to say “45 > exsect [D> exsect GT, which taken with ‘43 gives “46 > exsect [= ¥ exsect GT, and implies for all positions of « “AT ea(a, G‘T) = ea‘(az, I). 17. The classification by means of excluding angles. Just as sectors fall into three classes, composed respectively of those whose angles are greater than 7, those whose angles are equal to 7, and those whose angles are less than 7, and the properties of a member of one of these classes for the most part differ widely from those of a member of another, so each point of a plane falls with respect to any set [ into one of three classes according to the value of the excluding angle of IT for it. We write 1711 UT =@fea‘(a,T) >a} Df, 12 VT =@ {eaf(a,T) =} Df, 13 WT =2 {ea(x,T)< 7} Df; of the three sets so defined it is the last which has the simplest and most important pro- perties, but the three sets are studied together. Wo, SOME AN Gy-4 OF 31 240 Mr NEVILLE, THE FIELD AND THE CORDON From 16°47 follow 7-21 NEARNG GA SULA Ue 22 DGATAGGhe SA A— Ver; 23 ReEATAGGshy > eA — Wel: propositions which would justify us in studying the sets U‘T, V‘I’, W‘T’ first on the hypo- thesis that [is complete, a course which we do not actually take. Since “31 vey —zdea(a, Uy v U'z)=T, "32 xe tridom“(u, v, w) Dd eaf(a, bu v tv Uw) < 7, we have from 16°42 33 ae ST D ea‘(z, T) a 7, B34 ve TT D ea'(z, 1) < o, that is, "35 SAG Vee We 36 Pew: also “37 we ACD Deas(a, 1) <7, that is, 38 ZONES IS and therefore “39 Teo eke Gye. From 16:24 we have “41 qi D—cw.ea(2,T)>0.5 Ne“(sg‘les)(a, [') & 2a + ea“(x, T) which has the corollaries “42 qi 0—ta. ea{(a, 0) >7. 2 Ne“(sg‘les)(a, T) = 1, “43 ea‘(x, )=7 3: Ne“(sg‘les)(a, Fr) =1.v. Ne“(sg‘les)(w, F)=2. A special case of the first of the corollaries just enunciated gives “bl ea“(z, [)=27 3: (qa).aefae.T Cue va, against which we put the converse 52 aefa.0TCtrua. dea (a, T) =27, of which 16°31, 16°32 are particular cases; ‘42 itself may be written in the form 53 weUeloriNer(seeles) «(a i))— iV. Dy—lec Corollary *43 can be simplified, for 54 ang’> =7 .ang‘T=q7.¢vx >.a@vxT.3:q!SaT.v.T=rixS, and therefore 1393) ceVT.Ne“(se‘les)(7, T) =2.5:(qh). he Stl. Ch, and so from °52, 9°64 56 ve VT. Ne(se‘les)(a, T)=2.3¢eST; thus “7 ceVT.9:. Ne(sg‘les)(v,T)=1. vive ST (qh). he Stl. CCA. (sg‘les)(a, ) = «Oh. OF A PLANE SET OF POINTS. 24] It is worth while to notice that, if ea‘(#, [) is equal to 7, there is only one line which can be the boundary of a limiting excluding sector whether the number of such sectors is one or two: 58 ze VSD). Ne“B‘(sp‘les) (a, T) =1. One consequence of 16°25 is ‘61 Sles(w, T)D:q!T a rfix'> .v. rflx‘S les (2, 1); writing this in the form 62 Slesi@; Dy Saqw E artix’> .v.. 1 C CS uriixs) we can apply 9°66 if ang‘= is greater than 7 and ‘56 if ang‘> is equal to 7, and we have 63 zeUT.Sles(a,T).d:q! CarixT.v.T = cc, ‘64 eeVl > lesen eb a rhx<). vice Sl. be Bes: the most interesting application of ‘61 occurs when ea‘(x, TI’) is less than 7, but before pro- ceeding to this application we complete the deductions which have to be made at the present stage from the hypothesis that ea‘(z, I) is greater than or equal to 7. With regard to the set U‘T, we have only to point out that 11-42 implies “(Al ceUM Drea iwavia(g © ate Chlf. ren. han=A. 18. Points for which the excluding angle is equal to 7. If = is a sector of angle 7 and « is a vertex of &, and if w is any point not in = or on its boundary, then in order that the ray issuing from «# in the direction of the ray through w from a point wu distinct from « and not contained in > should neither contain w nor lie in = or B‘S, the point w must lie either on the boundary B‘S or in the strip bounded by B‘> and the line through w parallel to BY} and must not lie in z+ w; the region to which wu is thus restricted we denote temporarily by P,‘w, the nature of = and the conditions as to the positions of « and w being implied: 1811 = P,fw=%t fangsl=7.2vxd.we OG we CS: 4 ycod uw D, ye GS —w} Dft [18], ‘12 P,Sw=sth {hprl BS. qtihacew}—c#ew. If w belongs to P,‘w, there are four sectors which have for one bounding ray the ray from £ in the direction of ~—»w and for the other bounding ray a bounding ray of ¥, and of these four there is one and only one which contains = and does not include w; this sector we denote for a time by R,‘w: 13 R,u=F (ue P,Aw . (qa, c).a,cefz.acodu>w.cbry.a,cbrT.=CT. we CT} Dft [18]. The properties of sectors of the form R,‘w relevant to our purpose are only two, namely 14 q!h,ud.(qa).aefe.abr>.aC R,u, 15 ve R,u-LIqlu-—vaweew. 31—2 242 Mr NEVILLE, THE FIELD AND THE CORDON The first of these, with 16°21, gives 18°21 ceVT.Sles(z, T). uve Pw. dq! Ta R,u—t, and so from the second we have 22 veVST.Sles(¢, VT). qi! Pa P,w.dqiSTazrcew. We have no reason to suppose that for every position of w in C*G@‘S points of I’ are to be found in P,‘w, and the alternatives we consider are hyp 18a (qz):ze CGS .wex—z),q!Tn Pu, hyp 18) (TY, 2%, w) ry, 2€CGSS wy tr z.vex—y.wexr—zZ. G&P, fv = GP, fw. Ta Pv=A.TaPw=A; the form adopted for the second of these is designed to shew that one of the hypotheses is necessarily fulfilled, but this second assumption is equivalent simply to hyp 186 (qv, w):v, we OG .2@ 0+ aw. GP, = Pw. Ta (P04 P,w)=A; in both forms, the condition G‘P,‘v=G‘P,‘w is a method that happens to be simple notationally of expressing that the line through v and w is parallel to B‘S, a condition required in °32 to ensure that «—v is contained in P,‘w and «—w in P,‘v. From ‘22 and 12°42 we have 31 zeVT'.S les (x, T). hyp 18a.32¢ DST; on the other hand 32 veVT.>les(2,T).v, weCG .2r a+ 27. GP, v = G'P,w. >. Sve&v Pv P,w e Ciilf, 82 TDast=A.la(’,0u Pw)=A.d:ece0.v.Ta(vuieu Pv Pw)=n, and therefore 34 zeVT. > les(a,T). hyp18b.3:a2e0.v. (qT). Te Clilf.ceT.[aT=A; ‘31 and °34 imply ‘Bd e2eVTD:¢2e DST .v.(qT).TeClilf.ceT. PaT=A, since T’ is contained in D‘S‘T if T has more than one point and V‘T is null in the case excepted. 19. Points for which the excluding angle is less than 7. Turning to the set W‘T, we have first to conduct an investigation in some respects analogous to that leading to 18°35, but with a result ultimately of more value. An immediate deduction from 16°22 is 1911 O les(#,T).qtI'a rflx‘S, hyp 196 ang’= < a. > les(#,T).rfix‘S les (a, P) . = sect (a, c) . qila CG(e v rflx‘= vave), OF A PLANE SET OF POINTS. 245 which we distinguish as hyp 19a and hyp 196, writing (q=) hyp 19a if there is a limiting excluding sector such that the first hypothesis is satisfied, (q>=) hyp 196 if there is one satisfying the second. From 1762 and ‘11 12 ea(a, T) << r=:(q>) hyp19a.v.(q>)hyp19b.v.ce XT, and though the possibilities are not mutually exclusive we can extract the information we require by treating them separately. We notice that 13 q !rfix*> D0 < ang‘, so that in hyp 19a@ we have actually 0 their bounding rays and their vertices is to complete = urflx‘=, but if = is a null sector no bounding ray consists of limiting points of ©, and indeed G‘(= vu rflx‘=) as well as = itself is null. If > is a sector with angle between 0 and 7 and vertex 2, then in order that the reflex of a ray «—»y may be neither a part of = or rflx‘= nor a bounding ray of &, the point y must lie outside both = and rflx‘= and must not belong to the boundary of rfix‘=; y must belong to C“(> v Gérfix‘S). The constructions we have to make in relation to a point y which when ang‘> is between 0 and 7 require y to belong to CZ vu G‘rfix‘=), we can make if = is a null sector provided then that y does not lie in the line containing the bounding ray of ©. The regions concerned in the two cases are covered by the one definition 21 P, > =§9 {a, cefx. Ssect(a,c).re—y C C(Z v rfix*> vavec)} Dft [19]; if ang‘> is not less than 7, then C( urfix‘2 vavuc) consists of the one point « and can- not contain any rays: hence ye P,‘> is false unless ang‘> is less than 7, and it is super- fluous to introduce the condition ang‘= << 7 explicitly into the definition. If y belongs to P,‘X, then the ray « sect (a,c). ye P= . T sect (a,ve—y). SCT. yeOT} Dft [19]. The use here of sectors of the form R,‘(y, &) depends on the propositions 23 ea“(z,T) << 7. les(@,F).dq!0n P,S, "24 ea(a,T) <7. les(¢,T).3:(y).q!I'n {R,(y, 5) — 5}, consequences of 16°21 and 16°35; these imply 25 hyp19a 3: (qu,v,w).uv,wel.werfix'S.we R,(v, >)—3, 26 hyp19b 3:(qu,v,w)-u,v,wel .ve R,(u, >)—>.we R,(u, rfixs>) — rflx‘S, and shew the kinds of properties of the sectors of the form R,“(y, =) that concern us. 244 Mr NEVILLE, THE FIELD AND THE CORDON From the definition 19°31 R(y, =)— aC Ps for all positions of y in P§>, and therefore 32 ze Ry, 2)—=Iq! R.(z, =); moreover 33 ze R,(y, )—TD ye R,(z, =) —X, so that 34 ze Rz‘(y, &) -—L=ye R,(z, >) — = and the relation ze R,“(y, =)— %, which -25 and -26 shew to be connected with the use of excluding angles, is symmetrical between y and z. We are now in immediate touch with the proposition we wish to establish, for “41 werflx'> .we R,“(v, >) — >. De tridom“(u, »v, w), in which ang‘= cannot be 0, and “42 ve Ru, 2) —E.we R,(u, rfix‘>)— rflx*= . D ve tridom“(u, v, w). and from these with ‘12, ‘25 and ‘26, we have “43, eaf(z, T)< wo reTT vu XT, that is “44, EECA (SED which taken with 17°39 gives 45 HOM wi NOI VOL and implies, from 14°33, “46 WT ¢ Dom. The last property of W‘" which we wish to mention is deducible immediately from 16°23, 11:14 and 11:15: ‘51 ze WT. dles(2,T).q!t>.dIqiST ak; this result emphasises the possible discontinuity of the excluding angle regarded as a function of the position of «, for it gives immediately ‘52 De Cvx.2e WT. Sles(a, Tl). DS=A, that is 53 T ¢ Cvx D: eaf(z, 1) << 7. =. ea‘(a, [)=0. 20. Excluding leaves and the points outside the field of a set. We say that a leaf M excludes a set IT if no points of T° belong to the leaf, and we write 20°11 exlf=Mf{MeLeaf. [a M=A} Df, from which we have at once “12 TCA.MexlfA.3MexlfYl, ‘13 MeLeaf. MCN .Nexlff.3 Mexlfl; OF A PLANE SET OF POINTS. 245 the leaves excluding a set form a class (sg‘exlf)‘T which may be null but otherwise contains an infinity of numbers; from 12 14 TCA JD (sg‘exlf)‘A C (sgfexlf)‘T. We say that a point is outside the field of a set if there is a leaf which contains the point and excludes the set; the points outside the field of a set I’ compose a set which we denote by HT: 21 EV={(qM).Mexlfl.2eM} Df, which is equivalent to 22 EBT =s(sg‘exlf)T. From ‘21, 8°31 and ‘13 comes 23 ce HTD:(qN).Nexlfl.zcpvtN; on the other hand 24 zpvtNozeN, and therefore "25 Nexlff.apvtN.ove HT, so that 26 EV=2{(qM).Mexlfl.2pvtT}, or more compactly PH ET = pvt*(sg‘exlf)‘T, an elegant but not as far as we have found a useful result. 21. The field and the cordon of a set. It is the set complementary to H‘T which is of value in analysis: indeed, it is the known importance of this set, which we call the field of I, that justifies our whole study; we write 21-11 MV SCVIA De but sometimes we make use of the equivalent 12 ICTs. A direct definition of FI’ is of course ‘13 FT=2{MeLeaf.ceM.IyqilaM}, but it is usually easier to deal with ‘I’ defined by 20°21 than with #*IT’ defined by ‘13. The boundary of the field of I’ we call the cordon of I and denote by Q‘T; thus 14 QT=BFT Df, 15 QT = BET, 16 QU =FLTa DET VET a DFT, the last embodying the definition of the boundary. Every set possesses both a field and a cordon, but it is not every set that can serve in either of these capacities, for the fact of being a field or a cordon itself implies properties; it is convenient to write ‘17 Fild=f {(qA).0=F*‘A} Dft [21-26], A 18 Cdn=P {((qA).P=Q‘A} Df 246 Mr NEVILLE, THE FIELD AND THE CORDON but the first of these is a temporary definition, for we shall presently identify the class of fields with the class of convex sets. From 20°21 and 20:11 we have 21°21 ET CCT which from ‘11 is equivalent to 22 ern: hence if T itself is not null neither is F*T’: 23 ee rede The existence of E‘T cannot be asserted; for example, if I’ consists of a pair of inter- secting straight lines every leaf in the plane contains points of I; but we can write con- ditions for the existence of #‘T’ in the forms “24. q!#T=:(q7T).TeChif. cr, 15 q!ET=:(qh, K).heStl.Kex‘Ch. CK, the second of which merely embodies the definition of a clipped leaf, but permits of a simple translation into words: the field of a set IP does not occupy the whole plane if there is a straight line which has all the points of I’ on one side of it; what is in fact an equi- valent statement is that the field of T does not occupy the whole plane if there is a straight line which has no points of I on one side of it, which is a translation of 26 q!£C=:(q7T).TeChif. PaT=A, the distinction between this condition and the former being that here we allow points of the set to lie in the bounding line. As with any other boundary 27 qiQV=a:q! FT .qi £T, that is, in virtue of °23, 28 q!:QT=a:q!il.q!£T. 22. Elementary properties of the field and the cordon. For the construction of proofs it is useful to note that from 20°14 and 21-11 the sets E‘., F‘Y have the properties 7B Aa | TEADEACET, ae, Gr Ay) hale Gur A. From 21:22 and ‘11 ‘21 SHIA RMT Tf Ba on the other hand, from 20°21 and 21°12 22 Mexlfl.@a@eM.5,r%¢CF'T, that is 23 Mexlf fs. Fla M =A, so that 24 M exlf PD M exif FT, OF A PLANE SET OF POINTS. 247 which is equivalent to “25 (sgfexlf)‘T C (sg*exlf) SFT, and implies, from 20:22, 26 HO GEE. from ‘21, -26 “27 BET = ET, which gives immediately the important result 28 ee shewing that the operator F possesses the property expressed by 29 jell Boo From the preliminary result 8°33 combined with the definition 20°21 we have 31 vey—zn BT) .yeHVvze ET, and therefore BY vey—Z.y, 2eFT. Dre FT, that is 33 SoHE Geer : or in other terms 34 FET ¢ Cvx, that is 35 Fld C Cvx; for immediate use we have to note that 21:22, 12°68 and ‘33 imply “36 EAM GEST and we can sum up propositions 34 and 36 in words by saying that whatever the nature of I, the field of T is a convex set containing all the points and all the primary chords of T. The two propositions ‘34 and 15°16 imply “41 Heel Ds: that is “42 IP STLCAT 6 ACID SIDS 3 unless [ has only one member, F*T has no isolated points. Again “43, Mexlff.2eM.dJz2e DM, and therefore “44 ceHT Dac DET, that is “45 ET eDs: E‘T has no isolated points. The results -42, -45 fall far short of expressing what we really Wiis OE 1S \aseo.4 IF 32 248 Mr NEVILLE, THE FIELD AND THE CORDON know of the nature of F‘T and E‘T, but they are sufficient grounds for asserting the pro- positions 22°51 (Pei 7 5G H/05 DID a 52 GET = DET, 53 Ret vagus = 6=— DFT an Deas “54 Tel.v.- QT «Ds, 55 Peds ve Qoie et “56 elev ones — RS Oo ‘BT DET = ET ve OT, of which the last two may be put into the forms “58 Tel.v. DFT =C(ET—-QT), 59 DET= (FT — QT): 23. The domains inside and outside a cordon. With reference to a set I’ the points of the plane may be divided into four mutually exclusive sets, namely, F*"+Q‘T, the region of the plane inside the cordon, "Ta Q‘T, the part of the cordon which belongs to the field, E‘I'a Q‘T, the part of the cordon which does not belong to the field, and H‘T—Q‘T, the region of the plane outside the cordon; there is a close relation between this division of the plane and the division by means of excluding angles into the sets U‘T, VT, W‘T. The two sets FY —QT and £T-QT are necessarily domains, and in connection with each of them we use the principle ex- pressed by 23:11 AeDom.ACT.SACT—-BT, which gives 12 AcDom. AC FT.IAC FT-QT, 13 AeDom .ACET.JACET—-QT. From 11:22 follows ‘21 URE Du Lr. Since 22, we VT. Z les (a, I). D:.(qy, z)-y, ze. = sect (ty, ez): V:(qa).aefe.abr=.Tana=A, we have from 9°64 ‘23 ce VT D:.ceTuST:v:(qa, ).aefe.abry.angsS=7.-Ta(2vavis)=A, so that from 11:23 24 Ve FE aT; OF A PLANE SET OF POINTS. 249 moreover, 21°22 and 22°36 shew that in -21 and -24 the partition is into mutually exclusive sets. Again, from 11:26 5) ve HT Dea‘(x, [)>7, that is 26 JERIDXS (OA AI Bs and this is equivalent to 27 Wale rein We can now make our applications of 12 and +13. From 8:43 we have 311 TeChlf.zeT.TanT=A.D2c ET, and so from 841 and 13 312 TeChlf. PaT=A .dTCET —QT, which implies 313 TeChif.ceBT.TaT=A.d26 DET, whence, using 11:24 on the one side and 22°57 on the other, 32 eaf(2z,T) Srdre HT QT, that is 33 CA OMVING een oOo which is equivalent to “34. FT —QUC WT. On the other hand, from -12 with -27 and 19:46, 35 WTECFT-—QT, and this taken with °34 gives the important result 36 Hi Ose — Wile of which an equivalent form is 37 JIA) OES OID LLC We have to notice that ‘36 implies “41 HMOs — Wel Gory and that 22°36 and 22°56 give “4.2 IM DISSED 791 DIA he that is “43 Du DST CWT yu OF. From 17:71, 18°35, and 12°37, we have shill caus Oo Vly. Diavie la DEST .vs(qt).tveChifscet inl aT =A. since W‘T is the complement of Ut UV‘T, °51 and ‘43 give 52 ceUTuVT—Q09:(qT).TeClilf.ceT.TaT=A, which from ‘37 is equivalent to ‘53 ET — QT C sf (T eClilf. Ta T=A}; 250 Mr NEVILLE, THE FIELD AND THE CORDON on the other hand, °312 is equivalent to 23°54 oT (Te Chif. Pa T=A}C ET — QT, and this combines with the preceding result to give 5B ET —QT=s‘T (Te Clilf. 'a T =A}. From 842 we have “56 Te Chilf.ceT.PTaT=A.)5:(qh)-heStl-h part (la, I); since also ‘57 heStl.h part (Sx, T).3: Ka, Ch) e Clilf. Ta K (a, Ch)=A we have 58 (qV).TeChilf.ceT.TaT=A:=: (qh). he Stl. h part (a, VP), and °55 is equivalent to 59 ET — QT =2 {(qh) -he Stl. h part (cx, T)}. It was in anticipation of °53 that the preliminary propositions 11°37, 11°38 were proved, for we have from these three results ‘61 qi! O0T=.£ET-QT=UT- GT, 62 UT=A.q!FT.=:T, OcChlf. TaT=A.TaQ=A.)9 BT pri BO. It is easy to prove that the conclusion of ‘62 implies that the cordon Q‘T is either one straight line or two straight lines: in the first case either I is contained in this line but not in a ray contained in the line, F‘I coincides with Q‘T, and #T—Q‘T is the complement of Q‘T, or FT—QT and HT—Q‘T are the parts of the plane lying one on each side of the line; in the second case F*‘[—Q‘T is the strip between the lines and ET—Q‘T is the part of the plane complementary to the sum of the lnes and this strip. It is convenient to speak of all these cases and of the case in which H‘T does not exist as abnormal, writing 63 Abnicdn =f {(qA). T=Q'A.T=AveTCStl} Df, “64 Nledn =Cdn—Abniedn Df, definitions not justified until it is shewn that the cordon cannot consist of a number of intersecting lines; then we can write 65 QTeNiedn=:q!i#T. ET—-QT=UT —GT. 24. Analysis of the cordon of a set. There remains the consideration of the points of the cordon itself. One simple ex- pression for the cordon comes directly; from 23°37, 23°51, and 23°55 2411 OF GUE VED)in(u DSS); and 23°43 gives the converse inclusion; hence “12 QT =(C8LRVVD) a (Tv D'S‘), and we may appeal to 12°51 to substitute for this 13 Pel.v.QT=(UT vu VL) a DIST. ee OF A PLANE SET OF POINTS. 2 wo — More detailed results, which follow from 17°71, 18°35, 23°55, and 22°36, are 14 ARID ay (OID Gy JAS DS UI ay Dy, 15 VT AUT on ET =UT a (D'T—TL), 16 OsinVelingeEa— VA a Een, ltl VT aVE a BL=VT an (DST — LT); it must not be forgotten that points of all the classes T, DOT —T, L'T, DIST — L‘T may lie inside the cordon although none of these classes have points outside. From 23°36, 17:23 “211 IBS YAN 5 ANS GOI Ss SIGNS (ONS S(O DE and from 23°59, 3°65 “22 PeAPNGGA DEA — OSA = FE — Os whence 23 NE IDES) (QU AN) (9G Die the cordon of a set is unchanged if the set is enlarged by the addition of any part of its boundary. The cordon of the completed set G‘I° belongs wholly to the field of G‘T, and in terms of this field Q‘T is given by "24 Oot (USE OVEN) nH Gale while corresponding to 14, 16 we have 25 OT AUT=UT a GT, 26 Ooo Velen sG a 25. The nature of a cordon and of a convex are. Enough has been done to prove that the definitions adopted enable us formally to establish the properties which a cordon obviously possesses. A normal cordon is a complete curve, that is, a united set identical with its own derivative and contained in the derivative of its complement, it has at each point a pair of tangential rays, bounding rays of the limiting excluding sector of the set for that point, which may or may not determine a tangent at the point, and it is a Jordan curve and has a definite finite length between any two of its points. No line that has points of the cordon on both sides cuts the cordon in more than two points; if a line contains a tangential ray of the cordon, the points of the cordon on the line form a stretch, and the remaining points of the cordon are all on the same side of the line. To constitute a single curve, a set of points must be united. It can however be proved that a set contained in a Jordan curve must be united if it is connected, and therefore a convex are in a plane may be defined as a connected set contained in its own cordon; such a set need not be complete, for if from a complete convex arc which does not extend to infinity we take an end-point if the are is not closed or any point whatever if the are is closed we obtain a convex are which does not contain one of its limiting points; but a convex arc has at each point which is not an end-point a pair of tangential rays and has between any two of its points a definite finite length. bo or bo Mr NEVILLE, THE FIELD AND THE CORDON 26. The fundamental properties of the field of a set. Returning to the subject of the field, we have from 23°36 26-11 FT n(UTvVVT)= FT aQT a (UT v VT), and so from 2414, 2416, and 17°35 12, es (OSGI EI) — Eleni ( Un aiValays and since also, from 23°36, 13 FT aWr=WiT, we have “14 FTH=aLTUWT, which may be expressed, in virtue of 19°45 and 1422, in the rhetorical form ales (RSP SAR) A ie the points composing the field of a set are the points of the set, the points of its primary chords, and the points of its secondary chords. If we take 15 with 13°41 we have 16 MAN FEI or in terms of operators alone ‘17 v= IT The fact that #‘T is convex can be written in the form ‘21 LEFV=FT, and so by ‘17 gives "22 T=, and implies that Z has the curious property, possessed also by B, the operator giving the boundary of any set, that its complete effect is produced in two operations, though not as a rule in one operation : 23 ne2=2391=L’, a proposition of which 22°29 is a part; we see now that when we defined primary and secondary chords but not chords of a higher order the limits were not imposed arbitrarily. In consequence of *16, we can write 15°58 in the form 31 T=6v®.06,®c«Cd.3FT=LT: the field of a set which is connected or is the sum of two connected parts consists of the points belonging to the set and those belonging to its primary chords; the result may be analysed by means of 12°14 and 12°83 into 32 Peds FT =Tu ST, ‘33 r,AceCdd. F(T vu A)=LTvAvuSTv SA v S(T, A), and we may use the first of these results in the second and write “34 Tr, AceCdd. F(T vA)= FT u FAvu S(T, A). EE OF A PLANE SET OF POINTS. 253 It follows from +15 and 13°31 that “41 FYT=TuSTu ST: also for all values of n “42 SEAMS IHC and therefore the set S"‘!’ is contained in FT for all values of n, and if S,‘I denotes the class of sets whose members are [ and all sets of the form S”‘I’, the sum s‘S,‘T is con- tained in F*T’; *41 asserts the converse inclusion, and we have “43 TAG Sor a result whose significance will presently be briefly considered. The value of F‘T’ in analysis comes largely from a theorem now to be proved, that the field of any set [‘ is composed of all the points that belong to every convex set containing I’; in logical terms, F‘T is the product of the class of convex sets containing TP. From 15:12 “bib er CyvaeD 2 oy— sn that is, in virtue of ‘16, “52 IN a Oxee 2724) ae since, by 22°34, F‘T’ is convex 53 IN = J341P SIMO and combining this result with the preceding “54 IMeChx oS o/h 1s a convex set is a set which coincides with its field. Again, from ‘52 and 22:12, G55 NaCvxe iG Ay Sse GA which is equivalent to 56 FT Cp‘A {Ac Cvx.T CA}; also 22°34 and 21:22 together are equivalent to ‘3T FTA {A Cvx .T CA}, which implies 58 pA (Ac Cvx .TCA}CFT, and ‘56 and -58 give the desired result, “59 FT =p‘A (Ac Cvx . °C A}, a formula by which #‘T’ has sometimes been defined. In order not to interrupt the argument, we did not point out an immediate consequence of 52 which we announced in 21; from °52 ‘61 PeCvx >. A=PF) FA=T, and in 62 TeCyvx 3: (qA). T= F*A, 254 Mr NEVILLE, THE FIELD AND THE CORDON a weak deduction from “61, we have authority for the assertion 26°63 Cvx C Fld which combines with 22°35 to give “64 Fld = Cvx, whence also 65 Cdn = B“‘Cvx; after -64, use of the contraction Fld is entirely superfluous, but in spite of -65, Cdn remains of service. 27. Fields, cordons, and convexity with respect to relations in general. It is not only for plane sets of points and for the relation denoted in this paper by S that a field is of service, and the merit of 26°43 and 26°59 as definitions of FT is that they suggest extensions of the theory of convexity to classes and relations between classes of a very general type. If R is any one-many relation which determines from a class a of any kind another class R‘a of the same type as a, we can call « convex with respect to the relation R if Ra is contained in a, writing 27-11 evx=GR{Re13Cls. RiaCa} Df. If we denote by hyp 27a the hypothesis that R is such that hyp 27a aCy) Rac Ry it can be seen at once that 12 hyp 27a 3. RiaC py {ycvx R.aCy}. e Moreover, with the notation already explained in connection with 26°43, Ry‘a is a class of classes, and therefore s‘Ry‘a is a definite class derived from a, and s‘Ry is like R itself a relation between classes. Again, 13 hyp 27a 3. ycvx s‘Ry = y cvx R, and therefore 14 hyp 27a D.s‘Ry‘a C pF fy evx Rua Cy}. Whatever the nature of R, the class a is a member of the class of classes Ry‘a and is contained in s‘Ry‘a, but the assumption hyp 27 is not sufficient to secure (s‘Rya) cvx R, that is hyp 27b Rés‘ Ry fa C sf Ry 6a, We may secure the last condition by a hypothesis ad hoc, and enunciate the theorem 15 (a,7)-aCyD Ria Roy: (a) Ris‘ Ry6a C ‘Ry fa:. D. 3 Ry a= py {ycvx R.aCy}, but if we secure the condition hyp 27b by the assumption hyp 27¢ Ravy)C Rav Ry, and observe that this and the original assumption hyp 27a together are equivalent to hyp 27d Rav y)= Rav Ry, OF A PLANE SET OF POINTS. 255 we have the less general but more useful theorem 16 (a, y) R(avuy)= Rav Ry... s‘Rya= py {fy cvx R.a Cy}, sufticing to shew that with respect to any relation with the property expressed by hyp 27d the field* of a class a may usefully be defined either as a sum or as a product of a class of classes, and the idea of the field may be connected with the idea of convexity with respect to the same relation. Further, if the classes between which the relation ‘R holds are sets of points in any space in which boundaries exist, we may define the cordon of a with respect to R as Bés‘Ry‘a. For example, if [ is a set of points in any reduced+ space in which distance is numerical, the field of I’ with respect to the operator D that connects I" with its derivative is the completed set Tv D‘T, and this is the product of the class of complete sets containing [’; the cordon BYG‘T of T with respect to D) is not necessarily the same as the boundary B‘T of I’, but is in fact composed of those points of BT’, the boundary of the boundary of I’, which are not isolated points of the complement C‘T. The utility of a field is certainly not confined to cases in which the relation concerned satisfies hyp 27 d, for the relation S defined in 12°11 does not fulfil this condition, and whether the two classes s‘Ry‘a and p‘y {fyevx R.aCy} are equally important in cases in which they differ, experience alone can decide. 28. Fields in Euclidean space of more than two dimensions. Returning more nearly to the main subject of this-paper, we observe that in Euclidean space of any finite number of dimensions properties of the field and of the cordon—with respect to the relation of lying in a chord—may be investigated precisely as we have investigated them for a plane, the initial definition of the field being not of either of the general forms 26°43, 26°59 but of a form similar to 21:11, adapted to the geometry of the space and shewn ultimately to be equivalent to a definition in a general form. In three dimensions, the excluding angle is the measure of a dihedral angle, and the points outside the field are defined by means of a standard geometrical figure composed of a leaf together with all the points of space on one side of the plane containing the leaf. It is sufficient to say that in the whole of this theory everything that is obvious is true, but we add that the appearance of the number 2 in 26:16 and 26:25 is associated with the fact that in those propositions plane sets only are in question. In Euclidean space of m dimensions 28-11 FT =L1"T; whereas the operator B determining the boundary of a set has the same property§ 12 Ba Bian s=\2 ) BY = B * In work of this general character, the field of a a neighbourhood that does not contain the other; in relation has a definite meaning, and although a relation a reduced space with numerical distances the distance is not a class and there can be no real confusion between between distinct points cannot be zero. the field of a relation and anything which we choose to § Note in formulae 12, -13 that to write R+S of describe as the field of a class, the use of the word field is two one-many relations R, S between classes means open to criticism. ~ (ql).-RD+ST, and does not mean (I) RT+ST; the + A space is reduced if of every two points each has relations are not asserted to be mutually exclusive. Vou. XXII. No. XII. 33 256 Mr NEVILLE, THE FIELD AND THE CORDON in space of all dimensions, the use of Euclidean space of m dimensions makes us acquainted with a simple operator LZ requiring precisely m applications to secure its ultimate effect, that is, having the property 28:13 no=mo2 "=I". n a", where @ is a primitive root of the equation 2@—1=0. These sums are obviously of very great interest, and a few of their properties have been discussed already*. But, so far as I know, they have never been considered from the point of view which I adopt in this paper; and I believe that all the results which it contains are new. My principal object is to obtain expressions for a variety of well-known arithmetical functions of n in the form of a series = asc; (n). s A typical formula is =n i) CG 3 (n a(ny=" \" ) a5 ae + me ) +... , where o(n) is the sum of the divisors of n. I give two distinct methods for the proof of this and a large variety of similar formulae. The majority of my formulae are ‘elementary’ in the technical sense of the word—they can (that is to say) be proved by a combination of processes involving only finite algebra and simple general theorems concerning infinite series. There are however some which are of a ‘deeper’ character, and can only be proved by means of theorems which seem to depend essentially on the theory of analytic functions. A typical formula of this class is ¢,(n)+4e.(n)+4e;,(n) +... =0, a formula which depends upon, and is indeed substantially equivalent to, the ‘Prime Number Theorem’ of Hadamard and de la Vallée-Poussin. Many of my formulae are intimately connected with those of my previous paper ‘ On certain arithmetical functions’, published in 1916 in these Yvansactions. They are also connected (in a manner pointed out in § 15) with a joint paper by Mr Hardy and myself, ‘ Asymptotic Formulae in Combinatory Analysis’, in course of publication in the Proceedings of the London Mathematical Society. * See, e.g., Dirichlet-Dedekind, Vorlesungen iiber Zahlentheorie, ed. 4, Supplement vir, pp. 360—370. Vou. XXII. No. XIII. 34 ~~ 260 Mr RAMANUJAN, ON CERTAIN TRIGONOMETRICAL SUMS 2. Let F(u, v) be any function of u and », and let (2'1) D(n)= = F' (6, 8), where 6 is a divisor of n and 66’=n. For instance Diy Tr); D(2)=F (1, 2)+ F (2, 1); D(3)=F(1,3)+ F(3,1); D()=F (1, 4)+F (2, 2)+ F(4, D; D(5)=F (1, 5)+ F(5, 1); D(6)= FA, 6) + F (2,3) + F (3, 2)+F (6,1); ....... It is clear that D(n) may also be expressed in the form (2°2) D(n)= = F(8, 8). Suppose now that (23) 3 (n)= Si a ‘ so that »,(n)=s if s is a divisor of x and 7, o- =) otherwise. Then * (2:4) D(n)=5- SF nen) F (0? “), where ¢ is any number not less than 7. Now let : 277 (25) s(n) = & cos =n where 2 is prime to s and does not exceed s; eg. ¢,(n)=13 c.(n)=cos nr; ¢s(n)=2 cos Zn ; — - —e 2 S 4 . c,(n)=2cosin7; ¢;(n)=2 cos nz + 2 cos £n7 ; c,(n) = 2 cosini; ¢,(n)=2 cos zn + 2 cos na + 2.cos $n ; c,(n) =2 cos tur +2cos2nm; ¢(n)=2 cos tnm + 2 cos {nm + 2 cos Frm ; Co(n) = 2 cosinm+2cosinr; ....... It follows from (2°3) and (2°5) that (2°6) Ns (n) == c3(n), where 6 is a divisor of s; and hence+ that (27) c.(n)= 2 #(8') ns (x), where 6 is a divisor of s, 66’ =, and 1 28 Ys, oe EC) ¢(s) being the Riemann Zeta-tunction. In particular © (n) =m (n); G(r) =m (n)— m(M); Cs (2) = 5 () — (ays c,(n) = y(n) — no (N)5 Cs (MR) = 95 (2) — Mm (NM); vreeeee t (e} * > is to be understood as meaning S, where [t] de- + See Landau, Handbuch der Lehre von der Verteilung 1 1 notes as usual the greatest integer in ¢. Gar rena ilen eS AND THEIR APPLICATIONS IN THE THEORY OF NUMBERS. 261 But from (2°3) we know that ms (n)=0 if 6 is not a divisor of n; and so we can suppose that, in (2°7), 6 is a common divisor of n ands. It follows that Cs (n)| < 28 where 6 isa divisor of n; so that (2°9) c, (n) = O(1) if nis fixed and vax. Since Ns (Nn) = Ns (N+ S$); Cs (NM) =e; (n +8), the values of cs (m) for n = 1, 2,3... can be shown conveniently by writing ¢(n)=1; ¢.(n)=— 1,1; 6(n\==Dh=1 2: ¢,(n)=0,=2)0)2; ¢(v)==1, 1, == 1,4; Cad) = Lei (77) | et enn 6: ¢s(n) =0, 0, 0, — 4, 0, 0,0, 4; ce (n) =0, 0, —3, 0, 0, —3, 0, 0,6; (Snes IS = 7h, = 1 Se ounce the meaning of the third formula, for example, being that c, (1) = — 1, ¢, (2) = — 1, ¢, (3) = 2, and that these values are then repeated periodically. It is plain that we have also (2:91) c, (n) = O(1), when v is fixed and nn. 3. Substituting (2°6) in (2°4), and collecting the coefticients of ¢, (mn), ¢.(n), ¢;(n), ..., we find that i tell oi ie seal & n (3:1) D(n)=¢,(n)3= Fv, =) +ea,(n)>— F Qv, eens 3 F (3, 3) + 1vV Vv 1 2Qv 2 1 3p 3v where ¢ is any number not less than n. If we use (2°2) instead of (2:1) we obtain another expression, Viz. “al xt] n val a) => pies > a 5 ys 3 yee : eae, (82) D(n)=e,(n) SFE, v) + a(n) = a5 F (3, ; 2) +c (n) = : 35 F( 3v) + where ¢ is any number not less than n. Suppose now that F, (u, v) = F (wu, v) log u, F, (u, v) = F (u, v) log v. Then we have D (n) logn= e F(6, 6) logn= > F (6, 6’) log (88’) > 2F, (6, 6’) JEG, OY) where 6 is a divisor of n and 66’ =n. 34—2 262 Mr RAMANUJAN, ON CERTAIN TRIGONOMETRICAL SUMS Now for = F,(8, &) we shall write the expression corresponding to (3°1) and for = F, (6, 8’) 5 3 the expression sce gecd to (3°2). Then we have 2 “log 3 (3°3) D(n)logn=o.(n)3 a v, *) 4 a5(n) 5 log 2 Loe 5) + +e,(n) 3 6 F (30,5) + log v n log 2: 2Qv ‘log 3y sans ERE» ») + ex(n)® it et 2p ren F(> By) tes, 2p 3y where r and ¢ are any two numbers not less than x. If, in particular, F(u, v)= F(a, u), then (3°3) reduces to sige it log 2 tlog 3 (3-4) 1 D(n) log n= c, (x) = 8” F(v,") + o,(n) 395 F (20 50) + ) teak nye “= 5 (80, 5.) t--3 where ¢ is any number not less than n. 4. We may also write D(n) in the form (41) D(n)= = F(8, 8) + = F(8, 8), $=1 s=1 where 6 is a divisor of n, 68’ =n, and u, v are any two positive numbers such that w=, it being understood that, if w and v are both integral, a term F(u, v) is to be subtracted from the nght- hand side. Hence (with the same conventions) D(n)=> S = ne (n) F(, a2 ars Se n(n) F (5, v). Weeita v v © Applying to this formula transformations similar to those of § 3, we obtain (42) Deseo: ”) + ean) 5 F (2, 5-)+-. aie v ll n GTN Se S ‘ +e(n)B 5 F(T, v) + e,(n) 5 3p ee where wu and v are positive numbers such that uww=n. If wu and a are integers then a term F (u,v) should be subtracted from the right-hand side. If we suppose that 0= - iF (», ") +F(*, »)} +¢,(n) & S 5 1 (2, 5. } +F( , 2v |b +... ivi v v 2pv | 2y/ 20g If n is a perfect square then F'(V/n, 7) should be subtracted from the right-hand side. 5. We shall now consider some special forms of these general equations. Suppose that F (u,v)=%, so that D(n) is the sum o,(n) of the sth powers of the divisors of n. Then from (3:1) and (3:2) we have (5'1) oy = n)s +¢,(n)& + * _ = C. a pHs i] ar a ni’ Cy (n oat Co ( 1 (Qn) Cs ( 7 (By) Me (5'2) o,(n)=c, (n)3 vv + ¢, (n) Sry! (n) = (Buy + set SS . . ——— AND THEIR APPLICATIONS IN THE THEORY OF NUMBERS. 263 where ¢ 1s any number not less than n: from (3°3) (5°3) os(n) logn =e (nm) & Sv logy + ¢4(n) & (20)! og 20 +. 1 ‘ j & logy | x < log 2y Bel is (n) = Soa Cy (n) 7 (2) s+ atte + zl where r and ¢ are any two numbers not less than n: and from (4'2) 2U hu (5'4) a, (n)=¢, (n) 3 vs + Cy (n) > (2v)s + ¢3 (n) > : (3v)s4 + 1 1 el + ns {0 (n) 00) 5 a ya tol) = ya te f where w=n. If w and v are integers then uw’ should be subtracted from the right-hand side. Let d(n)=o,(n) denote the number of divisors of n and ¢ (rn) =o, (n) the sum of the divisors of n. Then from (5:1)—(5‘4) we obtain Zl sa 1 1 (5°5) d(n) = cx + Cs (n)> gt Egt. (56) a(n) =, (n) [t] + c2 (n) te +¢;(n)[4t]+..., ~ logy ¥ log 2p “log 3y A SS , Ss 7) Paes (57) td(n)logn= aN) : a= oy ai Gai) = 3) tee eats v (2 Su 4u w 1 (au ] go] (5°8) d(n)=e,(n)|30 +3 +23 + C2 (2) \ooF + Saftam{s5,+egt+ ee where #>n and ww=n. If wand v are mtegers then 1 should be subtracted from the right-hand side of (58). Puttmg w=v =n in (5'8) we obtain aus 1 at unless n is a perfect square, when 4 should be subtracted from the right-hand side. It may be interesting to note that, if we replace the left-hand side in (5:9) by [4+ 4d (n)], then the formula is true without exception. (59) bd (n)=e,(n) Bo tea(n) 2 = +¢,(n) > 1 6. So far our work has been based on elementary formal transformations, and no questions of convergence have arisen. We shall now consider the equation (5:1) more carefully. Let us suppose that s>0. Then geo! na 1 S Be see ts ca a (kv)s*) jl (kv)? ate 0 (al=- js o(s ar 1) oF 0 (a) The number of terms in the night-hand side of (5'1) is [t]. Also we know that ¢, (n) = 0 (1) as y—o. Hence 264 Mr RAMANUJAN, ON CERTAIN TRIGONOMETRICAL SUMS Making to we obtain (61) as(ny=m E0041) {A 4g 4b ifs>0. Similarly, if we make t > im (5°3), we obtain r dr a; (n) log n =e, (n) > vs log v + c,(n) = (2v) log 2v +... 1 log v 2 log 2v + nt fe, Ob s pt G(n nS Goon But = lo kv logk 1g, pyres ee SCD ~ peal! 6D It follows from this and (6:1) that (6:2) «a,(n) {re log n| =e, ()> S ve log v + Cs (n)5 S (Qn) log 2v+. +n® f(s +1) {" gs ze ee 2 ees oo = a : where s>Oandt>n. Putting s=1 in (6:1) and (6:2) we obtain o(n)= =n fa Ces ae, + (63) 12 22 opr (6:4) a(n) {ES + log nh =7n (elo el+— 2 tog +. 4) + ¢, (n) [t] log 1 + ¢, (n) [At] log 2+... + ¢, (n) log [t]!+ ce. (n) log [$4] !+..., where t>n. 7. Since (7-1) a; (n) = n'o_, (n), we may write (6'1) in the form . as(m) _,(n) , e2(n) . ¢,(n) (7 2) f(s+1) 1s Qs41 in eH a5 wey where s>0. This result has been proved by purely elementary methods. But in order to know whether the right-hand side of (7-2) is convergent or not for values of s less than or equal to zero we require the help of theorems which have only been established by transcendental methods. Now the right-hand side of (7:2) is an ordinary Dirichlet’s series for 1 C_s (n) x f(s+1)° The first factor is a finite Dirichlet’s series and so an absolutely convergent Dirichlet’s series. AND THEIR APPLICATIONS IN THE THEORY OF NUMBERS. 265 Tt follows that the right-hand side of (7:2) is convergent whenever the Dirichlet’s series for 1/€(s+ 1), viz. : sy H (n) (7 3) =) FRE 2 is convergent. But it is known* that the series (73) is convergent when s=0 and that its sum is 0. Tt follows from this that (7-4) ¢, (n) + $c, (n) + de, (n) + ... = 0. Nothing is known about the convergence of (7°3) when —4< s<0. But with the assumption of the truth of the hitherto unproved Riemann hypothesis it has been proved+ that (7:3) is convergent when s>—4. With this assumption we see that (7'2) is true when s>—4. In other words, if —4 1 —s> lop d+ ...; where 6 is a divisor of n. But : p : = log 6 = log & = & log (60’) = ne hone 3 3 8 where 66’=n. Hence (8:2) o_;(n)=d(n)—4sd(n) logn+. Now equating the coefficients of s and s* from the two sides of (72), and using (8'1) and (8:2), we obtain (3°3) ¢ (n) log 1 + $c, (n) log 2 + te, (n) log 3+... =—d (n), (84) ¢, (n) (log 1)? + $e, (n) (log 2? + 4¢, (n) (log 3)? + ... =—d (n) (2y + log n). 9. I shall now find an expression of the same kind for ¢ (x), the number of numbers prime to and not exceeding n. Let p,, p,, ps, ... be the we divisors of 2 and let (9:1) ds (n) =n (1 = _ ahi 2) (se) kee so that ¢,(n)= (xn). Suppose that F (u, v) = p (u) v*. * Landau, Handbuch, p. 591. + Littlewood, Comptes Rendus, 29 Jan. 1912. + Landau, Handbuch, p. 594. 266 Mr RAMANUJAN, ON CERTAIN TRIGONOMETRICAL SUMS Then it is easy to see that D (n) = o; (n). Hence, from (371), we have

0 we can make t— #, as in§ 6. Then we have: (1) f (v) = ) PO) = 0, (n) 3 oe + Cs (7) 5 Ee (9°3) But it can easily be shown that (nv) b(n) 9-4 Se oe 1 ov f(s) —-p)d—p.*) (1 —p)... where p;, Ps, Ps, --. are the prime divisors of 7. In other words 9:5 SO ae oe ede ds (7) €(s) It follows from (9°3) and (9°5) that bs(n)F(st1)_ wal) , w(2)e(n) , w (3B) 6 (n) Ge) ifr Tiger an RESIDE DULITAGNA (2): = GORENG) In particular ‘ 7 Cy (”) a Cs (2) E C; (n) (97) +6 $@=4()-—5 4-4 BHT C, (n) C; (n) re Cio (1) sagas?) 721° (#—YGt=1),~ 10. I shall now consider an application of the main formulae to the problem of the number: of representations of a number as the sum of 2, 4, 6, 8, ... squares. We shall require the following preliminary results. (1) Let ee ly Js ae 331 a D SS n — Sees eee Ne (10:1) D(n) a =X, tineti;et We shall choose F (u,v) =v, u=1(mod 2), F (u,v) =— v4, u=2 (mod 4), F(u,v)=(2'-1) vs, w=0(mod 4). Then from (3°1) we can show, by arguments similar to those used in § 6, that (1011) D(n)=n* (1 +37 + 5-*+ ...) {1% ¢, (nm) + 2c, (n) + 3c; (mn) + 4 c, (n) + 5-8, (2) + 6-8 eyo (nm) + 7-8 c; (nr) + 8 Cig (MW) +... } We fjsa lr (2) Let . 9 ee ee ae |, BPN (10:2) 2 D(n) a= X,= Sena ico Ue AND THEIR APPLICATIONS IN THE THEORY OF NUMBERS. 267 We shall choose Gay) w= 1 (mod 2), Hi (Ge) — ea, u = 2(mod 4), F(u, v)=(1 — 28) 7, w= 0 (mod 4). Then we obtain as before (10:21) D(n)=n*"(1-*+3%+5*+...) {lq (n)— 2 ¢,(n) +3 ¢; (n) — 4c, (n) +578 ¢,; (nr) — 6-8 ey (n) + Tc, (n) — 8 ey (NM) +... (3) Let j= L 23-1 ip 3341 xe : a \7n — .= cece (10°3) 2 D@)e@=X=7 e+ get iget We shall choose F(u, v)=0, u=0(mod 2), F(u,v)=v, w=1(mod 4), F(u, v)=—v-!, w=3 (mod 4). Then we obtain as before (10°31) D(n) =n? (1-§ — 3-3 + 5-8 —...) {1-8 e (n) — 3c, (n) + 5-8 6, (n) — ... |. (4) We shall also require a similar formula for the function D (n) defined by LG ie 2 amo lie lS ae (10-4) > Din) 2 — xe — The formula required is not a direct consequence of the preceding analysis, but if, instead of starting with the function 2anXr c, (n) = & cos , r r we start with the function 1 i 2 s,(n) == (—1)°°-) sin man ; A where 2 is prime to r and does not exceed r, and proceed as in §§ 2-3, we can show that (10-41) D(n)=4$ne (1-*— 378 4 578...) {1 8, (n) + 278 85 (n) + 37 Sy (N) +... }- It should be observed that there is a correspondence between ¢,(n) and the ordinary ¢-function on the one hand and s,(n) and the function n (s)=1*—3+5*-... on the other. It is possible to define an infinity of systems of trigonometrical sums such as c,(n), s,(n), each corresponding to one of the general class of ‘ Z-functions*’ of which {(s) and m (8s) are the simplest members. We have shown that (10°31) and (10°41) are true when s>1. But if we assume that the Dirichlet’s series for 1/n(s) is convergent when s=1, a result which is precisely of the same depth as the prime number theorem and has only been established by transcendental methods, then we can show by arguments similar to those of § 7 that (10°31) and (10°41) are true when s=1. * See Landau, Handbuch, pp. 414 et seq. Vou. XXII. No. XIII. 35 268 Mr RAMANUJAN, ON CERTAIN TRIGONOMETRICAL SUMS 11. I have shown elsewhere* that if s is a positive integer and 14+ 7, (n) a” = (1 + 2a + 2a + 20° +...) then Tyg (11) = Oog (NM) + Cos (”), where @,, (n) = 0 when s = 1, 2, 3 or 4 and is of lower order+ than 6,; (7) in all cases; that if s is a multiple of 4 then (11:1) (943545 +.) EB (nen = Kis if s is of the form 44 +2 then (11:2) (#438454...) E84 (n) a" = at ee if s is of the form 44 +41 then (113) (1-8 3-84.58...) 3 8 (n) a" = a (X,4+2->X)), except when s=1; and if ¢ is of the form 44 + 3 then (11-4) (1-*—3- 4.5 —...) 3 8,,(n) a2" = a= (X,-2-X)), X,, X., X,, X, being the same as in § 10. In the case in which s=1 it is well known that = x“ Prd Hid (115) 3 8,(n) a= 4( + 7) =% Zz e 7 (eee It follows from §10 that, if s is a multiple of 4 then (aie) OA) ea c, (n) + 2-* 4 (n) + 3-* cy (n) + 47 C5 (2) + 5 €; (Nn) + BO Cp (nr) + 7-*c,(n) + 87? ce(n) + -..}5 if s is of the form 44 + 2 then stl D! {1-8 @, (nm) — 278 ce (n) + 37 cs (n) — 4-8 cg (n) + 5-8 C5 (2) — 678 Cy (n) (ULB) Oe, (=e + 7-8, (n) — 8 C5 (nr) +... }5 if s is of the form 44+ 1 then z {1-* c, (n) +27" s, (7) = 3 ¢, (nm) + 4-* 5, (2) + 578 Cs (12) + 6 Syo (M) } _ mnt (11°81) 84 (®) = G4)! —7-*¢,(n) + 878 sy6(n) + ..-}, except when s = 1; and ifs is of the form 4k +3 then mn ; : p . (11:41) 6, (n)=— D! {1-* ¢, (n) — 278 s, (n) — 3-8 cy (n) — 4-* 8 (n) + 5-* Cy (2) — O~* Sp (2) —7-*¢,(n) — 878 8y5(m) + «e}. * Transactions of the Cambridge Philosophical Society, + For a more precise result concerning the order of vol. 22, 1916, pp. 159—184. @o, (n) see § 1d. AND THEIR APPLICATIONS IN THE THEORY OF NUMBERS. 269 From (11°5) and the remarks at the end of the previous section, it follows that (1151) ry (n) = 8 (n) = 7 {e, (n) — 40; (nm) + $e; (n) — ...} = m7 {k5,(n) + £8 (n) + ds (n)+...}, but this is of course not such an elementary result as the preceding ones. We can combine all the formulae (11-11)—(11°41) in one by writing rns) (116) 6.,(n)= coun {178 e (m) + 278 y(n) + 3-8 ©, (n) + 47-8 Cg (n) + 5-* ©; (n) + 67% Cyo (2) + 7~* Cy (Nn) + 878 Cy (1) + ...}, where s is an integer greater than 1 and Cy (n) = ¢,(n) cos $7rs (r — 1) —s,.(n) sin $78 (r — 1). 12. We can obtain analogous results concerning the number of representations of a number as the sum of 2, 4, 6, 8,... triangular numbers. Equation (147) of my former paper* is equivalent to Ge or. Ont 2 IE) (ay et gl 1 Wi | (Ga) een Ge) VEX) where K, is a constant and f(#)=(1—2) (1 — «#) 1 —@’).... Suppose now that G=e€5™, a =e 2nla Then we know that (12:2) Va(1 — 2a + 2a — 20° +...) = 2a’8 val +a ta%4+ a+ ...), (123) Vda) a2 f(a) = a2 f(a), awl? f (a2) = a"? f (a). Finally 1+ > 825 (2) (— «)" can be expressed in powers of a’ by using the formulae :— 1 5 4 Sa 5 13 928-1 3231 (124) a EG -2)+ oa +ea-eait f 133-1 925-1 33-1 = (— 8) }4 6(1 — 28) + a Se aRT ae tut, where a8 = 7 and s is an integer greater than 1; and 15 228 328 \ (125) — (2a)°* + if ansa 5 = lex + es erraee e82 + ¢g-% a 1° 325 52s =(- BY V28)) 8 1(- 2) + a4 9 a ee B— | eB —] where a8 = 7°, s is any positive integer, and 7(s) is the function represented by the series —— 3-4 5-*—... and its analytical continuations. Tt follows from all these formulae that, if s is a positive integer and (12°6) (+ a@ a3 + 8 +...) = Dog (N) B= D8'n, (0) 2” + Deo, (Nn) 2, then fies (2) > KG. (— ame ve (2) Sy \ pn —v¥ 2 €og(N) 2” = — 2 a =< 9 J” (&) 1 n. It is easy to see that, if n and & are positive integers, and hk odd, then (141) is equal to 1 if k is a divisor of x and to 0 otherwise. When k is even we have (with similar conventions) sin nr i) ee Sl PO (142) k tan (n7r/k) ; according as / is a divisor of x or not. It follows that ‘sin nar sin 17 _{ sin nr sin n7 \ Aye) Or) (22) = Ne SF ( : ) + 2-*/ 3°§ (= ) 4-8 ( - bisdie i a) sin 27 tan ae) =e sin 4 nr i tan aa) Similarly from the definitions of 6,;(n) and &,,(n) we find that = Time | eee SUL a) ee 8 oP eB aia erent) =e cooryy tN rags a 2-3 sin 177 ) ba a*( sin 27 ) wes sin 177 )+ l ~ \sin ($nm +4s7)/ °~ \sin (hn + 877) lea (tna + 3sr)/" “"'S if s is an integer greater than 1 ; (145) AOE O Se (3 me a = Th ) a (= nT ) ce a sin 27 sin dn sin Ln sin 177 sin 277 sin nr | = 4 — i ( —_— 4 {| — =|) = anoles u ee =) . ee te + (= 4 a) We 5 , sin (n+ 4s) 7 14 1=8 4-8 4 8 _ Ga a 4 be ‘S (n+4 ) Bea 8+ ON) (s — - yn + 4s) : sin (n + 4s) 7 par (aDot ide), 5(smot Lo) sn 4 (n+4s)7 sini (n+ 4s) 7 if s is a multiple of 4; 272 Mr RAMANUJAN, ON CERTAIN TRIGONOMETRICAL SUMS (47) (e849 +.) Ba = ET sn s—1 (4-8 Sy ey (Gr (Qn + 48) _,/ sm(2n + $s) 7 Las ( sin (27 + $s) 7 5: (= 1(2n ee Be \sin 1(2n +48) 7 =) a if s is twice an odd number; (148) (8-3-4 5-* =...) 8x (nr) = 2B (ns poy a ea) = ai ( sin (4n +s) 7 22 \sin 4 (4n +s) 7 sini(4n+s)7/) ~~ —3° if s is an odd number greater than 1; and ; Aen ere (Se Coen ie sin (4n+1)7 )\ sin (4n+1)7 aes =e O- ae. oar (@n 45D eae (ey acy oi ote In all these equations the series on the right hand are finite Dirichlet’s series and therefore absolutely convergent. But the series (14°3) is (as is easily shown by actual multiplication) the product of the two series r . IS*en(@v)-- 2=%'G, (nm) + .<- and ne (18 4-2-*-- 35° + ...): We thus obtain an alternative proof of the formulae (75). Similarly taking the previous expression of 6,,(7), viz. the right-hand side of (11-6), and multiplying it by the series USS (8) 7 ol Ca we can show that the product is actually the mght-hand side of (144). The formulae for &.,(n) can be disposed of similarly. 15. The formulae which I have found are closely connected with a method used for another purpose by Mr Hardy and-myself*. The function (151) (1 + Qa + Qa* + 2a°+ ...)* = > re; (nm) 2” has every point of the unit circle as a singular point. If @ approaches a ‘rational point’ exp (— 2p7i/q) on the circle, the function behaves roughly like 1 (Wp,9)° 52 PY ; ad {= (2p7i/q) — log a} where «,,,= 1, 0, or — 1 according as q is of the form 44+ 1, 44+ 2 or 4k +3, while if q is of the form 4k then ,,,=— 2i or 27 according as p is of the form 44+ 1 or 4k +3. Following the argument of our paper referred to, we can construct simple functions of « which are regular except at one point of the circle of convergence, and there behave in a manner very similar to that of the function (15'1); for example at the pomt exp (—2pri/q) such a function is 7* (Wp, 9) (s—1)! x "S ‘Audi e2Rprilg yn Tez: (15°3) * “Asymptotic formulae in Combinatory Analysis’, Proc. London Math. Soc., ser. 2, vol. 17, 1918, pp. 75—115. AND THEIR APPLICATIONS IN THE THEORY OF NUMBERS. 2 —Y ie) The method which we used, with particular reference to the function af 15-4 - == p(n)2", GP) (1—a#)(1 —2*)(1—2%)... NG) was to approximate to the coefficients by means of a sum of a large number of the coefficients of these auxiliary functions. This method leads, in the present problem, to formulae of the type Tog (2) = 8.5 (n) + O(n2*), the first term on the right-hand side presenting itself precisely in the form of the series (11:11) ete. It is a very interesting problem to determine in such cases whether the approximate formula gives an exact representation of such an arithmetical function. The results proved here show that, in the case of r,,(n) thisis in general not so. The formula represents not 7.,(n) but (except when s=1) its dominant term 6.,(n), which is equal to 7,,(n) only when s=1, 2, 3, or 4. When s=1 the formula gives 26, (n)*. 16. We shall now consider the sum (161) o, (1) + 6, (2) +... +o, (n). Suppose that (162) 7,(n)=45 — at Uva mx sin {(2 + 1) 7A/r} =1), a) = sin (7X/7r) sin (7A/7r) where X is prime to r and does not exceed 7, so that T,.(n)=c,(1) +e, (2) +... +¢,(n) and U,(n)=T,.(n)+46(7r), where $(n) is the same as in §9. Since ¢,(n)= 0 (1) as r > x, it follows that (16:21) Tn) 0); 4) Ui (2) = O1(r); asr—>o. It follows from (7°5) that if s>0 then (163) o_.(1)+o_.(2) +... +0_s(n) = E(s +1) sn+ { Qst1 3341 4sh T,(n) _, Ps(n) a hae Since if s > 1, (16°3) can be written as (16°31) o_,(1)+o_,(2)+...+0_,() U.(n) U,(n) U,(n) =€(s+1) n+ + Osh ac 36H = Agti + of =B E06), ifs>1. Similarly from (8°3), (84) and (11°51) we obtain (164) d(1)+d(2)+...+d(n)=—347,(n) log 2-47, (n) log 3 —1T7,(n) log 4-..., (165) d(1)log1+d(2)log2+...4+d(n) logn = 47, (n) {2v log 2 — (log 2)?} + 47, (n) {2v log 3 — (log 3} + ..., (16°6) r.(1)+72(2) +... +72. (n) = 7 {n — 47, (nm) +1T, (n) —4T, (n) +...}. * The method is also applicable to the problem of the and in particular of five or seven’, Proc. London Math. Soc. representation of a number by the sum of an odd number (Records of proceedings at meetings, March 1918). A fuller of squares, and gives an exact result when the number of account of this paper will appear shortly in the Proceedings squares is 3, 5, or 7. See G. H. Hardy, ‘On the represen- of the National Academy of Sciences (Washington, D.C.). tation of a number as the sum of any number of squares, 274 Mr RAMANUJAN, ON CERTAIN TRIGONOMETRICAL SUMS Suppose now that 2rr 4 2 T,,.(n)=% (Ico = 250 cre eines aa, A where X is prime to rT and does not exceed r, so that T,,,(n)= 1% c, (1) + 2%, (2) + ... + n*¢,(n). Then it follows from (7°5) that (16°7) o,(1)+o5(2) +... +o9(n) Ls ME Go =f(+1) {r+ +254+...+n8)+ - ae. is T,, 5 (%) Ash a ifs>0. Putting s=1 in (16°3) and (1677), we find that (168) (n—1)e_,(1)+(n—2)o_,(2)4+... +(m—n) a(n) Sr fee = Dee eOree ae) Cie. 2 a ge a r/r) che - yer a eee where V,(n) nen r) —n, d being prime to 7 and not exceeding r. It has been proved by Wigert*, by less elementary methods, that the left-hand side of (16°8) is equal to Wns nm 2 or (169) 5 n? —4n(y—1+4+log Qn) — 3 where J, is the ordinary Bessel’s function. —) J, {dar af(vny, 17. We shall now find a relation between the functions (16:1) and (16°3) which enables us to determine the behaviour of the former for large values of n. It is easily shown that this function is equal to (17-1) = ( (184 2843°4.. +" oy )+S [2] tua Sw. Now “ k+4 , 194+ 2°+...+h=C(— s) + F a + 0 (ke) for all values of s, it being understood that (k+4) SS se aeare denotes y + log(k+4) when s=—1. Let nh i |; |--tte, [Vn] =t=Vn—4t+e. vas Then we have 8+ 8 s-1 1424... e Se (2) wv (n al \= pio ad s+] iat 1 Olay n and vs H =n — fy? + er’. “ Acta Mathematica, vol. 37, pp. 113—140 (p. 140). AF Got bo Sf oOo AND THEIR APPLICATIONS IN THE THEORY OF NUMBERS. It follows from these equations and (17:1) that 2 e t s) t { : 1 n\ sti ae (72) @,(1) +4,(2)+... +a,(n)= }6(—9) + 5 (“) + } n\* ; ne Fe(=) ter —(ynte)i" +0 (a) : y p*—1)}) t 7981 s+1 (17°21) 8 {o_,(1) + o_, (2) +... +o_s()} = = jm (s) + + (7) +e,v° v=1 =. Changing s to —s in (17:2) we have pet, | By It follows that t (173) n® {o_.(1) + o_,(2) +... +a -2(n)} — {o,(1) + 0, (2) +... +o, (n)} = S es) -S-8) v=1 +e, (5) -wa+9(™ ) +0/ Sen 0 \ae 8 UN? mom BEN) — / iS sf ae ale cane. + (n+ €) vf — (4/n +6) 5) + OS ee Suppose now that s>0. Then, since v varies from 1 to ¢, it is obvious that . pst ns} n po fpr ~ {ns\ and so 0 ( n ) =0 \ pel ) ° t Also = {n*F(s)—(—s)} =(V/n—$ +6) {n? f(s) —F(—5)} ; = (F) is Seen 6 +s)- Wnt ey ‘+ 0 (n**); Vv Ss n - ls i =e(1- s)+ pay Wvn teh + O(n); (/n+e€)prs l+s ies), S (tu ene CEC a) t ) alg >; (in aenle ae (ln Pelee) (/n +e) + O(n**); vol l-s t né 1 and D>, ‘Soltt NE Eo v=1 where (17°4) m =n (s<2), m=nlogn(s=2), m=n*"(s> 2). It follows that the right-hand side of (17°3) is equal to 2+3 __ » stl —3. Ty il ts) +, £1 5) — dnt oe) ¢ EE Wnts) +S s 8 se , rWnte) n§ (4/n +6) NOG: l—s / 2+5 __ »sti —s 5 , f But (Wn + €)Pt8 = °F (s/n + €) = ent +9) 4 Q (nd): l+s n(f/n+ ee ile eae Jen 1+) 4 O (nd), 276 Mr RAMANUJAN, ON CERTAIN TRIGONOMETRICAL SUMS, ETC. Tt follows that (17°5) 0, (1) +o, (2) +... tog (n) =n! {o_, (1) +o_5(2) +... +o_2(n)} — +ins 6(s)- €(1—s)+0(m) if s>0, m being the same as in (17'4). If s=1, (17°5) reduces to (176) (n—1)¢_,(1)+(m—2)o4(2)+...+(n—n) o4(n) = >—In(y—1+ log 2n7) + O(W/n)*. From (16:2) and (17-5) it follows that (177) as) +o5(2) +... +9 (= G1 +s)+3n' E(t 60 —s) (T.(n) _ T; (ay “?, (nr) | Qst1 31 4sHi for all positive values of s. If s>1, the nght-hand side can be written as . U, : (7s) ¢(-s)+wS(.+s) {e+ 4+ = + eae + a = + 0(m). +n°E(1+4+s) = +..b+ O(m), Putting s =1 in (177) we obtain (17-9) o,(1)+.4,(2)+... +6; (n)= put hn (y—1+ log 2nzr) (Ts E T, +o ws — o) +f + O(n) Additional note to §7 (May 1, 1918). From oy it follows that {1-*o,_, (1) + 203 (2) +)... = 172 1 Cy, (1) 22 > Mae Gn (2) eee 1 a cr)" E(s) On +8 eg se Cm (n) or = Pa ' f(r) 11 mn from which we deduce (8) 3 w(8) a" = ~ sal?) = eee Peay 8 being a divisor of m and & its conjugate. The series on the right-hand side is convergent for s >0 (except when m= 1, when it reduces to the ordinary series for §(s)). When s=1, m>1, we have to replace the left-hand side by its limit as s— 1. We tind that Cm (1) + 4¢m(2) + $¢m (3) + ... =— A(m), A(m) being the well-known arithmetical function which is equal to log p if m is a power of a prime p and to zero otherwise. * This result has been proved by Landau. See his a more transcendental method, replaced O (,/n) by O(n) report on Wigert’s memoir in the Gdéttingische gelehrte (l.c, p. 414). Anzeigen, 1915, pp. 377-414 (p. 402). Landau has also, by XIV. Asymptotic expansions of hypergeometric functions. By G. N. Watson, Se.D., Trinity College, Cambridge. [Received June 11, 1917. Read Feb. 4, 1918.] 1. The hypergeometric function F(a, 8; y; “) presents two distinct problems to mathe- maticians interested in the theories of analytic continuations and asymptotic expansions. The first, and simpler, problem is that of finding the analytic continuation of the function beyond the circle |~|=1, which is the circle of convergence of the series by which the function is usually defined. More generally, the problem is that of finding the analytic continuation of qu, beyond the circle |z|=1, and of finding the asymptotic expansion of »F, for large values of |z| when p0,—az2; and then, taking the path of inte- gration to be the real axis, we have 1—¢t<|z—1)|. Hence, on expansion, we get = P(a+A+n) /1—t ri =(z-— —a—A ‘7 — F\ata— \y—8+A-1 > ee) ja ey: n=o 2! D(a+A”) G —2Z n ) dt. Integrating term by term, we have at once _ 2D (atr—y+ DI (y—-B+%) (2 = RG =A PAD) \ Shy, 2 25 ; ) Platdatd—7+lia—B+ 2441372): Since J, is analytic and one-valued throughout the plane (when cut from +1 to— co), this equation, proved when | z— 1| >2, persists throughout the cut plane. ‘ Next take the integral J,, defined by the equation “1 L= | (1-8) (1 4. £)7-8+> (2 — t)-=A dt, = in which the path of integration passes above the point t=z when J(z)>0, and below it when I(z)<0. Then J, is analytic in each part of the plane when the plane is cut along the whole length of the real axis. Deforming the contour in the manner indicated in Figure 1, we see that (2) ome i (1 —t)***-7 (1 4 £789 (2 — tA dt, 1 where the path of integration starts from ¢=1 and returns to it after encircling the point t =z negatively or positively according as J (z) is positive or negative. -1 1 Fic. 1. To evaluate this loop integral, take |z—1|<2 and write 1—t=(t—1) e**, so that arg (t—1)| <7; and then put u=(t—1)/(z-1), so that t—-l=u(e—-1), z-t=(l—u)(z-1), t+1=2+u(e-1), where arg (1 — 1) is zero on the first part of the loop and } 27 on the second part. OF HYPERGEOMETRIC FUNCTIONS. 281 Then Gre & 2 D(B-rA—y+14+n) uw (1—z)" <= a(S Grae R SY Da OS Se ee See ne I, ee om! T(B—-A—y+1). 2 x {(1 — u)(z — 1)}-2 4 (2-1) du a T(a+rA—y + 1)PA—a-d) /z-—1\!7 ie + CEOS —B p¥rt(atA—y) ee 5 Ns oe ON [ee ety. ; r2-y) \ 2 ) x Fa+r¥-yH+1, B—-A-yH+1; 2-7; 4-42) _ Qrriet™72* FT (a+ X—y+1) /2—1\7 = = —-r- ; 2—y; 4-432). Ga) ee, PPE BRS y els 29 892) Since it follows from the formulae of analytic continuation* that ey ot A) = uae Pee 2235) oe Nd —p +X) ae P@tA yt DEG) (Za) a, | = Sis ee oe eee EGS) i ) "F@+trx-7+1, B-r-y+ 1: 2-75 4-49), it is easily shewn that F(a+ar, B—-A; ¥3 $-—42)= BCSEEALEC) PIP (y— B+) strictly speaking, this result has not been proved when y is a positive integer, in view of the factor ['(1—-) which occurs in the course of the reduction; but, since both sides of the final equation are analytic functions of y (except when y is zero or a negative integer), the equation holds also for the exceptional values of 1, 2, ... of y, provided only that 2) is so large that the integrals J, and J, are convergent. {+ ety I, F e*7I,}; 4. The contours provided by the method of steepest descents. We now apply the principles of the method of steepest descents to the integral fa —t)2-7(1 + t)Y-#> (2 —t)-* exp (- nV log = = =) dt, with a view to determining asymptotic expansions of J,, 7, when |X) is large. The stationary points of log {(z—#)/(1—#)}, qua function of ¢, are given by the equation t? — 22#+1=0, and so they are t=¢é, t=e°%. The contours which are provided by the method of steepest descents are consequently ares of the curves + Tog 2! = Tlog $=", lee 2 — = Tog — 1- es 1- 1 Writing? t= X +7Y, and supposing that ¢,, ds, d; are the ae which the vectors ¢—1,¢+1, t —z make with the Y-axis, we see that each of the curves is such that ¢,+¢.—@; is constant on it. These curves are portions of circular cubics. * Barnes, Proc. London Math. Soc. (2) v1. (1908), cedure involves very great labour in obtaining terms of the p. 147. asymptotic expansion following the dominant terms (cf. + Riemann takes the real part of the logarithm constant Cambridge Philosophical Proceedings xrx. (1917), p. 45). and then applies the method of stationary phase; this pro- + Of course X and Y are supposed to be real. 282 Dr WATSON, ASYMPTOTIC EXPANSIONS | ete rite i a Js V(X*+ Y24+1)—sinh €sin» Ss - = —1)+ 2X ¥ cosh Ecos Li | a ie eee! X (X?+ Y?—1)-—cosh Ecos » (X?— Y?—1) —2X Ysinh & sin» the equations of the two cubics are . Y (X2+ Y2?+1)+sinh £ sin n (X? — Y°—1) —2XY cosh £ cos n = + tan 7 LX (X2+ Y2—1)—cosh & cos » (X?— Y?—1)—2XYsinh Esiny], and the cubics have nodes at (ef cos 7, e sin) and (e-‘ cos n, — e~ sin n) respectively. . We shall consider fully (§§ 5-7) the properties of the cubic (S,) obtained by taking the upper sign in this equation, and deduce (§ 8) comparatively briefly the properties of the other cubic (S,); the simplest mode of passing from one cubic to the other is by changing the signs of £ and » throughout the work. 5. Properties of the cubic S.. To express the coordinates of any point on S, as rational functions of a parameter p, we write the equation of the curve in the form (Y cos n —X sin n) {|X (X — e cos n) + Y(¥ — ef sin n)} — e*(X — e cos n)(¥ — e sin 9) =0; and if we now put Y —e§sinn =p (X —e§ cos 9), we find after some straightforward algebra that HEE body ate RISES 20 8) (sin y — 4 cos») (1 + 4’) uw {u2e® cos n sin n + w (ef cos 2n — e~*) — e cos 7 Sin 9} (sin 7 — / cos 7) (1 + p*) The only real asymptote is Y cos y — X sin n= e~£ cos 7 sin 9. Y=efsinn+ The curve degenerates (into a straight line and a circle) only when 9 is: 0) = fee and, in each of these cases, the curves C, and C, coincide. It will, however, appear later that the degeneration when » = + 47 does not affect the analysis; but in the cases »=0, +77, the difficulty has to be surmounted by taking as contour not a portion of the degenerate cubic, but a slightly different curve (§ 7). To return to the non-degenerate cubic, we notice that the effect of changing the sign of 7 is to reflect the curve in the axis of X; while the effect of writing +7—y for 7 is to reflect the curve in the axis of Y. We can consequently derive the shape of the curve for any admissible value of » by considering the shape of a curve of the family for which is a positive acute angle. We now construct the following table of values of t(=]X +7Y) and p: 90 0) 1 2 3 - 4 5 6 | | ‘= HL z 1 e§ cosy | iefsiny | —1 | esinyn | | efsiny | p= tany coth €tany e§ cos n — | ef cosy +1 The parameters and complex coordinates of these six special points will be denoted by attaching the suffixes 1, 2,... 6 to w and ¢. OF HYPERGEOMETRIC FUNCTIONS, 283 It is easy to see from the table that (when » is a positive acute angle) as jw increases from tan» to + 2, and then from — 2% to tan 7 again, it passes in succession through the values* Hy, Pos (Ms OF Ms), Bs) Mes ba To determine the positions of the node relative to these six points, we observe that the parameters of the node are the roots of the quadratic g («) =0, where 9g (hw) = we cos sin n + pw (e§ cos 2y — e~*) — e§ sin 9 cos 7; now g (ps) > 9 > g (ps), and 9 (4s) = 2e- sin n (cosh & — cos y)/(e§ cos 7 — 1)? > 0, 9 (w,) =— e-* tan n < 0, 9 (us) = e-* sin 7 cos n (coth? € tan? y + 1) > 0. Hence one of the parameters of the node les in the intervals (us, u;) and (44, “;), while the other lies in the interval (4, /42). Hence, if a point starts from infinity and traverses the entire length of the cubic, it passes through the points of interest in the following order: co, e(node), z, 1 or e§cosy, es(node), ie?sinn, —1, © (the points being specified by their complex coordinates); if 7 is a negative acute angle, the points are traversed in the same order; while if 7 is obtuse (either positive or negative), the order is: #2, es(node), z, —1 or ecosn, es(node), itefsinn, 1, x. The curve is shewn in Figures 2 and 3 in two cases, 7 being a positive acute angle, and e' cos 7 > 1 in 2 and e' cos < 1 in 3, while in Figure 4 »=37. The portions of the curve from eo es a \ 1 S Se Asta HI Dian Fie, 2, Fic. 3. Fic. 4. which the contour has to be selected are shewn in continuous lines; and it is obvious (even in the critical cases when 7 = + $77) that the cubic has an are which passes from —1 to 1 through the node without passing through+ the critical points z and #; and further, this are cannot * The sign of e&cos—1 determines whether fz comes is true if cosech cos »1, we prove that My pre- + This is not true in the other critical cases 7=0, £77; cedes x, by proving that coth ¢(e6cosy—1) d.7*) = 0 (7"+4)+0(e%!7h, s=0 s=0 when 7 is large, K being a positive constant (7.e. independent of +) which depends on a, 8, y, 2, and r being any fixed positive integer. * Since —7<7€z, no one of the points 2¢+2k7i is nearer to the origin than 2¢. Wornexexstinn Now XLV. 37 286 Dr. WATSON, ASYMPTOTIC EXPANSIONS - 7. The asymptotic expansion of I,. Tt follows, by applying a general theorem* to the result of § 6, that we have the com- plete asymptotic expansion Pea a (1 —#)*-7(1 + t)-8"! (zg —t)-* e-™ (dt/dr) dt ~ + C S (o- | rte dr + S d, | Te dr, / 0 s=0 J0 é s=0 “0 and so I~ 2 Cer 3 6, P'\(s4+ 4) a7? 03, s=0 the expansion being asymptotic in the sense of Poincaré when |X| is sufficiently large and jargX <}a—46; provided that » is not equal to 0 or +7, as the integrand would then have a singularity for a positive value of 7. We shall now examine to what extent these restrictions can be removed. It can be shewn that the expansion is valid when n =0 or + 2; for suppose that 7 is slightly greater than 0 (or — 7); then instead of taking the contour to be the real axis in the t-plane, we take it to be+ the ray argt=—48; the modified integral is an analytic function of X when — 37+ 80 (or —7). Hence, making € assume the real value & (or the value €—-i), we see that, when »=+0 or —7+0, I, is equal to the modified integral, and also the asymptotic expansion is unaffected. To discuss the cases 7 =—0, 7—0 we proceed similarly, but we swing the contour round in the opposite direction; and we note that the expansion is the same whether 7 =+0 or —0. Secondly, to extend the range of values of arg X, we observe that the process of swinging round the contour can be carried further, as shewn in Fig. 5. Take the two of the points 2+ 2kzi e e e e e Fira. 5. * Proc, London Math. Soc. (2) xviut. (1918), p. 183. deformed contour, is one which passes from —1 to 1 and + The contour in the ¢-plane, corresponding to this it is of a spiral form near each of these points. OF HYPERGEOMETRIC FUNCTIONS. 287 which are nearest to the real axis (one on each side of it), and let the rays joining the origin to them be arg 7 =— @,, arg T= @», So that @,, w, are positive (or zero) acute angles. When arg >0, we take the contour to be the ray argr=—@,+46, and the modified integral (which has the asymptotic expansion already given) provides the analytic continuation of J, over the range for which arg < 47+, —6, provided that |X| is sufficiently large to make K\r\|—R (ar) < 0. Similarly, if arg’ <0, we swing the contour round to be the ray arg t=, — 36, and we get the asymptotic expansion of the analytic continuation of /, over the range arg \ >— 47 —@,4+ 6. Hence, when |X| is sufficiently large, and — 47 -—o.+60: and by the formulae , w, = tan {(n + 7)/E}, —o,= tan (n/é), when 7 <0, the symbol tan~™ in each case denoting an acute angle, positive or negative. [The method obtaining the asymptotic expansion of J, when 2X does not lie in the specified sector of the plane is indicated in the first footnote to § 2.] 8. The asymptotic expansion of I,. We next consider the second contour (S,) of § 4, namely — Tlog 5 aioe f= See—1: The analysis is derived from the oa. analysis by writing (— &, —») for (&, »). If Y+e*sinyn=y(X —e-f cosy), we can construct (as in § 5) a table of corresponding values of ¢(= Y +7Y) and wu. he '| 8 ie | 10 | 1 ae | |. t= ao | | —ie-— sin n 1 | z e-£ cos n e-— sin -F si n | _e Fsiny ee | L= | —tan 7 SICSGEE cei | 0 (= 32F oS = | coth € tan 7 00 We denote the parameters and complex coordinates of these six special points by attaching the suffices 7, 8,... 12 to w and ¢. It is obvious from the table that, when » is a positive acute angle, w;, fls,.+» pa are m ascending order of magnitude. Bi 288 Dr WATSON, ASYMPTOTIC EXPANSIONS The parameters of the node are the roots of h(w)=0, where h (mw) = we cos n sin n — p (e~* cos 2n — e) — ef cos n Sin 7; now h(ps) <0, A (40) = 2e-€ sin 7 (cosh & — cos n)/(1 — e-§ cos n? > 0, A(uy2)>0, h(u;) =— e tan n < G, and so a point which traverses the entire length of the cubic passes through the points of interest in the following order: 0, —1, —vzefsinn, e%(mode), 1, 2, e cosy, e-S(mode), a. Hence the are of the cubic (Fig. 6) which passes from —1 to 1 through the node is an admissible contour which lies entirely on one side of the real axis while the point z lies on the other side, as shewn in Fig. 6. Fic. 6. By suitable reflexions, we see that the cubic possesses these properties for all values of n from — 7 to 7. Hence, writing : log a4 + log 2eS= 7, ro we get - i (| + [) (1 —#)2-7(1 +t) (2 — t)-*# De" en = dr,. ca “0 7 Also, when ¢ is e~$, we have l1-—t=l—-e, 1+t=l+e% z-t=te(1—eS)(1 +e), where . jarg(l—e)|<7, jarg(1+e-)| <7. We now proceed as in § 7, and we find that the asymptotic expansion of J,, for large values of |X|, is given by the formula Tw 2YCe-* E 6 T (s+ Darcie ; s=0 eye CO! = 28-1 ¢-25 (1 — e-S)8- 11 + e-$)¥— 4 -B-4, and the value of «;' is 1, while the general coefficient ¢,’ is derived from ¢, by changing the sign of € In particular ¢ = 3 (L + Mes + Ne-*)/(1 = e~*s), where L, M, N have the values given in § 6. OF HYPERGEOMETRIC FUNCTIONS. 289 There is, however, an important difference in the range of values of argr, for which the asymptotic expansion of I, is valid. For the singularities of the integrand are the points 7 = 2hrri, 2k — 2E, and: the points 2h7i — 2¢, being on the left of the imaginary axis, do not hamper the process of swinging round the path of integration; we may therefore swing it round so as to be either of the lines arg7,= + (47-46), according as [() 20; and therefore the asymptotic expansion of I, is valid over the sector jargrX | <7—6, provided that || is sufficiently large. For brevity we shall describe the sector for which jargr| clGeepae at ben Ne MrGs Det Dart, ) s=0 s=( valid when | 2) is large and —}7-—0@,+6 ee) In the special case when z= cos 9, Qn" (2) is defined + as 3 {Qn™ (cosh (0 + im)) + Qn™ cosh (0 — in)}, where we may take 0 << 7. f D) is B(e- Dj (m+ ntl, n+]; 2n42; —). =m —4}+(m*—}4)cothé If we write in turn €=0+ a and €=0—1, we get ef = etn /(1 —e-%) =e 7 7+ 47! V2 sin 7), since /(1 —e-*) is positive when ¢ is a pure imaginary. We thus obtain the complete asymptotic expansion T(n+1) sin(n+m)r ~ » (, _m?—4 m eth! ss = ; 3 a 20S ae ae a ad T(n—m+1)_— sinnr al (x5 sin y . | cos{(n+3)9 +47} ie Ga *—})cotn . _ (mn Ss sin |(n+4)n+47} + a ‘ valid when 0< 9 <7, jargn|<7—6. * This is in accordance with the definition given by (p. 114). It differs from Hobson's definition, Phil. Trans. Barnes, Quarterly Journal (loc. cit.), pp. 100, 107. (loc. cit.), p. 471. + This is also in accordance with Barnes’ definition OF HYPERGEOMETRIC FUNCTIONS, 291 11. The asymptotic expansion of P,(z) when \n| is large. From the well-known tormula* 1 1 tm : P,” (2) = rasa (Z 1) F(n+1, —n; 1—m; 4— $2), we at once derive the asymptotic expansion T(n+1) e 3 pa Oe D(n—m+1) (nw)? (1 — en3)8 x [otra 2 GED , greiteng iene § &EG+D] aon, L (3) n° s=0 iD () nr valid throughout the incomplete range of values of arg” given by —tr-—o,+d Py” (cosn) ~ cos {(n-+})n+}mm—}rr} VW 9 el erg + cot sin (ut) n+ dur te) +...], It is worthy of note that the asymptotic expansion of P,’"(cos 7) is valid over a more extended range of values of argn than is given by Barnes’ method (p. 155). The results of this section are otherwise equivalent to those given by him on pp. 152-161. 12. Asymptotic expansions of Legendre functions whose order m is large. These expansions need a rather more lengthy investigation than the expansions given in §§ 10, 11, since the cases R(z)>0, R(z)<0 have to be considered separately. The formulae+ which will be employed to obtain the expansions are gm (22-1) 3™ P,” (z) = esa) F(4n+4— 3m, —tn—tm; 1—m; 1-2), Q(z) =(2—1)-3 FD 2-71 gn+3m+s)CGnt+3m+l1) NT a T (n+) Ee 1 x SOTOT (nt im+h, L—1im+4; n+3; =.) sin nr fe = See Fe = 1-2 which are valid when R (z) > 0. * Hobson, p. 451; Barnes, p. 102. + Barnes, pp. 120, 123. 292 Dr WATSON, ASYMPTOTIC EXPANSIONS If in the first of these formulae we write —m for m, and then in the formulae of § 3 | we write a= 8B =4n+}, y=n+3, X=4m, we see that ’ Q-EMTBNT3 2) 2hae Pm (2)= ed) T(gm—$n)T(Gm+}n4+1)” | where, in 7, we have to write (1—2)~ in place of }(1—2z) wherever z occurs; so that € is now defined in terms of z by the equation 4 (1 —cosh €) =(1— 2)", 2 - and therefore cosh f= SST? sinh [= = and, since R(z)>0, the upper sign must be taken in order that we may have |e7$| <1. The asymptotic expansion of P,,~""(2) is therefore Peas) lew a = Dae ei 1+ S 26cy I’ (s+ 4) , : T(ém—3n)T4m4+hn41) \e41 m | sar TG)m where ¢,’ (1 + 2)*(42)~* is the coefficient of 7* in the expansion of f1—$(0. +2) T}-4"-19 44240 42) TH" (1-429 (2 -) T-4*3 o\2 ( oP 72 -s-4 x | Core log j1 + ses 1] ; 42T* 42 —2(2—1)7-—(1+2yP 7? and the asymptotic expansion is valid, when R(z)>0, over a complete range (§ 8) of values. of arg m. By using the second asymptotic expansion given in § 9, we find Qn (2) m Qm-1 sin (n +m) 7 T (4m+4n+4)P(4m—}n+4$) (: + a FA < 2%e, D(s +4) ‘ . sin nar V(2mz7) B= = Tam rnxi [2 ilt zm 2 28 st if " —e "i ) i+ 2 (+a), ee eat Ch) me) this is valid, when R(z)>0, over an incomplete range of values of argm; the upper or lower 2+ z—-1 sign is taken according as J ( ) £0, we. as I(z)2 0, and ¢, is derived from c¢,’ by changing the sign of z. From the formula (Hobson, p. 462; Barnes, p. 109) = Aall P,-™ (2) P," (2) 2 ni (z wen 25 s ee 0S eee : Qn” (2) TP (—m — n) r* sin mr sin nT P(l—m+n) [TQ+m+n) 5 mir\ ot amt anth) Cl Gm—3n+h)[ e+ 1\3™ S 2c, T (8 +4) wefind P,™(z)~ — Salim) sin mr ( = 1! \1 + 2 Te Gn te Te GY Lat 2 2%¢, T(s+4) iis FMT > Tas Bid e In nT (Si i) 1 + 2 Tam This result is simplified by the disappearance of the first series in the special case when m 1s a positive integer. The general formula is valid, when R(z)>0, over an incomplete range of values of arg m3 the special formula is true over a complete range. OF HYPERGEOMETRIC FUNCTIONS. 293 Since Qn” (2) D(— m= n) = Qn-™ (2) TV (m — n) (Barnes, p. 105), the asymptotic expansions when # (z)>0 are completed. We next consider the expansions when F(z) <0. From the formula (Hobson, p. 463; Barnes, p. 106) P,™ (— 2) = Pi™ (2) e™™™ — 2Q,™ (2) 7 sin nt, we see that, when R(z)> 0, 27 T (4m+4n+4)0(dm—h4n+})[ . g— aya 2 Mat) a V(2mm) OWe snc at 1 eee mmCL nt Jee (—2) f 4m wo Os 1 eed a=) | 2 eat e sin nr ee i 1+ 2 Seay : Changing the sign of z, and noting that, when this is done, c, interchanges with ¢,) and e”™™ with e~”™, we see that the expansion of P,’(z) has the same form for all values of z in an incomplete range of values of arg m. Next, taking the formula (Hobson, p. 463; Barnes, p. 106) Onn ( Ee 2) ——— etnn age (2), we obtain the asymptotic expansion of Q,""(— z) when R(z)>0; writing — z for z, we see that, as in the case of P,(z), the asymptotic expansion of Q,""(z) has the same form whether R (z) be positive or negative; and from formulae already given connecting P,*™, Q,*™, it is evident that the same is true of P,~”(z) and Q,~”(z). The expansion of P,~”(z) 1s valid for a complete range of values of argm when R(z)>0 only; the other expansions are valid for an incomplete range, as is that of P,~™(z) when R (z) <0. The permanence of the form of the expansions when f(z) passes through zero might have been anticipated* from the fact that, when m is positive, (z+1)™ and (z—1)™ have the same modulus when R(z)=0, and that (as was pointed out by Stokes) discontinuities in asymptotic formulae usually occur in terms which are negligible in comparison with the dominant part of the expansion. Part III. MiIscELLANEOUS PROPERTIES OF LEGENDRE FUNCTIONS. 13. Definite integrals representing Legendre functions. We shall now obtain important and interesting definite integrals for P, (cosh £) and Qn (cosh €); they are derived from the formulae of §§ 3, 6,8 by putting a=1, 8=0, y=1,A=n; this substitution gives I, = ( fs 35 if ) (cosh €— t)~ 2”e"Se—* (dt/dr) dr, J @D JQ / t— cosh € where 7 =log ae log 2eS, r 2 — cosh so that dz __(#—1)(¢—cosh $) dt —2tcosh €+1 * The permanence of the expansions may also be seen F(n+1, —n; m+1; 4-42), from the fact that P,—”(z) is expressible in terms of which is of the form described in Part IV below as type B. Wor exXexenn, “No! XT Ve 38 294 Dr WATSON, ASYMPTOTIC EXPANSIONS Let t,t; be the values of ¢ corresponding to any assigned positive value of +, of which ¢, is on that part of the contour S, in the ¢-plane which joins the node to the point ¢=+1. Then, since 4, t, are the roots of the quadratic tf —1= 2e— (¢ — cosh €), we have cosh €=(1+4#,)/( +4), and so (cosh €— t,)~ (dt,/dr) — (cosh £— t.)+ (dt./dr) ib ta a fg St - t2—2ét,cosh€+1 t2—2t,cosh +1 EGG Ais h-t h-hh = des~/(t — t). Wor ty p= $(1—e7) (1— eS), where the upper sign refers to ¢, if arg (l—e%) > 0 astT> om. a —(n+1) Tr Hence we have Te 2 gts | a : -0 (l-e7?(1—e%)P provided* that 7 is not zero or + 7. In like manner E=2 Hees i = 5 0 (1—e7)? (1—e-)2 provided* that 7 is not +7. It is desirable to modify these results slightly, by observing that 1— e% = e& { —(1 +77) sinh €+ (1 —e™) cosh ¢} =e? ef {(1 + e-7) sinh £— (1 — e~*) cosh €}, according as J (£) 2 0, where arg (sinh ) lies between + 7. Similarly 1 —e-*%-* =e-$ {(1 +e") sinh £+ (1 — e~") cosh €}; and so we have the formulae re g—(n+1)r T= 2nrie M4HE | Aaa (1 +e-7) sinh £4 (1 —e~*) cosh €} “dr, 0 —e'p - 2 et (nt)r = T,=arrrettris ines [© 7 4 o*)sinh & — (1—e-*) cosh g} bar, 0 (1—e7")3 The first of these gives the definite integral for Q, (z), namely =e (+4) (2) =e ——— Qn (2) ke aan valid except when z+1 is real and negative; when —1<2<1, the mode of approach to the real axis has to be specified. = e7in+i)r {(1 +e7*) sinh € + (1 —e7*) cosh §} “4 ar, * Allowance can be made for the exceptional cases by a suitable indentation of the contour at the points where eT = eS, OF HYPERGEOMETRIC FUNCTIONS. 295 When z is on the real axis, however, we define Q,, (cos 7) as the mean of the values on either side of the cut; so that Q,, (cos 7) = $Qn {cosh (0 + %7)} + $Qn {cosh (0 —%)} ; since sinh (0 + 77) =e*2™ sin» when 0<7 <7, we see that, for values of » between 0 and 7, we have sf — (n+1) 7 : E a Qn (cosm) = gel" Din+dr | ~__{(] + e-*) sin 7 — i (1 —€“") cos n} ~ 4 dr Jo (1—e*)? Ants 5 [pat — (n-+1)F : ; +t¢e- (+ t)in-at | ne Se \(1 + e-7) sin n +7 (1 — e™*) cos nf 3 dr. ; Jo (l—e)3 Similarly, from the formula P, (2) = + U2.—L)/(2"" 771), we have oa — (n-+1) Tr Pe) = arate +4) 5 | eee {(1 —e-") sinh £—(1 —e~*) cosh §} ~t dr 0 (1—e-*) e aire — (n+1)T + mate (@F4ys fie ose \(1 +e-7) sinh €+ (1 —e~*) cosh ¢} —4 dr, 0 (1—e-*y3 provided 7 is not zero or 7. Making & — 0, we see that, if 0< <7, we have mi, (n td) eyiaa | eRe 563 eae 2 P,, (cos n) = 771 Pommeenn |! chen) Sin 7 (len) cos | 2dr. o (l-—e’p : - (2 e-(ntr 5 foarte (Bintan | 2 (1 +e-*) sin 7 +4 (1 —e-*) cos n} > Pdr. Jo (l1—e)3 These integrals are fundamental in the subsequent analysis. 14. Some properties of the zeros of Legendre functions. We shall now shew how to obtain roughly the positions of the zeros of P, (cos @) and Q, (cos 6). When 0 <@ <7, we see that, as r increases from 0 to «, arg {((1 + e-")sin @—7(1 —e7*) cos 0} varies monotonically from 0 to @— 32; and so arg {(1 +e77) sin @—7(1 —e7*) cos 6-3 varies monotonically trom 0 to 4a — 28. Hence, since e~"1)"/(1 — ee is positive, we see, by considering the definition of an integral as the limit of a sum, that the value of 7 ee {_ + e-7) sin 6é—-i(1 —e-*) cos 0} 3dr 0 (l—e*)2 arg pe g_ (n+1)r 2 | lies between 0 and 4a — 30. 296 Dr WATSON, ASYMPTOTIC EXPANSIONS We may consequently write [ eon ; (1 + e*)sin @ F t(1 — €™) cos OF tdr= ln Wer, /0 (l—e")? where W is positive and @ is an angle (depending on n and @) between 0 and i7— 40. We thus obtain the formulae Pe (cos 0) = Weos {(n+ 4) 0—47+0}, Qn (cos 0)=47 W cos {(n+4)0+47+ a}. Next it will be shewn that (n+4)@+@ is an increasing function of 6 when n is fixed. We hi piaks Foe tan {(n+4)0+47 + a}, and so ? = [(n+ 4) 0+ 44 + 0} =2r 1Q,(c0s 8) ee SP(can 6) ra / G {Py (cos 6)}2 + 4 {Qn (cos a) | ; But, by applying a well-known theorem, due to Abel*, to Legendre’s equation, we deduce that sin? @{Q,P,'—P,Q,’} is constant; and, on writing @=}7 and making use of the values of P,, (0), Pn‘ (0), Qn (0), Qn’ (0) given by Hobson, p. 469, and Barnes, pp. 121, 124, we get sina On Ore} — le Therefore J {n+ })0 +40 + 0} =2/(msin OW? >0, which gives the result stated. We can now obtain limits for the zeros of P,,(cos@); when 0<6@<4m, we observe that (n+4)0—47 + certainly lies between (k — 3) and (k + 4) 7, k being an integer, if (k—3) 7 <(n+3)0—4Fm, (n+ 3)0—t7+(t7—30) <(k+4)7, (4h -—1) 7 2k+1)a4 oo, In this range of values of @, cos {(n + $)@—427+.} has the sign of (—1)*; therefore, as 0 increases from (24 +1)7/(2n) to (4443) 7/(4n+2), (n+4)@—427+o steadily increases from a value between (k + $)7 to a value between (k+1+44)7; hence its cosine changes sign once and only once. Thus the only zeros of P, (cos @) in the range 0 <@<47 are in the intervals [(2k + 1) w/(2n), (44 + 3) 7/(4n 4 2)]; and there is one zero and only one in each of these intervals*. 1.e. if When 37 <@<7, o is negative, so that the inequalities are replaced by (k—4)m <(n+4) 0-40, (n+4)0-4da<(k+4)7; and we get one zero and only one in each of the intervals [(44 + 3) 2 /(4n + 2), (2h + 1) /(2n)] and none outside these intervals. The function Q, (cos @) can be dealt with in a similar manner; we shall not give the details as the reader will have no difficulty in constructing the analysis. * Crelle u. p. 22. See also Forsyth, Differential Equa- internal point unless n is an odd integer, in which case the tions, $65. The dashes denote differentiations with regard _ corresponding interval is evanescent, and we have the known to cos @. result that P,, (0)=0. + None of these intervals can have the point 47 as an OF HYPERGEOMETRIC FUNCTIONS. 297 Part IV. ASYMPTOTIC EXPANSIONS OF A SYSTEM OF HYPERGEOMETRIC FUNCTIONS. 15. The system of hypergeometric functions with large constant elements. We shall now determine the asymptotic expansion of any hypergeometric function in which — one or more of the constant elements is large, provided that, when more than one of the constants is large, the ratio of the large constants is approximately +1. The Jacobi-Tchebychef functions discussed in Part I are the most obviously important functions of this nature, but others seem to be of sufficient interest to justify the very brief account which we shall give. The functions which will be considered are of the form F(a+er, B+ear; y+e,A; 2), where a, 8, y, w are assigned, || is large and e,, &, €; have the values 0, + 1. There are obviously 27 sets of values of (€,, €, €;), but of course the set (0, 0, 0) has to be omitted; and nine other sets may also be omitted on account of the symmetry of the hyper- geometric function in its first two elements; thus (1, — 1, 1) is effectively equivalent to (— 1, 1, 1). We shall take the hypergeometric equations associated with the surviving seventeen functions and obtain asymptotic expansions of a fundamental pair of solutions of each equation. It will appear that the equations fall into four distinct types, according to the values of (€,, €2, €,) shewn in the following table : Case & €5 €3 | ypemn| | | | it 1 -l 0 ‘) 2 1 1 0) We uN 3 -1 Jt 0 f 4 0) 0 d 5 0) 0 al 6 0 a 0) 7 0 —] 0 8 | 0) il 1 B 9 0) =I =| 10 il 1 1 1] -1 fen— 1 —1 lee? 0 1 —1 13 (0) —] 1 14 1 =a ] | c 1403} 1 —] -—1 | 16 1 | 1 = Il ‘| 17 -1 - 1 J = | 298 Dr WATSON, ASYMPTOTIC EXPANSIONS 16. Hypergeometric functions of type A. The reader will have observed that the functions of type A are those already investigated in Part I of this paper, in yiew of the fact that the function of case 1 is F(a+A,8—2; y; 2). We shall merely give a table, indicating the nature of the order of magnitude of the constant elements in the twenty-four hypergeometric functions connected with the equation which is satisfied by F(a+x, 8-2; y; 2). By expressing any one of these functions in terms of the two funda- ‘ mental integrals J, and J, introduced in Part I, the asymptotic expansions of the twenty-four solutions are at once obtained for a range of values of arg \ greater than the half-plane |argA| <7; for values of X outside this range, we put X=—A,, and then we obtain the asymptotic expansion of the function under consideration in terms of ),. The complete set of functions of type A is given in the following table, the numbering of the solutions being that adopted by Forsyth, Differential Equations, §§ 120-121; the first three columns in the table give the coefficients of X in the corresponding elements of the hypergeometric functions connected with the solutions shewn in the fourth column. Coefficients of X Functions | Case | 1 -1 0 | —(VIIT) 1 1 1 0 | (XVII), (XTX), (XXT), (XXIV) | 2 | —1 -1l 0 (XVIII), (XX), ( (XXII), (XXIII) 3 1 1 2 (IX), (XID), (XII), (XV) — =i = 2. | (X), (XD, (XIV), (XVI) | — The simplest method of procedure is to express the function to be investigated in terms of the two fundamental solutions (I) and (IX) by means of the formulae given by Barnes, Proc. London Math. Soc. (2) v1. pp. 141-177, and then to use the equations connecting these solutions with the integrals /, and J,. 17. Hypergeometric functions of type B. This type consists of the twenty-four functions associated with the equation for which solution (I) is the function F(a, 8: y+; x); the coefficients of X in the constant elements of the twenty-four functions are as follows: Coefficients of \ Functions Case 5 0 0) | 1 | SD ar | 4 0 0 1 (QUAYS 5 0 l 0 (XI), tern), (XIII), (XIV) 6 0 —] 0 vee (X), (XV), (XVI) 7 0 ] l (2 X VIN, (XVIII), (XXIII), (XXIV) 8 0 —1 —] (XTX), (XX), (X XI), (XXII) 9 ] l l (11), (VII) 10 | 1 = oF (III), (VT) 1 4) OF HYPERGEOMETRIC FUNCTIONS. 299 The two solutions which will be regarded as fundamental are the solution (1), namely F(a, 8; y+; 2), and (VIII), namely 21-7 (1— a)Y*4-2 8 F(1—a, 1-8; y+X—a—B+1:1—-2); it is evidently sufficient (since both are included in case 4) to obtain the asymptotic expansion of one of these functions. The reader will easily verify that these solutions form a fundamental system when || is large. We now investigate the asymptotic expansion of F’(a, 8; y+X; x); it will be found that, in the case of this function, the saddle-point, which is usually characteristic of the method of steepest descents, does not appear in the analysis. 1 We take the integral* J,=] ¢?7(1—t)1+*-#> (1 — at)--dt, 0 and we observe that (when 2 is positive), (1 —t)* decreases steadily from 1 to zero as ¢ describes the path of tegration. Accordingly, writing 1 —t=e-", we have Ths =| (a emnrates a?) (I a+ ae*)-*) es der. é Now, when 7 is sufficiently small, we have (1 —e=*)P e+?) (1 — a + we-*)-* = 78 & 78, s=0 where k= 1. Hence, when R(@)>0, we have Iw SV (8 +8) kg[rPt3: 0 S= the singularities of the integrand are at the points t = 2n7, 2n7i + log (1 — 2), and so, as in § 8, the expansion is valid over a complete range of values of argX when |1—«2—|<1, and overa certain incomplete range (greater than a half-plane) when | 1 —a#7?|>1. Diy+aA) 24,0 (B+) L(y+rA—B)AP eo T(R)AS When &(8) <0 this result may be obtained by taking a Hankel-Pochhammer contour (x; 0+) in the t-plane in place of the real axis. Hence F(a, 8; y+; 2)~ The expansion on the right may be obtained formally by taking each term of the series for F(a, 8; y+X; «), expanding in descending powers of A, arranging the sum in powers of \, and multiplying by the expansion of I (y+A—)/E' (y+) in descending powers of A. 18. Hypergeometric functions of type C. This type consists of the twenty-four functions associated with the equation for which solution (I) is the function F(a, 8+; y—2; x); the coefficients of X in the constant elements of the twenty-four functions are as follows: * If «>1, an indentation has to be made at ¢=1/x in the path of integration. 300 Dr WATSON, ASYMPTOTIC EXPANSIONS } Coefficients of X Functions Case = = al (I), (IX) 12 (IV), (XD) 13 (XIV), (XIX) 14 (XV), (XVIIT) 15 (11), (XII) = (,() | | V), (&XT) — (VIIT), (X-XITT) — (XVI), (XX) = (XII), (XVII) a (VI), (XXT1) = [Avi (XXIV) — | lel | | | ol OO PSs POD DOOR OS) oe i mae COC OR Re RK OCS | The two solutions which will be regarded as fundamental are the solution (V), namely F(a,B +r; a+8—y+2rX4+1; 1—2), for all values of 2, and* (IV), namely wi-y*A (1 — v)y-*-8-* F(1—a, 1-8-2; 2-yH+A; 2), inside the circle | #|< 1. We take the integral f(— 08> (L—#)r #29 (1 — at) det. Writing the integral in the form Si | (— t)8 (1 — t)y-8> (1 — at)~* exp > log ast dt, we find that the contour to be taken is given by the equation I(r) = I log {— 4t/(1 — #)} = 0. This curve consists of the negative part of the real axis, together with the circle |¢ =1. The saddle-point is t= — 1, and 7 vanishes there. We therefore consider the integral+ 0 t a Nae ae (1 — at\—* ex I, tex #81 (1 — YF (1 — at) exp {Alog gy aa ae It is easy to shew that P(B+A)T (a—y+rA+1) T(a+B— y+2rX+4+1) when the «-plane is cut along the negative real axis. If g, is the coefficient of Z** in the expansion of 2 -s—4 2% (1 — 7')8- 11 -47)r- B— ‘(1- =) {plo (1+5-7p)} F i F(a, 8B+XrX; a+B—y+2X4+1; 1-2), l+a Fe 4—47 j we find in the usual manner that gs U(s +4) I,w2-P-A(142)-*(n/ryi S B_T2 ( Sle Bienes ar e's 5) * In part of the plane (viz. | z|>1) we take (XI), namely + The integrand does not converge at t=1 when al -¥*A(] — x)¥—9-B-2A Bd gy —a—ds1-a+8+A31/2x), | 8"8\|<4n, and hence we do not take the circle as contour. instead of (IV). OF HYPERGEOMETRIC FUNCTIONS. 301 It is found that the only finite singularities of ¢ gua function of 7 are at the points in the t-plane for which ¢=—1; these are the points T=2s77. The finite singularities of the other function si Pg : : — 4a are at the points in the r-plane at which ¢=1/z; these are the pomts t= 2s7i — log (., 1 2s7r1. One of these points is on the real axis if |z|=1 or if w is negative; and one iG the points approaches the origin if, and only if, 7>—1; hence the expansion holds for an incomplete range of values of arg except at ec=—1; and it holds for a complete range of values of arg’ when 4g |>|a—1 |. The domain of the complex variable # for which this inequality holds is shaded* in Figure 7. 7] W/ Eic. 7. When «=~—1 the expansion assumes a different form, since (1 —«t)~*, when expanded in ascending powers of 7, has its leading term 7 #* instead of (1+.2)-*; it is easy to make the necessary modifications in the analysis. Next take the integral T= @-1*4 (1 — x) 8 j (= £)-8-A (1 = 8-17 (1 — at) dt. The method of steepest descents gives the same potential contours as in the case of J;, but now, in order to secure convergence, we take the circle ¢ =1 (taken counter-clockwise starting from t=1) as contour; if | “|< 1, by expanding in ascending powers of «, we get T(1+8—y+2a) T(2—y+a)T(8+2) i Bint * (1 — 7) ¥aeabaes F(—a, 1-8-2; 2-—y+A; 2). If | z|>1, however, by expanding in descending powers of a, we get 7 f*. a—l T= at-¥* (1 — a yr-e-8- | f)e-#1-a (1 tft (a - “) dt T(1+8—y+2a) TN(l+a—y+A)Pd—a+f8+r) x Fl —a,y—a—aA; l—a+f4+%; 1/2). SS atin (1 ae) es * This is not a scale drawing but it affords a rough the points where the curve meets the real axis are —1, indication of the region in which the inequality holds. 0-172, 5-828. The small loop should be drawn very much smaller, as Vou. XXII. No. XIV. 39 302 . Dr WATSON, ASYMPTOTIC EXPANSIONS To obtain the asymptotic expansion of J,, we write (l—t)/(— 4) =e, and then, if g,’ is the coefficient of 7* in the expansion of k ey » al fi coat (-)F 2 a7" a3 (1- SE slog (1+ gay saat) ; we get r —y+1+9. al pl—y+ ee 3 ge VED) ) Tyme 19F-1 (Lf aa Layee tmp BS The domains of values of # in which this expansion holds for a complete range of values of argX are the unshaded portions of the plane in Fig. 7. 19. Hypergeometric functions of type D. This type consists of the twenty-four functions associated with the equation for which solution (I) is F(a+2, 8+; y+3X; z); the coefficients of A in the constant elements of the twenty-four functions are as follows: | Coefficients of X | Functions | Case | ; | (IX), (X) 1 0 OF 8) CX), Catt) OF FST), EV) 0 | (XV), (XVI) 3 3 1 1 3 © Pees (Tr) it oe = = yy (LET) (ie al | el ae =3 | (IV) = | Eg HCV) 16 a) ts) Baan rs [ote (vn) aot a | (VIII) won (XVI), (XVILD) eXEX), (EX) | (XXI), (XXII) | (XXII), (XXIV) TS OS CS CS CS CS CS CO a COT ll cell Ot Oe | ee OO Ol a Od Od The solutions which will be taken as fundamental are (1), viz. F(a+A, 8+2A; y+3r; 2), and either (X), viz. a F(B +X, 8S—y—2A4+1; B—at+1; 1/2), or (XXIV), viz. wb-y- (] — x)y-2-8 +4 FLT — B—-2A, y—-84+2d; y-—a—B+A41; (e—-1)/a}. The analysis is somewhat similar to that employed in Part I; we take the integrals* 5 = I;, I,= | PASE (Li — UN) alae (1 — at)787A dt. * It is supposed that, near t=0, | argt|< and arg(1-t) and arg(1-2t) are small. OF HYPERGEOMETRIC FUNCTIONS. 303 The path for J, is reconcilable with the real axis without crossing over the singularity t,= 1/2; the path for 7; passes above or below this point according as the point is above or below the real axis. It is readily verified that _TW@-B+2a)P(B +) f T (7 +3) 3S F(a+nr, B+r; y+ 3A; 2), P(8+r) T(a+rA) Td —a+ fp) xF(B+2, B—y—2041; B—atl: 1/x) +e? T., where the upper or lower sign is taken according as J (x) 20, and it is supposed that | arg.) <7. Tie =+ Qariermlata) m-B-A In order to employ the method of steepest descents we have to determine the stationary points of (1 — at) t~'(1 —t)~* qua function of ¢; they are given by the roots of the quadratic 2at? — 3t+1=0. Put 9— 8x =z, it being understood that |argz <7 [the cut from «=1 to r=+ x in the «-plane insures this inequality being satisfied if we define arg (9 — 8x) to be zero when x = 0], and the stationary points are t, = 2/(8 + 4/2), ty = 2/(3 — /2). The values of (1 — zt)“ (1 — t)~? at 4, t are respectively e(V2+3R(V2+1), (2-3) (V2 —1). We shall now discuss, by electrical methods*, the topography of the contours in the ¢-plane (for all assigned values of z) which are supplied by the application of the method of steepest descents. 1—at #1 —?) where V and W are real, it is evident that V is the potential at the point t due to a two- dimensional electrical distribution consisting of line charges through the points 0, 1, 1/x in the t-plane, the charges per unit length of the lines being in the ratios 1:2:—1. If we write =el iW, The points ¢,, ¢, are the only points of equilibrium; and the curves on which V and W respectively are constant are the equipotentials and the lines of force. By straightforward algebra it is seen that the equipotentials are bicircular sextics and the lines of force are portions of circular quartics ; the quartics pass through the points ¢=0, 1, 1/2, and have a node at f=1 and two real perpendicular asymptotes; the curves on which W is constant consist of portions of the quartics with end points at the points 0, 1, 1/#, »; and it is these portions of quartics (ending at the points 0, 1 only, when R(A)>0, to secure the con- vergence of 7; and J;) which are required by the method of steepest descents. We can now consider the topography of the different equipotentials obtained by varying V from + 2 to —cx. * I should have preferred to have employed the algebraic (in general) only two nodes, they are not unicursal curves, methods of Part I in discussing the forms of the contours and so the algebra appeared intractible. The investigation instead of this combination of geometrical and electrical actually given is, I think, quite rigorous. theories; but as the contours are portions of quartics with 304 Dr WATSON, ASYMPTOTIC EXPANSIONS When V is large and positive, the equipotential consists of two small ovals* surrounding the positive charges at 0,1 respectively. As V decreases, the ovals increase in size, until we reach an equipotential through that one of the equilibrium points at which the potential is higher; this equipotential has a node which may arise in one or other of two ways, (I) by the two ovals uniting to form a figure-of-eight or (II) by previously distinct parts of the same oval bending round towards one another and uniting. Case (I) is shewn in Fig. 8 and case+ (II) in Fig. 9. Fic. 9. First take case (1). As V decreases further the equipotential becomes a single oval sur- rounding the figure-of-eight and this form persists until we reach the equipotential through the equilibrium point with lower potential; the node at the equilibrium point can only be formed by distinct parts of the oval uniting to surround a portion of the plane not previously enclosed ; this area haying a portion of an equipotential as its complete boundary must contain a charge; this can only be the charge at t,=1/#. Subsequent equipotentials consist of two ovals, a large one surrounding all three charges and a small one inside the former surrounding the charge at ¢, only. * The word oval is used, in the absence of a more suit- could join up with itself; but, as will be seen later, this able term, to mean a closed branch of a curve without nodes phenomenon does not oceur when the charges have the or cusps; it is not supposed that the branch has no inflee- _ proportions of those under discussion. Figs. 8 and 9 are tions. not drawn to scale, but merely indicate the general topo- + It might have been anticipated that the left-hand oval graphy of the plane; the dotted curves are lines of force. OF HYPERGEOMETRIC FUNCTIONS. 305 Next we take case (II). As V decreases further, the equipotentials become tripartite, consisting of an oval round the charge at 0, another round the charges at 1 and ¢,, and a third, inside the second, round the charge at t, only. This form persists until the first two unite at the remaining equilibrium-point to form a figure-of-eight ; and subsequent equipotentials are bipartite, consisting of a large oval round all three charges and a smaller one round the charge at t, only. If now we regard # (and therefore z) as a variable, the transition from case (1) to case (11) can only occur when z passes through a value which makes the nodal equipotentials coincident ; ze. when z satisfies the equation | (v2 + 3)?) _ (V2 = 3)*| eee \fz— 2 |" The curve in the «-plane on which this equation is satisfied is shewn* in Fig. 10; the simplest form of the equation of the curve is obtained by writing z= re (r>0, —7<0<7), when the equation of the curve reduces to r= 6/3 cos }@—9, together with the coimcident rays cos $6 = 0 (.e. 0= + 7). sh et Fie. 10. It is easy to see that when « is outside the curve of Fig. 10 and fairly near the origin (so that /z is comparatively nearly equal to 3), the potential at ¢, is higher than that at ¢,. And, when || is very small the charge at t,(=1/z) has little influence on the form of the equi- potentials moderately near 0 and 1; and so the equipotentials moderately near 0 and 1 have nearly the form which they would have if the only existing charges were at 0 and 1. Hence, when « 1s outside the curve of Fig. 10, the equipotentials have the configuration of case (1); and the node of the figure-of-eight is at ,, while the node of the other nodal equipotential (which may be described as a closed crescent) is at tp. When |#—1| is small, so are |t,- 1 and |z—1); and, if we consider the special case in which ¢,—1 is positive+, the equilibrium point ¢, is on the right of ¢, and the potential there, viz. log {4 (3 — /z)3/(4/z — 1)}, is much higher than the potential at ¢, and ¢, is near the point 4; hence we have the configuration of case (II) as shewn in Fig. 9; and so we have the configuration * Fig. 10 is drawn to scale; the dotted curve is + So that x is just less than 1, and z is just greater r=9-—6,/3|sin 36 , which will be required subsequently. than 1. 306 Dr WATSON, ASYMPTOTIC EXPANSIONS of Ftg. 9 whenever « is inside* the curve of Fig. 10. It is to be noted that ¢, is always the node of the figure-of-eight. [As a confirmation of these results, take | | large, when the equilibrium points are both near the origin ; when we take = positive, ¢, is nearer the origin than /,, and the potential of ¢, is higher than that of ¢,; thus we obtain the configuration of case (1). If now we vary arg z from 0 to +7, keeping _2| fixed, ¢, describes an are round the origin, driving the node of the figure-of-eight round the origin in front of it with approximately half its own angular velocity. When argz becomes +7, the two parts of the figure-ofeight unite behind ¢,, and we get a degenerate equipotential with two nodes, near the imaginary axis, which are symmetrically placed with respect to the real axis. ] We shall now shew that part of the line of force through t, always passes from 0 to 1 and is reconcilable with the real axis without crossing over ts. First take the configuration of case (I). The line of force through ¢, has a node there, and one of its branches (in the neighbourhood of ¢,) lies inside the loops of the eight. This branch of the line of force cannot emerge from the loops of the eight before passing through an equilibrium point, and no such point exists. The line of force therefore terminates at 0 and 1. Further, when |) is small+ the line of force nearly coincides with the real axis and ¢, is at a great distance from the origin, while ¢; and ¢; are on the opposite sides of the real axis. Therefore the line of force is reconcilable with the real axis. Now vary 2, and we see that the line of force remains reconcilable with the real axis so long as ¢, does not cross the line of force or the real axis. But as « varies, t; cannot cross the line of force without entering the figure of eight, i.e. without « crossing the curve of Fig. 10; and since the variation in # may be supposed to take place without « crossing the real axis (since the initial value of «, with | «| small, may have a positive or negative imaginary part, as we please), we see that whatever be the position of « (outside the curve of Fig. 10), a line of force passes from 0 to 1 through ¢,, and this line of force, so far as the point ¢, is concerned, is reconcilable with the real axis. Next take the configuration of case (II). In the special case when 1 —« is positive, t; is on the right of 1, and one branch of the line of force through ¢, passes straight from 0 to 1. As we vary « continuously the form of the line of force through ¢ varies continuously except when « passes through such a value that the line of force has another node; but the quartic of which a line of force forms part can only have two nodes (other than the point t = 1) if ae (Vz+ 3) _ ang (vz — 3) Vz+1 Vz—1 is zero or an even multiple of 7. Now this difference is a multiple of 7 only when r=9+6/3sin }6, or when sin }@=0, where, as previously, z= re (r >0,-—2<@<7); the difference is an odd multiple of on the branch r=9 +6 /3|sin $6| near z =9, and so the difference is an even multiple of + only when sin $¢=0 or r=9—6/3 sin $@; this curve is shewn by a dotted line in Fig. 10, and it lies wholly outside the curve r= 6/3 cos $@0—9. It follows that, as varies inside the continuous curve of Fig. 10, the general configuration of the branch of the line of force from 0 to 1 through ¢, does not change, but always les inside the figure-of-eight and * And so no ease arises in which the closed-crescent justifies the statement made in footnote +, p. 304. equipotential contains the charges at 0, 1/a only; this + Whether I (x) be positive or negative. —— OF HYPERGEOMETRIC FUNCTIONS. 307 does not go near* t;. Also, as we may suppose that ¢, and ¢, do not cross the real axis, and as the branch of the line of force is reconcilable with the real axis, so far as t, is concerned, when I (1 —2) is very small and R (1 —~) is small and positive, it follows that, for all positions of « inside the continuous curve of Fig. 10, the branch of the line of force under consideration is reconcilable with the real axis. It is now a simple matter to apply the method of steepest descents to obtain the asymptotic expansion of J,. Writing (1 — at) (7 (1 — t)* =2 {24+ 3) (V¥z4 1} e, where T-1s positive, we get a {8 v2 +1))* _,, dt fs (J: +{") Eo ceage y 8CL = 2), ( (vz +38)3J Ce dr and dt/dr is expansible (near t=0) in a series of ascending powers of 7* commencing with a dr, term in t~# whose coefficient is £4274 (2+ 1)/(v2 +3); and hence, after the manner of Part I, we obtain an asymptotic expansion for J;, in descending powers of X, of which the dominant term is given by the formula PA se. am oy, (8 (2+ 1))4 (or\3 Tw 2228-1 2 * (/e+1)-2 8 (V/2z4+3) en (= This formula is valid for a complete range of values of argd provided that R(7) is negative when t=; te. provided that the potential of ¢, is higher than the potential of t,. Consequently the formula is valid for a complete or for an incomplete range of values of arg according as x 1s outside or inside the continuous curve of Fig. 10. We shall finally shew that a branch of the line of force through t, either starts from 1, encircles t,, and returns to 1; or else it starts from 1 and ends at 0. The former is the case when « is inside the dotted curve of Fig. 10, and the latter when ~ is outside it. First suppose that « is inside the continuous curve of Fig. 10; then the equipotentials have the configuration of case (II). Consider the branch of the line of force which enters the horns of the closed crescent at f,; it cannot cross the boundaries of the crescent without passing through an equilibrium point, and no such point exists; hence both ends of the branch must terminate at the point 1. Now the configuration of the lines of force only alters when « crosses the dotted curve of Fig. 10. Hence, whenever « is inside the dotted curve of Fig. 10, there is a branch of a line of force which starts from 1, goes to t,, and returns to 1, obviously encircling ¢,, which is in the region surrounded by the crescent. When « is outside the continuous curve of Fig. 10, the closed crescent contains the point 0 as well as 1; and the only possible change of configuration of the line of force is th» ne of its ends+ should be at 0 instead of both being at 1. _ and of Clasi. > * It can only go near t, by assuming a form in which it sideration is zero, and since the cha passes through tj, and we have justseenthatitcannotassume curve, it would have also to contain this form when 2 is inside the continuous curve of Fig. 10. at 0 (these charges being numeric + Both ends cannot be at 0; for suppose the line charges in sign). Therefore the quartic (991-949, replaced by surface distributions on circular eylinders of forms part would have a cusp at ( very small radius; since, by Gauss’ theorem, the total charge _it from 0, it would bifurcate befor inside the (closed) branch of the line of force under con- neither of these events actually 40 R merican Journal of Mathe- : the Carnegie Institution of vis er 308 Dr WATSON, ASYMPTOTIC EXPANSIONS OF HYPERGEOMETRIC FUNCTIONS. To see that the branch of the line of force has one end at 0 and the other at 1, when z is outside the dotted curve of Fig. 10, take |x| large (greater than 9/8 is sufficient) and consider the limiting case when « is positive, so that argzis #7. In this case the nodal equipotentials coincide and form a curve consisting of two ovals crossing one another at 4, & (which are con- jugate complexes); the left-hand oval contains the charges at 0 and 1/#, and the right-hand oval contains the charges at 1/z and 1. And obviously the branch of the line of force through t has its ends at 0 and 1; hence, whenever R (# — 9/8) is positive and J (2) is very small, the line of force must pass very near 0; and so it must actually have its end at 0, in view of the manner in which lines of force radiate from the charge at 0. | Hence, whenever « is outside the dotted curve of Fig. 10, a branch of a line of force passes from 0 to 1 by way of ¢,; moreover ¢; lies in the region between this curve and the real axis ; . for t, is in the region surrounded by the closed crescent, and is consequently inside the region bounded by the line of force and any curve joining 0 and 1 and lying wholly inside the crescent ; | and, since ¢, and ¢, are on the same side of the real axis, the curve just mentioned is reconcilable | with the real axis so far as f, is concerned. Hence, when ~« is outside the dotted curve of Fig. 10, we get Ae E BN Zy) T, wv 2*8 (— /2) ed — vz)yreF (3 — zy Gees ae G a and it is easy to shew that —/z, 1-1/2, 3—¥/z have to be taken to have their arguments numerically less than 7. If, however, « is inside the dotted curve of Fig. 10, the function which possesses the asymp- totic expansion of which the dominant term has just been written down is (1/z=) fs #8#A-2 (1 — tyr 842-1] — at)-= dt, BL where the contour is described counter-clockwise or clockwise according as J (x)20. By writing t= 1 —u(#—1)/a, this is easily found to be T(y- B+ Qn) et? — a-B+A— = WC oem ECE She + Qari — fp YR pre ae x F(y—8 +24,1-8-2; y-a-B+r41; *—), x and so we get wT (y — B + 2) —a—B+A »8—y—2A Ce soy a CE ae orp ba it ee = 2 eae y 8 We=1 ™ Na where z=9—8a. By considering the potentials of ¢, ane t., we see that, when the first type of asymptotic expansion (viz. that involved in J,) is valid for a complete range of values of arg A, the second type is valid for an incomplete range, and vice versa. XV. Asymptotic Satellites near the Equilateral-Triangle Equilibrium Points in the Problem of Three Bodies. By Professor Danie, BucHANAN, Queen’s University, Kingston, Canada. [Received 30 March, 1918. Presented by Professor Baker.] 1. INTRODUCTION. If two finite bodies are subject to the Newtonian law of attraction and move in circles about their common centre of gravity, then there are five points, as Lagrange has shown*, at which an infinitesimal body would remain fixed with respect to the moving system if it were given an initial projection so as to be instantaneously fixed with respect to the finite bodies. Three of these points are situated on the line joining the finite bodies and these are called the straight line equilibrium points of the problem of three bodies. The remaining two points are situated at the vertices of the equilateral triangles having the line joining the finite masses as base. These points are called the equilateral-triangle equilibrium points of the problem of three bodies. If the infinitesimal body is given a slight displacement from one of these points of equili- brium, and initial conditions are so determined that it will move in an orbit which is closed relatively to the moving system, it is called an oscillating satellite. If the infinitesimal body is disturbed slightly from an equilibrium point or from the periodic orbit about the equilibrium point, and initial conditions are so chosen that it will approach the equilibrium point or the periodic orbit, respectively, as the time approaches infinity, it will be called an asymptotic satellite. The orbits which are asymptotic to the straight ine equilibrium points were determined by Warren} in 1913. Those which are asymptotic to the periodic oscillations about these equili- brium points have been determined by the author of the present paper and appear in another memoir. [Proc. Lon. Math. Soc., vol. xvu. (1918), p. 54.] The periodic orbits to which they are asymptotic are the orbits of Class A and of Class B as determined by Moulton in chapter Vv. of his Periodic Orbits. * Lagrange, Collected Works, vol. vi. pp. 229-324. Problem of Three Bodies,’ dmerican Journal of Mathe- Tisserand, Mécanique Céleste, vol. 1. chap. v1. “Moulton, matics, vol. xxxvm1, No. 3, pp. 221-249. Introduction to Celestial Mechanics (New Edition, chap. yut.). } Publication No. 161 of the Carnegie Institution of + Warren, “A Class of Asymptotic Orbits in the Washington. Wor exc. No. XV. 40 310 Pror. BUCHANAN, ASYMPTOTIC SATELLITES NEAR THE EQUILATERAL- The object of the present paper is to make the discussion for the equilateral-triangle points of equilibrium which corresponds to the two papers mentioned above. The periodic orbits of the oscillating satellite which are approached by the asymptotic satellite are those determined by Buck *. Two classes of periodic orbits are determined in Buck’s memoir. One class of orbits is of two dimensions and lies wholly in the plane of motion of the finite bodies. This class exists only when one of the finite bodies is relatively small in comparison with the other body. The other class of orbits is of three dimensions, but there is no restriction as to the relative masses of the finite bodies. The treatment of the problem under consideration is divided into two parts, Part I being devoted to asymptotic orbits of two dimensions, and Part II to asymptotic orbits of three dimensions. The orbits considered in Part I are asymptotic to the equilibrium points themselves and not to the two-dimensional periodic orbits about these points. These asymptotic orbits exist only when the masses are more nearly equal than in the case of the two-dimensional periodic orbits. It is therefore doubtful if orbits exist which are asymptotic to the two-dimensional periodic orbits. The three-dimensional orbits considered in Part II are asymptotic to the three-dimensional periodic orbits. The same restriction as to the relative masses of the finite bodies must be applied here as in Part I. Only the formal constructions of the asymptotic solutions are made in this memoir. It has been shown, however, by Poincaré+, that if certain divisors which appear in the construction of such solutions do not vanish, then the solutions will converge for all values of the time ¢ Now t can occur explicitly in the solutions only when such divisors vanish and, further, if ¢ does not occur explicitly then these divisors are different from zero. Hence, if the solutions can be constructed so that t does not occur explicitly their convergence is assured, by Poincaré’s theorem. 1 Bay Rad beet Two-DIMENSIONAL ASYMPTOTIC ORBITS. 2. THE DIFFERENTIAL EQUATIONS. Let the motion of the infinitesimal body be referred to a set of rotating rectangular axes £, , €, of which the origin is at the centre of mass of the finite bodies, and the &y-plane is the plane of their motion. The £ and 7 axes rotate in the same direction as the finite bodies and with the same angular velocity. The masses of the finite bodies will be denoted by « and 1—y, * Buck, ‘Oscillating Satellites near the Lagrangian chap.1x. This paper will be cited as ‘“‘ Oscillating Satellite.” Equilateral-Triangle Points,” Moulton’s Periodic Orbits, + Mécanique Céleste, vol. 1. p. 341. TRIANGLE EQUILIBRIUM POINTS IN THE PROBLEM OF THREE BODIES. 311 and the notation chosen so that «<4. The distance between the finite bodies will be chosen as the unit of length, and the unit of time will be so chosen that the gravitational constant is unity. If the coordinates of the imfinitesimal body are denoted by &, », and & and if derivatives with respect to ¢ are denoted by primes, then the differential equations of motion for the infi- nitesimal body are* oU oU 0U TIAL > thy Se a ” 2 fy Se Uf ep aed E =?) dE? + 2& on ? g ale * 7 Doi, eee ee Oe cose (1) Tn (SA) eee sy (+7) PAS Fy p, and p, being the distances from the infinitesimal body to the bodies 1 — 4 and yp respectively. The points of equilibrium are the solutions of the equations+ Uae 30 0 OF On OC There are two sets of points which satisfy these equations, but those with which we are concerned in this paper are I &=$-p, m=+4Vv3, &=0, IL &=4-p, m=-4V3, 6 =0. These two points lie in the rotating plane and at the vertices of the equilateral triangles having the line joining 1 —y and w as base. Obviously, the coordinates of the points differ only in the sign of V3. The asymptotic orbits will be discussed only for the point I, for on changing the sign of V3 we may obtain the corresponding results for the point IT. Let the origin be transferred to the point I by the transformation Serre ENO EY, C= 2 me mace eeeeeensecsnconass so8 (2) Then the right members of the differential equations (1) can be expanded as power series in the new variables #, 7, and z. These expansions converge only up to the singularities of the functions 1/p, and 1/p,, that is, in the region which is common to the two spheres having their centres at the finite bodies and radii V2, but which excludes their centres. Let a parameter e be introduced into the differential equations by the substitutions THC THQVp BHC) coooccwoor coesscescoae ocsegedsoaccuobad (3) where x, y, and z are the new dependent variables. Then as a consequence of (2) and (3) the differential equations (1) become x’ —Qy! =X, + X.e4+X;,E+..., | yf + 2a’ = Y,+ Yoe+ Vee Se toe a =4,+Zoe+ Z.e+ Oy * Moulton, Celestial Mechanics, p. 280. + Ibid. p. 290. Charlier, Die Mechanik des Himmels, vol. xt, pp. 102-111. { “Oscillating Satellite,” equations (4). 40—2 312 Pror. BUCHANAN, ASYMPTOTIC SATELLITES NEAR THE EQUILATERAL- where X,=$[e+V3(1—2u)y], Yi=3[V3(1—2u) 2+ Sy], X,=4+-8,[7 (1 — 2p) 2+ 2V3ay —11 (1 — Qu) y+ 4(1 — 2p) 2"), Y, =— 3, [V322 + 22 (1 — 2p) cy + 3V38y? — 432°], Z4,=-2, Z,=3[(—2p)az+ V3yz], (5) X,= 35 [— 370° +. 75V3 (1 — 2p) ay + 1232y? +. 45 V3 (1 — 2) y3 — 1222? + 6V3 (1 — Qy) yz"), Y, =a, [— 25V3 (1—2y) 28+ 1230%y +135 V3 (1—2y) wy? + 3y° — 60V3 (1—2y) wz? +132y2"], Z, =— 3 [ate + ly%z — 42° + 10V3 (1 — 2p) yz]. The remaining X,, Y,, and Z, are polynomials of degree n in a, y, and z. — 3. THe CHARACTERISTIC EXPONENTS. For e=0 equations (4) become a” — 2 —32—3V3(1—2u)y=0, y + 2a’ —8V3(1— 2p) x—2y=0, done seecceevcntiocesccescessres (6) The first two equations of (6) are independent of the last equation and can be integrated by putting mehe,. y= Le. where KX and LZ are arbitrary constants. The characteristic equation for the determination of 2d is M+? + 224 (1—pw)=0, and the resulting values of \? are 2 —14+¥V1—274(1—p) Ss) ‘ ; y ~~ “I — b For small values of » the expression under the radical is positive and numerically less than unity, and therefore the four values of X are purely imaginary. The limiting value of w for which 2 is purely imaginary is that solution of 1—27n(1—p)=0 which does not exceed 4. This value is found to be = y= 0385... For « > the four values of X are complex. In order to construct asymptotic solutions of (4) it is necessary that at least one characteristic exponent shall be real or complex. Now in problems of dynamics in which the differential equations of motion do not involve ¢ explicitly, the characteristic exponents occur in pairs which differ only in sign*. Hence, if one value of 2 is real or complex, there must be another real or complex characteristic exponent which differs from the former only in sign. Further, if one pair of exponents is real or complex then y« 2 wo, and it follows from (7) that the other pair of char- acteristic exponents is also real or complex, respectively. In view of the fact that the two- * Poincaré, Mécanique Céleste, vol. 1. chap. tv. p. 69. TRIANGLE EQUILIBRIUM POINTS IN THE PROBLEM OF THREE BODIES. 313 dimensional periodic orbits exist only when « the characteristic exponents are o, — a, , and —o, where o=a+18, c=a—i8, ) a=$[V27(1— »)—1}, B=4[V27e0—4) + ry The quantities o and a, like most of the constants in the sequel, are conjugate complex. The notation adopted is such that a symbol having a stroke over it is the conjugate of the same symbol without the stroke. 4. CONSTRUCTION OF ASYMPTOTIC SOLUTIONS. We shall construct in (A) the solutions of (4) which approach zero as t approaches + 0. In (B) we shall show that the corresponding solutions which approach zero as t approaches — 0 may be obtained directly from those obtained in (A). According to Poincaré’s* definition of an asymptotic solution, each term must have the form EXP (i), where 2X is a constant and P is a periodic function of ¢. The solutions approach zero as ¢ ap- proaches — 0% or + © according as the real part of is positive or negative, respectively. (A) Solutions in e~*'. In making the construction of asymptotic solutions it is convenient, although not necessary, to transform into the normal form the terms of the first two equations of (4) which are independent of e. In order to obtain the transformation for the introduction of normal variables, it is necessary to know the solutions of the first two equations of (6). They are a= a,e" + ae + ase” + aye, } (9) pa OOleeti eo? bat + ae aT Tai ile where a,, ..., @, are the constants of integration, and b,, ..., b, are constants so determined that (9) shall satisfy the first two equations of (6). It is found that 8a —3V3 (1 — 2p) | 6, = ; | haa We Nel is te tual a Pe ECE ee (10) 5, = 82 + 3V3 (1 = 2p) AP 4a? —9 , Normal variables «,, 22, 2;, and 2, are introduced by the substitutionst+ B= I+ Lo+ y+ Ls, ) Yy = da, + bya, + b, 2; + b.x,, (11) uv =o (x, —%)+6(x3— 24), y =o (b,x, — b,2,) +E (b,3 — b.2,). * Poincaré, Mécanique Céleste, vol. 1. p. 340. + Ibid. vol. 1. p. 336. 314 Pror. BUCHANAN, ASYMPTOTIC SATELLITES NEAR THE EQUILATERAL- The differential equations (4) then become a —ox,= A,(X.e+ X,2+...)+B,(Yoe+ Yee ...), a,’ + oa, =A,(Xret+ XsE+...)+B(Yret+ Vse+-.)f (12) &y — Gr, = A, (X.e+ X,2¢+4+...)+ B,(Yoe+ Y3e+...), ay + on, = A,(X,e+ X,64+...)+ B,(Yoe+ Y3e+ a where A= ae, (b= 1,2, 34 eee ae (13) A being the determinant of the transformation (11), and A; the minors of the elements of this determinant, 7 referring to the row and k to the column. The computation shows that A a ee, B= 2, B= Be The equations (12) will now be integrated as power series in € by the method of undeter- mined coefficients. Accordingly let 2 = 2, + M+ ...+ ay @) aa faiats (k == Dee i (14) ede eas tee Then from (11) it follows that 2 and y are likewise power series in ¢ of the form BONE 9) =, yD — 2s Rs eee cseae sees ee (15) 5=0 j=0 where LI = 2,9 + 2) + 2) + 29, 1 (16) y) =b,a,9 +d, a) + bar) + boa.) When equations (14) are substituted in (12) and the coefficients of the same powers of ¢ are equated in the resulting equations, we obtain sets of differential equations which define the various coefficients of « in (14). In order that the solutions of these equations shall be asymp- totic we impose the condition (C,) that each term shall contain the factor e~*” or e~**, where k is a positive integer. This condition disposes of the two constants of integration which are associated with the exponentials e” and e. There still remain the two constants associated with the other exponentials e~** and e~*, and in order that these shall be uniquely determined we impose the conditions (C,) that “z=a, y=0, at t=0, that is, we suppose that the infinitesimal body is initially displaced from the point I at the distance a on the z-axis. When these conditions are imposed on (15) we obtain 2%(0)=a, #4(0)=0, (j=1,... 0), y2(0)=0, (GO .000 ). The terms which are independent of « when equations (14) are substituted in (12) are a —o2,9 =0, x," + o2, =0, r4 ae ou,” =— 0, ao + ox, = 0, See eesancccsesccecccccsccecns (17) 2 + 2 =(), The solutions of these equations are a0 = d, ©) get, 1 = d, eg, = dy e**, 7,0 = ad,” rat (18) Mae aii4 dmc: la Aa ee TRIANGLE EQUILIBRIUM POINTS IN THE PROBLEM OF THREE BODIES. 315 where d,”, ..., ds are the constants of integration. From condition (C,) it follows that all these constants are zero except d, and d,®. When the resulting values of (18) are substituted in (16) we obtain A = d, est er d, et y= bd, e-% + bd, | From conditions (C,) it follows that d,. +d,% =a, bod, + b.d, =0, then d.% =ad®, d, =d,. =ad™, where d” = i, = Ri With this determination of the arbitrary constants d, and d,”, the solutions (19) take the form 2O=@ [do ent + qd el, y =a[b.d e-* + b,d o The differential equations obtained by equating the coefficients of e in (12), after equations (14) are substituted, are a”) o) ox, = X,", Xo) ae ox," = Xo, fe EMOTO ee OA ce OV 2 NON caters ae rwerasare ens eas (21) 2/70) 4 20) —0, where X,) = a? [My et + MyM e- tt + Mya” e-™], Xi = a? [Nye + Ny, ett 4 Noo e-**], Xo =%.0, Xo =X, My” = A,M,+B,N,, No = A.M, + BoM; M.” = 4,M,+B,N,, No” = AM, + BoM), MM, = A,M,+B,N,, Ny = A.M, + BM,, M,=+ 3,2 (d (7 (1 — 2p) + 2V3b, — 11 (1 — 2p) 6], N, =— #02 (d [V3 + 22 (1 — Qu) b. + 3V3b."], M, = + Sad d® [7 (1 — 2) + V3 (by + b.) — 11 (1 — 2) baba], N, =—8a?d d® [V3 + 11 (1 — 2p) (b, + bs) + 8V3b,b,]. The constants 4,, A., B,, and B, have the values defined in equations (13). The general solutions of (21) are x, = dy er + a? [mo e- 2" + my" E— FF) + MQ C—™*], Iq = dg” e—% + A? [Mog C= 278 + ry MEX FE Mg BE, 40) = dy” 6 + a? [gg - 288 + My VE FFE F Ahgg C—F], bose eevenrenes (22) x," = d,™ eat +a? [Mig e—20t ora Ny) e~ e+2) ty Noy") e—™*], 20 =d," sint +d," cost, 316 Pror. BUCHANAN, ASYMPTOTIC SATELLITES NEAR THE EQUILATERAL- where d,"), ..., d;) are the constants of integration and a) a) () "leh _ Ms my") ie M,, : Mo” 2a Me - 30” Qo = Q+6 (23) a) a) @ ase a) UE o—— Nee Noo » My = 02 = co c 2a-—a From condition (C,) it follows that d, =d,® =d,” =d,” =0. Then, on substituting (22) in (16), we obtain 2) = d.™e—°t +d,%e—* +a? [An emt Aes once + Ag” e*], y = b.d,™ e—st = bd," e—* +a Be este + B® e—(oraty Bo™ Gmc. eae (24) Zz = 0, where Ay” = May + Noy™ + Moo™ + Tog"), A,” = My," + Ny, ae mM," + N,, By = by im” + dat)” + by Migs” ats baFioo™, By® = bom” + bry + bm," + bin”, Awe = Ay”, By” = Bay", A," = Ay”, By” = B,”. It is observed that the constants A,,") and B," are real. The only undetermined constants in (24) are d,” and d,). When conditions (C,) are imposed, it is found that these constants can be uniquely determined and that they are conjugates. Since they carry the factor a? let d.) =A”, dd =a@Aq”, bd," =a?B,", bod,” = 0B": then the solutions (24) may be eas in the form aM =a 5 AVe Getla)t Q SA uC e— Go+ks)t O Be —ictekoyt: «sa ER pRB ese Sel ae (26) jan ke 0 PA) = 0 h (I where A yr and Bie are constants such that Ul h) I Ae Bo Be If 7 #* these constants are pate” bata but if j= they are real. We proceed to show that the solutions have the same properties when h =v. TRIANGLE EQUILIBRIUM POINTS IN THE PROBLEM OF THREE BODIES. 317 The differential equations which define a,”, 2,”, 2”, and #,” are the same as (21) except the right members, while the equation in 2” is the same as in (21). Let the right members be denoted by X,”, X,”, X,”, and X,” respectively. These right members are functions of ax, ...,@*~, which are assumed to be known. They have the form x y+1 v+1 ‘ X=a | S > M™ e- iets) ( . g=0 x=0 ik vt+1 v+1 ={¥) y NG atta ee Ne Ge TENE Pee, coeee tach sur coeesen (27) j=0 k=0 7 x, = X,), x,” = X,”), 2sj4+khsv+l1, where the coefficients of the exponentials are known constants. Since j7+k22, the right members will contain no terms in e~* or e~*, and therefore the particular integrals will have the same form as the right members. The complementary functions are the same as at the previous steps, and therefore the complete solutions have the form v+l v+1 (v) ape \ x”) = d,”) ert 3: arti >» = Min e- Gat )t | j=0 k=0 | v+1 v+] () , XZ) = d.“e—% ate avr} >: s Nip e— Got kat | j=0 k=0 | vt v+1 : \ (28 2,” = d,”) et + qt Shs: m\”) e- (jo + ka) t | ) j=0 k=0 ' v+l v+1 (») | — \p—T sy 0) A — 0 = z,”) = d,” ett 4 qt > Sy Ty; e— Ja+ke) a | j=0 k=0 | z” =d,;”! sin t + d,” cost, ) where d,”’, ..., d;” are the constants of integration, and the remaining coefficients are known constants. From condition (C,) it follows that d,” =d,” =d,” =d,” =0. On substituting the resulting values of (28) in (16) and imposing conditions (C,) that a (0)=y” (0) =0, it is found that d.” and d,” are uniquely determined. As they contain the factor a”*' and are conjugate complex, we may put d) =a" Ay”, d=at1A,”, bd =a" Bo”, bd. =a"By™, in which case the solutions for 7” and y”) are of the same form as (26) if h=v+1 in (26). This completes the induction. Thus the integration of the differential equations (12) can be carried on to any degree of accuracy desired. It remains to be shown that the solutions which have been determined are real for real values of a. This will now be discussed. Consider a typical term AD es is Fee (0)