eae h Cale, (CRN MIE Sue sh pone ‘ AO mA, RAs AMER use aaa) ey Bas a4 CeNrenene te Sy sheey Sealy Pea * Fates I by easier cy é shige Ne i on : ss scoot ss ; es oar ; WABN E ; : =e Sur Y a a yee t Ne ESR Bead 5 9 ar ket ete Boar hore wear sare tei yt Waresa hae ‘ OMENS its ae ee prergetree try better S 9 res NIAAA eet Si Sed Ae ena nt wae NS Seat Tic aearicer hie ree Perey santero ar LaF anFap Bae 2 oP ao inte Digitized by the Internet Archive in 2009 with funding from University of Toronto http://www.archive.org/details/transactions19camb TRANSACTIONS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY. VOLUME XIX. CAMBRIDGE: AT THE UNIVERSITY PRESS, AND SOLD BY DEIGHTON, BELL AND CO. AND MACMILLAN AND BOWES, CAMBRIDGE; Cc. J. CLAY AND SONS, AVE MARIA LANE, LONDON. = ‘ m.pcceetv. — 404 ADVERTISEMENT. Tue Society 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. Tne Society takes this opportunity of expressing its grateful acknowledgments to the Synpics of the University Press for their liberality in taking upon themselves the expense of printing this Volume of the Transactions. Wil VIII. XVI. XVII. CONTENTS OF VOL. XPX. PAGE On Differential Equations with two Independent Variables. By A. C. Dixon, Se.D., Professor of Mathematics, Queen’s College, Galway .................ccscccececcneuenenenes 1 On the Calculation of the Double-Integral expressing Normal Correlation. By W. F. SERIO BAUR SAV eA re sue Miya aes elope cersteinn ie aeietslaisiamteiesictiels saiselSisincisatslnslcsie ais isjsisiaitiewn. gale duoade su ete 23 On the simplest Algebraic Minimal Curves, and the derived Real Minimal Surfaces. By Herperr Ricumonp, M.A., King’s College, Cambridge...........................05. 69 Diophantine Inequalities. By G. B. Matuews, M.A., F.R.S. ...........0.0.2.. cece eee 83 The Diophantine Inequality kx >py. By Major P. A. MacManon, R.A., D.Sc., PRES eee on se Viernes Gammon Nilay SOG eee eter ane ae eee mne orator see eserasicrois spake wis sion esedive 111 Rational Space-Curves of the Fourth Order. By Hersert Ricumonp, M.A., King’s (Collemena @amibrid re yay ee aes se se ece aes ce es saiean one tome tee alsin abswecmseebvestiss tear oteatas 132 On Divergent Hypergeometric Series. By W. M°F. Orr, Professor of Mathematics, ruoyals Gollememot Sciences Ub linicree: Jaa tanniesssceee ance cnassicececeeeeacre aceesicaes oe 151 On the Fifth Book of Euclid’s Elements (Second Paper). By M. J. M. Hitz, M.A., D.Sc., F.R.S., Professor of Mathematics at University College, London ............ 157 Some Problems in Electric Convection. By Gitpert T. Watker, M.A., Fellow of iRrintt ya Collemers Camlbrd 86 qc .<-5. 0st. ceac case racoceas -etinne ads c0coacns paabee bed eeeeesoensdope card RacnEsncnsedeo. 426 lL. On Differential Equations with two Independent Variables. By A. C. Drxon. [Received 27 November 1899.] § 1. In a paper published in the xcitird volume of Crelle’s Journal (pp. 188— 214), Herr Hamburger has given a method by which solutions of partial differential equations with two independent variables may be found if certain systems of total differential equations have exact integrals. The present paper contains an investigation of this method from another point of view which leads to some practical simplification. The number of systems of total differential equations used is equal to that of the independent variables. In Herr Hamburger’s paper each of these systems is supposed to have an exact integral without reference to the others. It appears now however that if any one of them has an exact integral this may be used to reduce the rest, and that if after the reduction another has an exact integral this may be used in the same way, so that when we arrive at the last system the exact integral desired is found by solving a system of ordinary differential equations. Although the scope of the method is thus increased it is even now by no means of universal application. The theorem on which it depends is however extended so as to hold generally. The result (§ 18) is theoretically interesting but it is doubtful whether it will be of great use in practice. A note is added on the relation of Singular Solutions to a certain determinant which plays an important part in the investigation. A discussion of Ampére’s equation Rr + 28s + Tt + U (rt — 8?) =V on somewhat similar lines was presented to the London Mathematical Society at their last meeting (Noy. 1899). (Proc. L. M. S. vol. XXxXI. pp. 3832—350.) §2. Taking a system of n total differential equations in n+3 variables, in the form dy; = pada, + pida, + pisdars (CSI, Bsco ft) bool Ul): where the quantities p are known in terms of a, a, #5, %,...Yn, let us consider how a solution consisting of n+1 integral equations is to be found, if one exists. Vou. XIX. Parr I. if 2 Mr DIXON, ON DIFFERENTIAL EQUATIONS Let $(a;, %2, 2s, Yi,--- Yn) =O be one of the n+1 integral equations; then when it is known, if it is not an exact integral, the other integral equations will be found by substituting in (1) for one of the variables, say a, its value in terms of the others as given by ¢=0, and then integrating; and the function ¢ must satisfy such con- ditions as will make the equations resulting from this substitution a completely integrable system. Let D,(a=1, 2,3) denote the operator Ci) Oia a. E Dies 0 . it (OY then by substituting from (1) in the equation dé=0, we derive ID Re E ASE ID jai hey SID KS SCHR Depo conaonnonosesasasoossoassessc (2), so that the reduced system is dy;= (pa = DS Po) da,+ (pe ~ D5 Ps) dans (Uae a) dx;=— Dip dx, — Lae | with the one known exact integral ¢=0. The system (3) must be completely integrable; for this it is necessary and sufficient that Dib ip _Pd _ Dd (D.- Dg 22) (Pa - Dies) = (D, pig Ps) (Pe DgPe)G=1 Aine: ») (p,- 7 D,) De _( Db p,)® Di ae D6”) 0.6-\?~ De”) Dg | The first of the two equations just written is simplified by subtracting from it pis times the second and the result is Dd (D. pis = Ds pix) 55 Dh (Dspa = Dy pis) a5 Db (Dy pis = Dypir) U0) sasnascoasbe (5). The x equations (5) are equivalent to the n+1 equations (4) since the second one written under the head (4) above reduces to = a [Dip (Dzpis — Dspiz) + Duh (Dspia — Di pis) + Dob (Dipia — Dapin)] = 0, and is therefore a consequence of the system (5). § 3. Now the equations (1) are supposed not to be completely integrable, since otherwise there is no poimt in the question under consideration; thus the equations (5) are not illusory; they may reduce to a single equation, or to two, or to the three D.d=0, D.p=0, Dp =0. In the last case ¢=0 will be an exact integral of the system (1) and may therefore WITH TWO INDEPENDENT VARIABLES. 3 be used to diminish the number of equations in it as well as that of the variables, so that we shall have the same problem after the reduction as before, only with fewer variables. We therefore suppose that @=0 is not an exact integral. § 4. Assume then that two of the equations (5), say the ith and jth, are inde- pendent and that the others are all consequences of these. Then from the two equations P.,D,6 ate P;, Did te PDs =0, PD, + P, Dif a PDs =0, j j j where P,; denotes D,p;;—D;pz2 or (D,D;—D;D:)y;, and so on, a third can be formed, namely, (P2Dit PaDi + PD.) (Pe Did + Pa Dib + PuDs$) — (P2D.+ PaD, + Pus) (PDib + Pa Dib + Pre) = Oeve secre (6’). w] J The left-hand side of this contains the three terms of the form (Fa ia PuPa) (D.D; — DD.) ¢ ; i 3 7 a a but D,D; — D, Dz = & Pos =~ > k=1 k OYk and therefore the sum of these three terms vanishes, since by hypothesis the kth equation in (5) is a consequence of the ith and jth, so that yar? Pa, » |=0. k k z By | Pov Pa, Pr The coefficient of D,p in (6’) is (Pod: AF 12 Dp. See) Iz > (P=? 3° J? 1D}s Fass) IE. i i i j j ) i or Ps, (PPs a D3 Ps IF Ds Pry) i j j j oa Pr. (Ps a D, Ps, ct DP») = D, Ps; Ps = Ps, Ps a D; PoP aa Ierolege . ACS ns ) (Gaa a) If then Q,, Q, Q, are the determinants of the matrix 31) Je | i i | || ||P 239 JP; t 239 Ps ? 1B ) 7 &. 4 Mr DIXON, ON DIFFERENTIAL EQUATIONS the equation (6’) becomes Dy (D.Qs — DsQz) + Dob (Ds Qi — Dy Qs) + Dyh (D1 Qo — Do Qi) = Ov -eeeeeeeees (6), the other terms disappearing on account of (5). Now the condition (6) must not be a new one or we shall again have D,¢=0, D.dp=0, D,p=0. Hence it is necessary that Q, (D.Qs — Ds Qe) + Qe (Ds Q: — Di Qs) + Qs (Dir Q2— Ds Q1) = 0... eeeereeeeeeeee (7) even when the system (5) reduces to two equations only. § 5. When the condition (7) is satisfied identically, there will be »+1 independent functions ¢,, ¢ ... dni satisfying the conditions (5). Choosing one of them, ¢,, we may take ¢,=a, as one of the integral equations, transform the system (1) by means of it and then integrate. Suppose the integrals to be TSO, THC} cto LS Om and that a, does not occur in these equations, ¢, being substituted for it if necessary. The functions 14, uw ... %, must satisfy the conditions (5) just as ¢, must; they must therefore be functions of ¢,, ¢....¢ni:, and in fact the solution reduces to di=h, Gr=M. -- Gnu = Anu: The same thing may also be seen thus. The auxiliary total system used in the solution of the two equations (5) may be written | Gh hee CER. soc dfn, 5 ace || =x | Ps, Ps, Pr», tee Pm Ps + PmPa + PnsPre, t t t 1 v t | Ps, Py, Pry, tee Pm Pos + PmoPs + PsP», Jj 7 J J J J it therefore includes the equations (1) and one other, and the expressions such as dy m — Pm da, — Pme dt. — Pms dx, must be linear combinations of dq¢,, dd. ... dbni:. Hence if dd,=0, the n equations (1) give dd, = dg; = ddx=... = Adni= 0, and the integral is di=h, Go=My -. Oni = Ani The existence of such an integral, involving n+1 arbitrary constants, is therefore ensured if the equations (5) reduce to two and if the condition (7) is identically satisfied. To find the integral we must take with the system (1) the new equation Qida, + Q.dax, + Q,dx; = 0 WITH TWO INDEPENDENT VARIABLES. J which is the other equation included in the auxiliary system of (5): the integral of these +1 equations is the solution desired. If (7) is not an identity it may be possible to take it for one of the n+1 integral equations and thus arrive at an integral with n, or fewer, arbitrary constants. § 6. Suppose lastly that the equations (5) reduce to a single one only; this will be the only restriction on the form of ¢ Let ¢,, go... ni. be the independent solutions of (5); take ¢,=q, as the first integral equation. Then the others may again be taken as od» = As, ds = dz ..- dni = Ant by making a proper choice, and thus the equations (1) will be equivalent to such a system as Abin = Andy. (m= 2, 3...N2+ 11) Now the equation (5) must be covariant for all changes of the variables on account of the property of g which it expresses; let ¢, @, ... gni2, 2 be taken as the new set of variables, z being any function independent of ¢,, ¢)... bas, then (5) becomes 06 Dn _ 8 Mm _ Ome loreeor Oi, (m=2, 3...n+1) These n equations for ¢ are to reduce to one and to be satisfied by ¢,,., hence (m=2, 3...n+1) so that Am is a function of ¢,, do... dais. It is now clear that the n+1 equations of any integral equivalent will involve $;, do --- n+» Only, and thus these functions will all be determined in terms of one of them. The first of the n+1 equations connecting them may be chosen arbitrarily, but there will be no loss of generality if it is taken to involve only two of the functions and in fact to be f (di, od») =0. We then have === “0g. Od, if ¢;, d, are not constants, and the other n—1 relations are found by integration from the equations Api =m gy. (m= 3, 4... n+1) Thus the primitive contains n—1 arbitrary constants and an arbitrary function of one variable. 6 Mr DIXON, ON DIFFERENTIAL EQUATIONS § 7. Before applying these results it may be well to state an important lemma. If we have m+1 independent variables a, 2...%m, t, and n dependent, 7%, Yo--. Yns then a solution of the 2n differential equations —— = aj, (GS, Baca) n Y BMAVy,=B", (G=1, 2 ... 2) 0 m : Dali, Sif 5 where A denotes > ¥,” an and the quantities a, 8, y are known functions of the r=1 r principal variables, is given by Uy, = Cy, Ug=Cy .-- Un=Cn, if for each value of j du; is a linear combination of the determinants n ! CHER (he ang Cha DF B® (dy; — a;dt) i=1 il ym”, 2”), cee Ym”, B® It is supposed that 4, % ..- Uw, all involve the dependent variables and that the deter- minant of the coefticients 8; does not vanish. The truth of the lemma is then seen as follows. ™ n We have duj= > p,%day+r9 = B;% (dy; — adt), r=1 =] where S Yr pyr +29 BO =O fT=1 Thus from the equations du,=0, du=0 ... du,=0, on the supposition that #, x... are constant, we find S S a (Y- ai) = 0. Gel, Seen From the » equations of this form it follows that ays, zs ag = Oe GS lly 2 soo 10) Again, on the supposition that ¢, 7 ... #4, %4, «.. %m are constant, we find nv 3 Bo & os pr) = 0, Ga: ei) whence by summation after multiplying by y,! Faun r m rn” > BPAY; —— >> fey Dey, = BO), r=1 t=1 Thus the 2n differential equations are satisfied. WITH TWO INDEPENDENT VARIABLES. G A further point is that if 1, has been found, the equation w%=¢, may be used to simplify the expressions from which du; is constructed, for if this is done the only effect on the process of verification is that m n SE py day, +029 YB; (dy; — a:dt) r=1 i=1 is equal to duj+edu, say, instead of du;, and this will not affect the argument. More generally, the proof will not be affected if each expression of this type is equal to a linear combination of Git 0H, con Cine it being of course understood that A” does not vanish. § 8. There is an intimate connexion between the results we have established and Hamburger’s method of solution* for systems of mn equations in x dependent and two independent variables. The solution of such a system is equivalent to the integration of equations of the form dy; = pada, + pirda, (C=, B00 10) cod(ts))) where the 2n derivatives p, being connected with the principal variables a, a, %... Yn by n+ equations, may be considered as known functions of these and of n other variables, SAY Ws, @y...nyo. The integral equivalent desired for the system (8) consists of 2n equations, giving 4, Yo-.. Yn, X3, %+-- Trt. In terms of a, a. For the sake of symmetry we shall take the system n+2 dy; = > Dimd&m @=1, rd eet) work (2) m=1 Now if »—1 equations of the integral equivalent are supposed known and are used to eliminate a, 2, 5 see £,4. from the equations (9) the resulting system is of the type (1) with an integral equivalent consisting of n+1 equations. The n—1 equations assumed must therefore be so chosen that the equations corresponding to (5) are equivalent to two at most. If they are actually equivalent to two the further condition corresponding to (7) must be satisfied and the solution can then be completed by the process indicated. This will be the ordinary case. If however the equations corresponding to (5) reduce to a single one, the condition corresponding to (7) disappears, and the problem is much simplified. It is this case that I wish to discuss. §9. Since a,...%,4. are supposed expressed in terms of a, 2, 23, Y,... Yn we have as the modified form of (9) n+2 n+2 2-2 dy; = (Pir abe > Pimbam) dx, i (Piz + > Pim bom) di, SF ( Pis Te ~ Pimtsm) dx; (@ = i, 2 tee N). ..(10), m= m=4 m= * Crelle, vols. 81 and 93. + This is the ordinary case when the equations are of the first order. For the extension to higher orders see below (§ 21). 8 Mr DIXON, ON DIFFERENTIAL EQUATIONS to which may be added dat, = Lym, + fom dL, + Hymdars (m=4, 5...n+2)...(11). Here pam (a=1, 2, 3) is determined from the system of equations n+2 Dad; =F = Ham Ding = 0, (F= ile 2m — 1) ¢-=0 being one of the n—1 integral equations assumed, and D,, the operator a us : 02m it = Ne Oy; , Let A.(a=1, 2, 3) denote the operator n+ 2 Dat > Ben ins m=4 then the equation corresponding to (5) may be written Aid (4,434; — A, Ayy;) + Ard (ASA: — A Avy:) + Ash (A,A,y; — AAW) =0...... (12), and this must also hold when 2,, a... OY 2m. is put in the place of y¥;. Now A.A, — A,A, 2 n+ n+2 = (D.D, = D.Ds) aF = Hon (D,,Ds = DD) ar p2 Kam (D.Dy, =~ Di, Ds) m= m=4 n+2 m2 n+2 Ss Se) echoes Bsj (DD; = DD) — & (Anttsm a A;Hom) 1D), m=4 j= m= Let P nj denote (D,,D;— DD) yi, then (AAs — A,A,) ies Spe (A.A; = A;A3) Lm m=1 n+2 N+2 n+2 = Px + (eins sr ji Jee) Se > = Ham [sj mj v m=4 t i m=4 j=4 i =e, SAY: Thus from the conditions (12) we deduce Qes : Ad ap Oe . Asd =P Oro . A.d = 0, G—e Ds ss n) z i i and in order that these conditions may reduce to one it is necessary that the ratios Qes : Qs : Qio v t t be the same for all values of 7, or that Qn = Qe Qe = Qa vereerserieerseereeeercereereensen (13), where 6, 7 are suitably chosen multipliers. WITH TWO INDEPENDENT VARIABLES. 9 These conditions are sufficient as well as necessary, for we have identically, if 4, 8 are any two of the numbers 1, 2, 3, A.¢, = 0, Agd, = 0, (sy PA Gan (el), and therefore (A, Ag — AgA,) $, = 0, = or Pr (AM UNA) cenit S OOF (AvAgA,A,) y= 0: m= a) 02m = Be Thus in any case x —1 identities can be formed from the equations (12) by taking different linear combinations and the system thus includes only n independent equations, but these are reduced to a single one by the conditions (13). This single one is of course Aid + OA.d + nAsd = 0. § 10. Now the expression for fam 1s much simplified if the differentials of a, a;.. are absent from the original equations (9), that is, if Inte Pim = 90 (m=4, 5... + 2). We then find 02m we Om + = Kam >= > = Pia O%_ = i=1 OY: 2 the derivatives on the right of this equation being formed on the assumption that Wy... Uy. are functions of a, #2, #3, yy... Yn defined by the assumed n—1 integral equations g¢; = 0, siele ono =U) We may therefore write pam = Dit. In the case of the equations (8) there is a further simplification, since p;,=0 for all values of 7; thus P,,;=0 unless m or j is 1 or 2 and i n+2 Qus becomes — D,pj.— = Hom DmPis Z a nee while Qn becomes D,p;,+ = = Ham DnPir» 3 > 02m and psn 18 simply a’? Des - for the values 3, 4...n+2 of m. Hence (13) gives Ds pix + ODspin + Se Pare CO eae OD a Dies) — Ones ceusenseirnsecsesssalnence (14). m=4 WOi, XODS IAT IE 2 10 Mr DIXON, ON DIFFERENTIAL EQUATIONS So then @ must be a root of the equation Dspy 4 ODs py», Dp ae OD pr», tee Dn4oPn ae ODnisPrs | Ds pn + ODspx, SEE SCOR AC OOO LOUD CCTICAE COO IOC OOM OORaO See ee (15), and when any of the n roots of this equation have been found, the values of py, Mss +++ Hs,n42 are given by the system (14). §11. Denote the roots of (15) by 0%, 0®...0™; let the value of jm corresponding to the root 6 be pm/ps and let »,", »... rv, be quantities proportional to the minors of the constituents in any column of the determinant. Then a familiar argument shews that if 0%, @” are unequal, nt2 n DP Pon yi) Dm pir =0, m=3 t=1 nt2 n and SD, Lm @) (7) Dyn Pir —=i( 0} m=3 i=1 t=1 z n and therefore =v" Qw = 0, t=1 zt n n+2 or yy” {Pr -: (Dam, : DinPir — Dyam,. Dupe) =0, t=1 G m=4 2 n n+2 or again > vy,” {Px + > (Dytm + 0” Det). Dupo} = QE pace secneotemenees (16). t=1 i m=4 If all the roots 0”, 6%...0™ are different, the form (16) will include n—1 equations which with the set Die Pera ee fess caeiesccisiaaente anctnepeareccetaceaseeee (17) will make 2n—2 equations in all and take the place of the conditions (13). §12. Now let r be any one of the numbers 2, 3... and suppose that for each value of r a complete differential du, can be found by adding multiples of the following*— dz, — 0 dz,, dy; — pada, — pda, (@=1, 2...n), = a n+2 Se . 2 ue |” Pdr + (43) Chik az fem™ dz) Dna ’ a 4 a m= * JT shall refer to these expressions as the list A”, WITH TWO INDEPENDENT VARIABLES. of which the last may be replaced by n nt+2 zy; | Pade, + Dn pudien| > .—3 i m=3 to which it is in a constant ratio. Then the system of relations Up = Cr 11 (PESPA, Biicca 10) 5 by the lemma of § 7, will give values of 4%, #,...%n4. in terms of 4%, &, %, Yi --- Yn; satisfying the conditions (16), (17). Hence under these circumstances the process indicated in §6 will give a solution of the equations (8) involving one arbitrary function and 2n—2 arbitrary constants. § 13. The next step in the process is to find a function ¢ satisfying the single equation (12) which may be written Aidt 0% Ash t+” Asp = 0. The differential df must be a linear combination of the expressions dx, — 0 dz,, daz; — "da, n+2 dy; = > Dim bin, m=1 dE, — Fam dx, — Kom di, = Pear Cites and here the last series of n—1 expressions may be replaced by (hi, Chis con Gh which are all supposed to vanish. Again, an expression for 7" is given by the equation n n a ZH 1 > vi! Oro — apie? > y;| Qos, i=1 i t=1 i and thus the second expression in the list just written may be replaced by n > ys") |Qxdz, = Qua! , =1 t i (m=3, 4...n+2) or by n n+2 n+2 2 py”) | (Pn + [les 1D) Pi) dir ate {Pp + (Ham Pin Dis = Han Dm Pa) de, | . t=1 B=4 i m=4 n Since = vi" (Dn pia + O° Dn Pz) = 0, — 2—2 12 Mr DIXON, ON DIFFERENTIAL EQUATIONS this expression may be written i DAs n+2 yy | Pde, + D, Pineda, += {(Him +6" Hom) dx, + Hamas} Dn P| > i=1 i m= and may be replaced by n [ n+2 = yp; Py.da, a5 > Se alee : t= m=3 Dr pudtrn | , which differs only by multiples of other expressions in the list. In accordance with the former notation this list da, — 0" da,, dy; — pada, — pdr, (i=1, 2... n) n n+2 >> yy | Pads a7 > Dnpistn| t=1 i m=3 will be denoted by A”. §14. If we have n partial differential equations if; (Li, oy Yr) Yor «++ Yn» Puy Pr» +++ Ps Pne) =9 (7= il, 2 seed) see CLS) the foregoing results apply, but are not in a convenient form. Instead of transforming them it is perhaps best to start afresh and form the corresponding expressions. In virtue of the equations (18) we treat py, Px, +. Pn aS determinate, though implicit, functions of a, %, Yr) --- Yn 20d Pu, Pn, »-+ Pm, and we consider the determination of the last n quantities in terms of a, M%, Yi, +. Yn as the first stage in the solution of the problem. Any one of the equations (18) gives, on differentiation with respect to 2, the suffix j being dropped, There are n equations of this type, to be satisfied by py, pa, .-- Pm}; their solution accomplishes the first stage as just explained. We assume that by properly determining n—1 of the quantities py, pa, --- Pm In terms of the nth and the principal variables, the conditions affecting the nth may be reduced to a single one. Let p denote this nth quantity which is last determined, then when py, py, .-. Pm all have values assigned them in terms of p, 2, 2, %, -.- yn the equation (19) becomes t=1 an & (a of D+ 31 Diva toh Dip} ~ (Of Op: OF Op; ) +2 Ee tp D, p+ ds oes D, aya! «acca eeeeaeE CRO (20). WITH TWO INDEPENDENT VARIABLES. 13 It is clear that the symbols D, pa, Dzp; have not the same meanings in (19) and (20). For instance what was represented in (19) by D,pj; is in (20) denoted by + Pit Dy, Dd, Pa dp We have then, corresponding to (13), as conditions that the system (20) may reduce to a single equation, = Of Opa _gps OF Opa é 2s oe eG ee (21), So el | _n & Af Pa oy and Dit » = = Dy pir + ane D. pat = Dh: a. (22), § 15. The equation (21) may be written of of r) Opin =a) Meveccteeae ita sessamasadeewasetane 2 i= (a7? Opal Op (28) and there are n of this type to determine @ and the quantities 7 of which one is by hypothesis unity. The determinant of these equations takes the place of the determinant (15). Let 1,7, 1. ... 1, be proportional to the minors of the constituents in any column when 6” is put for 6, so that En (2 gn Ti) <9 (Gal, & n) j=1 Pia Opin Then the equation corresponding to (16) is = Ad) Ex +2 na (Dy pa + 0 D.ps) | (1). Sea ua See ant earaeee (24), and the list A” is as follows: dx, — 0” da,, dy; — pada, — prdas, (Gly B coc @)) 2 5 D, fy. dx; + 2 — aps} In this last expression dp does not occur in an unsymmetrical way since s S 1 2 Opin _ a 2b; . 6 j=14=1 Opin op and this is the multiple of dp which should be subtracted according to the lemma (§ 7). Another expression may be added to the list, namely Ag |D. fide, += ae ap. j=1 14 Mr DIXON, ON DIFFERENTIAL EQUATIONS but this is not independent of the others, since dj;, df, ... all vanish. These expressions are substantially the same as those given by Hamburger (Crelle’s Journal, vol. xctt. p. 193 (15)), and those corresponding to A” are of the same form, for the single condition to which the system (20) now reduces may be written of eh } Of; Opin af; Opa ! ) > Io s \ 95 . ij ee Ue Dj+3 4 Dip Z2 Depa +31 Ps Dp+ 2 T ipl | =o, i Ms j and here the coefficient of D,p;,, namely Ss 1,0) Of raed | aed, j=1 CPi2 is equal to gu) S A CA ; 4 j=1 Opa also the coefficient of D,p is @% times that of D,p. Thus the list A becomes dz, — 0 dz,, dy; — pada, — pixdt., (=a eeu): y ie oF; OPir Sy A = e : ce ‘Op i=1 Replacing @%da, by dz, in the last expression, and p,da,+p,dx, by dy,, we have the shorter form 3 1,0 |, fide, + & Hi aps | j=l i=1 Opir as given by Hamburger. His other expression may be replaced by ie i | Defi ge S 2% aps. ; and is again a combination of the rest with the vanishing differentials df, df, ... dfn. J § 16. The method of integration as stated by Hamburger is substantially as follows. From each of the lists A”, A® ....A™ form a complete differential by linear combination ; let these be dw, du... dup, then solve the equations (2) oo (HS) HSC}. THE Ey, coe WSO for the 2n derivatives p and integrate the system dy; = pada, + pi.dxo, @=1, 2... 1); after substituting for the derivatives their values thus found. WITH TWO INDEPENDENT VARIABLES. 15 The above investigations however lead to a modification of this rule, for it appears that if the functions w,,... w, have been found, then the equations Ug = Co, Ug = Cz, +--+ Un=Cy may be used to reduce the number of variables in A”. The number of variables will thus become n+3, or one more than the number of expressions in A", so that wm is to be found by solving a system of n+2 ordinary differential equations in n+3 variables. The complete primitive of this ordinary system will consist of n+ 2 equations, say wy=C), Un+i = Cn+19 erele Uon+1 => Con+1+ Any one of these, say the first, may be chosen and a complete solution thence derived for the partial system by integration of n exact equations, as in Hamburger’s method. The n equations thus found by integration may be taken to be Cin = Crsigpoo US Gans then the most general integral of the system fi=9, ... fn=0, y= Cy, «+. Un =Cn is derived by means of the equation EG, Uns) =0} where F denotes an arbitrary function, and it will involve n—1 more arbitrary constants _ of integration, as explained in § 12. § 17. The theory is capable of further extension in the same direction. If one of the functions, say u., has been found, the equation u.=c, may be used to modify the other lists A®,....4™, and in fact to reduce the number of variables in them. This is in accordance with the remark made after the lemma in § 7, and im fact the functions wz, ... Ww, need only be such that from each of the lists A®, ... A™ an expression can be formed by linear combination which shall be equal to a linear com- bination of du, ... du,, and from which the differentials of the derivatives p shall not all be absent. This suggests a further possibility, namely, that the functions 4, w, ... U, might all be put on the same footing, and that if from each of the lists A”, A®,... A™ an expression could be formed by linear combination which should be a lmear combination of du,, dus, ... dup, then the values of the derivatives given by the equations Jip cco fp =Uy Th [Gg con Win = Gpsonsossosoossnnesvoonoceaaccs (25); which must be supposed such as to give them determinate values, would perhaps render the system dy: = pada, + pda, (Gi WF ee 0) sooeeesee (26) 16 Mr DIXON, ON DIFFERENTIAL EQUATIONS integrable. The verification given by Hamburger (Crelle, vol. xctt. pp. 194—197) does in fact still apply*, and therefore this is the case. $18. But we may now go further and say conversely that if the equations (25) give such values for the derivatives that the equations (26) are an integrable system, then a linear combination of the expressions A\)(r=1, 2, ... ) is equal to a lear com- bination of dij, dus... dup. ; For since afi eon t (Hin eed = [ela Sy Ae) SEE 1 OPiz Jak, Opa i Az wy we have from the equations of the type (19) EY Df Ss 2508 pe (Diy FiO O Das) = 0220-022 (24) = a j= J=1i=1 D,px and D,p; being now formed on the supposition that py, pn... Pm are given in terms of 4, Yo--- Yn, %, by the new equations w%=¢,, U.= C2... Un =Cn- Now multipliers 4, A»...An, Mi, Mo... My can be so chosen that 53 Ca een es (i=1, 2...n), the index (7) being dropped for convenience. The equation (24) then becomes 8 2 GDsfi+ ss (4 CET D) (Dp + 6D, Pi) j=1i=1 Opin () n n Ou; e +E 3 (a Som, £2) (D,pa + OD. pe) = 0, j=1i=1 \ Pi2 whence it follows that n nr n 6 = LD, fj = Sy Oj; Dy =F m;D, f;) + é = Aj Deu; 3P m; Ds f;) eee ever eeneces (27). = = — Hence n n = Adu; += madfj, J=1 yeu * Another verification, much the same in principle, + Note that @ cannot be zero, since the original may be given by reversing the steps of the following inves- equations are supposed to determine pj, ... Poo in terms of tigation. : the other variables x, y, p WITH TWO INDEPENDENT VARIABLES. I7/ n / ls of; wo UF} Ou; of; or = > si nape *) dpin + > (% ee ») dps j=1i=1 ’ ODin nm nN Ou; Of; n Of; = > (A / 2 —— i ea a (x i ty Miles , dy; + (ue Bi aE my - =) da, a Ou; Of; y aus Vi + & (a a, +m; 2) de, is equal to OS Sip CE RD Si) ace = Ehe = - = (A; D.u; + m;D, f;) (dx. — Odzx,) Jj + = = (a; aa +m; a) (dy; — pada, — pizdx.), which is a linear combination of the expressions A”. Now df; is zero for each value of j, and therefore n > (7) 2 = Ww” du; is equal to the expression just written. Hence, in order that the equations Up (Cy) Ug Cy) 21-2 Ung = Cy may belong to a complete primitive, it is necessury and sufficient that the equations (25) define the derivatives algebraically* in terms of a, 2, 4%, Yo--- Yn and that for each value of 7 from 1 to m it be possible to form a linear pombination of the expressions 4") which shall be a linear combination of the differentials du,, du. ...du,. We must now examine how this theorem is modified if the values @%, 0% ...@™ are not all different. § 19. If for any root @” all the (a—1)th minors of the determinant of the equations (23) vanish then this root counts at least a times, and there are a series of quantities taking the place of A TA) coo tie accordingly the root @” gives a lists of expressions in the series A”, A® ... so that unless the root is of a higher order of multiplicity the modification in the theorem is very slight. Now it is a known property of systems such as (23), easily recognized from the canonical form of a homogeneous linear transformation}, that if the root @” occurs, say a+1 times, then a series of quantities Ie tits ooo (Be * That is, without differentiation. + See for instance a paper by Prof. Burnside in the Proc. L. M. S. for 1899. Wr, IDG IBA TE 3 18 Mr DIXON, ON DIFFERENTIAL EQUATIONS can be found such that = of, Of; n af \ J OF) Ss" Se ewe j=l (ann : =) ae Opis (t=1, 2... n)...<...-.(28), the quantities J,”, 1,” ...1," being a set such that n of; af; SOY La Vy | ete 7a a ded Vs - a eo j=1 4 (ap. s a e a n) (29). If the root occurs 2+2 times then another series [ea OS wae | can be found such that eof; afi\_ & >. oi SS J (r) J I = SS, 4) = 5) yi a a0 = hae G21, 24) while h,, k,...k, satisfy the conditions (28), in which the quantities /;” are again definite combinations of the a sets which satisfy the conditions (29). Similarly for a greater number of repetitions. § 20. Let us take for simplicity the case of a root counting a+1 times and drop the index (r). From the equations of the type (19) we have 2 2 bDifi+ & 3 3 (2 Dora Of; 2 Op: ar —- Ds a)=0, Opin that is SbDifit +3 3% sg Dip + OD.pa) + 0% E1 5a Dips =0 HA (31). Suse =P j i=1 The last term in this equation is equal to n nm (/_ Ou; of; ou; of; > } J aD) n- er] peed N:. j=l = (sp Opin thee =| Depia Ai (x Piz Oey a D.psh i or to aS (A; Du; + m; D> f;), j=l rj, mj; being again determined by the equations a Ou; of. Of; j Ss wo Jaa 31 NE) = jal (sap Opi a i io a j=l 1 Opin’ ( = 1, 2... n); n ra) rf] = (age +m 2) =o, (i=1, 2...7). Thus the equation (31) becomes is 5 i, 7 yg (D,pi + OD; pa)+ 3 > Shy Dippy Diy mj D, ff =O). (32). j WITH TWO INDEPENDENT VARIABLES. 19 Now let us restore the index (7) and suppose that 0° =@” and that ES WAC) pa NASI aOR ee are so determined that £ Ou; Of; \ Lo Vie > (rr; wed +m; ~/\=@8 > hk ; j=1 0 z Opa S Opin j=1 q Opa S Ou; of; r;() = + m;®) J =(; j= a E(x Opir al thus for the list A we have, by the same method as before, da, — 0") da,, dy; — pada, — psd aa, ((=1,2...n), 3 {Si re dpa + (kD, f; — mj D. fj — r;' Dau) di. This method will apply however many times the root is repeated. The list A® in- volves the coefficients of the expression formed from the list A” and the same holds at each successive stage. Thus n lists can be formed in all cases and the theorem as now modified holds always, though it is perhaps less likely to have practical value in the case just investigated than when all the quantities @ are unequal. § 21. The above work applies to partial differential equations of higher orders with the following slight changes. The number of independent variables is still two. In the system (8) of § 8 some of the derivatives p are included in the series y’ 3 Pp Yr Yo «+» Yn and the rest are functions of 2, 2, y, ... yn and other variables whose number is less than n, say n’*. The integral equivalent includes n+ n’ equations and the first stage in the solution is to find n’—1 of these. The number of variables y; whose derivatives are included among %, yz ... Yn is n—mn and for any one of them Ping =0 for all pairs of suffixes m, j, so that Qos, Qa; Qu vanish identically. Hence the determinant (15) has only n’ rows and columns and the equations (16) are only n’—1 in number. The final solution when the method of §§ 8—12 applies depends on an arbitrary * For instance in the case of an equation of the second order with one dependent variable, the system (8) is in the usual notation dz=pdx+qdy, dp=rdx+sdy, dq=sdx+tdy, and n=3, n’=n"=2. 20 Mr DIXON, ON DIFFERENTIAL EQUATIONS function and n+n’—2 arbitrary constants, of which n—1 are constants arising out of the final integrations; these cannot in general be accomplished until the form of the arbitrary function has been assigned. Compare § 12. § 22. When the differential equations are given in the form (18) they will be 2n—n in number and will include n—W Owe NG) WM py == IA coggasocueconcdocossupsostessesise (33), n—n GP eS! I 7S Saf; socos casoocnssboocosoosueesesaas (34), n—n’, say, of the form Pj = Pha -os--sseecescesenessevecosrenees (35). We may take it that pa, pi, py do not occur in other equations than such as those just written. The equations corresponding to (19) in these three cases are noel (88) IDAG rs = oA soncsasscbenoosgspsnauonsomobHsoeecosed (36), iaeoin, (8%4) JD Aji Yar conocsonesomcasaconaeroocoscucwesccbos (37), frowns) (35)! Dap — Oy Pin acres eens ee cee ale eee (38). Now the expression for pj; in terms of 2, ®, J, ++. Yn, p is not at our disposal for it is yj; thus (36) becomes an identity and (37) follows directly from (35). These do not give any condition to be satisfied by p, but (88) becomes Op; a Dz pj + oe D, p= Dy pia + e DD since te tee eee (39), whence He =-0 a seals uae mar tae oniadte tae cia) ese eee (40), 0 and D, Prt Dd, Pra= oa ence eect eee e eee e nese e neces eeeeene (41), if we suppose all the conditions that p must satisfy to reduce as before to a single one. Thus of the equations corresponding to (14) or (23) n—mn” are of the form (40) and there are n’+n’—n others included in the general form (23), since this is the number of differential equations other than those of the types (33), (34), (35). In all then = CPi. of these quantities there are Pap. eB n’ also, for if pi; occurs in such an equation as (33) it will not occur elsewhere and pj, will there are n’ equations linear in such quantities as only oceur in (34); thus the coefficient of = in any one of the linear equations such as (23) will be zero. Hence the determinant corresponding to (15) will have n’ rows and columns, and n—n’ of the rows will consist of 1, @ and n’—2 zeros. Each root of the equation formed by equating this determinant to zero gives a list of differential expressions which may be used in the same way as one of the lists A”... A™, WITH TWO INDEPENDENT VARIABLES. 21 Nore oN SINGULAR SOLUTIONS. § 23. I have not met with the remark that singular solutions of the system (18) are characterized by the vanishing of the determinant of the equations (23) for all values of 6. Here I understand a solution to be singular when it is an aggregate of elements at each of which the ordinary method of solution in series fails* through the dis- appearance of the successive coefficients in the series from the equations which should determine them. In the present case for instance assume a solution of the form Ys = Yo + Y™ (Ge — #2) +4Y;" (Ho — Ho P +... (Gere); yi, yi... being functions of 2, of which the first is arbitrarily chosen; the second is found from the system (18) by putting and solving for pi, which is then y;". To find y;*) we must differentiate the system (18) with respect to #, and make the same substitutions, the result being a system of n linear equations for yy”), Yo") Aa Yn. These equations will not determine the quantities if the determinant O( fis Je --- In) ey; 0 (Piz, Poo --- Dns) vanishes for the particular system of initial values that has been chosen. Notwithstanding this it may be possible to find a solution in series of integral powers of a#,—a,% or of some other variable &, a function of a, a, the coefficients being functions of & another function of 2,, 2. If however after any transformation whatever of the independent variables the determinant corresponding to J still vanishes for any element, the method just sketched will fail to give a solution in series with this particular initial element, and it is therefore qualified for a place in the singular solution. But any change of the independent variables leads to a simple linear transformation of the derivatives Pa= Pat pP is, P2= yPy + (alr * This does not mean that no solution in series can be found for the given initial element. 22 Mr DIXON, ON DIFFERENTIAL EQUATIONS, etc. r, #, v, p being functions of 2, 2, the same for all values of the suffix 7 Thus ange Dit OP Opa P apn’ and the transformed determinant corresponding to J is practically that of the equations (23) with —y/p in the place of @. The determinant is of the mth order and the nth degree in 6, so that in general the conditions characterizing a singular solution are n+1 in number, since the coefficient of each power of @ must vanish. The elimination of the derivatives then gives a relation connecting the principal variables. Also expressions are given for the derivatives and by substituting these expressions in the equations dy; = pda, + pdr, (Ge, sea), and integrating when possible, the singular solution when it exists may be found. The problem will be rather more complicated in any case in which the number of conditions falls below n+1, since then the derivatives cannot generally be eliminated, and if the number falls below » cannot be found by merely algebraical processes. By similar considerations the relations characteristic of singular solutions may be found when the number of independent variables is greater than two. II. On the Calculation of the Double-Integral expressing Normal Correlation. By W. F. SHepparp, M.A., LL.M. [Received 25 January 1900.] Preliminary Formulae. 1. THE integral considered in this paper may be written, in its primary form, nt 2. [Ye P A A e~ + (x?—2xy cos D+y*) cosec?D dady TONG. GRRE a> D) = ——— ti, a Yu» Yai D} 2 sin D Ja.) y, and we require formulae for calculating its value for any given values of 2%, 22, Yi, Ys and D. The integral arises in the theory of normal statistical correlation. Let X and VY be the measures of two correlated organs, and let their means, mean squares of deviation, and mean product of deviation, in a homogeneous community, be L, M, a’, b’, and abcos D. Then, if [X,, Xe; Y,, Y,] denotes the proportion of individuals, in this community, for which X lies between X, and X,, and Y between Y, and Y., the distributions of X and Y are said to be normally correlated when* [X,, X.; VY, V.J=F {(X,-D)/a, (X.—LD)/a; (Y,—M)/b, (Y2—M)/b; D} ...(2) for all values of X,, X., Y,, and Y,. Since this gives 1 X, _1(X¥-L) Prrere aon 4-est] = | CP ae (3), V2r al x, the definition implies that the values of X and of Y, taken separately, are normally distributed. The integral (1) would also occur in the theory of heat, in considering the changes of temperature in an infinite homogeneous conducting solid: the initial conditions being a uniform temperature within an infinite prism bounded by two pairs of parallel planes, and another uniform temperature elsewhere. * Edgeworth, ‘Correlated Averages,” Phil. Mag., vol. 34, p. 190; Karl Pearson, ‘Regression, Heredity, and Panmixia,” Phil. Trans., vol. 187 (1896) A, p. 253. 24 Mr SHEPPARD, ON THE CALCULATION OF THE DOUBLE-INTEGRAL 2. There are two classes of cases for which we have to consider the value of the expression (1), in regard to normal correlation. In the first class of cases the values of X and of Y are taken at equal distances along the scales of measurement, and a double-entry table is made out, for as large a number of individuals as possible, showing the numbers falling into the different classes determined by these values. The values of Z, M, a’, b*, and abcos D are calculated from this table, and we have then to compare the different numbers shown in the table with the corresponding numbers given by (2), in order to see whether the correlation may be regarded as normal. In making the comparison, we have to take account of “errors of random selection,’ due to the limitation of the number of individuals actually observed. The effect of these errors is considered elsewhere *. In the second class of cases we have only one value of X and one value of Y, so that our double-entry table has four compartments; and we require to determine the value of D, on the assumption that the distributions of X and of Y are normal and normally correlated. Here our data are [X, 0; Y, 0], [X, ©; —o, 0], and [—w, wo; Y, wo]. The second and third of these, by means of (3), give us the values of (X —Z)/a and (Y —M)/b, without any information as to LZ, M, a, and b; and the problem is therefore to determine D from given values of 2, y, and F{z, 0; y, 0; D}. The value of D will, as in the first class of cases, be subject to a certain error of random selection. 3. The expression (1) contains five arbitrary constants. But, if we write bi (ays DD) = Na 00): Ay, 00 sD) Nae enone ctecieceneme eee eae Rete (4), we have F{x,, 2; yi, ys DD=o(m, wm, D)—G(m, y, D) SON Cain CDRS CAS Why LD ecbaosacencscacse (5), and we therefore only require formulae for calculating p(x, y, D) for given values of a, y, and D. The values of w and of y may be either positive or negative, and D may have any value from 0 to 7. 4. If we write, under the integral sign in (1), c= 2 sin) yay) —7 sine): we have, by (4), re 1 $(x,y,D)=5- | i! xcosee D i e-He?-2ry cos D+y sin D . da’ dy!s.s..eece0e- (6). y cosec D * «On the Application of the Theory of Error to cases of Normal Distribution and Normal Correlation,” Phil. Trans., vol. 192 (1898) A, pp. 151, 152. EXPRESSING NORMAL CORRELATION. 25 Now consider the solid generated by the revolution, about its axis OZ’, of the figure bounded by the curve and the base 2=0. The volume of this solid will be unity. In the base-plane of the solid take two straight lines OX and OY, including an angle D; and draw OX’ and OY’ at right angles to OY and OX respectively. Then the angle X’OY’ is r—D; and the equation to the surface of the solid, referred to OX’, OY’, OZ’ as coordinate axes, is i allie e—} (%*—22'y’ cos D+y'*) Qar Zz Hence ¢(x, y, D) is equal to so much of the solid as lies on the positive side of a plane at mght angles to OX, at a distance x from O, and on the positive side of a plane at right angles to OY, at a distance y from 0. 5. Let the first of these planes divide the solid into portions whose volumes are respectively $(1+4) and $(1—a); and let the second divide it into portions whose volumes are respectively }(1+ 8) and $(1— 8). Then, as in (3), 4) to) 4(1-a)= Bahay eos DY sin Didi ay = azcosec DJ —a sel so that and similarly For any particular values of a and £, the values of # and of y are given by the t — — tables of =| edt, where t=a/V2 or y/V2; so that we may regard ¢(«,y, D) as 0 7 being a function of a, 8, and D. Or, if # and y are given, « and 8 may be found from the tables. Since the solid is a solid of revolution, it is easily shown that p(e,—y, D) =e —4)-o@y, i | 6(—2,y,D) =40—-8)—OG,Y, T—D))} ....cecceeceecnneceseees (10). $(—2, —y, D)=t(a+8)+ 4, y, D) Hence we only require formulae for calculating the value of $(#,y,D) when « and y (or a and §) are both positive. Vor, XIX. Parr I. 4 26 Mr SHEPPARD, ON THE CALCULATION OF THE DOUBLE-INTEGRAL Conversion of Integral into different forms. 6. Let the planes mentioned at the end of § 4 cut OX and OY respectively in NV and n, and let their traces on the plane of XOY be »’Wn and &’ WE respectively Fie. 1, (Fig. 1). Let their sections of the solid intersect in the ordinate WR. Denoting this ordinate by z, and taking w# and y to be positive (§ 5), we have ON=2, NW=y cosec D — x cot D On=y, nW=z2 cosec D— y cot D 0 Ww? = (2? — 2Qay cos D ate y) cosec? D | TR eee eee (11). WR=z= as e734 (2x cos D+ y*) cosec? D The expression ¢$(«, y, D) denotes the portion €WRn of the solid. Now suppose that D varies, the values of w and of y remaining fixed. Then it may be shown, from the elementary properties of the curve (7) and the surface (8), that* 7 Ml (ped) 0H Bee Se OPE PER EE SHAasGo8OR (12). Also, when D=7, (a, y, D) vanishes. Hence $(c, y, D)=| 2dD ~D Lge ; ; 3 = — | e~HeP Bey cosets) C0860 de eee acccscceeeesesesenes (13). =7 JD * Loe. cit. p. 146. EXPRESSING NORMAL CORRELATION. 27 7. Let the plane through OZ and WR meet the base-plane YOY in OW¢ (Fig. 1). This plane divides the volume €WRn into two portions EWRE and {WRn. Now, since the solid is a solid of revolution, the magnitude of the volume ¢WRy depends only on OW and the angle €Wy; i.e. it depends only on ON and the angle NOW. Similarly the magnitude of the volume WARE depends only on On and the angle nOW. The volume €WRn is half what ¢(#, y, D) would become if we took y=z, and D=2NOW; and the volume €WR€ is half what $(z, y, D) would become if we took «=y, and D=2n0W. Hence, denoting NOW and nOW by 4D, and 4D, respectively, we have* (2, ¥, D)=4O(@, z, Di) +4h(y y, Dy) ..0..coeeccccrreevecees (14), where y — xcos D «sin D tan $D, = =" an 3D — “FZ, cot $D ae z—ycos D uD ysin D =o tan 4D —"5™ cot 4D We therefore only require formulae for finding $(, 2, D) in terms of « and D; « being supposed positive. The value of D may range between 0 and 7. But if either y—xcos D or e—ycos D is negative, D, or D, will be negative. This only means that the angle Wy or EWE is greater than a right angle; and, for such a case, it will be seen that we must use the formula o(2, x, —-D)=1—¢4(-2, —2, D) 8. Before considering the formulae for $(, 2, D), some special cases of (a, y, D) should be noted. (i) Let D=0. The volume Wy then becomes 4$(1—a) or $(1—8) according as is greater or less than y. Hence $ (2, Y, )=5-] e—t(@?—2zy cosw+y") cosec*w day ( ) 1 e@D = Ca day at =a | N20 Jaz abe a Hl Sine eee a eee T - V7 * This is a general formula, for any function of the form f(x —22'y' cos D+y’). sin D dx'dy’. (2, y, D)= ee zcosec D J y cosec D 28 Mr SHEPPARD, ON THE CALCULATION OF THE DOUBLE-INTEGRAL (ii) Let D=47. This represents the case of independent distributions, and we have p(x, ¥, in)= == “eM de. a= |e Wdy =$ (1 — a). (1 —B) conc. se rsteecerecswersene case ren nerat (18). (iii) Let y=0. Then W becomes the intersection of OX’ and 7'Nn, and we have $(a, 0, D)=4$(2, x, 2D - =) Satay es Ge eee: Sabon adosascaqsaaareo tor (19). (iv) Let z=y=0. Then $(0, 0, a er LE (20). This last result follows at once from the fact that the solid is a solid of revolution of volume unity. 9. From (5), (10), (14), and (16) we see that, for positive or negative values of %, %, y;, and y,, the expression TOURS Coe Tar nee DE See pep Beacon asco eSScoAceA aC os eaGbc (1) can be split up into eight terms, with or without additional terms involving the class- indices denoted by a or 8; each of these eight terms being of the form d(x, x, D’), where « is always either 2, 2, y,, Or Ys, taken positively, and D’ is given by an equation of the form iene AID (ap oreo sD) Gris/D)) Sopsnacaetaccagnesc9a00505" (21). For caleulating the value of (1), for any given values of %, ™, y:, ys, and D, it would therefore be sufficient to have a double-entry table, giving the values of p(x, x, D) for all positive values of w, and for all values of D from 0 to 7. It would be inconvenient, however, for general purposes, to use « and D as the arguments in this table. The determination of D’ from (21) requires the use of an Inverse tangent table; and this is not to be found in the ordinary collections of mathe- matical tables. The difficulty is avoided by using as argument either tan4}D or some function of « and tan}. It will be seen presently that there is a reciprocal relation which makes it convenient to use « and xtan}D as arguments. If then we write ONC 2h ID) == FGA, 21))5 son oszaccasdaponeoagosoosaNoeCossece oCoSH (23), the relations (14) and (15) become Pa; y, DY = LAG (Gy Deepa U a ca ctvectionicst con ceesavaeeresuntee (24), EXPRESSING NORMAL CORRELATION. 29 where : & =ycosec D — x cot D =t(atye-t(e—-y)e n =x cosec D— y cot D =h(aty)e+t (aye?! The values of & and of 7 are easily calculated from (25); and therefore, if a complete table is made out giving S(@ &) for all positive values of w and of &, the calculation of (a, y, D), and thence of the double-integral in (1), is very simple. It should be noticed that a+ & =(a — 2axy cos D + y") cosec? 1D)| Pe I a, OY ec sce atalantsicl (26), =7ty | so that the ordinate WR in § 6 is pa ea) 2 | Ne oraT eee ites ee...) Oa (27). ! = old ety +n?) | Qa 10. For shortness, denote o(a, «, D)=f (x, xx) by V. The value of V in terms of x and «, or of « and E=xk, can be expressed in a good many different ways. (i) Omitting the lines OX’, OY, &’n€, in Fig. 1, and replacing D, by D, we have (p. 30, Fig. 2) ONS ace NOW D: and V2 wold CWajsarsereroneeebeecnen conn csanscewarrancss (28). Also the volumes X Nn, XO, €OY’, are respectively }(1—a), }D/7, }(7—D)/7. Hence we find V=4(1-a)—2 vol. XNWE D VE AOI area 4 0 2\(0).01 ee (29), v= a 2 vol. nWOY' | Each of the volumes (Wn, XNWE, NOW, »WOY’ can be divided into elementary 30 Mr SHEPPARD, ON THE CALCULATION OF THE DOUBLE-INTEGRAL prisms either by planes at right angles to OX and planes at right angles to OYAnor by planes through OZ’ and circular cylinders having OZ’ as common axis. The first method of division will give a double-integral. For the second, let y be the angle 1 Fie. 2. between two consecutive planes; then the wedge lying between these planes will be divided by a cylinder of radius r into portions (1 —e7!").wW/27 and e-}"./27 respec- tively, so that the volumes in question can be expressed as single integrals. For instance, €Wn becomes ed a: a Qa +D v; or, changing the independent variable to # tan y, x iy e-t@+£) 2ar xtani}D w+ & : Hence, from (28) and (29), we have the following formulae: T= - Me [ie GSD ATO gs oe sees eae gee (30) == , aE cao e - (30), V=4(1-a)-1 i eee cists (31) =3(-a) 2 fo OI ee (31), EXPRESSING NORMAL CORRELATION. 3] a (KO V=30-2-2++/ | GUE OIO UD ete saceneancanensss( (32) cH 7/0/70 Da [fl —enHerte) BS Oy eae ee ¢ Glee) atl. Fee aro vee (32 a), m= D 1 r. Veet F Ve = ~-h Le ACEO (On a aBE OE aE cacao ceooosdcaepeanecons (33) =D « [?1—e tere 2 oe ap Seeaees ks aiclelatola\ulalatstviateletelejoleveiclelaicieteleteretersiainte (33 A). (ii) Again, draw a plane through WR at right angles to » NWn, cutting OY’ in n, and dividing the solid into portions $(1+p) and $(1—p). Then One oie te ecen eeentacoceiictn dela: sasews a eeanslews (34), and therefore 7 2 fs = = Gas 35 p Ve ik ORIEN. Srom cea cic nae caleeesrotinaie nae iat aisles (35), the value of which is known for any given value of & Let the plane meet the base- plane XOY' in y’n'Wy (Fig. 2). Then, since y’Wy and 7’ Wn are at right angles, WO GCN) ee US) cee(Ik = /m)) cescaconcocopsopoonddeK0qpo000006 (36). But this volume is the sum of the volumes Wn and yW€; and the latter of these is got from the former by interchanging w and & We have therefore* HES DEINE DH CSE) C= 9)’ oo00000000000000090 cno0nea0006 (37), so that HG, Ds (Weyl = 7B, 2)" coonscc00Accons500nn0000008 (37 A). The interchange of « and & involves the substitutions Ge ge ID ~m—-D €E p\ le Fats) Hegre: 2) SR SR Sotsai aise pais (38). Hence, taking (37 4) with each of the formulae in (1), we get a further set of formulae :— Vana) === |. [. gate eey DODO sai tet le. soeetus (39) p> ph (E2+22) =30-aa-p)-2 oe hao Pe erations AS Seta s Soe ose Bae (39 A), Tesch llceays) EEe-Sal lt hE TLL a 40) OSD) aera Os $ ( =—ta(l—p)+ px eg hie +22) E | g Ge Se et Bene ah tk nadie oes! (40 4), ro £24 go? * Cf. Glaisher, ‘‘On a class of definite integrals,” Phil. Mag., vol. 42 (1871), p. 426. 32 Mr SHEPPARD, ON THE CALCULATION OF THE DOUBLE-INTEGRAL B= ae ees.) — =| [ee+aeag iene eer te (41) ="? ~400-p)-* [ ae Pieri te a, (41 4), V =agse Gea = =f I eHOTONGOGD ..ccceseecseeeceeees (42) =A aide = a : f. z SS ie eee (42). 11. Before considering the formulae obtained in the last section, a few points with regard to the tabulation of V= f(a, &) need attention. (i) From (304) we have dV eter ae) seen sens sac erin owornvonweesonearowaenn 4 dé Ls te ($3), and from (39 A) dV 2 h et (+22) AV 40-9) -n/ 2m sb cece cece cece teeescessnes (44). Suppose # to remain constant. Then, when & becomes very great, V tends to become equal to £ en? (z2+ &) m E(a+&) Thus the values of V become very small, while the differences of V are relatively great, and change very rapidly. The table of values of V would therefore be inconvenient for purposes of interpolation. The difficulty is similar to that which arises in the case of the simple probability-integral, and it can be avoided in the same way. Instead of tabulating V, we can tabulate either log,V or e”’**)V. If we write Fae) eh eB ee) eer mnre eee teenn eespneeaseneeneee (45), and tabulate the values of ¢, (x, &), we have by (24) and (26) b(x, y Dy =flet*™ g(a, E)te ree” Sy, 0)} = fet -axy cos D ty) cosec*D (h(a, £)+- ily, 7)}----2c00ceececsescreeeces (46), the values of & and of 7 being given by (25). (ii) When « and & are very small, dV/dé and dV/dzx are very great. The differences at the commencement of the table will therefore be great, and interpolation for small values of # and of & will be difficult. The reason of this will be seen by reference to the formulae (324) and (414). The integral in either of these formulae has small differential coefficients for small values of « and &; but V contains in addition a term in 7—D=2tan—a/£, and this becomes indeterminate when « and & are both zero. It EXPRESSING NORMAL CORRELATION. 33 is obvious, however, from Fig. 1, that the angles D, and D, in (14) are connected by the relation 1D SEUSS 1) Se eee tes (47), (ahem me ete FCS 2 fae ea ey ae ae hap (48), wwe Tere, lip OO) tee D) =" =” = $du(2, yo ONE) Me co ete (49), the values of & and of » being the same as before. It will be seen from (37) that ho (x, E)+ do(E, v) = F(a + p — ap). (i) From (35) we have dp 2 —1ee e dea oz Cra EE re, ineorasiseitelen se omee eames esleeaasasnenesliwsel (50), and therefore, by (43), dV 1 =A 1 doe ition” a+ & eielslaislaloleloletelute(alalalsicivlslelolslalcis\ela)vlsis/eieisysie (51). Similarly dV lege fae eal a ie = Si) eras Ee HI 6 Ete sialelelelele.elele\cleleicieleicicicicessieee.cs (52). The second differential coefficient of V with regard to p will contain e as a factor; the third will contain e*; and so on. But, if & is small, these differential coefficients will not be very great. We might therefore, for certain portions of the table, take p instead of € as the argument, the value of & being found from that of p by means of the ordinary tables of the probability-integral. (iv) Thus, on the whole, it would be desirable to have three tables. From (37) we see that if # ranges from 0 to », & need only range from 0 to a. For small values of «, we should tabulate 1/7. tan2/E—/f(a#, &); the arguments being either x and € or a and p. For moderately great values of , or for great values of « and small values of & it would probably be simplest to tabulate f(a, &) in terms of a and p. For great values of # and & we should require a table either of log, f(z, &) or of BTN KGa this table bemg in terms of w and &. Won, XD, IPAv Ie 5 34 Mr SHEPPARD, ON THE CALCULATION OF THE DOUBLE-INTEGRAL Adaptation for Numerical Calculation. 12. To obtain V in a form suitable for numerical calculation, the appropriate integral has to be expanded in a series. We may consider that we are taking a definite value of z, and that we require to tabulate f(«, &) for different values of & (or p). There are then four cases to be considered, according as w and & are small or great. (i) First, suppose that # and €& are both small. Then obviously the proper formula is (382) or (824). Expanding e74* in (82) in ascending powers of @, and performing the first integration in the double-integral, we find 1 f2 ; 1 303 1°60 F(@, F=40 -—4)—5 —— ={e# (RS eee a? =(1-a)— * tan£/n+ CE C8 ad Geee eas nccte cues teas eeeee (53), where (replacing @ by 2) C; =i [re 42? oda, TX mel Wipe o-? Cs = eel = Ht C; wel 5 wan et? oda, dondanoonoopoaponuocedad (54). TS. x 1 1 i Nae Ohne = ee | e73@ on da w2.4...2n—2.2"—J0 Integrating by parts in these last equations, we have C; == (1-e Pex *e m (Sto at: saete)s C; =" - (1+ 5) wien ane le ae qare taa.eet ), em Coal ae +otey a eee =) aa ae re ("+ a PE Sc le 7 2.4...2n 2Q.4...2nt2 2.4... 2n4+4 =) From (55), or directly from (54), we see that the successive coefficients are connected by the relation / 1 x Gs — —— e-i@ on+1 Ons = é 2.4 “se EXPRESSING NORMAL CORRELATION. 35 The actual values for any given value of # are easily calculated from (55) when z is less than 1, or from (56) when is greater than 1, When n becomes very great, the ratio of the (n+1)th term to the nth, in the series in (53), 1s ultimately 2n—1 EF 2n+1 2n+2’ and therefore the series is convergent for all values of £&, whatever the value of x may be. When « and & are both small, it converges very rapidly. (ii) Next, let 2 be small and & great. Then from (40 A) we have e- Het) f(a, )=-ta—p)+ ef a INGSES 2 Ose =—tha(l UD) |. Tea Expanding 1/(1 + 2*/&*) in ascending powers of 2°/&?, we get 2 1 ee : z if (2, &) =— 4a(1—p) ar = e-te+e)) (G,E> = G.E= + G,E~ ——leeey Wssecte areata (57), where rz G, =e | e- 3 da, 0 G, =| e- a° da, : rr DM Manat Me Blea tei ta (58), Gon (7 | : e732? gn da 0 or, integrating by parts, Zz une iy Cae ips! aaa 7 Era a’ Ge 3 te68 sosn7 TP a ai ee Se 7 ee Se a ae eee 59 Caran para , 2n—1 y2N+1 r2N+3 ae x wf x x mS Oa Se lO eoe Toe The coefficients are connected by the relation Goncra = (2n —1) Gh fet OS no RO ORO C RR GOR ARC oR RCE (60), the initial value being 36 Mr SHEPPARD, ON THE CALCULATION OF THE DOUBLE-INTEGRAL The terms in (57) are alternately positive and negative; and it is clear from (59) that Gini, is less than 2G ,,, but becomes more nearly equal to it as m increases. The series is therefore convergent so long as & is greater than or equal to =. (iii) Again, let x be great and & small. Then from (394) we have fepeta= oop tee | =e : Po Jp ae Expanding 1/(x?+ &) in ascending powers of & we get f(a, £) = (1 — a) (1p) — 4 eH HE — Hl + HB.) eeeeceecnee (61), where el H, = | e 3” oda | x “2 H, =| et” ada \ x Seas csisciinb den see ed teas Soe ceeeeeeee (62) Ele — Oe | ea da . a . These integrals cannot be expressed in convergent series. But if we write Fea e'dz= We EFF (iain. coos ameeecnseetecee sees (63), Ee ; 2 the coefficients are connected by the relations H, = = = 8 2 Hi, => i (= zy Ht) SING Res sss sient eee (64), 1 il Fons — m+1 == Ba? £2) + (a2 + BEY)? -E* (a? + E*) (a? + BE*) + (Bat + 10°F? + 15E4)’ £2(a2 + £%) {E2(w? + £2) (at + 202? + BEY) + (5a? + 1Tat E? + Q1arE4 + Q1E*)} E8(a2 + EP (at + QE? + BES) + E2 (a? + E) (6x® + 220 E* + B0z2E* + B0E*)’ + (Bae + 240° E? + 5QatE* + GOa2E® + 4588) etc. When « and & are nearly equal, we can use a symmetrical expansion. From (87) and (68) we have F(v, &=41-2)1-p)-FE @) ar ane ny L BeTRHE (| Bett | dat + 10 + BE =4$(1—a)(1—p)— 7 a(a+&) ~ #(e+F) e(e+ BP = Multiplying (68) and (69) by &/a and 2/£ respectively, adding, and dividing by &/x+a/€, we find f@, )=40-)0- aoe 1 ed ee L503 + 632°E* + 12024&* + 63a7E* + 15E* Vr E+E) ZE(G+E) HE (e+ BP _ 105. + 540uE* + 1239a°E* + 1800z°E" + 1239.x4€ + 5400°E° + 105E" (70 ae ). If we write e2+8=P, c&=Q, this becomes aes e = ae &-x ei (oe BP? + 4Q° if (z, &)=401 (l= p) a +P TF ak (@ + Ey? | PQ? 15P!+ 8P°@+24Q 105P* — 90P*Q? + 24P7Q + 1920 + PY PQ , 946 PS— 1785 P°Q? + 45 PQ! + 240 P2Q* + 1920Q8 PAGE 10395 P® — 30240 P8Q? + 9765 P*Q + 540 P#Q® + 2880 P2Q* + 23040Q' sh - PMS eee (70a) EXPRESSING NORMAL CORRELATION, The convergents to the series in brackets, expressed as a continued fraction, are PQ PQ (BP? + 4Q°) + (6P! — 21P2Q! + 8Q) > PE + (BP? + 4Q)’ PE (BP? + 4Q") + (5 P! + BP2Q? + 24Q%)’ P2Q: (6P— 21P2Q: + 8Q) + PY (42P* — 120P*@' + 0. P?Q + 64Q") P2Q*(6P*— 21 PQ? + 8Q) + PQ? (60P* — 159 PQ? — 60 PQ! + 96Q°)’ + (90P* + 60P°Q? — 1017 P!Q* + 528P2Q)° RN To get rid of the exponential in (70), we have 2 Al 18 Sal yam eee yea at ( ae pee 2 4..), via a x x Be 2 A l 2 1 wa WOR EGY sf ze a=n(l—a (1+ 5S Sk See = eee T Fg x a wv z jee whence Similarly 2 10 Ven F-») (1+ 5 a EB oF Multiplying the two together, we get i e7Ke+) — 19€(1 —a)(1— p) {1 eet eee ee 10P — 32PQ? Tv (? Q Q° 74P4 — 306 P?Q? + 164! = Q , 106P* — 3604P'Q + 3732PQ Qe @E ~ 8162P§ — 49678 P!Q? + 76134P2Q* — 175400 +. and thence, substituting in (704), 5 i bad —-# Fe, )=30-0) 0-9) | at Cope U| where ee 2P?+ 4Q? ‘ve! 10P! + 4P? 20)? + 240+ | T4PS — 54P*Q? + 20.P244 + 192.0% 7 PQ P2 Q P3Qs 4, T06Ps — 1264P%q: — 48P*Q' + 168P°Q" + 19200 ess meclezey — 23074 P2Q? + 6078 P'Qt+ 148 P3Q%+ 1920 P2Q3+ 23040Q" | Pe o0n The successive convergents to U are ; PQ PQ? (P? + 2Q*) + (BP! — 6P?Q? + 4Q') "PO + 2(P?+:2Q)’ PQ*(P? + 2) + GPt+ 2P2Q? +124)’ P2Q (3P*— 6P2Q? + 4Q4) + PQ (21 P* — 51 PsQ? — 6P*Y + 32M) P2Q!(3P*— 6P2Q? + 4Q°) + PQ (27P* — 51 P!Q? — 22 P24 48Q")” + (24P® + 54P°Q? — 336P*Q! + 136P2Q° +.96Q") 40 Mr SHEPPARD, ON THE CALCULATION OF THE DOUBLE-INTEGRAL 13. The choice of the suitable formula for calculating V=/f(#, &) in any _par- ticular case depends mainly on the value of a*+&, If this is small, we can use (53), or the corresponding formula obtained from (874) and giving f(#, &) in a series of ascending powers of # If a*+& is very great, we must use one of the continued fractions considered in the last section. If #7+& is only moderately great, we must use either (57) or (61), according as & is greater or less than 2; or we might use the formula derived from one of these by means of (37 A). A difficulty, however, arises when x and & are moderately great, and nearly equal, since the convergence of the series in (57) or (61) is then intolerably slow. For use in such cases, the formulae require modification. (i) First, let & be greater than x Then, substituting from (59) in (57), we have F(a, §)=—4a(1—p) 1 any ME a a 2 p- Het) i Fa NG Tee sh) f aah ised it: tl eRe SRR aoe eae (73) But = a + & —...=tan™ @, 3 5 7 rh fete =]. tan. ddd cine a ieee (74), =4(1 + 6) tan“ 44) whence & = = + aS = tan w/£ | 23 | a / ES 7 | £5 \ ; ce ee Frcieilaa eeaie saabnc (75). Thus we can take out different portions of the series in (73), and convert them into finite terms. The radius of convergence of the series which is left is the same as before, but the terms are individually less, and their initial convergence is more rapid. The resulting formula is therefore more convenient for calculation. (1) If we only take out the terms constituting $(7—D), the result is the same EXPRESSING NORMAL CORRELATION. 41 as if in (57) we substituted for G,, G,, ..., in terms of G;, G;, ..., by means of (60). Thus we get fla, B)=—4a(1—p) tre tere) {tan W/E + GE — GE 4 EGE ees) (2) To find the general formula, write a ech aie F()=| pate rb F.(#)= |" F.($). dag F, (@)=[ Pi($) bd 0 P tettttesreteeeeeeeeeeees (77). F,(6)= | Fra($)- be It is easily seen that F,() consists of a term 1 ———— (1+ ¢*)? "tan ¢, nono $ $ together with terms involving odd powers of ¢ up to $¥”-*. Hence, writing 1 a ae | an eX Re (78), where 67 (G2) =) eeasac ape candbe doconeccnnccne ven CEcreeae eae (79), we have, from (77), y 1 2) p—2 Wp (b) = eps (seus ek Ginza (@) cospanncooesacsoenoce (80), by means of which the successive values of ,() are easily found. It will be noticed that all the coefficients in y,() are positive. Also, by expanding 1/(1 + ¢*) in ascending powers of ¢, and performing the successive integrations in (77), 1 Fy (8) = 3 ea es Pt ee 5... 2p—1 3.5.7...2p+1 5.7.9...20+3 WO, XODK, IRArin Te 6 42 Mr SHEPPARD, ON THE CALCULATION OF THE DOUBLE-INTEGRAL Now, by integration by parts, ro eho? ) eae [ rea te= [or ro ag 0 = es" Fg) + e[” eH" bF, ($) db = 8 (F, ($) + OF. (6) + BF (G) +... EP Fy (G)} + BY |e SE, ($) a Substituting from (78) and (81), and writing @ for &$ under the sign of integration, we have pe er dpa eta easey+ i eatey [ ee peem et [14 FeO tots Ghd + or. 1 p—2 2\p—1 = 2 4 2 + py pent + 9 | ta 8 Gh @) + VAG) HEV) + FEM WOT} i 1 a ga [ae ge qg—-—____! __g¢s i ei” oP+? do + sy ne GaN 0 3 .2p4+1 0 1 fl ee Se. . 29-1 Hence, taking ¢=~/£, and substituting in (404), f(«, £)=-}a(1 —p)-+ 2 eteere {| $5 +E) +5 4 (+h +... : a2 2 1 ee oo +6) | tan 1 e/E — [Wi (@/E) + Ee (@/E) + Eps (w/E) + --- + EP Wy (#/E)] + [P,&> - P,é? + PES - er wieisetuetpen wenaedeeemecateee (S4), where, with the notation of (57)—(59), 1 a —— G, 1.3. D2 1 [= Gays 3. S.ieepase Joka cil ebee UNE (85) P : Cpa | The second expression in square brackets in (84) will be an algebraical function of x and & of degree 2p—2, containing 4p(p—1) terms. EXPRESSING NORMAL CORRELATION. 43 (3) Suppose, for instance, that we take p=5. Then F,($)= tan ¢ F,($) =) (1+ ¢') tan 6 — 5 2 1 1 5 F,($)=5-q (1 + gy tan" $— 5 $- 579 eG) 1 2)3 = 1 3 a 5 TiC ammo at o 4g?~ 19% 240% Ree re ee eee ee ee Se SL | F(¢)= 946.9 (1 + PY tan} — 353% 1750? s7E0? — aas0? ) and therefore 1 1 ates yates earns 1 ee F(a, &) =—4a(1—p)+ Zevere{ly +a l@t Oh) ty get +54 eet & where or, expressed as series, 1 tog g BO tey | tong 1 Be Ae tals. = ‘11 a 5+ (qet gl) +(sa0% + peel tage) pele 75, WS. Meee eee +\aas0” * 5760" > * 11527 © * 384 )| wk + [PE — PE + Pye. 1 Heptertheed coh nh (87) 1 Pi=73.5.7.9 1 Y ean PETER (2) ER Ae (88), 1 Y Ps=57.9.11.18 Pp Fines a8 a SIE Os a SASS Oe 111s | LeSeam MoS 15 Pp 3 pe gi ee eon) © Sop 7c OaThasels aenamOmuIse is.i7 y\ .0¢aGy. P, ps git as = & Will 1H 2 oe eT NaI nly? Gin 1s En LL es ce Ones — 44 Mr SHEPPARD, ON THE CALCULATION OF THE DOUBLE-INTEGRAL The values of P,, P;, P;..., for any particular value of «, may be obtained from those of Gj, Gy, Gs... by means of (88), or they may be calculated successively, by the relation qmn+s 1 15 3= Qn = 1 ; On ci 1 =e In 49 + n= 1 Bers ee (90), starting with the last coefficient which we require, and working backwards. Or the last expression in square brackets in (87) may be written We IS —y ol CH k2) at el 4 £3) eral Bqocopsccocododoooopaonusosoce (91), where 10 / Ls az 1 x? 1 Hid ) gies EC Eb RATE I oe (sty re ee ) E = SiSeTeOetes1s\ 2 1b) Lae) See (92), a Ae \ Ps — arom Ses (eae ape =! the relation between the coefficients being 1 ( Pa) 2 Pana = =a \2n ZEKE asia 2 +2 Pas} aivla'sttoieis sloleceleleievateisters (11) Next, let & be less than « Then, expanding the integrals in (62) in divergent series by integration by parts, and substituting in (61), we find F(a, £) = 41 = a)(1—p)— Lerner (E28) wet ae 2 SC: é = cee acne (94) The process to be adopted is the same as before, except that instead of removing sets of terms we introduce new sets of terms, so as to make the coefficients of successive powers of & tend to the complete series i eee at ofl ere RO tere Code SIC; (95), th ths oe a Ns each such series being set aside as it is completed. EXPRESSING NORMAL CORRELATION. 49 (1) If we introduce the terms Ie areere ih E/a — 3 ES/a + 58/2 —... = tan &/2, the result is the same as if we substituted from (64) in (61). Thus we get ee) — }(1—a)(1—p)—~e- da 8) {tane/ o£ HE+s 5 hE — 5 Het +. SORnORen EEC (96). (2) To find the general formula, write FAD) AD ae Hn LUD) weet eee cee stoltjetinc 2 peiclesielielss desc (97), so that ty (o) = a ena=D + G?)? cot h — PPP Wy (1/P) «2. -eeeeeeee (98), where y,(1/¢) has the value given by (79) and (80). It may be shown that the values of fp(@) are given by the relations pie A =|, ree oo ry Ora : anal as ts (¢) = ere — bf, (6)| a (99). aa | es ee aoe as pag Me Oh a If we write EC ae GC Pit cenene ee Seb ache RPE (100), then we have, from (98), : 1 lO) SA eps (1 + £?)P 1 cot 1h = Xp(P) .....ceseeeeeeeees (101), and it may be shown by (99) that = ee 2+ bxpa(G) -----(102) ee (S05 lapse 294.6. dpe Mae a ar This may be used as an alternative formula to (80), and it serves to check the values of ,(¢) obtained by that formula. Also, by expanding 1/(1 +?) in descending powers of ¢, and performing the successive integrations in (99), or directly from (97) and (81), , 1 1 il AG ee ececeetpeara fh =a ————— Gas = = Ot (103). 3.5...2p—1 Bawa ll coo 47) 4e Il B60 68) con Ase 6) 46 Mr SHEPPARD, ON THE CALCULATION OF THE DOUBLE-INTEGRAL Now, by integration by parts, * eh Pe? ai “18 £1( ihe i 1+¢° dp= oe Si (¢ g = en he A (o)- an e—t#¢" dd = e[ eW hes? fi (o) do =e | A (pb) + & fa (b) + Gifs ($) + --. + EP fo ($)} 1 1 1 pris =—( 24-8. EP 3236? (Sos at tase PoE... Spa Vee ? —e| ST TREN Ee ee Mee nee Te ghee (104). Cs) Substituting from (101) and (103), and writing @ for &$ under the sign of integration, we have ® phEgt g1+¢ = ei? {la +5 +5 P(1+6) +5 GE Ut eye + wae == (bas wor] cot™ 4.6... Ip—2 — [xa () + Exo (f) + E*Xs ($) +--+ + EP MND er / 1 le 1 2p 18 dg BE tage hy ang fee home an £ We ae 1 ee pen | e- 3” 6-2 d@ — ees — E278 Is e-3” 9@-4+dO + Maen [Spon e2osal & SIO ssepse dl) ) Hence, taking ¢=/&, and substituting in (39 A), fle, D=4 0a) (1 p)— 2 et {14 5+ Ete Hert. Si Le eS S ig avec epee adel Ela = [xa (a/E) 3 E*y. (w/E) =F Etys (a/&) Foe. 2S Xp (x/€)] / 1 1 1 = = 3 ee a) 2p—1 H..(E + grat pemeeen oe ea eae + eae re Hers — i HE? 4+... |f (106), oy 2p+1 SET omaop ans ee where H_,, H,, H;,... have the values given by (62)—(64). EXPRESSING NORMAL CORRELATION. 47 (3) Thus, if we take p=5, we have filg)= cots fa(d)=5 (1+ 62) cot 6-5 4 ; 1 aye 5 1 (QS Go EO sya re Oren (107) 1 : il 1 1 EC) mama os or Pao? 1g? = age | 1 cries 31 (Se WH pe 1 DO) ggg + PON gaan? 570% — 1152” — 384 ® and therefore F(a, €) =1(]— ) = 2 =42 fi es peel 5 1 1 aro) p+r/ ae ests se imaeney rimawsey coe) 1 PeT; U 1 2 35 2 i, a n2 2 11 4 -|5+(5" Fa eRe igs teaye) < a ae we as 78 Bot a F) + [QE — QE + Qe I POI 25 Sets MO eR OR (108), where Q=s 57 9.11% G57. r 11 137 Nil Ot eR a (109). 1 Or, if the last expression in square brackets in (108) is written liga (S/ a)? — 45 (e/@) 2 19s (EG) a acid ete ences enseicasenaisi ve «= (110), and if we take SD ia SOL aur faa (111) (= N/io SETS Reha Parana ; = age) 1 2) 1 Po) 2)2 1 72 2 1 2 2 = = Za 1+ 5(@ +) +54 eat gmeg reg lee ie) Womaeramele te): tan™ &/x 48 Mr SHEPPARD, ON THE CALCULATION OF THE DOUBLE-INTEGRAL we have q my we Eb \ JT sD een gC mae a 1 / ge P ; Par acenaais ot ee es gels (112), 1 a eh eS 1 ( Plas: ee) by means of which the coefficients can be calculated. (iii) In calculating the value of ¢(a, y, D), without having a table of f(a, &), one of the above formulae would be necessary if either y/z or a2/y were nearly equal to cosD+sinD. But if x and y were nearly equal, and if D were nearly equal to tn, both f(x, £) and f(y, 7) would require to be calculated in this way. We might then simplify the calculation by the method adopted in § 11 (ii). Thus, if we required to use (87) in each case, and if we wrote it in the form f(a, &)=—ha(l—p)+ FP (@+2®). = tan 2/E + $;(a, &), we should have also s aaa: JAWS n)=—tB(L—o)+F(y+7).— tan y/n+ psy n), where o corresponds to 7 in the same way as p to & Hence, by (24), (26), and (47), a~—D (x, y, D)=—4a(1—p)—48(0 —¢)+ F (a? — 2xy cos D + y’) cosec? D} . =— p(z, y 4 p)—- y y on +4 \ds(a &)+s(y, 7)}- Thus the calculation of 1/7. tan7a/E and 1/7. tan y/n is avoided. A similar formula can be obtained for using (108) in both cases, or for using (87) in one case and (108) in the other. We might also, of course, take out the common factor 1 — e-hH2et+) — 1 e-hytr) = 1 e—} (2—2zy cos D+ y*) cosec? D T 7 7 from $;(#, &) and ¢;(y, 7). Calculation by Quadrature-Formulae. 14. For dealing with the second class of cases mentioned in § 2, a quadrature- formula can be obtained from (13). Cases of this kind arise in questions of marriage selection (or “assortative mating”*) with regard to particular faculties or tastes, and in questions of transmission of particular faculties, tendencies, or defects from parent * Karl Pearson, ‘‘ Regression, Heredity, and Panmixia,” ubi supra, p. 258. EXPRESSING NORMAL CORRELATION. 49 to child*, or their correlation in the individual+; the characteristic observed being one which is not capable of quantitative measurement. The general result of tabulating a collection of statistics of this kind would be expressed thus :-— Not-X x Total Not-Y J2 Q P+Q 4 R S R+S Total P+R Q+8S P+Q+k+S We must assume that the existence or non-existence of X or Y depends on the greater or less development, according to the normal law of variation, of some physiological factor, and that the variations are also normally correlated. To find, on this assumption, the divergence (D) of Y from X, or the coefficient of correlation (cos D) of X and Y, we divide the number in each compartment by P+Q++/S, and treat the arrangement as equivalent to:— Not-X xX Total Not-Y 4 (a+B)+(a, y, D) | 4(1-a)-¢$(a, y, D) 4 (1+) Total ¥(1+a) 3 (1-a) 1 | The values of a=(P—Q+R—S)/(P+ Q+ R+S8) and of B=(P+Q—-—R-S)/(P+Q+R+S8) give us the values of « and of y. Adopting these values, we have to construct a table of values of }(a, y, D) in terms of D; and then from this table we deducef{ the value of D for which ¢$(a, y, D)=S/((P+Q+R+S). The required table is readily constructed by a quadrature-formula. Thus, if we calculate the values of C= 40. eH 22Y COD HY) COSC D a. seesecesseessensenees (113) * Galton, ‘‘ Natural Inheritance,” passim. Prof. Pearson has for some time been using a method, but + Yule, ‘On the Association of Attributes in Statistics,” a different one, for determining D from ¢(z, y, D): see Proc. R. S., vol. 66, p. 22. Proc. R. S., vol. 66, p. 157.] + [Since the present paper was written, I find that Wit, IDK TBA Ie 7 50 Mr SHEPPARD, ON THE CALCULATION OF THE DOUBLE-INTEGRAL for D=470, D=370,... D=(p—4)70,..., Legendre’s central-difference formula* gives us, for D/wr=p8, = 1 na. SGlo 27859. ‘ (2, y, D)=C- (1 tag ao a where of is the sum of the values of € from D=47@ to D=(p—4)7O, and & cf, d4.cf,... are the second, fourth, ... central differences of of; C being 4(1—a) or $(1—8), which- ever is least. If we have no tables of f(2, &), we ought to find ¢ for all the values of D from $70 to 47—478, or from $7+470@ to t—476; the calculations being checked by od (x, Y 0) =C Beta er eae =a) penne ate see (113), $ (2, Y 7) =0 If our tables of f(z, &) have been formed, we can obtain a rough value of D by a graphic method, and calculate ¢(#, y, D) for this value from the tables; we then only require to apply (114) to the formation of a table of (a, y, D) for neighbouring values of D. To get the rough value of D, we adapt a method explained in the paper already mentioned+. Taking any point O on a plane surface, we divide the surface into n? areas, symmetrically distributed about O; each area being the base of a prism whose volume is 1/n? of the volume of a normal solid whose axis is at O. Drawing a line at a distance x from QO, there will be $n?(1—a) areas on the one side of the line. Then, keeping another line at distance y from O, we turn it round O until the number of areas included between the two lines is n?$(#, y, D). The inclination of the two lines gives the value of D. 15. In the absence of tables of f(#, &), we can also apply a double-quadrature formula to the treatment of the first class of cases mentioned in §2; but it must be admitted that the process is a tedious one. Having found the values of ZL, M, a’, b%, and abcosD, from the data, we take Z and M as our zeros, and a and 6b as our units of measurement. The values of e=(X—JL)/a and of y=(Y—2M)/b will proceed by constant intervals, which we may denote by h and & respectively; and, to test whether the distribution is normal, we have to compare the data with a table giving the corresponding values of 1 z+h pytk é m an | e—4 (2? - 2xy cos D+y") cosee"D dlasdiy x y for the different values of w and y. To form this latter table, we must first calculate the values of c= «ail ead (x? —-22'y' cos D+y'*) cosec? D 27 sin * CO. W. Merrifield, “Report on Quadratures and Inter- + Phil. Trans., vol. 192 (1898) A, pp. 141—2. The polation,” British Association Reports, 1880, p. 358. The formula for the probable error in D, as calculated by c-notation is explained in a paper recently communicated the method of this section, will be found on p. 149 of to the London Mathematical Society. the same paper. EXPRESSING NORMAL CORRELATION. 51 for each pair of the intermediate values w# =a+th, y’=y+4k. Each of these is hk times the corresponding ordinate of the solid of frequency, the planes of reference being taken at right angles. For each column separately, we take the differences of ¢, and perform the operation 1 24 Le elles 1+ & — G9 ot on each value of ¢«. This gives a series of quantities, each of which is h times the portion, included between planes at distances y and y+hk from O, of the section of the solid by a plane at distance #+4h from 0. Arranging these in a fresh table, and applying a similar process to the rows, we get the volumes included between the sets of planes at distances ...z, wx+h,... from O, and at distances ...y, y+h,... from 0. To illustrate the method by a simple case, let us take D=47, and suppose that the values of z, and also those of y, are 0, +°4, +°8, +12,..., so that h=k="4, We then have to calculate the values of ex 16 paw ey ie EVE for # =+°2, +°6, +1°0,..., and y’=+°2, +6, +10,.... The calculations are checked by observing that the sum of each column (or each row) is the corresponding value of ae ee”, N 2ar the sum of all the columns being unity. Taking e to five places of decimals, and multiplying by 100,000, we have first the following table of ordinates :— 52 Mr SHEPPARD, ON THE CALCULATION OF THE DOUBLE-INTEGRAL r=|-4:2 -8|~s4|—s0|-26/-29 ~1:8)-1-4 -1:0 nen 6 10 | 1-4] 1:8] 2:2] 26] 3:0] 3-4 err el Ele el a Wale -3°8 1.) clin alin? yt dumsel Die pe ~ 3-4 Tl et |e Sd [aes 2 Mat Re) 7 al 9} 1 - 3-0 Tess 7) wel 24) 81) 38) 27 ao) Ud) Th, 2) aa - 2-6 2| 5/|15] 32| 58| 85| 100] 94} 72) 45] 22) 9) 3) 1 _o9| 1| 2| 8 | 24] ssliizliss| 246] 260| 221] 152| 85] 3s] 14 4] 1 -18| 1 2} 9 | 31 | 85/188}]339} 493) 578) 548] 420) 260) 130) 52) 17) 4) 1 -14 2| 9 | 32 | 100} 246) 493/ 796 1039 1096 934) 643) 358) 161) 58) 17) 4] 1 -1:0 font! 7 | 27 | 94] 260} 578 |1039]1509)1772|/1680|1286| 796} 398) 161} 52) 14] 3] 1 - 6 1 4 | 19 | 72| 221 | 548 |1096)1772|2313/2440/2079]1431| 796) 358/130] 38) 9] 2 - 2 2] 11) 45) 152} 420] 934/1680)2440/2863|2714/2079/1286) 643) 260/ 85} 22) 5) 1 2 1 5 | 22] 85| 260} 643]1286/2079|2714)2863)/2440/1680} 934) 420|152} 45) 11 | 2 6 2 9) 38/130] 358] 796)1431|2079|2440/2313/1772)1096| 548 | 221| 72) 19); 4] 1 1:0 | 1 | 3] 14) 52] 161) 398] 796)1286/16801772)1509/1039| 578 | 260) 94) 27} 7 | 1 1-4 1 4) 17} 58} 161) 358) 643) 934/1096/1039| 796) 493 | 246/100, 32 | 9| 2 | 18 | 1 4) 17] 52} 130) 260} 420) 548) 578) 493) 339) 188] 85) 31] 9}] 2 L 2:2 | 1 4) 14) 38] 85] 152) 221) 260) 246)188)117] 58) 24] 8} 2) 1 2-6 | 1 3] 9) 29) 45) 72| 94) 100} 85] 58) 32/15) 5) 2 30 TS) Bi TT) 19) 27) 32h Si |) 241) 5) oe ee EXPRESSING NORMAL CORRELATION. 53 Then, taking each column separately, and performing the operation mentioned above, we get a table, half of which is as follows :— i 2 6 1:0 1-4 1:8 2°2 2°6 3-0 3-4 3°8 4:2 y=—44 — 4-0 = 36 1 ~ 3-2 6 2 1 i —2°8 23 10 3 1 —2-4 90 4] 15 4 1 — 2:0 269 134 55 19 4 1 -16 654 367 167 60 17 4 1 =f-9 1294 805 405 165 55 15 3 1 Sos 2073 | 1432 800 362 132 39 10 2 = ek 2693 | 2067 | 1282 644 262 86 22 5 1 0 2838 | 2418 | 1667 928 419 152 45 11 2 “4 2424 | 2296 | 1757 | 1086 543 220 72 19 4 1 8 1681 | 1766 | 1500 | 1032 572 257 93 27 7 1 1-2 945 | 1102 | 1039 793 491 245 99 32 9 2 16 430 558 584 496 340 187 84 32 9 2 1 2-0 159 228 266 250 190 118 59 23 8 2 1 2-4 ; 48 77 99 103 87 59 32 15 5 2 2:8 12 20 29 34 32 24 16 ai 3 1 3:2 2 5 7 10 9 9 5 3 1 1 36 1 2 2 2 2 2 1 1 4-0 2 1 4-4 Completing the table, and repeating the process with each row, we get our final table, half of which is shown on the next page. The entries in this table are checked by observing that the sums of the successive columns must give the differences of 54 Mr SHEPPARD, ON THE CALCULATION OF THE DOUBLE-INTEGRAL 4(1—a) for «=0, “4, ‘8, 1°2,..., and also that a double median division must give 10°. (7 — D)/2m = 33333 in two compartments, and 10°. D/2a7 = 16667 in the other two. 3°6 4:0 4-4 2060 2673 2813 2403 1668 Total |15542 13272 1:2 16 20 2-4 2°8 3°2 3-6 4:0 4-4 1 | Sale ieaal Tah pceealiMnd B74| | agu| pecpal ea yah | ey I) eI TURN Tigi oie, Sie) Se) goo | 371| 137| 42/ w| 2 1289 | 655| 271/ 91] 23/ 6 | 1 1668 | 939| 429] 159| 48] 12 | 92 1751 | 1092 | 553| 997| 76| 20 | 5 | 1 1491 | 1082 | 580| 2963/ 98| 299 | 7 | 2 1032 | 791] 493] 249] 102| 34 | 10 | 2 580| 493| 340/ 189| 86) 32 | 9 | 2 | 9 263 | 249 | 1891 118 €0| a4 | 1% | 9 | a 98| 102} 86| 60| 33| 16 | 5 | 2 99.) 34] /3g0\| ewe te || derma se Wad Z| 16 |’ ) (a, x,7—D) a—D eee ie D }(1—a)—¢(2,2,D)| 2 Gang! Be ONE arti (ceo 2D) Y, to M + (x, 2, D) pa —4¢ (a, 2, 7—-2D) — $ (x, 2, 7 — 2D) — > (, x, r—D) + p(x, x, r—2D) a —D D —(l-a ees ese) | Pe alee a D) M to Y, +(x, a, D) ka — > (x, x, r—D) — > (a, x, r—- 2D) —4¢ (x, x, r—2D) + (x, x, r— 2D) i p(a,a,7—-2D) |4(1-a)—¢(@, 2, D) Y, to > (a, x, r—D) $ (a, x, D) £(1-a) — > (x, x, 7 —D) —4¢ (a, x, 7-2D) Total $(1-a) da ka 3(1-a) 1 56 Mr SHEPPARD, ON THE CALCULATION OF THE DOUBLE-INTEGRAL Hence, in order to compare the actual table with the corresponding theoretical table, it is only necessary to have a complete table of values of (a, a, D) for values of D from 0 to 7. The value of D to be used in making the comparison is to be found by a double median classification of the data, which will correspond to Values of X Values of Y Below L Above L Total xr—D D Below Jf on on 4 D ax—D Above M on = 4 Total 3 3 1 17. For applying this method, it seems simplest to take a=, so that X,, X,, and Y,, Y., will be the quartile values of X and Y. We have then x= ‘67448 97501 96081 74320 22, w= 45493 64231 19572 75194 25, ee” = "79654 77421 05315 68819 36, e* = 125541 75313 54680 35601 44. The formulae for calculating V=¢(a, x, D) in terms of D are obtained from (58), (57), (76), and (87), by writing eae where «=tan$D. EXPRESSING NORMAL CORRELATION. 57 From (53) I find Wares +° 06476 08650 54292 19040 « — [2] 286 21742 25076 89463 x3 +'[38] 10 53965 83660 94520 « — [5] 42315 22750 04354 x7 + [6] 1485 70373 24375 «° — [8] 45 82977 18023 + [9] 1 25493 05017 «* — {11} 3082 51001 «® + [13] 68 56588 x” — [14] 139255) <2 + [16] 2601 x — [18] 45 + [19] Me rete pee cen Sy fa (116), where the numbers in square brackets denote the numbers of 0’s after the decimal point. The accuracy of this formula can be tested by putting ia—ale which should give V=4(1—a)?=-0625. The formula gives V to ten places of decimals for values of D (x 180°/7) from 0° to 139°. For larger values of D, denote the coefficient of (—)" "> in (116) by Ain: Then, the logarithms being to base 10, log A, = 2°81181 26412 83 log Ay =17°41511 37400 log A, = 3:37331 19263 85 log Az; = 19°65270 22627 log A, = 402282 65338 25 log Ay; = 21°85891 72699 log A, = 6°62649 66801 26 log Ay, = 2203580 97181 log A, = 717193 22144 07 log Aw = 2418518 82998 ae log Ay, = 9°66114 76947 91 log Ay = 2630865 79320 log A,,; = 1009861 96751 72 log Ay, = 2840765 10756 log A, = 1248890 44949 79 log A,; = 30°48345 32801 log A,, = 1483610 80854 49 log As, = 3253722 41227 log Ay = 1514381 13042 91 log Ag = 3457001 44750 8 Von. XIX. Pan I. 58 Mr SHEPPARD, ON THE CALCULATION OF THE DOUBLE-INTEGRAL These values will give V to ten places of decimals for values of D (x 180°/a) up to 152°; or even further, if we calculate the logarithms of the terms up to Aw«®*, and continue the series by extrapolation. Also, from (57), (76), and (87), cee 5 CRS ay eS ONO he Rakes SN Re RR (118), where Die ite ae p=] edt Nao t= «x x 47693 62762 04469 87338 \ O= (14+ «*) x 22746 82115 59786 37597 © log,e= (1+ «*) x 09878 81890 88816 70245 and © may be expressed in the following different ways :— Q, = "37127 26047 12654 07100 «7 — 11641 78461 80988 23277 x + 06801 75312 12357 95927 xn — 04786 99193 36564 50013 «7 + 03688 32837 28088 29524 «-°* — 02998 14802 15355 31680 «™ + 02524 83547 18124 64219 «8 — 02180 24423 80443 78105 «* + 01918 23496 19371 66744 «™ —°01712 33977 97235 02851 «® +°01546 29781 35355 67215 «™ — ‘01409 57160 88647 79770 «8 +°01295 03454 89396 36594 nc — ‘01197 69505 67632 08409 «7 +°01113 95326 57167 82205 x — 01041 14786 88817 22561 «™ + 00977 26911 79696 49749 x-* — ‘00920 77102 05900 03240 «™® + 00870 44492 84487 45848 «—” —°00825 33232 14922 12027 «® ee eee OOO EO = EXPRESSING NORMAL CORRELATION. 0. =(4— D)/20 + 05296 27185 28863 39946 «7 — 01031 45507 86391 34226 4-3 +°00435 55539 75599 82496 «> — 00239 70784 53165 83277 «7 +°00151 55185 96555 99840 4-9 — 00104 42178 35010 71029 «—™ +:00076 29788 57833 05207 «-™ — 00058 17833 01524 40295 x» + 00045 82386 67383 98088 «-” — 00037 02458 92824 99317 x + 00030 53645 07556 11636 x — 00025 61558 19787 33372 «-® + 00021 79500 42044 73907 «- — 00018 76955 23787 98514 «—” + 00016 33296 85312 97131 «-” — 00014 34178 44178 81685 «—™ + 00012 69370 52914 96198 «* — 00011 31420 29220 29892 «-™ + 00010 14793 87628 28114 «-* 00009 15312 61491 59023 «—™ =tok OU Ciie atstasai sola cisleleialels nics crarelera sterol etcetera re sec neach one mee (Dye 60 Mr SHEPPARD, ON THE CALCULATION OF THE DOUBLE-INTEGRAL 0 =(1+ 044024 4O0*+ 1.0%). (7 — D)/20 — 08759 83387 94983 86404 « —-01007 28244 93693 26300 « —-00075 45919 58375 91295 ~ 00003 55076 34252 22487 x’ +6] 6182 65204 77409 «2 cig 473 32798 10453 x + [8] 94 32180 42602 «= — [8] 27 66224 98599 «7? + [8] 10 16774 77228 «= — [9] 4 34927 49796 «™ + [9] 2.07675 01899 «® = 70] 1 07843 50295 x + [10] 59842 55655 x" — {10] 35044 18869 4” + [10] 21459 37438 «= — [10] 13645 09043 «-™ +11] 8960 40792 «~ — [11] 6050 47405 «” + [11] 4186 38337 x» —-(11] 2959 55817 «—™ oe: RERRRER ER RETR t po pcobntn cOESOCCOC LARGO aE Aen a eeeRe SEERA Saade (122). EXPRESSING NORMAL CORRELATION. 61 The coefficients in this last formula may be tested to 12 places of decimals by taking «=1, which should give V=:0625. The general accuracy of the three formulae (120), (121), and (122) may be tested by giving « a particular value, greater than 1, and comparing the results got from (118) and (116). If we take K=2, we have* p= °82265 64493 47648 05654 . © = 1:13734 10577 98931 87986 i e-° = 32067 05336 68834 82010 D/2m7 = °35241 63823 49566 72582 | \ From (122), taking a few additional figures, I find QO, = 17289 55703 70922 67141, which agrees with the value found from (120) or (121). This gives V =-01110 66271 56733 37258. The formula (116) would give the same value, though to a less number of places. Tables I. and II. in the Appendix are calculated from the above formulae. Table I. gives V to seven places of decimals for values of D/w from ‘00 to ‘80. For larger values of D the differences of V become unmanageable, unless we take D/a by smaller and smaller intervals. The values can however be found from Table II, which gives ate) VY to seven places of decimals for values of D/w from ‘75 to ‘90. It is not necessary to continue the table beyond this last value, as D/r=-90 gives V=-0000002; but the differences shown in the table carry it to ‘91, and it could be continued with sufficient accuracy by extrapolation. To find V from Table IL, the quantity given by the table has to be multiplied. by en} +e2) — 1 ()—"19757 63782 (1+ cos D)_ The values given are correct to seven places. They were found originally to ten places, partly by direct calculation, partly by interpolation, and partly by a quadrature- formula based on Ig et a e7 2? sec? 20 ay, 2a D V These ten-place tables were tested by taking differences, and where the seventh (corrected) figure was doubtful the value was calculated directly. The differences given in the tables * For a first approximation, we find, from the ordinary We therefore use the formula tables, 5D =Tr/2+ tan (-075) + tan (132/125351). D/27="352416..., Since (-075)?=34,+4.s},, and 132/125351 is very small, whence the calculations are easy. The angle r—D is an angle of 5D=7r/2+:07591.... the triangle whose sides are 3, 4, and 5. 62 Mr SHEPPARD, ON THE CALCULATION OF THE DOUBLE-INTEGRAL are taken from the ten-place differences; but of course the final figure, where doubtful, has not been specially tested. ; 18. As an example of the use of the tables, take the following data*:— Correlation of chest-measurement and height in American-born white men. | Girth of chest at expiration, in inches Height in inches | » Total Under 29 29—31 31—33 33—35 35—37 Above 37 Under 61 456 535 347 219 93 24 1674 61—63 1451 | 3737 2953 1234 390 106 9871 63—65 2372 10002 13592 8245 2303 475 36989 65—67 | 1698 11385 | 26859 24787 9366 2062 76157 67— 69 729 7179 27497 36207 18119 4719 94450 69—71 229 2789 14053 | 25409 16638 5473 64591 | } 7i—73 63 662 4092 oS 7991 3375 25500 | | Above 73 8 113 744 2084 2172 1267 6388 | | | Total 7006 36402 90137 107502 57072 | 17501 | 315620 The tabulation, as is so often the case with official statistics, is incomplete, and we cannot therefore apply the best method of determining the means, mean squares of deviation, and mean product of deviation. The preliminary question, whether each distri- bution separately may be regarded as normal, would also on this account have to be treated by a special method, the discussion of which would be out of place here. To avoid this difficulty, we may regard the measures XY (chest-measurement) and Y (height) as being functions respectively of other measures X’ and Y’, such that dX/dX’ and dY/dY’ are always positive; and we can then test whether the correlation between the distributions of X’ and of Y’ is normal, on the assumption that each distribution, taken separately, is normal. Denoting the quartile values of X and of Y by X,, LZ, X., and Y,, M, Y, respectively, and applying a suitable method of interpolation, I obtain the following, as approximately representing the actual distribution :— * Statistics, Medical and Anthropological, of the Provost-Marshal-General’s Bureau, compiled by J. H. Baxter; vol. 1. (Washington, 1875). EXPRESSING NORMAL CORRELATION. 63 | Girth of chest at expiration Height Total Below X, X, to L L to X, Above X, Below Y, 38049 20140 12965 | 7751 78905 Y, to H 20481 22663 20232 15529 78905 M to Y, 12930 20498 22901 22576 78905 Above Y, | 7445 15604 22807 33049 78905 Total 78905 78905 | 78905 78905 315620 The sixteen numbers in the interior part of this table are connected by seven numerical relations. By choosing D so as to fit the table, we are left with eight inde- pendent numbers, which are to be compared with the corresponding numbers for a case of normal correlation. _ These numbers will be slightly different for different methods of determining D. If D is found by a double median classification, the corresponding theoretical distribution will be :— Girth of chest at expiration Height ; — a j |) Lotal Below X, X to L to X, Above X, | Below Y, | 35121 21908 14536 7340 78905 Y, to & 21908 22396 20065 14536 78905 M to Y, | 14536 20065 22396 21908 78905 Above Y, | 7340 14536 21908 35121 78905 Total 78905 78905 78905 78905 315620 The numbers in this table are calculated as follows. The double median classification gives Below L Above L Below M | 101333 56477 | Above UM | 56477 101333 64 Mr SHEPPARD, ON THE CALCULATION OF THE DOUBLE-INTEGRAL Hence, by (20), we have From Table I. we find (a, «, D) = 1112765 ¢ (x, x, w—D) = 0232565 + , $6 (x, x, t—2D) = 0693129 and then, calculating the different expressions shown in the table on p. 55, and multi- plying by 315620, we get the theoretical distribution. The difference between the two tables is so great that we may safely conclude that the data do not represent a case of normal correlation. It is possible, however, that the latter are subject to some defect. They do not give the measurements of all the men examined; and the exact reasons for the omissions are not clear. It will be noticed that, although the actual table is very unsymmetrical in one direction, it is fairly symmetrical in the other. This symmetry would not be disturbed by the addition of a number of men at the lower end of the scale. 19. When the correspondence between the two tables is closer than in the example given above, we have to consider whether the discrepancies are such as might be accounted for by errors of random selection. It has been pointed out, in the last section, that, of the sixteen numbers in the actual double-quartile table, only eight are independent ; the value of D being supposed to be obtained by the double median classification. We have therefore eight numerically-independent* discrepancies, each of which has to be compared with the corresponding “probable discrepancy.” The most convenient arrangement for selecting the eight values is that shown in the following diagram (fig. 3), representing a case in which the correlation is positive. With the notation of §§ 1 and 18, we have, for the portions of the actual frequency- solid as shown in the figure, (DSS Ss Ga 2 WG, a] ) (2) SN XG coco] (8) = Ni? scone, 5) 201] (4)=N[—», x4; —0, Yi] sb naggbaboadbedamcoocas cosoose (123), (5)=N[—o, 44; —wo, M] } (6)=N[—o, L; —o, Y,] | (7) = N [X, Be Oe ie od Y,] (8)=N[-o, X; Y, ? | } where JV is the total number of individuals. * The independence is only numerical. The discrepancies are correlated; but the correlation may be disregarded. EXPRESSING NORMAL CORRELATION. 65 Also, since Z and M are the medians, Nicos IE - — cok MWe, co: Mi 2» ]| BOOT ODAC oR TOtS 124). Nios i: Me ol=N[L, 0; —o, MI) ra If we take D so that N[—o, L; —-wo, MJ=N[L, ~; M, ©)=N.(@—D)/27 ioe N[-o, L; M, wo)]=N[L, 0; —o, MJ=N.D2er f ae? we have for the calculated values, with which the actual values have to be compared, (1) = (4) = V9 (a, «, D) (2) = (3) =(5) = (6) = V¢ («, 0, D) =1N¢(a, x, 2D—-T) =iN-iN¢(2, x, r—2D) (7) =(8)=N$(a, «, wD) Taking the differences between the two sets of values as the discrepancies, the expression for the probable discrepancy in each case is rather complicated. It will be found* that the probable discrepancy in (1) or (4) is where Q = x = 674489750 ..., * For the general formula, see Phil. Trans., vol. 192 (1898) A, p. 152. Vor. XIX. Parr I. 9 66 Mr SHEPPARD, ON THE CALCULATION OF THE DOUBLE-INTEGRAL, etc. and . (72) — 272 PP. v+5a -p) H} 1 Pat ys + 7g —pP+5e (+p) V—V* V=¢(, z, D) ee eiae serch aeecoeee sce (128). H= as — (a, «, 2D—7) z= ai e-ssec*kD Qa . —_— — — OO FY The probable discrepancy in (7) or (8) is a similar expression. The probable discrepancy in (2), (3), (5), or (6) is where =2ne | oe + oy 50+) Ww} toate a ‘ bag gtoy ao tag? +78+ 8) re +1. —9— 298) W— W W= ¢ (a, 0, D)= 30 («, xz, 2D—7) a © [rr ese ao x cosec D é6= Vt eo” a 0 Z= A oo-w cosec? D 7 ST It would not be difficult, by means of the tables in the Appendix, to tabulate the values of ®? and ®* in terms of D/z. APPENDIX. TABLES OF V=¢(a,2,D) IN TERMS OF D/z, FOR a=}. TasLeE I. VALUES oF V IN TERMS OF D/a (D/r=-00 TO °80). A A? | As | as | A Az | a3 D/m |180° x D/x V (2) (+) | (4) 1 C4) | Die [180° x Dix V (5 We) 00 | 0 0) -2500000 Bae OP #00|, 72°70 | 0961665 | g.179 | 280] 4, | ‘Ol 1 48 | 2460173 | so.50 4 | 0} 41] 73 48 | 926486 | 5,0... | 294 | 4, 02 | 3 36 | -2420351 | o,15 9] 4| 0] 42] 75 36 | 0891601 | 5,,7,| 308 | 4. 03 | 5 24 | -2380538 | go999| 13| 5 | 0] 43] 77 24] 0867025 | 445.5 | 324 | 1, 04 | 7 12'| 2340738 | 5575) | 18] 5 | 0] 44] 79 12 | 0822772 | 49,5 | 339 | 1, 05 | 9 0 | 2300956 | g5759| 23| 5 | 04-45) 81 0] 0788859 | 4,.,,| 356 | 1. 06 | 10 48 | 2261197 | 4474,| 27 | ,| 0| -46| 82 48 | 0755302 | 5,,.,| 3874 | 19 O7 | 12 36 | 2221465 | 479) | 32) | 0} 47 | 84 36 | 0722118 | 5.79, | 392 | 49 O8 | 14 24 | 2181764 | s95¢,| 36] ,| 0] 48 | 86 24 | 0689327 | 5...) | 412 | of 09 | 16 12 | -2142100 | 59604] 41 | , | O| 49 | 88 12] 0656947 | gio, | 432 | 04 Hom 180) 2102477 | corr | 46 | = | 01) 50") 90 0625000 | s1494 | 454 | 93 11 | 19 48 | -2062900 | g4,5¢| 51] 5 | O} 51 | 91 48 | 0593506 | 5.519 | 476 | o4 12 | 21 36 | -2023374 | o5,,)| 56] | 0] 52) 93 36 | 0562488 | 5...) 500| 4. 13 | 23 24 | 1983904 | g9499| 61 | 5 | 0] 53 | 95 24 | 0531970 | 50993 | 525 | o¢ 14 | 25 12 | -1944494 | ,4,,,| 66 | 5 | 0} 54) 97 12 | 0501977 | oo445| 551 | og 15 | 27 0| 1905151 | g95,5| 71) | | 55 | 99 0] -0472534 | ,.04, | 579 | ag 16 | 28 48 | 1865879 | 59:9, 77] ¢| 0| 56 | 100 48 | 0443671 | 4.054] 607 | a9 17 | 30 36 | 1826684 | 45,,,| 82| g| 0] 57 | 102 36 | -0415414 | 47.14 | 688 | 5, 18 | 32 24 | -1787571 | go995| 88] ¢ | 0] 58 | 104 24 | 0387795 | oeq44| 669 | 5 19 | 34 12 | “1748546 | 5453,| 94| g | © | 59 | 106 12] 0360846 | 4,547 | 702 | gx 20 | 36 0 | 1709614 | 44,5, | 100| ¢ | 0] -60| 108 0 | 0334599 | ,..19| 787 | 5¢ 21 | 37 48 | 1670783 | 44,5, | 106| ¢| 0 | 61 | 109 48 | 0309089 | 4,75, | 773 | 5, 22 | 39 36 | 1632058 | 4..,4| 112] 7 | 0 | 62 | 111 36 | -0284353 | J.55, | 810 | 56 23 | 41 24 | 1593445 | 5a49,| 119| 7 | 0} 63 | 115 24] -0260426 | 4,575 849 | 59 24 | 43 12 | 1554951 | 4.4,,| 126] 7 | 0] 64 | 115 12 | 0237348 | 5.19 | 888 | 4o 25 | 45 1516582 | go55, | 133] 7 | 0] -65| 117 0 | 0215157 | 1565 | 928 | 4 26 | 46 48 | 1478347 | 5.59, | 140| 5 | 0] 66 | 118 48 | -0193896 | 45593 | 969 | 4y 27 | 48 36 | -1440252 | 479,, | 148] g | 0] -67 | 120 36 | -0173603 | joo95 | 1010 | 4, 28] 50 24 | 1402304 | 5,79| 156 | 9 | 0 | 68 | 122 24 | 0154321 | 2595 | 1051 | yo 29 | 52 12 | -1364512 | g765,| 164| 4 | 0 | 69 | 124 12 | 0136089 | ,-1 4, | 1090 | 5. 30 | 54 0 | 1826884 | 4,,,,| 172| g| 0] -70| 126 0 | -0118948 | 5455 | 1128 | a5 31 | 55 48 | -1289428 | 475, | 181 | g| 0] 71 | 127 48 | 0102935 | 5,955 | 1163 | 5) 32 | 57 36 | 1252153 | go. | 190 | 19 | 0] -72 | 129 36 | -0088085 | ja6n6 | 1194 | on 33 | 59 24 | -1215069 | seo,, | 200 | 4) | 0] -73| 131 24 | -0074429 | 1,57 | 1219 | 4. 34) 61 12 | 1178184 | g675| 210 | 9 | 0| 74 | 133 12 | 0061993 | 54599 | 1237 |, 35 | 63 0 | -1141509 | o.,.,| 220| 4, | 0] -75| 135 0 | 0050792 | oo,, | 1245 |_, 36 | 64 48 | 1105054 | g¢55 | 231 | 1, | 0| 76 | 136 48 | 0040838 | 2/14 | 1242 | 4. 37 | 66 36 | 1068831 | 5.9., | 243 | j5| 1] -77 | 138 36 | 0032124 | _,., | 1224] 5, 88 | 68 24 | -1032850 | g.756| 255 | 45 | 1) 78 | 140 24 | -0024635 | gooq | 1190] xy Soni) 70 12)) 0997124 | 5454 || 267 | 15 | 1] -79 | 142 12 | 0018336 | 14 | 1136 | 7, 40 | 72 0961665 280 1] -80 | 144 0 | -0013173 1062 ee + ND fF WO DKF CO CO Hm wm mm a a a RR bo © TABLE II, VaLues oF e?*s8e*4D V In TERMS OF D/a (D/7 >°75). D/x | 180°xDjr | et sectD yy e ) ea 75 | 135 0 | -0240086 843 21290 ‘76 136 48 0218796 871 20419 Sk 138 36 0198378 900 19519 ‘78 140 24 "0178859 928 18591 19 142 12 0160267 955 17636 80 144 0 0142631 982 16654 81 145 48 0125978 1008 15646 “82 147 36 0110332 1032 14614 83 149 24 0095717 1053 13561 “84 151 12 0082156 1072 12489 85 153 0 0069667 1086 11403 86 154 48 0058264 1095 10308 87 156 36 0047955 1098 9210 88 158 24 0038745 1093 8117 89 160 12 0030628 1079 7038 90 162 0 0023591 1055 Ill. On the simplest Algebraic Minimal Curves, and the derived Real Minimal Surfaces. By Hersert Ricumonp, M.A., King’s College, Cambridge. [Received 15 February 1900.] THE following essay may be described as an examination of a portion of the first of the two classical memoirs upon minimal surfaces contributed by Sophus Lie to Math. Annalen, Bde. xtv. and xXv., in the light of the fuller knowledge we now possess con- cerning space-curves of the fourth order. Just half a century has passed since Salmon, in his “Classification of curves of double curvature,” (Cambridge and Dublin Math. Journ., Vol. v., 1850), announced the discovery by himself and Cayley of a second species of twisted quartic curve, which cannot be derived from the intersection of two quadrics. This curve, which is the only quartic connected with the present matter, now possesses a bibliography of about fifty publications, for the most part short papers in mathematical journals. and it may be claimed after a study of these that the properties of this rational quartic, the forms to which its equations may be reduced and the various types of the curve are now as thoroughly known as in any other curve, plane or tortuous, of the fourth order. Much of this knowledge has been developed since Lie published his researches. In the following pages I have attempted to supplement the list of surfaces which Lie derives from curves of the third or fourth order by giving the equations of the curves, in all cases except one referred to real rectangular axes; and generally also the equations of the derived surfaces. Incidentally I find that one of the surfaces Lie quotes does not exist, and yet it is only possible to establish the fact after considering in some detail the nature of a particular curve. This appears to me to detract seriously from the value of Lie’s method, for in curves of higher order similar cases would probably occur with greater frequency and be even harder to detect. [Darboux has given, in Book III. of his Théorie des Surfaces, so full an account of the history and properties of Minimal Surfaces that it seems better to give references to his work rather than to the original authorities: this I shall do by writing D. followed by the number of the paragraph in question. Lie’s researches are to be found in Mathematische Annalen, Xtv. pp. 331—416; they are considered by Darboux in Chapters VI. and VII, but see also D. § 218, foot-note.] 70 Mr RICHMOND, ON THE SIMPLEST ALGEBRAIC MINIMAL CURVES, The basis of much of the: work of Sophus Lie upon minimal surfaces is the study of the curves without length that le on such surfaces; curves that is to say which in ordinary Cartesian coordinates satisfy the differential equation (dx) + (dy) + (dz) = 0. These curves I shall speak of as curves without length or lengthless curves, or by Lie’s name for them, minimal curves. Lie shewed that from every lengthless algebraic curve one real algebraic minimal surface was derivable; that all real algebraic minimal surfaces could be so derived from algebraic minimal curves: and that the curves without length from which a given real algebraic minimal surface could be obtained were simply repetitions (by translation) of one of two conjugate imaginary minimal curves. It being always understood that the coordinate axes are real and mutually per- pendicular unless an express statement to the contrary is made, the equations aXe (G)) see Ye (0) ei) ees clever ne re anes eece eee (i), X, Y, Z being complex algebraic functions of a complex parameter @, define an algebraic curve without length, or minimal curve, provided that dX?+dY*+dZ*=0. Interchanging +7 and —7 we obtain the equations of the conjugate algebraic minimal curve, viz. 2 = XG (O)) ety — Wa (G3) ke Za (Op) emcee eee eee Seer (ii), X,, Y,, Z being the functions conjugate to X, Y, Z, and @, conjugate to 6, so that dX,?+ dY,?+ dZ? = 0. [The two curves may, (as Lie found), coincide, but only when the order of each is at least six.] The equations of the derived real algebraic minimal surface may now be written, (cf. D. §§ 218—225), Qu = X (0) +X, (6); 2y = ¥A(6) Y, (6); 22 =Z(0) +2, (A); or, more concisely, 2=R{X(ut+w)}, y= R{V(ut+w)}, z=R{Z(ut+w)} ......... (iii), u+iv being substituted for @, and the symbol R denoting that the real part of the quantity within the {} is to be retained and the rest dropped. Although from each curve and its conjugate one surface is thus derived and vice versa, yet it is well to observe that every curve (i) is a member of a family of lengthless AND THE DERIVED REAL MINIMAL SURFACES. 71 curves which furnish a family of associated minimal surfaces (D. § 210). For if (i) be without length so also is the curve x2=(a+wb)X(@); y=(a+2b)V(@); z=(a+%b)Z (8); and this leads to a new minimal surface 2=Ri{(a+wb)X(u+w)}; y=Ri(a+wb)VY(u+w)!; 2z=R {(a+2b)Z(w+w)}...(av), which is, like (iii), real and algebraic. Should 6=0 the new surface is merely the surface (ii) on a larger or smaller scale; should a=0, b=1, (iv) is the adjoint surface of (iii), discovered by Ossian Bonnet (D. §§ 210—217). On p. 397 of Mathematische Annalen, xiv. Lie gives a table of the real algebraic minimal surfaces, other than double surfaces, (D. § 224), whose order does not exceed 16. One type he derives from cubic minimal curves, the remaining six from quartic minimal curves; and he claims further (p. 393, foot-note) that his vestigations exhaus- tively determine the minimal curves of order four. I find several points for criticism in this treatment of the matter, the chief being that the type of real minimal surface of order 12 and class 18 does not eaist: and, although the claim to have discovered all minimal curves of the fourth order proves to be well founded, it is impossible to discover from reading Lie’s memoir that there are three minimal curves of the fourth order, that two of them are cusped, ze. are the same curve differently placed, and that the third is the equianharmonic curve discovered by Bertini in 1872 (Lomb. Ist. Rend. (2) Vv. pp. 622—638). In what follows my first endeavour is to obtain the equations of all minimal curves of order three or four referred to real Cartesian axes, and thence to derive the equations of the real minimal surfaces they lead to. The differential equations of these curves shew that their tangents all intersect, and their osculating planes all touch, the Absolute or imaginary circle at infinity. I. SURFACES OF ORDER 9, CLass 6. Minimal Curves of Order 3. The tangents of a cubic curve meet every osculating plane and no other planes in a conic: a lengthless cubie must therefore osculate the plane infinity at some point of the Absolute. Let the real plane containing this point, the conjugate imaginary point, and a real origin of coordinates O, (chosen at random im finite space), be taken as z=0; and let any two real perpendicular lines in this plane be chosen as axes of « and y. The cubic curve will osculate the plane infinity where Ema Operare 180 8.0) and its equations referred to this real system of axes will fall under the type ty — x =a, + 34,6 + 3a,0 + | A= 3D,0? + 3D.0 + Dg} ..e.ceceeeeene Boas vets emis ins (¥), ly +e = 30,6 + 30,0 + a 72 Mr RICHMOND, ON THE SIMPLEST ALGEBRAIC MINIMAL CURVES, the parameter of the point at infinity here being infinite, and all the coefficients possibly complex. The direction ratios of the tangent of the curve, which are denoted by J, m, n, are given by im—lL:n:im+l:: a2 + 2a,0 + ao: 2b,0+b, : 2c,0+ cs, and must satisfy /?+ m?+n?=0; hence c,=0 and Cs (@gO2-F= 20d 0 4 Oy) = (2010 = Og) a ce aeen ores eee eres este eeee (vi). Now we are still at liberty (1) to move the origin to any real point; (2) to rotate the axes of x and y about Oz through any real angle; (3) to take a new parameter p(@+q) in place of @, p and qg being complex quantities. By proper choice of g we can make 5b, vanish in equations (v): by (vi), a and a, also vanish, and QyCo = 4b. Nextly, by (2) and (3) we may alter c, to c,p(cosa+zsina) and b, to b,p* and thus can make a,=2b,=c,=2k, a real positive quantity. Finally by (1) we can replace a3, bs, ¢; by pure imaginary constants: hence The most general lengthless cubic curve is that given by the equations z=—k(®—30)+ai ale (OA SB) ON 1 eo dont sae se bes Some marten ee (vi), z =k(36)+ ct | the constants a, b, c, k and the coordinate axes being real. The derived real minimal surface is v=R{— k(u+iw)+ 3k(ut w)} =k Bu + 3uv? — uv’) y= R \-ik(w+ wy — dik (ut w)} =k (8v + 30 — 2°) z= R{Bk(ut iy} =k(Bw— 30) | This is the well-known surface (D. § 207) discovered by Enneper. Since all cubic curves without length may be reduced to the form (vii) we infer that the adjoint surface and all the associated surfaces are simply the surface (I) displaced and altered in size; see however D. § 215. Lengthless Curves of Order 4. Space-curves of the fourth order divide themselves into two genera according to their deficiency, which may be either 1 or 0. Those of the first kind, which are curves of intersection of two quadric surfaces that do not touch each other, provide, as Lie states, no instances of minimal curves. Among those of deficiency zero, we, which are AND THE DERIVED REAL MINIMAL SURFACES. 73 rational or unicursal, there are two curves whose tangents intersect a conic. The first is the curve which possesses a cuspidal point, the second is the equianharmonic curve discovered by Bertini; all the surfaces discovered by Lie may be derived from these two quartics, and I have verified independently that no others exist, but am unwilling to lengthen this paper by including work that leads only to a negative result. I pro- ceed to discuss the first of the two curves above; it will appear that the tangents meet not one but two conics, either of which may be made to coincide with the Absolute. The curve is therefore lengthless under two distinct conditions. The equations of a cusped quartic curve may be presented in the form BXGe Vara ce aE EO ul On culluseritc cise ascins sie cisemc aseen meanac (vi11), the point (YZW) being the cusp (@=«), and the line Z=0, W=0, the cuspidal tangent; the curve has one stationary osculating plane (XY) which meets it in four consecutive points where @6=0. We must have, in order that the osculating plane at the point @ should be ILX+MY+NZ+PW=0, Ls MN 3 P21 3 — 662: 863» — 36%: b) thus the osculating planes of the curve touch all the quadrics (MM? + 12LP)+ p(9N?- 32MP)=0, and among them two plane curves, (corresponding respectively to X=0 and ~=0), that he respectively in X=0, the stationary osculating plane, and in Z=0 the plane con- taining the point of contact of X=0 and the tangent at the cusp. The two cases must be discussed separately: I take the latter first, and subdivide it into (1) a particular simple form, (2) the general form. II. Surraces oF ORDER 12, Crass 12. (Cusped Curves of the Fourth Order.) The plane Z=0 of the foregoing paragraphs being at infinity, each coordinate of points of the curve, referred to any Cartesian system of axes, will be equal to such a function of 6 as A@+ BO+ C+ De. In the particular case to which we now apply ourselves the cusp, (@=), and the point of contact of the stationary osculating plane, (@=0), are situated at conjugate imaginary points of the Absolute. Any real origin O being taken in finite space, the real plane containing O and these two points of the Absolute is taken for z=0, the cusp being where Megas ledge, Wor. < UX. Parr I: 10 74 Mr RICHMOND, ON THE SIMPLEST ALGEBRAIC MINIMAL CURVES, and the point of contact of the stationary plane, where woyee sl 3—4 30: the ratios of the coefficients of @ and of 6-! in the three coordinates are therefore known. From the fact that the curve is lengthless we deduce that iy—2=4AG+C; z=HO+F; ty+e2=G+ He; where is the form of its equations; and we may still rotate the axes of # and y through any real angle about Oz, or write p@ in place of @ in order to effect further simpli- fication in A, E, H. As before we may make — A=H=H =2, a real positive constant. Lastly C, F and G may by change of origin be made pure imaginary quantities. Thus z= k(i@+6>) ) of = the ($0? — OA) 4 aD). cence sc secs-eeeeene naeestaen anaes (ix), z=2k@+%¢ is the final form to which the equations of the curve may be reduced. The derived real minimal surface is v=k jAw—w?+u+ ei} y =k {hu8 — vu? — v + (Ur + 0")} beeen eee c ec ee eee eeeeeeeeeees (II), z= 2ku and this will be shewn to be the only minimal surface of order twelve. As in Case I. the associated surfaces are merely the same surface displaced and possibly altered in scale. It will be clear that the constant terms in the expressions for the coordinates may always be reduced by real translation to be pure imaginaries which do not enter into the equation of the deduced minimal surface. For the future such constant terms will be omitted altogether except in the final form of the equation of the minimal curve. AND THE DERIVED REAL MINIMAL SURFACES. 75 III. Surraces oF ORDER 16, Crass 12. (Cusped Curves of the Fourth Order.) The only difference between this case and the preceding is that the two points common to the curve and the Absolute are no longer conjugate imaginary points. A real origin O again being chosen at random, the real plane through it and the cusp is selected for z=0; the real plane through O and the point @=0 cuts z=0 in a real line, (which is selected as Oy), and makes an angle 2a with it. Thus the directions of the axes are definitely settled; and, when 0= 0, m3 82 Al 3G ss when 6=0, L:y :2:: cos2a: —7: sin 2a. Taken in conjunction with the fact that the curve is lengthless, these give us the following formulae : wy — 2 = — AG" + 2B0 +1408, z=(— Ad + E6) cota, ty + «= (— AO) cot?a, where Ad (AO + 2B + C@*) = (AO + EF). Writing p@ for 6 we make =A, and therefore B= (C= A, and so have ty — ©=(k + ck’) (— 0 + 20 + 46) + constant z= (k + tk’) (— 0 + 6) cota+constant } ............cecceceseee (x), ly + x =(k + th’) (— 0~) cot?a + constant as the final standard form. The surface deduced is 2a = R {(k + tk’) (1 — cot®a) (uw + wv) — 2 (uw + wv) — 4 (w+ wv)'}} 2y = R {(tk — k’) ((1 + cot?a) (wu + ww) — 2 (uw + wv) —1 (wt wy} (IIT). 22 = RB f{(k + tk’) [— 2(w + iv) + 2 (w + wv) ]} cot a Corresponding to each value of a we have a family of associated minimal surfaces. We come now to the case when the stationary osculating plane is at infinity. IV. Surraces OF ORDER 16, Crass 8. (Cusped Curves of the Fourth Order.) It is convenient to suppose the parameter @ of the preceding sections replaced by its inverse, so that @=0 shall give the cusp and =o the poimt of contact of the stationary osculating plane. In the present case the stationary plane is to be the 10—2 76 Mr RICHMOND, ON THE SIMPLEST ALGEBRAIC MINIMAL CURVES, plane at infinity; thus with any Cartesian axes each coordinate 2, y, z will be equal to such a function of 6 as A+ B+ CO + Dz Let the origin be a real point taken at random in finite space: let z=0 be the real plane through O, the point of the Absolute at which the curve osculates the plane infinity, and the conjugate imaginary point of the Absolute. The tangent at the cusp, (9 = 0), also intersects the Absolute, and a second real plane may be drawn through O, parallel to the cuspidal tangent and the conjugate imaginary direction. This plane will meet z=0 in a real line which I take as Oy, and will make a real angle, denoted by 2a, with z=0. The directions of the axes are now fixed; but we may still put pé@ in place of @ if convenient. When @=%, we are to have w:y:2::1:%:0, since the curve is to touch the Absolute on z=0: we therefore may take for the equations of the curve ty — v= ta,0*+ 248+ 44.6 B= 4b, + $b.0° y+e= ta and the fact that the curve is lengthless gives the conditions C2 (oP? + 20,0 + dy) = (6,0 + b.)?. The direction ratios of the tangent at the cusp (@=0) being cos 2a : —7: sin 2a, we must have GQ, 2b, : Gs 2: 1 = cota = cotta; and lastly, since we may put p@ in place of @, we may without loss of generality suppose b,=b,, and therefore by (1) a=a,=a,. The final form therefore is iy —v=(k + tk’) (4 + 26° + 364) + constant z=(k+ tk’) (46+ 46) cota+ constant | ...............0 ccc (x1), iy + © =(k + th') (4) cot?a + constant and the derived surface is 2e= KR \(k + tk’) [} (cotta — 1) (w+ wP+3 (wt wy +4 (uta)'F} 2y= RK {(k' — ck) [4 (cot?a + 1)(w+ wP + 3 (ut wth (wt) }} (IV). 22 =2R \(k+ 2k’) [4 (wt wv)? +4 (ut iv)]} cota The case a=0 needs consideration. The point of the Absolute where the curve osculates infinity and the point where the cuspidal tangent meets infinity are now conjugate imaginary points. The reduction proceeds on the same lines as in (II); but although the final form of equations of the curve is as simple as (II), neither the AND THE DERIVED REAL MINIMAL SURFACES. eG class nor the order of the derived minimal surface is lowered. The surface is another instance of a minimal surface whose associated surfaces are similar to itself. The equations are v= k(66?—36*) + constant) P= = GH FEB) ASO ENG, GoongondeocaseocGrsnesscenooedec (xil), z= k(86*) + constant | and v= 6k(w-v*)—3k(ut+ot- Gee) y = 12kw + 12k (w — v*?) w z= 8k(u>— 3uv?) } We Pee aaacak cals siauieceetiioce (any) for the curve and surface respectively. It is to be observed that in the surfaces (IV) and (IV’) the origin is a quadruple point. This completes the cases when a cusped quartic is without length; there remain in Lie’s table of real simple minimal surfaces three of class 18 and orders 12, 14, 16 respectively, of which I say the first is non-existent. The quartic curve from which Lie derives these surfaces, and which I now investigate briefly, has the Absolute for a triple curve; it is the quartic discovered by Bertini as the intersection of a ruled cubic surface with its first polar with respect to any point, this curve being placed in a particular manner with regard to the plane infinity and the Absolute. The Equianharmonic Rational Quartic Curve. The homogeneous coordinates X, Y, Z, W, of any rational space-curve of the fourth order may be expressed as rational integral quartic functions of a parameter 7; and by substituting these functions in the equation of a plane, IX +MY+NZ+PW=0, we obtain a quartic equation in 7, whose roots are the parameters of the four points common to the curve and the plane. The osculating. planes of the curve, for which three of the four intersections coalesce, are distinguished by the vanishing of both in- variants of this quartic, (of degree two and three respectively), and therefore touch a surface of class two and another of class three. If the osculating planes all touch a conic, either this surface of class two must be such that its discriminant vanishes, or else the osculating planes must touch a family of surfaces of class two, and the curve itself be of class four. The latter possibility leads us, by help of the investigations given in Salmon’s Geometry of Three Dimensions, to the cusped curve and to no other: the former requires deeper knowledge of the nature of these curves for its discussion. The following method, while slurring over many details which can only be properly explained by reference to the general theory of rational quartics, leads fairly rapidly to the results we need. 78 Mr RICHMOND, ON THE SIMPLEST ALGEBRAIC MINIMAL CURVES, Supposing the osculating planes all touch a conic, we may take the plane of the conic as W=0. Let the equations of the curve be X:Y¥:Z: W :: (apa,.050,07, 1)*: (by --.0n, 1)* : (@--- 8m, 1) : (do... 50, The osculating planes all touch the surface of the second class (La, + Mb, + Ne, + Pd,) (La, + Mb, + Ne, + Pd,) —4 (La, +...) (Last...) +3 (Lat ...P =0; in which equation, when expanded, P will not appear. Hence d,d, — 4d,d, + 3d.2= 0, da, — 4d,a; + 6d.a,— 4d,a, + da, = 0, d.b, — 4d,b; + 6d.b. — 4dsb, + dbo = 0, dye; — 4d,ce; + 6d.c. — 4d,c, + dic, = 0. That is to say (dd,d.djd,§n. 1)* is an equianharmonic quartic, and each of the other three quartics in 7 is apolar to it. Now by introduction of a new parameter @ in place of » (such that end + fn+ge@+h=0, e, f, g, h being properly chosen) we may reduce any equianharmonic quartic to either of the forms k (6+ 6y@+1), where 1+ 3u7=0, or k (6 — 1). Each of the two has its advantages, but here those of the former predominate. If we suppose the plane W=0 above to be at infinity and refer the curve to any real system of Cartesian axes, Ox, Oy, Oz, each coordinate (#, y, z) will equal some such function of @ as (PoP P2 Ps ps0, 1)* + (H+ 6u@ + 1), where Po+ Pst Spo = 0, 143W=0. It is however inadvisable at this stage to restrict ourselves to the use of real co- ordinate axes: it is best to keep such coordinates (#, y, 2) in the background, and to work with coordinates &, 7, € each equal to 1&, n, C}=Ax+ By+Cz+D, A, B, CG, D being complex constants. If these constants be suitably chosen, we may reduce the equations to the shape 2-6) | __ 2(@48) . ,__(#=1) ” 6 + 66? + 1” 164 6ue+1° ~ B+ 6u+1 wee c erect ccccccns (xi) in which it is clear that, if (& 7, £) be a point of the curve, so also are the points (-& —n, 0), (-&£7,-), (& —7, —£), their parameters being (—@), (@7) and (—é@>) respectively. Also we may verify that (dé? + (dny + (do)? = 0. AND THE DERIVED REAL MINIMAL SURFACES. oe) Let the point common to the imaginary planes £=0, »=0, £=0 be called Q, and the lines of intersection of each two, Q& Qn, O£; these lines and points are probably all imaginary, but we have seen that it is a matter of little moment whether Q be imaginary or not: any change in the position of © produces a real translation in the resulting real minimal surface, and nothing more. We therefore suppose that Q coincides with O. In order that (xiii) should be a minimal curve it is necessary that our coordinate system should be such that (dé) + (dn) + (do)? =k [(da)? + (dy + (dz)"): the lines O£, On, Of, must be such that the points at infinity on them are conjugate two and two, with respect to the Absolute; they must in fact be mutually perpendicular. Three possibilities present themselves; O£, On, Of, may be all of them real, or all imaginary, or one real and two imaginary. There is however another feature that must be considered, the order of the resulting surface. The curve meets the plane infinity in four points, at which it is not hard to see that 5 Gps (eves alsbeyn g ileoByn 2) ee ay Bee Sl (Go: GPa) ie. at four points which lie on the Absolute and form a quadrangle whose diagonal points are on the axes. The order of the minimal surface Lie shews to be sixteen; but if two of the four points of the curve which lie at infinity be conjugate imaginary points the degree he proves will be fourteen; and if it were possible that the remaining two points at infinity were also conjugate imaginary points the degree would be twelve. Now, the Absolute being unicursal, we may express the coordinates of its points by means of a parameter s; thus for example to suppose that in— eG Gane 28 3 is one possible manner. The parameters of the four points of the Absolute where the quartic curve meets it are roots of an equation in s of the fourth degree of which the quadratic invariant is found to vanish; and this invariant of the quartic equation which gives the parameters of the four points of the Absolute situated also on the curve will vanish whatever coordinate system we employ. Suppose if possible that the four points fell into two pairs of conjugate imaginary points; with a real origin O and real axes we may suppose the four points to lie in two real planes through O whose intersection is Oy and which make equal angles a with the plane yOx; in other words we could always choose axes so that the four points would le on the planes et) CATINOe The four parameters of the points would then be (1 + cosa) +(+ sin 2), 80 Mr RICHMOND, ON THE SIMPLEST ALGEBRAIC MINIMAL CURVES, and being real could not satisfy the necessary condition (the vanishing of the invariant). The four points therefore cannot be conjugate imaginary points two and two: and The real minimal surfaces of order twelve and class eighteen in Lie’s table, (Mathe- matische Annalen, XIv. p. 397), do not exist. Consider next the conditions under which two of the points common to the curve and the Absolute are conjugate imaginaries. The real line which joins them is chosen as the line at infinity in z=0, and the values of the parameter s (determined by Sess ty esas 2Yit) are denoted by 0, ~, p, q, the equation PPro being the new form of the invariantive relation between them: we deduce that p=-—ogq where o=1. The four infinitely distant points of the curve le where mye swt: 2: O21 0 sg Ile 8. eos) esp BOP 9 Il 8 respectively. At the diagonal points of the quadrilateral they form we have wa: wtao:2:: pg: —1:0 Be yah el aya 20 Vili See 1h 8 respectively ; or wiy:z:pqt1l: t(1—pg):0 :: 1— pq : —t(1 + pq) : 2p 2: L—pg : —1(1+ pq) : 2¢. These points are generally all imaginary, but the first of them may be real; if so, it involves no loss of generality to suppose that it coincides with Oz; and therefore pq=1, p=tw, gq=Fto; surfaces of order 14 appear both when O£, On, Of are all imaginary and when one of them is real, but not when all three are real. We are now in a position to determine the real minimal surfaces derived from the equianharmonic rational quartic. AND THE DERIVED REAL MINIMAL SURFACES. 81 V. SURFACES OF ORDER 16, CLAss 18. (Bertin’s Equianharmonic Curve.) First Case. The axes O&€, On, O€ of equations (xili) are real, and may be supposed to coincide with Ox, Oy, Oz. The curve without length is then w= 2(k + ik’)i (6 — 0) + (6 + 267 ./3 + 1) + constant y=2(k+ik') (6+ 6) + (6 + 267V3 +1) + constant | z= (k+ztk’) (—1)+(@ + 26973 4 1) + constant and the equation of the minimal surface is derived as usual by substituting u+ dv for 6 and selecting the real part of each coordinate (a, y,z). The surface, like the curve, must be such that if the point («, y,z) lie upon it, so also do the points (x, —y, — 2), (—2, y, —2), (—2, —y, 2); it seems unnecessary to give its equation. Second Case. One of the three O£, On, O€ is real, the other two being imaginary. It is immaterial which of the three is real, but the simplest results arise from the ease when O€ is real, and is supposed to coincide with Oz. Without difficulty we show that real axes Ox, Oy, at right angles to it and to one another, exist such that a = E cosh 8 + im sinh £, y = cosh 6 — 2€ sinh p. On substitution we find for the equation of the minimal curve aw = 21 (k + ik’) (Oe? — be-*) ~ (64 + 26°7V3 + 1) + constant | y=2 (k+ 1k’) (G8 + Oe-®) + (64 + 2621/3 + 1) + constant z= (k+ik’)(6*—1) + (64 + 2673 + 1) + constant | when referred to real axes Ox, Oy, Oz. The equation of the surface may be deduced. Third Case. O£€, On, O€ are all imaginary. The conjugate imaginary lines form a second trio of mutually perpendicular lines through O; by an elementary result in Geometry, a cone may be described with O as vertex passing through O&, On, OF and the three conjugate imaginary lines. The equation of this cone referred to a real coordinate-system is necessarily real, and may be reduced to the form Aa? + By? + C2=0, by a real change of axes. But there lie on this cone sets of three mutually perpendicular generators; hence Ae BC — 0: and we infer that O£, On, Of necessarily lie on a real cone 2° = a? cos*a + y? sin?a, Von. XIX. Parr I. 11 82 Mr RICHMOND, ON THE SIMPLEST ALGEBRAIC MINIMAL CURVES, Ete. where the angle a and the coordinate-system are both real. In order to obtain the equations of the curve referred to these axes we must apply formulae for transformation of coordinates to equations (xiii): if (l,m,7), (l,m.nz), (lsmgns) be the imaginary direction cosines of O£&, On, O€ referred to Oz, Oy, Oz we have to substitute the values of & 7, € from (xiii) in x=LF&+1.n+1,€: y=etce.... and the quantities /, m...ete., besides the usual conditions that connect direction cosines of three perpendicular lines, satisfy 1? cos?a + m2 sin?a = 7,2, 1.2 cos?a + m,° sin?a = n,’, 1.2 costa + m,? sin?a = n,?: they may in fact be all expressed in terms of one imaginary or two real variables, The equations of the curve or surface may be written down if required. VI. SuRFAcES OF ORDER 14, Cuass 18. (The Equianharmonice Quartic.) These are derived from the second and third cases of V. when two of the points at infinity on the curve are conjugate imaginary points. In the second case of V., 8 must be such that cosh 8 = + /3 sinh 8. Lie shews in his memoir that no real double surface exists of order 12; it follows that the surface obtained in II. is the sole minimal surface of that order. It appears further that the degree of the minimal curves that lie on the surface is a very inconvenient basis of classification even in these simple cases. IV. Diophantine Inequalities. By G. B. Maruews, M.A., F.R.S. [Received 28 February 1900.] THIS paper was suggested by and has immediate reference to MacMahon’s memoirs on the partition of numbers, especially the one entitled “Applications of the Partition Analysis,” etc, Trans. C. P. S. xvul. pp. 12—34. Its object is to contribute towards the solution of a problem, which may be stated in general terms as follows :— There are m assigned sets of n integers a;, 0;, c;,...l;; it is required to specify all integral values of the n indeterminates «, y, z,...¢ which satisfy the m relations av+by+oz+...+1;t>0, (GaN, B Bh sco GO) Of the n coefficients a;, b;,...1; m any one of these relations, some, but not all, may be zero or negative; it may also be supposed, without loss of generality, that their greatest common divisor is unity. It will be assumed that the relations are con- sistent and independent, and that they involve, either explicitly or implicitly, the restrictions B20, G20, 2205 oo vee when these restrictions are imposed, there must be at least mn conditions altogether ; there may be more. The conditions #>0, ete. do not really interfere with the discussion of the general problem; for instance, if we require all integral solutions of e+y—z2>0, 2y—32>0, we obtain them by considering separately all the eight cases x20, y20, 220, qr+ay—ez>0, 26y—3e,2>0, where €, €, €, assume independently the values +1 and —1. It will be convenient to use the term “positive integer” in the sense of “non- negative integer” so that the value zero may be included. Unless the contrary is stated, every small italic letter used henceforth will denote a positive integer in this extended sense. 11—2 84 Mr MATHEWS, DIOPHANTINE INEQUALITIES. The unrestricted system of conditions may be called a set of Diophantine in- equalities. If we confine our attention to “positive” solutions, there is another way of looking at the matter. The system is then equivalent to the set of Diophantine equations axtby+t+e2+...+lt—w,=0, where wy, Wb», +... Wm are additional indeterminates, and the positive solutions only are required. The problem may now, in fact, be regarded as a special case of finding the compositions of an m-partite zero out of (n+m) assigned parts. For example, if the conditions are da—3y—42>0, y>0, 220, which are independent and consistent, and imply #20, the corresponding Diophantine equations are 5a — 3y — 42 —w, = 0, Y — W. = 0, z—w,=0, the positive solutions of which give the compositions of (000) out of the parts (500), (310), (401), (100), (010), (001), where dotted figures indicate negative numbers. The number of indeterminates (n) will be called the order of the system. A system of the first order, if consistent, reduces at once to #20, and requires no farther discussion; the theory of systems of the second order is less obvious, but may be said to be virtually complete; the third and higher orders present difficulties which have not yet been completely overcome. One form of solution, however, applies to every order, and may be stated at once. Each positive solution (x, y, 2,...t) may be associated with a symbol a*@%y’...r, and the complete solution with the “denumerating function” La7@¥*.... It is found that this denumerating function is the formal expansion, in ascending powers and _ products, of a “generating function” ING [S550 2) G (a, De A) inca B,...)’ where WV(a, 8,...) and D(a, 8,...») are finite rational and integral functions of their arguments. In particular, D(a, 8,...%) is the product of a set of factors of the type (1—a?87...r*). This form of the complete solution will be called the “normal” solu- tion. Other forms of solution, the nature of which will appear as we proceed, are ultimately connected with certain identical transformations of G. The solution of an arithmetical problem consists properly in the statement of an algorithm, or method of arriving at the solution by calculation. Such an algorithm is, in the present case, implicitly obtained by the determination of G@; the identical transformations of G lead to different forms of algorithm. —_ * <= quienes eae =) pe Mr MATHEWS, DIOPHANTINE INEQUALITIES. 85 In finding G and investigating its transformations it is very convenient to make use of the method of nets or lattices. For the principles of this method the reader is referred to Gauss’s posthumous papers in the second volume of his works, especially pp. 313—3874, to Bravais’s memoir “Sur les systemes des points distribués regulitrement sur un plan ou dans l’espace,” Journ. Ecole Polyt. xix. (cah. 33), pp. 1—128, and to Dirichlet, “Ueber die Reduction der positiven quadratischen Formen mit drei unbe- stimmten ganzen Zahlen,’ Crelle, xu. p. 209. For the sake of clearness, most of the geometrical theorems we require will be stated; the proofs, however, will be usually omitted. Taking a fixed pair of rectangular axes, the two sets of parallels Z=h, y=k, (h, k=0, +1, +2,...), divide the plane of reference into a lattice, the compartments, or “panes,” of which are unit squares. Any point where two lines intersect has integral coordinates and will be called an integral point. We shall be concerned only with points whose coordinates are positive; that is, points within or on the boundaries of the first quadrant. Let (a, 6) and (c, d) be any two integral points A, B not collinear with the origin; then ad—be is not zero, and we may, without loss of generality, suppose it a positive integer e. If p, q separately assume the values 0, 1, 2, etc, the points defined by (x, y)=(patqe, pb+qd) are the nodes of an oblique lattice formed by the two sets of parallels —be+ay= a = eee (Gi, GSO, Uy Py ae8) This lattice will be denoted by [OA, OB] or [a, b; .c, d]. Each pane of it is a parallelogram of area e; the pane of which OA, OB are adjacent sides may be called the first pane. It is convenient to reckon as points “belonging to” the first pane the points within it, the origin, and the points on OA, OB exclusive of O, A, B. If this convention is applied to the other compartments, every point of the lattice will belong to one and only one pane. Two points which are similarly situated with respect to the panes to which they belong will be called congruent or equivalent: thus every point of the lattice is equivalent to one and only one point of the first pane. The number of integral points belonging to any one pane is e; this number is called the norm of the lattice. The e integral points which belong to the first pane will be called a complete set of least residues with regard to the lattice. If a, 6 and c,d are pairs of co-primes, all the least residues of [a,b; c,d] except O fall within the first pane. Such a lattice will be called simple. In every other case [a, b; ¢, dJ=[fa', fb’; ge’, gd’, we may write 86 Mr MATHEWS, DIOPHANTINE INEQUALITIES. where f, g are integers and [a’, b’; c’, d’] is a simple lattice, from which the compound lattice [a, b; c,d] is derived by putting together fy panes so as to form one larger pane. A lattice such as [3,5; 1,2] of which the norm is 1 will be called a unit lattice. The integral points belonging to a unit lattice consist of its nodes; conversely when the nodes of a lattice, and no other points belonging to it, are integral, the lattice is unitary. These definitions and theorems are illustrated by Fig. 1, which shows the lattice [6, 3; 4,6] and its derivation from the simple lattice [2, 1; 2, 3]. Fie. 1. To express that a system of points (2, y) are the nodes of the lattice [a, b; c, d] we shall write, according to convenience, either x, y=0, [a, b; co, d], or a, y=L[a, 0; c, a). Each of these relations amounts to saying that a=patge, y=pb+qd, @, q=0) oo If 2, y=L{a, 6; ¢, d), then La% By = Laretae Bvbtad == (ap)? x = (a°8")4 1 ~ d= ap?) (1 — a¢8*)’ the equivalence here being of a purely formal character, as already explained. Mr MATHEWS, DIOPHANTINE INEQUALITIES. 87 We will now consider some simple examples of the second order, before proceeding to the general case. (1) Suppose the conditions are a—2y>0, y>0. The solution is evidently c=pt+2q, y=4q and hence 2, y= EL (1, 05°21) baie: a wal (1-4) (1— a8)" Lat By = So, more generally, if the conditions are a—fy>0, y20, the complete solution may be expressed in any one of the forms c=p+fq y=; #, y=LU, 05f, ; 08 = aw) (2) Suppose the conditions are Ba—2y>0, y>0. We have to specify the imtegral points which belong to that “sector” of the first quadrant which is bounded by the lines 3x2—2y=0 and y=0. Let us construct the lattice [1, 0; 2, 3] as in Fig. 2. Its least residues are seen to be (0,0), (1,1), (2, 2); and if we take all the points congruent to them, we have the solution required. We may express this conveniently by writing a, y=(0, 0), (1, 1), (2, 2), [1, 0; 2, 3]; this means, of course, that the solution is given by L=p + 2q e=p+2q+1) 2=pt+2q+2)\ y= 3g) y= gt Tf y= Bg 2S” and hence it follows that 1448 + a6? Sat BY = (1 + a8 + 2B?) Sar B4 = Ges diaas): The solution may be put (after MacMahon) into another shape which is very elegant 88 Mr MATHEWS, DIOPHANTINE INEQUALITIES. and suggestive. The lattice [1, 0; 1, 1] is unitary, and its nodes comprise all the solutions of Fic. 2. again [1, 1; 2,3] is unitary, and its nodes correspond to the solutions of —ax+y>0, 3e—2y>0; hence we obtain the complete solution of 32—2y>0, y>O by taking the nodes of (1,0; 1,1] and [1,1; 2,3] together, and then subtracting all the solutions of «—y=0, which will have been reckoned twice. Using the symbol ZL [h,k] to denote the system of points (ph, pk), we may write, in the present case, G,-y=[l,0; 1) Wedaile ds. 2, 3) Af, de The result is that the complete solution is given by eee a= p'+2q) =| a) pe) the only particular solutions common to both sets being those for which p=0O in the first set and g’=0 in the second set. Mr MATHEWS, DIOPHANTINE INEQUALITIES. 89 The umbral form of this second solution is 1 1 1 2a = (Cea aaar (h=any0 Sap)” Sas: on reduction to a single fraction this agrees, as it ought to do, with the value of G previously obtained. The new form of the generating function arises from the fact that we are able to find a succession of points C1, 0), (1, 1), (2, 3), beginning with (1, 0) and ending with (2, 3), such that each pair of two consecutive points defines a unit lattice, and that the lattices thus found do not overlap. Now a series of points of this kind can be constructed in an infinite number of ways: for instance we may form the series (1, 0), (2, 1), CL, 1), (3, 4), (2, 3), and hence obtain z, y=L[1, 0; 2, 1)+L[2,1; 1, 1]+Z0, 1; 3, 4)+Z[8, 4; 2, 3) —L (2, 1)-LZ{l, 1)-L[3, 4), or, in the umbral notation, al il 1 (@—a)(—a’é) * (1—a8) (1 —a8) af) — asp) of itt 1 il 1 (1 — a36')(1—028°) 1-w@@ 1—aB 1-08 La7ey = Solutions of this second type will be called “chain-solutions.” There is one chain- solution in which the number of “link-points” inserted is the least possible: this will be called the reduced chain-solution. In the present case the reduced chain-solution is derived from the linkage (1, 0), (1, 1), (2, 3). A simple illustration may be given of a more general transformation of a generat- ing function. All integral points of the first quadrant satisfy one of the two sets of conditions, GQ) y20, 34—2y>0; Gi) —32+2y>0, «>0; and these are exclusive, except when 3a —2y=0. The denumerating function for the second pair of conditions is 1+a? (Uae) (eye) = 1 : 1 Bees, ~ (1— 028") (1 — a8?) (1 — a8?) (1— 8) 1 — af?’ Vou. XIX. Parr I. 12 Lat By = 90 Mr MATHEWS, DIOPHANTINE INEQUALITIES. the generating function for the whole quadrant is a ee (1—a)(1— 8)’ hence, adding the generating functions for the two parts, and making the proper correction for Z[2, 3] we obtain the identical transformation ny _ 1 +e8s- oR i 1+ a8? pea (l—a)(1—8) (1—2)(1—a'p*) (1—ap*)(1— 8) 1—a'p® ~ (1—a)(1— a8)" (1-28) (1 — 6%) © (1 — 28") (1— a6") (1 — a8") (1 — 8) t: 1 Bu 1 or iE 1-aB 1-8 1-2" It is clear that results of this kind may be obtained in an infinite variety of ways. The transformations effected in the discussion of the present example are illustrated by Figs. 2, 3. (3) To take one more case, let the conditions be —7Tx+10y>0, 130—5y>0. Mr MATHEWS, DIOPHANTINE INEQUALITIES. 91 Constructing the first pane of the lattice [10, 7; 5, 13] as in Fig. 4, we observe that the least residues are (0; 0), 2192 (2) e14, 19), Fic. 4. hence the normal solution is Lar BY=(1+a8+a0?+... + a#8")/(1 — a8") (1 — 2583), where the numerator contains 95 terms. To find the reduced chain-solution we observe those least residues which lie nearest to the boundary lines OA, OB; we thus find by inspection the reduced linkage (10, 7), (7, 5), (4, 3), G, 1), C, 2), (2, 5), ©, 18), and hence deduce the solution i i! 1 228 (ane — alps tae) ae) + + aa eR) 1 il 1 Si Senge a Saas 12—2 92 Mr MATHEWS, DIOPHANTINE INEQUALITIES. We are now able to attack the general problem of the second order. The con- ditions may be taken to be of the form —be+ay>0, dx—cy>0, where (a, b), (c, d) are pairs of co-primes, and ad —be=e>0. Let the points (hy, k,), (he, k.), 22° (he, ke) be the least residues of the lattice [a, 6; c, d]: then the normal solution, in its umbral form, is Lat By = LaliBk/(1 — a%B?) (1 — a"), Arithmetically, the pairs of integers (h, k) are defined as those which simultaneously satisfy e>—bh+ak>0, e>dh—ck>0, and there is no difficulty in finding them in any particular case. The simplest way is to put «=0, 1, 2,...(a@+c) im succession, and determine in each case the corre- sponding limits of y. To find the reduced chain-solution we may proceed as follows. Let OABC (Fig. 4) be the first pane of the lattice [a, b; c, d], and let the line AO be turned round 4, in the negative sense of rotation, until it first passes through an integral point within the triangle OAC. In this position it may pass through more than one least residue: suppose it goes through P,, P,,....P;, these points being marked in order from A, Then AP,, P,P.,... PP; are the first links of the reduced chain. Next let the line P,O rotate clockwise about P; until it first passes through a set of least residues Py,, Pris, ... Py; then PyPyir, PyisPyis,-..Pg1P, are the next links. We now rotate P,O0 about P,, and so on; it is clear that we shall at last arrive at a succession of collinear link-points ending with C; the process is then completed. Thus in Fig. 4, where a=10, b=7, c=5, d=13, the first four link-points are GO; Dy Gs By (5 Bs cq LD). all of which le on the line 2e—3y+1=0; the next link-point is (1, 2) or P,, and the line P,P, is e—1=0; P, is (2, 5) and the equation of P,P; is 3e—y—1=0; finally the equation of P;C is 8x — 8y—-1=0. A link-point P, is distinguished from other least residues by the fact that lines can be drawn through it which separate the origin from all other least residues. The limiting positions of such separating lines are P,,P, and P,Psi1; if Psa, Ps, Poy are collinear, these limiting positions coincide. Mr MATHEWS, DIOPHANTINE INEQUALITIES. 93 Although the truth of these propositions may be thought intuitively evident, it is desirable to give the strict arithmetical theory. Suppose that (2, y’) is any integral -solution of ay —be =1; then the general solution is c=x' + at, y=y + bt, whence da — cy = da’ —cy' + et (e=ad — be). Now let the integer t be so chosen that e>dzx'—cy' +et>0; this can be done in one way only, and if «,, y% are the corresponding values of x and y, then e > dx,—cy, >0, so that (4, y%) or P, is a least residue. It remains to prove that P, is not outside of the triangle OAC; to do this, we observe that 2A OAP,; = 1, 2 A OP,C = dx, —cy, <¢, by the conditions imposed: hence 2(OAP, + OP,C) da,—cy,>0; LY. — Ly, = 1, da, — cy, >da,—cy.>0; LY3 — Yu; = 1, day — Cys > daz — Cys > 0 ; and so on. It is possible, of course, to construct the linkage in the reverse order; the algorithm is then day—cy=1, e>ay—bm>0; NRA LVYir=1, ay — ba, > ayr—bay,>0, and so on, The equation of P;P;,; is T+. Ysti 94 Mr MATHEWS, DIOPHANTINE INEQUALITIES. now for any point on the line L(Ys— Ysir) — Y (@; — Ty) = 0, drawn through O parallel to P;P,.,, the value of U is 1; hence no point between the parallels can be integral, and P,Ps.: is a “separating line.” The umbral form of the solution is SatzBy = = a ——_—. + 1 + *aBY= C1 = ap) (1 — ann) * am BH) — BH) 1 * = a7") (1 — a8") 1 1 = um ~ Tap l-amBy 1 —a%gu’ from which the corresponding algorithmic form may be at once inferred. Major MacMahon worked out, and showed me in manuscript, a complete algorithm for the solution of Ax>py, y>0:; this practically amounts to a reduced chain-solution for that . special case, and suggested to me the corresponding theory for the general conditions. The essential idea of a reduced chain-solution is accordingly due to him. When we proceed to discuss systems of the third order, an extension of the geo- metrical theory becomes necessary. We take three rectangular axes, and consider integral points (#, y, z); more particularly, positive integral points belonging to the first octant. There will be three kinds of regular distributions of points for which we shall require convenient names and symbols; I begin by explaining the terminology that will be adopted. A point-row [a, b,c] comprises all the points (2, y, z) given by Ly, 2— pa, po, pe: (p=0) aes) Each of these points is a node of the row. If d=dv(a,b,c) is the greatest common divisor of a, b, c, there will be (d—1) integral points within each segment joining two consecutive nodes. We reckon as belonging to the segment these intermediate points, and the first node of the segment. This gives d points altogether: we call d the norm of the row. The least residues of the row are the d points given by x, y, z=1a/d, ib/d, tc/d. @=0, 1,2, a=) A lattice [a, b,c; a’, b’, c’'] comprises all the points given by x, y, z=patga’, ph+qb, pe+ge. (p;'¢= 0,452 yee) This is simple if dv(a, b, c)=dv(a’, b’,c')=1. Its norm, in the sense explained previously, is N[a, b,c; a’, b', ¢}=dv(be' —b’c, ca’—ca, ab’ —a’b). Mr MATHEWS, DIOPHANTINE INEQUALITIES. 95 A “trellis” [a, b, ¢; a’, b,c; a”, b”, c”] comprises all the points given by x, Yy,2=patqa'+ra”, ph+qb’+rb”, pet+gqe +re”. (pq 17 Oy et 2) ae) These points are the nodes of the trellis. If we write Kea a Ww | ane b” Cc” | which we may assume to be a positive, non-evanescent integer, the three sets of parallel planes (b’c” —b’c')a+(c'a” —c"a’) y + (a’b" —a'b’) z=pA, (b’c — be") a+ (c"a—ca”)y+(a"b—ab")z =¢qA, (be' — b’c) w+ (ca’ —c'a) y + (ab! — ab) z =ANn, (a GHW, Ny ooo) define a set of equal and similar parallelepipeds which may be called the compartments or cells of the trellis. By making a convention analogous to that used for lattices we secure that every point in the trihedral “sector” occupied by the trellis may be con- sidered as belonging to one and only one cell. To each cell belong exactly A integral points; hence we call A the norm of the trellis. The A integral points belonging to the first cell are the least residues of the trellis. They are defined by the conditions A>(b’c")a+(ca")y+(ab")z>0, A>(b’c)a +(c"a)y +(a"b)z >0, A>(be')w +(ca’)y +(ab’)z 50, where, for brevity, (b’c”) has been written for b’c’—b’c’, and so for the other coefficients. I shall say that the trellis is simple, when di) (GybNc)!— au (anibc))—aui(@ s106..cs)—le reduced, when dv {(b'c’), (c'a’”’), (a’b”)}| = dv {(b'c), (ca), (a’b)} = dv \(be'), (ca’), (ab’)} =1; unitary, when A=1. A simple trellis is bounded by three simple lattices; a reduced trellis is bounded by three unitary lattices. A “rational” plane ux + vy + wz = 0, where wu, v, w are integers, passes through an infinite number of integral points: the distribution of these points in the plane may be realised graphically by drawing contour- lines in one of the coordinate planes (say zy) for the admissible integral values of the 96 Mr MATHEWS, DIOPHANTINE INEQUALITIES. other coordinate (2). Fig. 5 shows in this way the distribution of integral points on the plane 2a +4y—3z=0. The points belonging to any simple lattice [a, b, c; a’, b’, c’] may be enumerated either by finding its least residues, or by constructing a reduced linkage in its plane from (a, b, c) to (a’, b’, c’). Thus the points belonging to [2, 5, 8; 11, 2, 10] are enumer- ated by forming the linkage (see Fig. 5) (2, 558) e 12); (45 14), Clee 0): the denumerating function being Lat Buy? = 2 =a Z +- Mis — (1 — a2B%y*) (1—aBy*) (1—aBy*) (1 —o4By*) (1 —a’By*) (1 — 28 By") 1 1 ~1-aBy 1—aBy = {1 at aBry? oh (a8 + a? B*) y' =k (28? + a8 B®) y* 2. (8B? + abs su at B+) y8 + (a°B® ate a’ Bt aE a®B°) y° + (a8 + aB°) y? ats (a2B° + a°B°) ys + a Biry'5} + (1 — a B*y*) (1 — aB*y"). The expression last written is the normal form of the generating function, and its numerator exhibits the 17 least residues of the lattice. The existence of a reduced linkage for any simple lattice follows at once from the Mr MATHEWS, DIOPHANTINE INEQUALITIES. 97 geometrical theory for the second order which has been already explained. But if we translate the proposition into a purely arithmetical theorem, we obtain a remarkable and by no means obvious result, which may be regarded as an extension of the theory of continued fractions. Let (a, b, c), (a’, b’, c’) be any two triads of positive integers such that du (a, b, c)=du(a’, b’, c)=1. Let dv (be’ — b’c, ca’ — c'a, ab’ — a/b) =6, be’—b’c= 6a, ca’—ca=68, ab’—awb=8y; then it is always possible to find a set of “link-triads,” (Go in As (G5 Oh B) cools Oy Ga) such that Yr — YinG= a, i — 2442; = B, BYin — GVHyi=y for 7=0, 1, 2...(n+1), with the convention that (%, y, 2%) means (a, b, c) and (nti» Ynti, Zn41) Means (a’, b, c’). Moreover, the linkage of this kind for which is as small as possible is unique. This theorem undoubtedly admits of generalisation. I conclude these geometrical preliminaries by explaining an auxiliary diagram which will be frequently used in the sequel. A plane making equal positive intercepts on the axes meets the coordinate planes in the sides of an equilateral triangle ABC; in Fig. 6 this triangle is shown after being rabatted about BC into the plane yz. The lne Vora exe Arnel: 13 98 Mr MATHEWS, DIOPHANTINE INEQUALITIES. meets the plane ABC in a point P whose areal coordinates referred to ABC are proportional to a, b, c. When a, b, ¢ are integers the point P is easily constructed in plano; we may suppose dv(a, b, c)=1, and associate P with the “mark” abe. Thus the marks of A, B, C are 100, 010, 001 respectively. The plane wz+vy+wz=0 meets the plane ABC in a line whose homogeneous equation referred to ABC is ua#+vy+wz=0. A simple trellis [a, 6, c; a’, b’, e’; a”, b”, c”] may be associated with a triangle in the auxiliary plane, the vertices of which are at the points marked abe, a'b’c’, a’b’c’; similarly, lattices and point-rows may be associated with rectilineal segments and points in the auxiliary plane. The general form of the conditions for the problem of the third order is Ue t+ Uy + Wiz >, He A a0. 170) If we restrict ourselves, as usual, to positive solutions, the number of conditions (m) is not less than three; but it may be more, and this is an essential difference between systems of the third (or any higher) order and those of the second order. In fact if the m planes represented by the equations Ue + YY + Wz = V, enclose an m-hedral solid angle or “sector” within the first octant, and if any one (and therefore every) point within the sector satisfies all the conditions wx+ vy +wi2e>0, the given system is consistent, and the solution amounts to the enumeration of all integral points belonging to the sector. Now a polyhedral solid angle may always be cut up into an aggregate of trihedral angles; hence it will be sufficient (in the first instance, at least) to consider a system of three conditions le +my +nz 20, ” > Ve +m'y +z 20, a+ my + nz > y 0, or, say, U>0, V>0, W2>0. If these conditions define a trihedral sector within the positive octant, we shall have min”! — mn’ : nl’ —n'l Um" —U'm' =a:b :¢, mn — mn’: n't —nl’ Um —tm" =a :b': Cc, mn — mn: nl —wvl : lm’ —Um=a" : b":c", where a, b,...c” are positive integers, and du(a, b, c)=dv(a, v’, ¢)=dv(a", b’, c”)=1. Mr MATHEWS, DIOPHANTINE INEQUALITIES. 99 Thus the lines VW, WU, UV meet the auxiliary plane in the points marked abe, wb'c’, abc’; and the sector bounded by U=0, V=0, W=0 is that which is occupied by the simple trellis =o, Oy 02 Oh, WB, CS) Gia e Ab The normal solution of the problem is obtained without any difficulty. Let Gh UY og | a’ be c’ this is numerically equal to the norm of JZ, and we may without loss of generality assume that it is positive. Let (pi, dis iP) Gr 2. we LNs) denote the least residues of 7. These are defined by the limitations la ++ mb + ne >Ip;tmgtnr 209, Va t+mb +r’ > ptm qatnr; >, Va’ + mb" + ne" > Wp, + mG + wr; > 0. Then the normal solution, in its umbral form, is > Pi Bdiry"s el = a Bry’) qd = al’ BY”) a = at” BP'ry°") ® To take a simple illustration, let the system be Lar Quy? = z>0, y>0, —2e—4y+ 3220. Here G@ b G (0) @ Tl we WY Ces WO 2 EAS GE VU Be! IS ={9) The least residues of [0, 0,1; 3, 0,2; 0,3, 4] are defined by the conditions 8>a220, 3>y>0, 3>-—2e—4y4+32230; whence we find the 9 points (0, 0, 0), (0, 1, 2), (0, 2, 3), Gl, ©; 1), i, 2), (Gl, 2 2a) GO, By (@s i, BD, CyB, 2 Consequently the normal solution is il af ay + By Jb ary” sik apy? =f By? =f a Bry’ + ap?rys as a BPry* Sa" Binz = (1 = ay’) (I= BY) (= 9) 13—2 100 Mr MATHEWS, DIOPHANTINE INEQUALITIES. The labour of finding the least residues of a trellis may be somewhat abbreviated by various devices which I need not detail; im any case it is not greater than might be expected from the nature of the problem. As an example of a system with more than three conditions, we will take z>0, y20, 220, 24+ 4y —3230. We have here a tetrahedral sector represented on the auxiliary plane (Fig. 6) by the quadrilateral ABDE whose vertices are marked 100, 010, 034, 302. The equation of BE is 2x—3z=0: hence we may consider separately the two sets of conditions G) y20, 220, 24-3220, (i) #20, 24+4y—32>0, —22+3220. The least residues of [1, 0,0; 0,1, 0; 3,0, 2] are (0, 0,0) and (2, 0,1); hence the normal generating function for the conditions (i) is ane l+a’y 1 .@ >a) Ge) aaa) The least residues of [3, 0,2; 0,1, 0; 0, 3, 4] are determined by 3>a>0, 12>—24%4+3220, 4>2¢+4y—3220; whence we find the 12 points (O05 0) 5 V(OF ae i); S(O 252) (On Ss3)s Ge) Gale 2) a (Gites) een Ga 3 aA) (ei Dy (G2 3), Cre oy 58) oh Accordingly the normal generating function for the conditions (ii) is nie (1+ By + aBy + aBy? + By? + a By? + aB?y* + Bey’ + oF B?ry* + a2 By + aB ry + a°B3y*) : Gey 8 )Ke— By) ' The points on the (unit) lattice [8, 0,2; 0, 1,0] are enumerated by 1 ~ (1—ay*)(1—8)" Hence the denumerating function for O(ABDE) is given by La® Buy? = G+ G. — Gs Gs ik 2 J EGS ia ~ (1—a)(1—8)(1 — ay) (1 — By)’ where F(a, B, y)=1+ By + ay + aBy? + Bey? — By — aB*y? + aBty! + Bin? — 2°By — aBin/ = a? Bry! —a By — as Bey' —a B ryt — a Bry’. Mr MATHEWS, DIOPHANTINE INEQUALITIES. 101 The appearance of negative terms in this expression points to the existence of fundamental syzygies, such as ; (8) (an? — (4) (By*) = 0, and so on (cf. MacMahon, lc. p. 17). It is remarkable that all the coefficients in F are numerically equal to 1; whether this is accidental or not I am unable to say. The integral points belonging to the sector O(ABDE) may be enumerated in a different way, which affords a check upon the foregoing calculations. The points required may be obtained by deducting from all those in the positive octant the points belonging to O(#DC) and then adding those which lie in the plane sector 0 (HD). Now the integral points belonging to O(ABC) are enumerated by Seite ws 5 aR death (=a) (Fs) a9)" and since the reduced linkage for [3, 0, 2; 0, 3, 4] is (ef. Figs. 6, 7) G05 4) Geib 2h (yusk <5) 302 101 il2 034 023 ole oo! Fic. 7. the generating function for O(£D) is 1 1 1 1 + aBy? + 228%! (T= a8) (1 = 2B) * 1 — a8) — Bey) 1 a8? ay?) (1 — By’ as may also be found by calculating the least residues of the lattice [3, 0,2; 0,3, 4]. The normal generating function for O(#DC) has already been found: so that finally the generating function tor O(ABDEZ) is 1 Paulie aBry? + 02 Bry (l-a)(1—8)(1—y) © (1 — 4%) (1 — By") i 1 + ary + Bry? + a7y? + aBy? + By? + By? + By + 2°26 ?y4 (1 —a3B?) (1 — B*y') (1 = y) When this is reduced to a single fraction in its lowest terms, it agrees, as it ought to do, with the expression for the generating function obtained by the other method. 102 Mr MATHEWS, DIOPHANTINE INEQUALITIES: A third way of solving the problem would be to draw the plane OAD, and consider separately the trihedral sectors O(ABD), 0 (ADE). When the number of independent conditions is greater than three, some care is necessary in finding the positions and cyclical order of the edges of the sector to which the required points belong. In other respects the theory of the normal solution is fairly simple and complete, and may be applied, with suitable modifications, to problems of higher orders. It is when we try to extend to systems of the third and higher orders the theory of reduced chain-solutions that we meet with difficulties of a really new kind. In order to show the nature of the results that may be expected, I consider the two cases for which the normal solutions have already been found. The first is w>0, y20, —2x—4y+32z2>0, and we have in the auxiliary plane (see Fig. 6) the triangle CHD with its vertices at 001, 302, 034. In the face OCE, we have the reduced linkage (0, 0, 1), (1, 0, 1), (8, 9, 2) which is easily obtained from the table of least residues previously given. On CE we mark the point 101. In the face OLD we have the linkage (3; 10572), se 2); (0; 354): on ED we mark the poimt 112. Finally, in the face ODC we have the linkage (O;0355 4) (O02) 3) (OF9 ee 2)), OOS): so on DC we mark the points 023, 012. Now we observe that the determinants eae lye Te ea ae | RO, eSiacdiels 028, Mo aah =) 0:52) 38 {0c lee 0) Ona all have the value 1: corresponding to them we have the unitary trellises 20S aeVO 253s Vs 2 rn 2N orm Os deity [ae 2eeOs 2 ONOry| which together fill up the whole sector occupied by [1, 1, 2; 0,3, 4; 0, 0, 1). Again [8, 0,2; 1,1, 2; 1, 0,1] and [1, 0,1; 1,1, 2; 0,0,1] are unit trellises; altogether —————————— Mr MATHEWS, DIOPHANTINE INEQUALITIES. L105 we have five unit trellises which, without overlapping, fill up the whole sector O(CED). Hence the generating function for the whole sector may be put into the form fleet yi 1 1 (1 aby’) (l=) =a) (l-ay)( = ay?) GS B27) C8277) eee. Cee (1 — B*y*) 1 — By’) (l— By’) 1-7) 1 (al 1 1 1 naa | a, } = Ts . SS oS . (1 = apy) (L—ay” 1p" 1—- Bye" 1-9) This is clearly of the nature of a chain-solution; it is also reduced, in the sense that the number of unitary trellises introduced is the smallest possible. But it is not the only reduced chain-solution that exists. Thus Fig. 7 corresponds to the solution 1 i 1) Z 0, «+y—z20. With the help of Fig. 11 we obtain yee iene sete OOM ene eile Is T= Oni 0s a, 0, 1 file Tevkwds Oe 11-0] Anat eset sc ONe 0, (Ie ple MnO tal [—wa[ ll, 1: 1h Ov 1] Se )5|0% Te yh This solution is unique of its type, and it is to be observed that we have to introduce the link-point (1, 1, 1) which is not on a face of the bounding sector. The correctness of the result may be verified by the identity 1 2 1 ts 1 ele 1+ aBy Say eae) l= ay) e we (U>)8y) 5?) easy) Oya) (l= a6): the right-hand side of which is the normal form of the generating function. (2) Let the conditions be 2>0, y20, 220, 2(y+z)—a220, 2(2+2)-y20, 2(a@+y)—2>0. As shown in Fig. 12, there are 10 irreducible link-points; of these there are three sets of four, 14—2 108 Mr MATHEWS, DIOPHANTINE INEQUALITIES. (2), OF) (20); Co 20) Gas Oh ed) Ge TE Ona (Gly PaO (Ohers 1D), (Oy thy ah))s OFS DS (Oya) a Where XGlg What) = which le in the planes y+z2—-1=0, z+x-1=0, x+y-1=0 respectively, Hence there are eight alternative chain-solutions, of which one, indicated by the linkage in the figure, is a, y, 2= L[111; 110; 011]4+ L[111; 011; 101]+ Z[111; 101; 110] + £2 [101; 201; 110]+Z[110; 201; 210]}+ 2[110; 120; 011] +£[011; 120; 021]+ Z[101; 011; 012]+Z[101; 012; 102] —L[111; 101] —Z (111; 110] —Z[111; 011]—Z[110; 101]—Z[110; 011] —L{[011; 101] —Z[201; 110] — L[120; 011] — L[012; 101] = Tsay: (3) Let the conditions be y>0, 220, 102-—2y—52>0. The linkages in the limiting planes are Gh Os Ws aly i Oy Ges OF} Gh sO. (5 4h O}F (Gh & O) tm eS) C705 0) GEO) GOs 2) eine — 0) (1, 5, 0), (1, 0, 2) in 10x —2y—5z=0. Within the sector there are two irreducible residues Geeky! (Ol Pan abys altogether we have ten link-points which all lie in the “syzygetic plane” z—1=0. Thus there is quite a large number of conjugate reduced chain-solutions; any one of these may be easily constructed by means of the auxiliary diagram, which the reader will easily draw for himself. Mr MATHEWS, DIOPHANTINE INEQUALITIES. 109 (4) Finally, suppose the system is y20, 220, 1l3a—Ty- 52250 (an example suggested by Major MacMahon). The surface linkages are (is O OF sO, Dy Ch Oy Ey@ ay Gy O58) in the plane y=0; (50; OQ; Gib Oo} @&s0) © sO), Ge OF GY OF Gyn Oy (ae ey ©» in the plane z=0; and Gr Onl ee (Ammo) (Sy 25) (23a (7, 13510), in the plane 132—7y—5z=0, Besides these, there is (1, 1, 1) within the sector, The syzygetic planes are z—1=0, which goes through 100, 101, 102, 110, 111; 22—-y—1=0, which goes through 110, 111, 230, 350, 470, 590, 6110, 7130, 231; 8a — 4y—3z—1=0, which goes through 111, 231, 325, 419, 5013, 205; and 3a —y—z—1=0, which goes through 111, 205, 102. From these data the auxiliary diagram can be constructed, and the chain-solutions found. These examples, and all others which have been investigated hitherto, suggest the truth of the following propositions, which are given here with all reserve, though I have little doubt of their correctness. Given a system of the third order, defining a polyhedral sector; then (1) The integral solutions of the system may be enumerated as the nodes of a finite number of unitary trellises, the sectors of which fill up, without overlapping, the sector associated with the given system. (2) There will be a set of conjugate “reduced” solutions for each of which the number of unitary trellises is a minimum. 110 Mr MATHEWS, DIOPHANTINE INEQUALITIES. (3) The “link-triangles” of a reduced system of trellises lie on a set of “syzygetic planes” Ag+ py +vz2—1=0, where A, pw, v are integers. (4) Every reduced solution leads to the same set of syzygetic planes; and the conjugate solutions arise from the possibility of making different arrangements of link- triangles in the syzygetic planes. It will be observed that if (1) is true, (2) and (3) follow without any difficulty, but not (4). At present I have not succeeded in finding a strict proof of these pro- positions: it seems fairly evident that an arithmetical demonstration would involve a somewhat detailed discussion of determinants whose elements are real integers, and in particular of their elementary divisors. It is not impossible that there is a corresponding theory of chain-solutions for systems of any order; it is not easy to extend the geometrical theory beyond the third order, and the arithmetical theory looks very complicated. It is, of course, easy to see that reduced chain-solutions will exist for an indefinite number of systems of any order. — V. The Diophantine Inequality \x>py. By Major P. A. MacManon, R.A., D.Sc., F.R.S., Hon. Mem. Camb. Phil. Soe. [Recewed 7 April 1900.] §1. I waAvE shown elsewhere* that the theory of Partitions may be discussed by means of one or more inequalities in integers of the type MyM, + Avy +... + Ags > Pi aT fo. a OOD ae MBs where all the quantities denote positive integers, and Ay, Ae,... As, fr, Mo, -+» Me are given. _ Hilbert has shown+ that in respect of any such system of Diophantine equations, say of type Nid + Ave +... + Aghs = HP, ate HP. +... + BB, there exists a finite number of solutions , ft , , 4 G, Sh, as 5 1> Ba, Bie " ” ” um u” , %, %, as 3 1> By ’ Bi 5) A (K) An"*) a (k) B,™, 8,” B® such that the most general solution may be written a, = Aja; + Aga,” +... + Apa, ™, a, = Ajay’ + Agus” +... + Apa, a, = Aja, + A.a,’ +... + Aza;”, B:= A,B; + AB," +... + A,B, Bs = A,B, + A,B,” Sup 05 arr A,B", Br= A,B) + A.Bi +... + ArBe™, where A,, A,,... Ay have positive integral, including zero, values which may be assigned at pleasure. * Phil. Trans. R. S., Vol. 192, Series A, pp. 351—401. + Math. Ann. t. xxx. 112 Mayor MACMAHON, THE DIOPHANTINE INEQUALITY dx > py. The like theorem obtains also for a system of Diophantine inequalities; this is intuitive directly the observation is made that the inequality DM + Ane +... + AgAs > Bi + HoBo +... + eB: is equivalent to the equation AM + Agae +... + Ass = Bi + HeBot ... + eeBe + Bear for this shows that the theory of Diophantine inequalities is, in reality, a special case | of the theory of Diophantine equations. The inequality . Aa > uB is thus equivalent to the equation Aa= "B+ BP; and I propose to determine the fundamental solutions of this equation as a contribution not only to Partition Analysis but also to the Theory of Hilbert. The quantities A, u . are supposed to be given relatively prime positive integers and I seek the solutions from which all others can be obtained by addition. To reach the end in view I seek the sum «v8, the summation being for all solutions of the given inequality \@>y8; and thence it is easy to form the sum Lary8zy, appertaining to the associated Diophantine equation. The summation can be carried out in a variety of ways and the result exhibited in a variety of forms; but it is not every form of result which establishes, in an irrefragable manner, the whole of the fundamental solutions. The desirable method of procedure is not the most obvious one and the desired result is not the most compact obtainable though it is one of great elegance. The method of summation which first presents itself to the mind is first to sum > > a from E = 8B to «, where the symbol # denotes the smallest integer > than the quantity which follows it, and then to sum 8 from 0 to ». The result easily obtainable is | Qu 2a-1 a ae yt... ta} by rir iV ee ae Ay nae x, y products in numerator and denominator but does not indicate which of these are the ground solutions. The true method leads to a sum of algebraic fractions and not to a single fraction, as will appear. yd} l—«w.1—ax*y* % which certainly proves that the ground solutions are included in the exponents of the Mason MACMAHON, THE DIOPHANTINE INEQUALITY da Spy. 113 We require a preliminary lemma which may be thus stated :— Lemma. The relation a> ws may be made to depend upon a similar relation in which 2 is unchanged and > pm. We have sap a8 7 Care Wen where © is an operator which expresses that when the fraction is expanded in ascending 2 powers of w and y we are to reject all terms which involve negative powers of a and subsequently to put a equal to unity. Now if p be the greatest imteger in - the given relation implies the relation a >p8, and we may write Lary? =O ae 7 1—a%ba.1 -—5 where © operates upon both a and b. Eliminating Db we find > ee Sarye = O I > Sl —ae. 1 — PL * @Qb—AP which denotes also the sum Lat (aPy)e = Sartre yB, where a and 8 are connected by the relation Na > (u— Xp) B. We have therefore reduced the sum to the sum Lxy® for the relation a>" Ya2*By® for the relation Aa >(u— Ap) B; wherein, from the definition of p, X > — Ap. This proves the lemma. In the next place I say that there is a second lemma which involves a further reduction. Lemma. The relation ha > eB, wherein } >, may be made to depend upon a similar relation in which p is unchanged and X< pm. Won, XEX. Parr I, 15 114 Mason MACMAHON, THE DIOPHANTINE INEQUALITY dAx>ny. This is really the key to the solution of the problem before us, as will appear. Observe that the relation Aa >uB8, where A> yp, may be broken up into the two simultaneous sets of relations :— da > wf, and Aa > wP, B24; a>B; and that the second of these simultaneous sets, a> uP, a>B, may be replaced by the single relation since, X being > yw, the relation is implied thereby. We may therefore separate a portion of the generating function corresponding to a>, c @ V1Z. — —; PS Sr, and consider the remaining portion, corresponding to the first simultaneous set, viz. :— da > uBR, B2a. The crude generator is 1 @ eee el b a = r 2 ¥ Ann, ad l—-a@*ay.1 i which gives the sum = (wy) y?, for the relation (A- p) a> BB. If X—yw< we have done what was required, but if \—->p we may repeat the process; and just as we have found 1 1—a\ "ay .1— y at il x 0 ng A) = : a ss > 4 ah hee Ve ae Mayor MACMAHON, THE DIOPHANTINE INEQUALITY Az >py. 115 we shall tind 1 ry 1 y Vl—sy tikes ry? ot > 1—a@'ay.1— pe WS Ge Sears dh Vid sie 6 mn . and, if q be the greatest integer in —, we obtain finally . Kh fo) iI Zs Nes eae) at = eee vy et vy? ~l-@.l—ay 1l-ay.l—ay "7 1l—ay.1—ay! +2 : ; * 1-ad4ay?.1 gle a where the last written portion of the generator represents the sum =(#y%)*y® for the relation (A= pq) & > BB. Hence Lemma. The relation Xa>puP, where X>p, may be made to depend upon a similar relation, in which X< y, as shewn by the identity VS ward lh 1 a ay! = {0) 2 1 —q"9 wy? . 1— Y : ar : , SN wherein q 1s the greatest integer In —. be These two lemmas evidently yield a process of reduction which can be pushed to the last extent. In what follows X will be considered >. If w>X it will be merely necessary to write #—Ap for w and «x?y for y where p is the greatest integer in &- It will be surmised that the reduction depends upon the convergents of the con- : : : r tinued fraction 7 Xr 1h oh ul 1 Tet aT hag Gy a, eae 15—2 116 Mason MACMAHON, THE DIOPHANTINE INEQUALITY Az py. where n is uneven; this is always possible because a, may, if convenient, be written qu j nh ‘ ds ; Qn ; and form the ascending series of intermediate convergents, viz. :— Pa PatPo Pat2Pp Part(a—1)p Ga’ Gath Gat2q’ gat(a—1)q’ Pr PtP, Pit 2p, Pit(as—l) ps Hh hte Ht2q’ q+ (as — 1) qo’ Pn—2 Pn-2t Pn = =Pnot 2Pn-a ee Pn—-2+ (dn—1) Pra Pn. Qn—2 Qn—2 + Ina Qn-2 + 29n— : Qn—2 + (Gn = 1) qn : dn j 4 xX where of course “=~. n Applying now the second lemma to the crude generator fe) i > > l-a@e.1— Y we obtain oO z 13 ai 2 en: See 1 l—aye >. 1 —ay%’ at a aye where = al SC, I es a“ ty? Ge yA sal = By. i= aaa ae ats 1 = eal aaa 2 1 _— Gere git (aq—i)0 gp (ax—2) 1 = git(a—i)o pees ih gta GPa ape wi-1 ype wrt ¥ -1tPo ~ Late yP-v, 1 — a79-1+% yetPe 1 = art yPatPo , 1 — wF-124 yet eo 9-1 (G4?) G geart eer Po | — £9-17t (G1) % P_\+(a,—1) Po, va Py? x y —ahy APs Mason MACMAHON, THE DIOPHANTINE INEQUALITY ix > py. LILY, wherein the exponents are derived from the intermediate convergents to E, Pr qa inclusive of these principal convergents. The remaining part, viz. :— Oo 1 > ee me Pec must now be subjected to the process of the first lemma and becomes i) J Qo t > ja ah yP2 ? “1 — a4 bry® 1 — ath AL a eaghyh 1 — — at b% QEP2-42 and now the operation of the second lemma yields fe) i > wz yh = J — ah-#G1 — U3 (HP 2-AQ2) pt M42 YyPrtasP2 aJh= PS as QeP2—-A4q S = chy (zteyes) ta 7 ad : 1 i= ayes (vty? jst 1 —ahyh (vi yP2)% 1 SS wi yhs = 1 —ade-¥Pa Ts yPs tS Y QPP2—-4Va en yr BUT YP Ps chy ie awh yp sh LAT yPrtPa a il — Kt Ta yPrtPs : _ GAM PF De gut (d3—-1) 92 gPit(4s—1) D2 ae oy oF nt GN pt@-Dr | — ah yrs? and we have before us a new portion of the generating function corresponding to the intermediate convergents to the principal convergents 2 Ea the latter both included. " 4s The portion remaining, 1 2 > rhyPe ~ 1 = ahts—#Ps gays, | — aah QeP2- 42 is now subject to Lemma 1 and the Lemmas operate alternately until the final result is reached. The last remaining portion must be to) 1 > 7 Lahn HPn gnyPn , 1 — a“in- yPn-i Q#Pn-1—-AQn-1 1 avin yPn ve 1 = an yPn an ui = adn yen 7 118 Mason MACMAHON, THE DIOPHANTINE INEQUALITY Ax>py. Hence the complete generating function may be written Ey xy ah yr 1 + ——__ {oa SS : Ewes ¥ l—a#.l—ay 1—ay.1—2y* L— gh yr . 1 — ahyr ah ys enti yt Pe aa 1 = wh yes =a [ = ah yPrtPa a ce ata yet Ps ie LNT yt Pa sah grt (43-1) 92 yPrt (as) Pe Z 1 — at Gs) @ pat (as-)) P2 | — gis ys SP nonnas 2 TAn-2 yPn—2 ain—2t (AnV) In-1 apna (Qn) Pa 1 = an-2 yPn-a 1 = gn In-1 YPn-2 Pn Sioa T= ana (tn Gn-a yPn—2t nV Pua, | — gen yPn Zin yen ny 1—az2m yn and, from well-known properties of the convergents, every xy product occurring in these fractions gives a ground solution of the relation. Hence the ground solutions are T—T, Ps, where _ 8 in its lowest terms and is any member of the ascending intermediate series x of convergents to i As an example, take 7792 > 2078, od Cees Ses ae a re 207 ver aS a eae The principal convergents are 0 1 38 4 15 64 148 636 779 LO ie alegre geoee ae Go. 20: and the ascending intermediate series Ce ETT ay UES EP 2? Tee aL Se eS eee RE? PES SR e201. where, if the principal convergents are 2m+3 in number, the number in the inter- mediate series is mMH+t2+a +d3t ast... + Goma —m—lLl=1l+aq4+as;4+ dst ... + Gomi. Hence L278 is 2 x avy avy? if: = =: Pi onl —ay* 1l—ay.1 = ay l=ay.1—ay ee _- wy? asy™ oa — xy>.1 = ay? t 1l—a*y’.1 —ayat 1— ay". 1 —aly® ay? Zaye Toy 1 aye t Tay 1 — ay Ce Ps ye aP i gays | Pe aye t he qr , Mason MACMAHON, THE DIOPHANTINE INEQUALITY Ax > py. RLS, and the ground solutions of the relation are a B 1 0 1 1 1 2 1 3 2 7 3 11 4 15 PAT Th 48) 38 | 143 207 | 779 As another example, take 77a > 1048. Lemma 1 shews that we have to sum Lx (xy)8, for the relation 77a > 278; al 1 Peel plea a ET ee net | a a aud Weis wie oer Ie 5 The principal convergents are and the intermediate series ON 2s eek Hire 57 WS Ww BP BSP TR EIB}? 90)? ah Hence the sum z* (xy)? is Z ay ad 2 wy? ays + l-#y.1l—ay Bere aye. 1 — ory? 3 1—a¥y.1— aeyn ay #1 wey ; i ey" a if = geye 4 1 = xy 1 = ey z i = ay" i/ = ay i | = eye Heal Tal ay ety? Te ee aye re aitys ar T= gry, ree qiesy7 ar re ay 1h = 120 Mason MACMAHON, THE DIOPHANTINE INEQUALITY Azx>py. Hence the ground solutions are a |\28 1), 20 2). ea Sale? 7| 5 Li 28 15 | 11 19 14 23 | 17 50 | 37 77 | 57 The operation of Lemma 1 is however not necessary, for, since 77 1 ea 104-9 474947 we can form the series of intermediate convergents Dh 2p 455 $85 OL a iG gals cod Gl 7 D2 7 172 WL? GPIB) OBB” BO Wer Oe which lead directly to the generating function and to the fundamental solution. The generating function shews that every solution is of the form a = A,aln ae AP Oa B=A,8 + A,.R", Bo [oho where a? qi are consecutive members of the ascending intermediate series of con- vergents to X ‘The corresponding generating function, associated with the Diophantine Kb equation ra=pB+y, is obtained by multiplying each product 2*y’, in the generating function, by gene, Ex, gr. for the equation 77a= 1048 + y, Mason MACMAHON, THE DIOPHANTINE INEQUALITY Az> py. 121 we find SayFzy az # ayz” Vaz" Vey? V—aty2 1 — ay228 wyZ ayrz 8 eeyezi® 1 = a 2723 | 1 = a’y’2 WF 1 = ay?z ‘ it ae aysz' + il Soe yeeo < iL = eoyzn gbytgu 4 weyizt 4 ways ilk ei zPyigt i iL = ay agi a. = wy tg : il pee aypizs 1 =e ay zs 1 = wy? Zz a ysiz2 ay Z gph yi ears == 45 == = = 1 = vysiz? i 1 — Bae 71 as a a= Bey il 2 ey 2 + establishing the ground solutions CHS a) 1 digas Obl eter 2a S50 2 | 23 (fa oie) || we as 1b) | 19), | 47 O30 lial m3 SO esi. 2 Goaleever eal 104 | 77 | 0 There is a connexion between the values of « and vy peculiar to this case which will be made the subject of investigation elsewhere. It appears, from the form of the generating function, that every solution (a, B, y) of the equation must be derivable from, at most, two of the fundamental solutions, say (a, Bm, y"), (@i), [Sia yy), viz.:—we must have a=A,a"+ A, a?, B=A,8"+4,,8", y= Ary + Anny”, the two solutions being consecutive. Eliminating A, and 4A,,, from the three equations we obtain | a B Y | al” Bp” y” |= 0, arty Birt yet Vou. XIX. Part [. 16 122 Mason MACMAHON, THE DIOPHANTINE INEQUALITY Ax >py. a linear relation between a, 8, y which must be identical with X%w—p~8—y=0; hence ( +1 | r+1) | y” y” Ves) y” y" ) = => | B” pr |? B al) qirt) BY Be -—— al” qt) the latter relation verifying an elementary property of consecutive intermediate convergents. We have the identity eh pbs -y= pAlS fra”) = pe” = yi} SE ARS (reir) = parr = yet, so that im a sense the linear function Aa— pB—¥ is reducible qué the fundamental linear functions ra) — pBM—_, Ag) — pRB — yt, Ex, gr. consider 77a — 1048 = 815. In the generating function we find a term (yz) (aeyPz® P= aye, yielding the simplest solution, and since 77.43 — 104. 24 — 815 = 14(77. 2 — 104.1 —50)+4 5(77.3 — 104. 2 — 23), we find that the solution before us of 77a— 1048 —815=0 is a linear function of the simplest solutions of the equations 77a, — 1048, — 50 = 0, 77a, — 1048, — 23 =0; viz.:—we have a= 14a, + 5a,, 8B = 148; + 5fz, 815 = 14.5045. 23. The general solutions of 77a, — 1048,— 50 = 0, T77 — 1048. — 23 = 0, are a =2+1046,, Bi =1+776,, a, = 3+ 1046., Bo=2+776,, Mason MACMAHON, THE DIOPHANTINE INEQUALITY Az > py. 123 respectively, and these lead to the solutions a = 43 + 104 (140, + 58,), B=24+4+ 77 (140, 4 58,), of the equation 77a — 1048 = 815; the general solution of this last equation being a =43+4 1040, B=24+ 776; consequently the solutions derived are those for which @ is of the form 140, + 50. § 2. I next consider the simultaneous Diophantine inequalities Aya > LB, fo > rom, equivalent to the simultaneous Diophantine equations Mt = B+, pod =A.x + 6. I observe that, if the given relations can be simultaneously satisfied by other than zero values of the arguments a,, 8,, we must have | | > 0. fi [28 | It is convenient to reduce the relations to others in which 2, >, and p> As, an operation which is always possible, as I establish in the following investigation. I assume i. fa to be relatively prime and also ,, w., and I assume further that w,>r,. The crude expression for Lary, the sum being for every solution of the simultaneous relations, is If the fraction ©! be developed, in the form of a continued fraction, the first step 1 is to write (i Cy —=&+—, Ay Cee: 16—2 124 Major MACMAHON, THE DIOPHANTINE INEQUALITY dx py. where c;=A, and c.a8, and the crude expression may be written 1 ahd, te” br: - aeid& $ al © operating upon a, b and d. Eliminating d this is In this expression ~,— «A. cannot be zero unless A,=1. Suppose then Aa—ale Pa — QhAg = 0 = pn — G. The expression becomes 1 at aa UA Vid ee which is unity, except when , —aA,=0, when we obtain the generating function 1 1 x%y’ corresponding to the Diophantine equality a=a,f. Moreover, if p:—a,A. be negative, the whole expression reduces to unity so that there exists only the singular solution 2=8=0. Assume therefore bs — hry > 0, and putting =), fe—GA.=Cc,, write the expression il g a bez ; L— pt 1d If now ec’ >c,, we have effected the reduction, because c,>c., and we have the two relations Ca > cf, Gea, connected with the sum v eet nByB _" Mason MACMAHON, THE DIOPHANTINE INEQUALITY 2 > py. 125 If c/=¢/, or po=(a,+1)r., we must assume A,=1, because Ay, f. are relatively prime, and yw, >2,. The expression then becomes fe) 1 > ol 7 1 —qg@tacm gat y | wy Qa MA, 2 which denotes Seatatas yet, for the single relation {(q +1) A. — py} @ > (H, — Gr) B. Putting this aside, as a case already dealt with in § 1, we are left with in which ¢,’c,; and the expression must be reduced further. I observe that “ and have the same first convergents, and that ; : ’ He Write 1 2 Tes. - va where a,+ — is the second convergent to = and ¢,;’ a 1 Co d ’ pane aa from which, eliminating d, we obtain O peeeeeeets 21. ia 7 > 1-2 C2 a Gta tl 7 /8g peepee | i pa ac x y?. il ae uty ne Co : Since & = a where ¢, Cy 126 Mason MACMAHON, THE DIOPHANTINE INEQUALITY dx > py. whenever ¢,— a,c, is positive, and then where c, is not necessarily < cs. The expression becomes 1 Cs C3" a be ‘ = ‘ +1 Aa+1 yt , 1) 1 pea” y Poe | Wa) If c, be zero, c;,’ must be zero also, and the generator is simply 1 jl TAH) yl 4 corresponding to Ay = (Aya, + 1) B. If c; be not zero, we must have Ss, =O c, the reduction is complete because ¢,’>e, also. In this case Oa tas is the second convergent to = but is not the second convergent to es 1 If c,=c, we can at once eliminate a from the expression and obtain an expression that has been already dealt with. uy ; Nts Cusp Gayct= a the second convergent to both - and = and a further process 1 of reduction is necessary. To sum up the results so far reached:— (i) if Ky > Kay Ho > re, Ci > Kis uo reduction takes place ; @)p ak fly — UyAy > Az, Ny > fay — Gy, the final reduced form is Mason MACMAHON, THE DIOPHANTINE INEQUALITY Ax py. : Ga : s wherein “ = ep — eng — le ands ay =P is the first convergent to ap yy Cy nh Ay A Cx : > eG +—. and a, is not the first convergent to Fe. de CG No (ui) if a, is the first convergent to both Fa and fe. GS, CS Ale oe 1 form is Q ¢€ : b¢: J ’ > acs 2 Ne tye Lk =: zPi yh 3 Oke : nc: wherein aE ee Bee Ay A, + Co Ke 1 BS Eee aeety wag ee ne = fy + Cy I Ys : and a,+— =" is not the second convergent to = but is the second convergent to Mog Qa 1 1 he 2 ; : If aa is the second convergent to both = and = a further reduction 2 2 1 2 comes necessary. =p IL ail ¢ Writ ae peste ec yy aad: Qs + Ay + C3’ He 1 1 Cy i ee ee Ne (lz + Uy + C5)” . 7 r ep: WHEKGINE Cse> GC, 16,4 SO that a --— = ans Gz+d3 Qs not the third convergent to = 2 The final reduced form is then ra) 1 = as b% 1 Bea" aPryt, i= —— EP3 yt ? a4 which denotes the sum > (a2 yi?) (xs ys PB = SgP.2tp3B year ass for the relations C2648, CB SC; a. We proceed in this manner so as to establish the following general results. final is the third convergent to 2 but is 1 128 Mason MACMAHON, THE DIOPHANTINE INEQUALITY ix >py. CasE 1. If a = a, a I i h Cn © ny Ag+ Ag+... +Gn+ Cy Ms = a, + 1 i ED Cnt1 5 re Ag+ Ag+... +Ant+ Cn where Cy1;>Cy, but ¢p>Cnia, So that z is the nth convergent to = n 1 nth convergent to = , the final reduced form is 19) 1 ————— SS = arn bean — ion ZPn-1 yn , i= oe HPn yI which represents the sum s (ePi yin) (vPn yn 8 = > UPn-19t FB afin at+dnB for the relations Cn& > Chai B, , , Gisan/e! ZC pM, WHEreINCAE>Cnsa5) Ciao C ae Case 2. If 1 1 1 9 ee + ew conte yy Ag + Og + 20. + Oni t+ Ona Bs =a+ i 1 bt Cnts e re Ay + Ag+... + Ana +6 nar F , , Pan : Bs WHELCUGn es Gna.) Cia > Css SO) unab ie is the n+1th convergent to i n+l 2 the n+1th convergent to M the final reduced form is 1 fa) 1 > qcnt2 bean = Pati yInti — P; 1 — Fo wPaes yinet 1 ayn 5) which represents the sum >> (wPav yinn)* (gPn ym )P = LaPntiePnB yIn+i27InB, for the relations CnzoA > Crk; , , c ranyo) ZC nzod, WhereiN Gnis>>Cas, Cnei> C nia: but not the but is not Mason MACMAHON, THE DIOPHANTINE INEQUALITY dz py. 129 This is the complete solution of the problem before us. As an example, suppose we have to find Sa*y® for all values of a and 8 which satisfy the Diophantine inequalities The first process is the reduction as above to a standard form. We develope the 275 142 fractions G4 and 35 to the poimt where they fail to have the same convergents. Thus 275 YoU & Sears = 64+ 34247 142 folie 1 4 33 342437 4 13 s : and we observe that rT and “yz are the first and second convergents to both fractions, 30 275 142 7 2% 7 rah but not to 33° This is Case 1 above and shews that the problem is reduced to finding the sum but that is the third convergent to S (EE (ay? )8 = > gisa+30B Genre for the relations which are of the standard form. We are now able to give a complete solution of the problem of finding the fundamental solutions of the simultaneous Diophantine inequalities a 2 bh B, fo 8 > Not. If these be of standard form, viz. \,> 4, fi >Ae, consider the three systems NUL III. Aa > HB, 44> HB, Ma > 48, [2B > roo, 2B > rer, MB > rea, Bea, a>BP, a=B. If F,(«, y), F.(x, y), F,(x, y) be the corresponding generating functions it is clear that the generating function that we seek is F, (a, y) + F(a, y) — F; (a, y). Nor, XLX, Parr I. 17 130 Mason MACMAHON, THE DIOPHANTINE INEQUALITY Ax>py. Moreover we may reject as superfluous the second relation of I, the first relation of Il. and the first and second relations of III., reducing them to ie II. III. NZS 148, poB > roa, Bea, a>B, GS/s} Hence F; (2, y)= ! and we have l—ay Fy(a, y)+ Fi (@, y)— Fs(@, y) fa) 1 fe) 1 1 = > aX i ave l—«wy fg EGP Sete Be 0 it oo 1 1 => > =e —qn-h = 1 — uh — My—Ao yp a — gee Y iw avy a l—a ry. 1 aS We now apply the theorem of §1 to the two Q expressions forming the ascending Ha — Ae and to We can thus determine the : . : M4 = intermediate series of convergents to eae be 2 complete generating function and by inspection ascertain the fundamental solutions. As an example take the relations Ta > 58, 48 > 3a; we find Q i i (a) 1 J a Ol Gey Sh a ey p= fe ae : as : 5 2-3 Qo Ut 2 The ascending series to = 1s I? 3° 3) i Oa and to 3 3s I 3° Hence Lay? is ey wy" ey 1 a sa —ay.1—ayi 1-#y.1 x yt t 1 ay? . A Sey vy atys 1 fee ee ‘ [—ay.1—a%y 1—atys 1l-—ay’ and, without further reducing the expression, it is plain that the ground products are ay, wy, ay’, ay’; Mason MACMAHON, THE DIOPHANTINE INEQUALITY a> py. giving the fundamental solutions oop oo | 2 os If we now wish to solve 1428> 38a, 151 which we have already reduced to the question of summing =(a"y*)*(x%y")? for the relations Ta > 58, 48 > 3a, we have merely to write «¥y°, «y’ for x and y in the result obtained; we thus reach the ground products (ay’) (a*°y") = cep (aye)? (ay") = Bayer (a y®)! (ay?) = ay, (ayy (cy? Nye = Goye: and the fundamental solutions a |B 43) 10 159 | 37 142 | 33 275 | 64 In general the number of fundamental solutions of two simultaneous Diophantine in- equalities is the same as the number appertaining to the pair of standard inequalities to which the given inequalities are reducible. VI. Rational Space-curves of the Fourth Order. By Hersert Ricumonp, M.A., King’s College, Cambridge. [Received 3 May 1900.] HISTORICAL SKETCH. THE theory of space-curves is one of the many branches of mathematical knowledge whose origin is to be found in the wonderfully fertile correspondence that passed between Cayley and Salmon some fifty years ago: among the first-fruits of the joint labours of the great English and Irish mathematicians in the subject was the discovery in 1849 of the existence of a species of curve of the fourth order which differed radically from the previously known curve of intersection of two quadric surfaces. In the half-century that has elapsed these rational quartic space-curves have,—so far as I have been able to learn from studying the table of contents of the Jahrbuch iiber die Fortschritte der Mathematik year by year, and consulting all, or practically all, the papers mentioned there that seemed to bear on the subject,—inspired some fifty or sixty communications to the various scientific periodicals: I have here attempted to present the substance of these papers as a coherent whole, in the hope that the compilation may exhibit the nature of these curves in a clearer light and form a chapter not without interest in the history of nineteenth century geometry. I pass on to a historical sketch of investigations concerning these curves, in which numerous references are made to the bibliography following it, beginning with a short account of the circumstances which led to the discovery of these rational quartic space- curves. Cayley had shewn in 1845, in the memoir (2), how Pliicker’s equations respecting plane algebraic curves could be extended to algebraic curves in space; obtaining his results solely by considering the cones on which the curve lies and the plane sections of the developable of which it is the cuspidal edge, without regard to the surfaces which by their complete or partial intersection determine the curve. For a classification of space- curves Cayley saw that his results were insufficient; in the last paragraph of his article he says:—“ Le probleme de classifier les courbes a double courbure au moyen des surfaces Mr RICHMOND, RATIONAL SPACE-CURVES OF THE FOURTH ORDER. 13: que l’on peut faire [? passer] par ces courbes, ou de trouver la nature d'une courbe qui est [intersection de deux surfaces données, parait appartenir plutét & la théorie des surfaces qu’d celle des courbes. Je n’ai rien de complet A offrir sur cela.” Four years later appeared the article described as (1) in the appended list, in which we find Salmon writing:—“The immediate object of this article is to take the first steps towards a classification of curves of double curvature considered as the intersection of surfaces ;......the present investigation having been pursued in conjunction with Mr. Cayley, our results are for the most part common property.” A method of classifying space- curves of any given order had in fact been devised by the collaboration of the two mathematicians, and the classification was actually accomplished for curves up to the fourth order: incidentally too a new species of quartic space-curve was discovered, the curve of intersection of a quadric and a cubic surface which have in common also two non-intersecting straight lines. It is interesting to observe that, in the interval between the publication of papers (1) and (2), Salmon had, in the course of an investigation (3), met with an instance of these rational curves of the fourth order, had attempted to find a net of surfaces of the second order passing through it, and, as was to be expected, had failed to find more than one such surface. The properties and nature of the new curves ‘are very briefly dealt with in memoir (1). The facts that the curve meets one set of the generators of the quadric thrice and the other set only once, and that the cone whose vertex is any point of the curve has a double line, are pointed out: we are led to within an ace of an explicit statement of the important theorem that the coordinates of points on the curve may be expressed as rational functions of a parameter: and lastly it is shewn that through eight given points pass four curves of this kind, and a construction is stated. No further discussion of the curve was attempted for twelve years after its discovery by Salmon and Cayley, but in 1856 Steiner (4) also noticed the existence of two radically different species of tortuous curves of the fourth degree. The symbol R,” being now recognized and extremely convenient for the purpose of denoting a rational curve of degree n in space of p dimensions, I shall speak of a twisted quartic of the above type as an Rs, or simply as an R, when there is no doubt that space of three dimensions is under consideration. In 1861 Cremona published the fundamental memoir (5) concerning these curves, the first in which their properties were exclusively investigated, and still the locus classicus for those who wish to study their nature. Without any wish to detract from the great importance of the advance made, I must urge that an exaggerated notion of the amount of original matter im Cremona’s memoir has always prevailed, owing to the fact that the illustrious author contents himself with a general reference to Cayley and Salmon in his mtroduction, so that in what follows it is impossible to distinguish between the proof of a new fact and the verification of one previously known: it was only after making an abstract of the memoir that I realized how large a proportion, both of the results and the methods by which they are obtained, are due in reality to Salmon and Cayley. Pre- 134 Mr RICHMOND, RATIONAL SPACE-CURVES OF THE FOURTH ORDER. supposing only a knowledge of Pliicker’s equations for plane curves (which, be it noted, is just what Cayley had presupposed in his study (2) of all algebraic space-curves), Cremona establishes for our special kind of curve, the Rg, the results Salmon had given concerning its class, its rank, the number of stationary planes, the degree of the double curve of the developable of which the R is the cuspidal edge, and so forth. Apart from these repro- ductions we find in Cremona’s paper proofs by synthetic methods of many new facts of the highest value; we find a clear statement that a [1, 1] correspondence may be established between the points of the curve and a straight line and that this is the reason of the greater simplicity of our ,°’s in comparison with the other class of twisted quartics; some theorems on the intersections of rational cubics and quartics which le on the same hyperboloid ; new theorems on the surfaces which by their intersection determine the curve; some deductions from Salmon’s results concerning the cones on which the curve lies, obtained from the known properties of plane rational cubics and quartics; the theorem that four tangents of the curve intersect it again in a third (non-consecutive) point, and other genuine additions to the store of knowledge of properties of the curve. Cremona is not happy in his choice of method; most of his results may be established simply from the expressions for coordinates of points of the curve in terms of a parameter, and later writers have applied the methods of pure geometry to the curve with far greater elegance and effect; at the same time, in spite of the serious fault first mentioned and these minor ones, the advantage to writers who succeeded him in having the properties of the curve collected and methodically developed cannot be too highly estimated. Of the three next papers, (6) and (8) deal with a special type of curve and may be passed over, although they to a small extent anticipate the method of the parameter, adopted in (9), (10), (12), and (13); (7) calls for no comment. Papers (8) and (9) are the earliest we owe to the most prolific writer on these curves, Emil Weyr, and belong to a series of notes on rational curves both in two and three dimensions contributed by him to the Sitzwngsberichte of Vienna and other journals during many years. By 1871, the date of (9), the fundamental properties of curves of deficiency p=0 had become known,—[the classic memoir of Clebsch, “Ueber diejenigen ebenen Curven deren Coordinaten rationale Functionen eines Parameters sind,” for instance, had appeared in 1865 (J. fiir Math. Crelle, XLv1.)]—; Weyr therefore takes as his subject all space- curves whose points have their coordinates proportional to four quartic functions of a parameter, thus including (although apparently he did not at the time realize this) a case specially excluded by Cremona’s definition, viz. of those curves which have a double point; such curves are in fact particular forms of both kinds of quartic space-curves, their pro- perties appearing to be on the whole more closely allied to those of the general rational curve. From the expressions for coordinates of poimts of the curve in terms of the parameter , which I shall in future refer to as the parametric equations of the curve, Weyr deduces the condition that four points of the curve should be coplanar; viz. that their parameters should satisfy a rational symmetrical relation in which no one of them enters in a degree higher than the first: and proceeds for the remainder of his paper to ring the changes on this result, readily deducing from it the facts that the curve has four stationary Mr RICHMOND, RATIONAL SPACE-CURVES OF THE FOURTH ORDER. 135 osculating planes, four tangents that meet it in a third (non-consecutive) point; that one trisecant chord and three osculating planes pass through any point of the curve, the points of contact of the latter and the chosen point being coplanar; that two tangents meet a given tangent,—all of which were previously known, and pointing out that these pro- perties define certain involutions of the second or third or fourth order upon the curve. No attempt is made to simplify the parametric equations either by choice of new coordinate planes, or by transformation of the parameter by a homographic substitution, nor does the memoir reveal any inkling of the fact that the whole theory of the curve depends on the single quartic equation which gives the parameters of the stationary planes and its concomitants. An important investigation, due to Bertini (10), next claims attention: under Bertini’s handling we recognize that the curve has a distinctive character and certain pure geometrical properties which are peculiarly its own. Quoting Cremona’s theorem that the curve is the intersection of a cubic surface which has a double line and a quadric containing the line, Bertini propounds the question whether the curve of intersection of such a cubic surface with the polar quadric of any point is an R, of a general or special type. He arrives finally at the conclusion that it is of a particular kind, to which he gives the name equianharmonic, but in the course of his investigation establishes incidentally a series of far more valuable results. Foremost among these is the existence of three principal chords of the curve, which meet in a poimt (to be described hereafter as the centre of the curve), each of which chords is the intersection of the osculating planes of its extremities: various properties of these chords and of the curve are given, noticeably some concerning sets of four points of the curve which form tetrahedra whose three pairs of opposite edges meet the three principal chords. The contacts of the stationary planes and of the trisecant tangents of the curve are vertices of two such tetrahedra, and thus a quartic involution determining the parameters of any four such points is defined, its double members bemg the extremities of the principal chords. The very simple form of statement of these theorems which is given in the next paragraph but one was overlooked by Bertini, who breaks off his investigation to explain a method of establishing a correspondence between the points of the curve and of a conic, and a few results that follow. If it be thought desirable to exhibit the parametric equations of the curve in a form containing as few terms as possible, Bertimi’s equations ie, 2 ik, Ody 8 Gay 82 OY BOR(GP=SCP) 3 oF=— Il 2 @ 2 3 are as simple as any that can be devised, but the loss of symmetry and of the possibility of transformation of the parameter more than outweigh the advantage; nor is the above a form to which all curves can be reduced by real transformation. On the other hand it is made clear that the classification of rational quartics depends on the value of a single constant a*, e.g. that the equianharmonic curves above have a?=—3, and that the curves in papers (6) and (8) have a2?=9. Bertini gives also the equation of the Steiner's quartic surface of which the Rf is an asymptotic curve and the three principal chords the nodal lines; but here as throughout his memoir, misled by the illusive simplicity of the 136 Mr RICHMOND, RATIONAL SPACE-CURVES OF THE FOURTH ORDER. above parametric equations of the curve, he fails to express the result in its best form. The paper (11) which follows has no share in the development of the theory of the curves. What Bertini proved concerning the principal chords leads almost immediately to the following theorem :—with a proper tetrahedron of reference, of which the principal chords are three concurrent edges, if a point («, y, z, w) lie on the curve, so also do the points (x, —y, —2, w) (—2, y, —4, w) (—#, —y, 2, w). A consequence of this which Bertini over- looked is that this tetrahedron must be self-conjugate to every quadric which stands in a unique relation to the curve, such as the quadric on which it lies, or that which all its osculating planes touch: again, Cremona saw that the planes containing any point of the curve and the contacts of the three osculating planes which pass through that point, all touch a quadric cone; we infer at once that the centre of the curve is the vertex of the cone—(this Bertini proved otherwise),—and that the principal chords are mutually conjugate with respect to it. We may further, without affecting descriptive properties, suppose the plane w=0 at infinity and the principal chords mutually perpendicular; the curve will then consist of four exactly equal and similar parts;—a rough notion of a possible form being suggested by a wire originally in the form of an ellipse of which two opposite quadrants have been symmetrically bent above and the other two sym- metrically bent below its plane. Every property, metrical or descriptive, possessed by any one point of the curve is now necessarily shared by the three other points placed on the three other quarters of the curve symmetrically with the first point, and while no descriptive properties are lost certain metrical properties are acquired, and the four- fold nature of the curve, if I may so express it, is unmistakably shewn. Yet another writer, Armenante (12) and (13), attacks the problem of the RJ by means of the parametric equations, and this time the manipulation of the analysis is irreproachable. Weyr had ignored the possibility of transformations; Bertini, in using it merely to reduce the equations to apparently simple forms, sacrificed symmetry, and lost in other respects far more than he gained; Armenante is the first writer upon these curves who has realized that the parametric treatment of a rational curve is a matter of invariants and covariants, ze. of functions unaltered in form by a lineo-linear transformation of the parameter: he therefore employs Aronhold’s symbolic notation throughout his investigations with excellent effect. Having defined the curves by their parametric equations, written in the symbolic form TE ORC EU Sa, = ATS Lee ane 2 ANTS he points out that a new quartic function #F or k,* is uniquely determined by the conditions that (ak)*, (bk), (ck), (dk)! should all vanish, (that is to say, in modern phrase, kt or F is apolar to each of the four ag‘, bat, ca‘, da‘); and that the condition that four points whose parameters are d, uw, Vv, p should be coplanar is then expressed by ky kliykp = 0, Mr RICHMOND, RATIONAL SPACE-CURVES OF THE FOURTH ORDER. 137 a result from which at once follows all that Weyr established in (9). The roots of F are clearly parameters of the contacts of stationary planes of the curve; those of H, the Hessian of #, are parameters of the contacts of the four tangents which also cut the curve; those of G, the cubicovariant of F, are parameters of the extremities of the principal chords. The involution determimed by F and H gives the parameters of points of the curve which are vertices of a tetrahedron whose edges meet the principal chords, and the roots of G@ are the repeated members of the involution. The cross-ratio of the parameters of four points of the curve is equal to that of the four planes through the points and any one chosen trisecant chord. The quadratic invariant 7 of F’ vanishes for the equianharmonic curves which Bertini discovered, and these further have the contacts of the stationary planes coplanar; the vanishing of J, the cubie invariant of F, imphes that the curve has a double poimt; the simultaneous vanishing of J and J implies that it has a cusp. Armenante’s second paper deals mainly with surfaces related to the curve: concerning these I shall quote only two results. viz., that, if X and w be variable parameters, the equations fh Sy 2 HE 8 GH SS GT, 3 OSD, 8 OtOn 8 GGT, 5 and Hey 8 ty 2 Dy S Gh B Geant 6 see Geoe 3 Chedl?. represent respectively the osculating developable of the curve and the Steiner's quartic of which it is an asymptotic line. The present is a convenient place to speak very briefly of the modern applications of the theory of combinants to rational curves of any order (n) in space of two or three or any number (d) of dimensions. The coordinates a, #,, @...«q beimg propor- tional to rational integral functions of a parameter 2X of degree n, any property or locus is said to be invariantive when the algebraic forms which represent that property or locus retain their shape, (1) when the coordinate planes are replaced by any new system, (2) when the parameter X is replaced by a new parameter connected with it by any lineo-linear relation. Clearly then the equations which represent such properties or such loci must be combinants of the d+1 functions of X; and, if for the moment we confine ourselves to what may be called Pure Combinants of the d+1 n-ics in X, (ze. con- taming only parameters and the coefficients of these d+1 forms, not coordinates of points, lines, ete.), we find that Gordan in Bd. v. of Math. Annalen (Ueber Combinanten, p- 116: see also Stroh in Bd. xx.) has obtained the generating functions from which they are derived, and shewn how, if » be equal to d+1, they may be derived from a single n-ic in a single variable, (Armenante’s quartic # or k,* in the case of an R). An excellent explanation of the method is prefixed to Berzolari’s Sui combinanti dei sistemi di forme binarie annessi alle curve gobbe razionali del quart’ ordine, (48), Annali di Mutematica, Tom. xx. 1892, p. 101; but a glance at the memoir will shew the impossibility of sketching its contents here; the geometrical nature of the subject is completely lost. Consult also pp. 254—265 of Meyer's Bericht iiber den gegenwartigen Stand der Invarianten-theorie, (in the first volume of the Jahresbericht der Deutschen Mathematiker-Vereinigung); S§112—117 of the same author’s ‘ Apolaritiét und rationale Curven’ are also devoted to our curves, as exemplifying the general theory of Apolarity. Wom, IDK, IeAra Il {3 138 Mr RICHMOND, RATIONAL SPACE-CURVES OF THE FOURTH ORDER. Of papers which are subsequent to those of Armenante it is not necessary to speak so fully; those of Weyr should have first mention. In (14) and (15) he retains the parametric equations as in (9), reducing the relation which connects the parameters (\, #, v, p) of four coplanar points to the respective forms (1) Apvp=k; (2) X+u+v+p=0; in the cases when the curve has a double point or cusp, and obtains some new properties of these curves, the work being as in (9) straightforward and somewhat unattractive. In (16) he adopts a quite different method; he maps the curve, point for point, on a conic (ie. establishes a [1, 1] correspondence between the points of the curve and those of a conic), and obtains the condition that four points of the conic should correspond to four coplanar points of the quartic, viz. that the Ime joming two of them should be con- jugate to the line joing the other two with respect to a second conic. Although he was at this time not acquainted with Bertini’s paper (10), he discovers a property which is characteristic of the equianharmonic curve, viz. that sets of three tangents are con- current, and which Bertini had overlooked; also that the tangents of this curve which intersect it do so at the contacts of the stationary planes. In (18) he verifies the results Bertini had proved about the principal chords, having in the meantime read (10); and in the later papers deals as a rule with special curves. As a pure geometrical method the plan of mapping on a conic is far preferable to Cremona’s: but a very slight admixture of analysis would vastly simplify Weyr’s later studies on the curve. The inclusion of the name of Sophus Lie in our list is due to the fact that at one stage of his classical research on minimal surfaces he found it necessary to consider curves without length, and incidentally discovered all such curves of orders three and four. Considered projectively, Lie’s work determines all rational quartics whose tangents intersect a fixed conic; but he contents himself with proving the existence of certain types of these curves, and had no thorough grasp of their nature: his ignorance led him into one actual blunder. It is however noteworthy that Lie discovered independently, atter a lapse of less than two years, the property of Weyr mentioned in the last paragraph, viz. that in certain rational quartics the tangents intersect by threes on a fixed conic. (See pp. 77—82 of this volume.) The method of mapping the curve point for point on a conic, presented in a more definite and more attractive form, led Adler (25) to class non-singular rational quartic space-curves under four types, according to the relative positions and reality of the intersections of two conics, which determine also the reality of the stationary osculating planes, trisecant tangents and principal chords. The results are unfortunately by no means accurate ; for instance, the curve with four real trisecant tangents and real stationary planes mentioned in Class I. does not exist; and a curve with four real stationary planes and no real trisecant tangents is omitted. In conclusion Adler mentions the line of striction of a hyperboloid, an interesting case of these curves; but his classification of R,*’s is untrustworthy and therefore valueless. R. A. Roberts (27) obtains several new and interesting properties of the curve. By a 2 eo Mr RICHMOND, RATIONAL SPACE-CURVES OF THE FOURTH ORDER. 139 taking the stationary osculating planes as faces of the coordinate tetrahedron he obtains the coordinates of points of the curve as fourth powers of linear functions of the para- meter 2, and deduces that these four planes determine on any tangent of the curve four points of constant cross-ratio. He rediscovers the fact that the roots of the Hessian of the equation which gives the parameters of the contacts of the stationary planes give those of the trisecant tangents; indeed Roberts writes apparently in entire ignorance of what others had done, and gives no references. The classification of the curves, he sees, must depend primarily on the value of the absolute invariant J*+ J? of the above equation. The form of the condition of intersection of two tangents shews him that, if one skew polygon of m sides exist whose sides touch the curve, an infinite number of such polygons can be constructed: if =3, the curve possesses sets of three concurrent tangents, (a property Weyr proved to be true of the equianharmonic curve), and their intersections lie on a conic, the degenerate form of the double curve of the developable formed by the tangents of the curve. If n=4, J must =0 and the curve have a double point. Some considerations of a special case complete the first paper. The Steiner’s quartic of which the curve is an asymptotic Ime is obtained in the second paper in its simplest form, and parametric equation of the same surface is given in such a way that the constancy of a parameter defines an asymptotic curve of the surface. The equation of the quadric on which the curve lies is also obtained, first under any conditions, secondly veferred to the tetrahedron formed by the stationary planes. A later paper (31) is also wholly without references of the labours of others and gives an impression of aimless wandering; it is however possible that some results concerning rational quartics are to be found in §§ 9—42 which had not been previously stated. It is impossible to mention each of the next succeeding papers in detail; many deal with particular curves, as will be seen from their titles; others mark advances in the application of theory of binary forms to the curve, or tend to new fields of study, e.g. (33), (35), (40), (41); these are, I think, better considered as marking stages in the growth of geometry of rational curves as a whole than in the development of the knowledge of our particular curve, and are therefore passed over: of the others that of Bordiga (34) may be mentioned as taking a novel view of the curve, viz. as the projection of a quartic in space of four dimensions. In paper (44) the problem of classification of rational quartic space-curves is again attacked, this time with success. The author, Rohn, explains that his investigations were prompted by the notion of constructing models of the curves and their developable surfaces: and no better incentive to the study of any class of geometrical entities could be found. The accomplishment of Rohn’s intentions is announced in (47) and his models are now members of Brill’s famous series*. There are four chief types of real non-singular curves, and all imaginary curves are classed together, (perhaps rather too curtly), as a fifth type. The equations of the chief loci connected with the curve are in most cases found explicitly, and finally some special curves are considered. The curves which have a double point or cusp do not come within the scope of Rohn’s investigation. * Now to be obtained from Herr Martin Schilling’s Verlagsbuchhandlung, Halle a. S. 18—2 140 Mr RICHMOND, RATIONAL SPACECURVES OF THE FOURTH ORDER. The last paper I wish to mention is (48). Nothing could prove more clearly than this paper of Berzolari’s how impossible it would be to incorporate in a compilation such as the present a detailed account of some modern work on the subject of our curve. I wish at once to acknowledge my indebtedness to Berzolari for the use I have made of the intro- duction to his memoir, both in tracing the history of the development of our knowledge of the quartic, and in the sketch of the application of modern algebra to rational curves; but his object is so different from that of the earlier and more purely geometrical writers, and the interest of the algebraic formule so predominant over the geometry, that even if a detailed account of his work were given here it would stand apart from most of the results quoted. Berzolari himself expresses some such feeling as this in the title he has chosen, but some of his predecessors make no such distinction. It appears to be a legitimate view to regard these papers as developing a particular phase of the theory of invariants as applied to geometry, the value of which is undeniable, but whose aims are widely different from those of the earlier geometers. Moreover, whereas in the older geometry each curve presented characteristics distinct from those of all other curves, a feature of the modern method is the similarity of the treatment bestowed on curves of various types; for the last reason an exposition of the method in a discussion of a particular curve is the less necessary. BIBLIOGRAPHY. [The abbreviations of the titles of current periodicals in this bibliography are adopted from the Jahrbuch iiber die Fortschritte der Mathematik; and the date attributed to each paper after (6) is that of the year (Jahrgang) under which it is reviewed in that Journal. The date borne by the volume in which the paper appears bound up with other memoirs often differs by a year, and occasionally by more than one. The first few papers, and again the last few, do not fall within the period covered by the published numbers of the Jahrbuch.] (1) Satmoy. On the classification of curves of double curvature; Cambridge and Dublin Mathematical Jowrnal, vol. v. (1850) pp. 23—46. The memoir is dated July 1849; paragraphs 25, 28, 29, 30, deal specially with rational quartic curves. See also Salmon, Geometry of three dimensions, 4th edition, pp. 312—317: Salmon-Fiedler, Anulytische Geometrie des Raumes, 11 Teil, pp. 133—141, and the corresponding notes. ; (2) Caytey. On curves of double curvature and developable surfaces; Camb. Dub. Math. Journal, vol. vy. (1850) pp. 18—22. Translated with a few slight alterations from a memoir in Journ. de Math. (Liouville), tom. x. (1845) pp. 245—250. Collected Works, vol. 1. pp. 207—211; 586—587. (3) Satmon. Note on a result of elimination; Camb. Dub. Math. Journal, vol. 11. (1848) pp. 169—173. (4) Sverver. Ueber die Flachen dritten Grades; J. fiir Math. (Crelle) Bd. tin. (1856) pp. 1833—141; specially p. 138. Reprinted in Ges. Werke, Bd. 11. pp. 651—659; specially p. 656. Mr RICHMOND, RATIONAL SPACE-CURVES OF THE FOURTH ORDER. 141 (5) Cremona. Intorno alla curva gobba del quart’ ordine, per la quale passa una sola super- ficie di secondo grado; Aunal dv Mat. (Tortolim) N.S. vol. 1v. (1861) pp. 71—101. fend, (1861) pp. 58—63. Also Bologna (6) Cremona. Sopra una certa curva gobba di quart’ ordine; Lomb. Jst. Rend. pp. 199—203. [The stationary planes coincide two and two.] (1868.) (2) 1. (7) Srurm. Sur la surface enveloppée par les plans qui coupent une courbe gauche du 4%#me ordre et 2"™* espéce en quatre points @un cercle. dAnnali di Mat. (Brioschi), (2) 1v. pp. 73—86 (1870). (8) Weryr (Emin). Sopra una certa curva gobba di 4° ordine; Lomb. /st. Rend. (3) vi. pp. 144—146. [As in (6).] (1871.) (9) Weyr. Ueber rationale Raumeurven vierter Ordnung; Wien. Ber. uxi. pp. 493—505, A short paper with the same title is published in Math. Ann. (Clebsch) Bd. tv. pp, 243-244. pay 1% pp (1871.) (10) Bertini. Sulla curva gobba di 4° ordine e 2° specie; Lomb. Ist. Rend. (2) 2) Vv. pp. 622 —638. (1872.) (11) Darsoux. Sur les lignes asymptotiques de la surface de Steiner; Bulletin de la Soe. Philomatique de Paris, x. pp. 3i—38; L’Institut, (2) 1. pp. 142—143. (1873. (12) Armenanre. Sulle curve gobbe razionali del quarto ordine; Batt. G. x1. pp. 221—232. (1873.) (13) ARMENANTE. Same title; Batt. G. xit pp. 250—266. Continuation of preceding, with special reference to Steiner’s surface. (1874.) (14) Weyr. Ueber Curven vierter Ordnung; Pray. Ber. pp. 164—166. (1875.) (15) Weyr. Ueber Raumeurven vierter Ordnung mit einem Cuspidalpunkte ; Wien. Ber. LXxxI. pp. 400—409. (1875.) (16) Weryr. Ueber die Abbildung einer rationalen Raumcurve vierter Ordnung auf einen Kegelschnitt ; Wien. Ler. Lxxit pp. 686—706. (1875. 8 pp (17) Appett. Sur une classe particulitre de courbes gauches unicursales du quatriéme ordre ; C. k. uxxxiu. pp. 1209—1211. (1876.) (18) Weyer. Weitere Bemerkungen wher die Abbildung einer rationaien Raumcurve 4 Ord- nung auf einen Kegelschnitt; Wien. Ber, uxxin. pp. 203—220. (1876.) (19) Weyr. Ueber Raumcurven vierter Ordnung mit einem Doppelpunkte; Wien. Ber. xxv. pp. 168—174. (1877.) (20) Weryr. Ueber Punktsysteme auf rationale Raumcurven vierter Ordnung ; LXXv. pp. 458—462. (1877.) (21) AppeLy. Sur une classe particuliére de courbes gauches du quatriéme ordre; Griinert’s Archiv, uxu. pp. 175—182, [As in (6) and (8), the stationary planes coalesce by twos.] (1878.) (22) Weryr. Ueber die Abbildung einer mit einem Cuspidalpunk te versehenen Raumcurve vierter Ordnung auf einen Kegelschnitt; Wien, Ber, Lxxvit. pp. 396—398. (1878.) Wien. Ber. (23) Weyr. Ueber die Abbildung einer Raumcurve vierter Ordnung mit einem Doppelpunkt auf einen Kegelschnitt ; Wien, Ber, uxxvu. pp. 891—895. (1878.) (24) Liz. Ueber Minimalflachen; Math. Ann. xiv. pp. 331—416: specially pp. 388—398. (1880.) Cf. Archiv for Math. og Naturvidenskab, Bd. 2 ae (25) Apter. Ueber Strictionslinien der Regelflachen zweiten (und dritten) Grades; Wien. 142 Mr RICHMOND, RATIONAL SPACE-CURVES OF THE FOURTH ORDER. Ber. uxxxv. pp. 369—380. (1882.) Followed by Ueber Raumeurven vierter Ordnung zweiter Art ; Wien. Ber. uxxxvi. pp. 919—936; 1201—1211; 1212—1229. (1882.) (26) Sporriswoopr. Note on quartic curves in space; Lond. M. S. Proc. xiv. pp. 15—18. (1883.) (27) Roperts, R. A. On unicursal twisted quartics; Lond. M. S. Proc. x1v. pp. 22—34, 308—314. (1883.) (28) Bramprita. Sulla curva gobba del quarto ordine dotata di punto doppio; Lomb. Rend. (2) xvu. pp. 857—866. (1884.) (29) W. Srauz. Ueber eine gewisse Gattung von Raumeurven; J. fiir Math. (Crelle), xcix. pp. 154—160. (1885.) (30) Bramprtta. Ricerche analitiche intorno alle curve gobbe razionali del 4 ordine; Ven. Ist. Atti (6) m1. 1471—1490. (1885.) Sopra aleuni casi particolari della curva gobba razionale del quarto ordine; Nap. Rend. xxiv. 279—298. (1885.) (31) Roserts. On unicursal curves ; Lond. M, S. Proc. xvi. pp. 25—79. (1885.) 32) Reraui. Sopra la projezione immaginaria delle superficie del second’ ordine e delle curve ( p proj s Pp gobbe del quarto ordine ; Bologna Rend. (1886) pp. 21—33. (33) Srupy. Ueber die Raumcurven vierter Ordnung zweiter Art; Leipz. Ber. pp. 3—9. (1886.) There is an earlier paper in the Jahresbericht des Miinchener Polytechnikums fiir das Studienjahr 1883—1884, which I have not seen. (34) Borpica. Studio generale della quartica normale; Ven, st. Atti (6) tv. pp. 503—525, (1886.) (35) Sr Jones. Die Theorie der Osculanten und des Sehnensystems der Raumeurve 1y. Ordnung 1. Species; Aachen, Mayer’sche Hofbuchhandlung, tv. u. 24 pages. (1886.) (36) Wrrtrncer. Ueber rationale Raumeurven 4** Ordnung; Wren. Ber. xem. pp. 28—45. (1886.) 37) Brameitta. Intorno alla quartica gobba dotata di due tangenti stazionari; Genova G. 1 be) to} > Ix. 223—229. (1886.) (38) Caytey. On the complex of lines which meet a unicursal quartic curve ; {of the same special type as the preceding]; Lond. M. S. Proc. xvi. pp. 232—238. (1886.) (39) Esernarpr. Die Raumcurven erster und zweiter Species in ihrem Zusammenhang mit den Steiner’schen Schliessungsproblemen bei den ebenen Curven dritter Ordnung; Schlémilch’s Z. XXxII. pp. 65—S2, 129—144. (1887.) (40) Srant. Die Raumeurven vierter Ordnung zweiter Art, und die desmische Fliche zwolfter Ordnung vierter Klasse; J. fiix Math. ci. pp. 73—98. (1887.) See also vol. crv. (41) Meyer, Fr. Ueber die mit der Erzeugung der Raumeurve vierter Ordnung zweiter Species verkniipften algebraischen Processe; Math. Ann. xxix. pp. 447—457. (1887.) (42) Det Re. Omographie che mutano in se stessa una certa curva gobba di 4° ordine e di 2° specie, e correlazione che la mutano nella sviluppabile dei suoi piani osculatori; Torino Atti, xxi, pp. 901—922. Correlazioni che mutano la quartica gobba con due flessi nella sviluppabile dei suoi piani bitangenti; Napoli Rend. (2) 1. pp. 167—172. (1887.) Cf. also Wap. Rend. (2) u. 37—46. (1888.) (43) Berrzonart. Sulla curva gobba razionale del quarto ordine; Lomb. Ist. Rend. (2) xxim. 96—106. (1890.) Mr RICHMOND, RATIONAL SPACE-CURVES OF THE FOURTH ORDER. 143 (44) Roun. Die Raumceurven vierter Ordnung zweiter Species; Lez. Ber. xii. 208—244 (1890); and xi. 1—23. (1891.) (45) Meyer, Fr. Ueber Realitiitsverhaltnisse auf Raumcurven zweiter Species; Boklen Mitt. Iv. 99—103. (1891.) (46) Srann. Zur Erzeugung der rationalen Raumecurven: Math. din. xu. 1—54, (1892.) (47) Rony. Modelle der rationalen Raumcurven vierter Ordnung und ihrer Developpabeln ; Deutsche Muth. Ver. 1. 43—45. (1892.) (48) Berzotart. Sui combinanti dei sistemi di forme binarie annessi alle curve gobbe razionali del quart’ ordine; Annali d. Mat. (2) xx. 101—162. (1892.) (49) Berzotart. Sopra aleuni iperboloidi annessi alla curva gobba razionale del quart’ ordine ; Lomb. Ist. Rend. (2) xxv. 950—971. (1892.) (50) Forsyru. On twisted quartics of the second species; Qwart. J. xxvul. 247—269 (1895. ) (51) Brampinia. Di taluni sistemi di quartiche gobbe razionali annessi ad una superficie cubica ; Napoli Rend. (3) u. 171—176 (1896); and Napoli Rend. (3) 111. 203—-206. (1897.) (52) Bramprira. IT poligoni principali di una quartica gobba dotata di punto doppio ; Mapoli Rend. (3) tv. 335—337 (1898); and Atti Napoli (2* serie) 1x. no® 10, 33 pages. (1899.) As a preliminary to entering upon an investigation of the properties of the rational space-curves of order four, some thought must be given to the question what is the best method to employ in order to obtain and express results as simply as possible. Cremona’s method need not be considered; and that of Berzolari must be rejected for reasons already stated; it would not give the elementary properties of the curve in an elementary form, and again would fail us in dealing with special types of the curve. But apart from these we have the plan of mapping the curve point for point on a conic, a purely geometrical method of proved value: we have the ordinary mode of expressing the coordinates of points of the curve by a parameter, in which moreover we ought not to disregard the possibility of changing the parameter by homographic substitution; and lastly we wish to obtain explicit equations of loci connected with the curve in as simple a form as possible. This points to the desirability (1) of choosmg some fixed convenient tetrahedron of reference, (2) of making no attempt to simplify results by a transformation of the parameter: all the equations in the parameter that we have to deal with will, as we shall see, be covariants of a certain quartic equation, whose roots are the parameters of four important points of the curve. But further,—and this is a notion of which use has not, so far as I am aware, been made,—if we suppose the roots of the quartic equation just mentioned to be known, we may work not only with the customary concomitants of the binary quartic, but with certain simpler unsymmetrical concomitants, expressible rationally in terms of the roots. This method of dealing with the parametric equations of the curve has the 144 Mr RICHMOND, RATIONAL SPACE-CURVES OF THE FOURTH ORDER. effect of bringing into line with it the geometrical plan of mapping the curve point for point on a conic; for it will be found that a considerable part of the investiga- tion is performed by means of three quantities which may be looked upon either as functions of the parameter that determines the points of our quartic, or as the co- ordinates of a point of a plane curve of the second order, between whose points and those of our quartic a [1, 1] correspondence exists. The two modes of treatment may thus be pursued simultaneously. In conclusion it is well to point ont here a fact which will become more and more obvious as we proceed, viz. that the whole investigation is neither more nor less than a geometrical illustration of the theory of a single binary quartic and its con- comitants. I. General expressions for the coordinates of points of an Rj, in terms of a parameter ; the relation which connects the parameters of coplanar points. By definition, the coordinates of points of an &,’, (by which is understood, in accordance with a recognized and convenient notation, a rational curve of the fourth order situated in a space of three dimensions), when referred to any tetrahedron formed by planes 2=0, 8=0, y=0, 6=0, may be expressed in terms of a parameter 2 as follows :— &: Bry 82: (ApdytedstsQr, 1)': (DodibsdsbsVA, 1): (CocrcocsesQA, 1)* : (dodid.dyd, QA, 1)*...... (1). For the sake of later work I stipulate that, in these “parametric” equations of the curve, the coordinate planes and the twenty constants a, b, c, d, shall be real; so that the coordinates a, B, y, 6, of every real point, whether on the curve or not, are real; and every real value of X gives a real point of the curve: conversely, it may be seen that, (save for the one exceptional case of an isolated double point), every real point of the curve is determined by one real value of 2». A formal proof, that in making this restriction I exclude no curves possessing a real are or branch, may be suppressed. In order that the four points whose parameters are roots of (Go G29:9s0 5 1D): should be coplanar, it is necessary and sufficient that the determinant G Hh de Ws Ys | ie Os, Oana tke, by b, bs b, b, Cy 0, Cy Cy Cy age dienes laae Mr RICHMOND, RATIONAL SPACE-CURVES OF THE FOURTH ORDER. 145 should vanish; and, this condition being satisfied, the equation of the plane is known, inasmuch as the coefficients of 2, 8, y, 6, are proportional to the minors of a,, b,, c,, d,, for any chosen value 0, 1, 2, 3, 4 of r. I now exclude, again without giving formal investigation, the case when the determinant vanishes identically, (which may be shewn to imply that the curve lies in a plane, and is a two-dimensional rational quartic curve); and, expanding, write the condition in the form (Rs = LD TP OG — CHO AP ceseonsanoooneoooereaubencouse (2), using P)PiP>Psp; in place of certain finite determinate functions of the coefficients a,, by, ¢,, dy, all of them real and not all of them zero. If «, 2X, yw, v, be the actual parameters of the four coplanar points, then PokNpy + P; Apy + pve + VAN + KAM) + Ps (my + VA H+AW+KA+ K+ KV) Ps (GN et Y) + 4 — O) seeeesesese0csoec (2). From this follow, as Weyr has shewn in (9), a series of important properties of the curve, which must be brietly discussed: that Aronhold’s symbolical notation would enable us to write the results very concisely is plain, but it is hardly worth while intro- ducing it here. (i) Should it be possible im (2) to imagine x, X, mw, v equal to one another, the point of the curve will be such that the osculating plane contains four consecutive points of the &,, and may be called a stationary plane. Thus we may include (2) in the statement: The Rj has four stationary osculating planes; and, if the parameters of their points of contact be roots of the quartic FAO) SS (PUP Ps PIP PONAOL) 2 oc core ncosoddnedasccowewscecteus (3), all quartics ‘apolar’ to F have as roots the parameters of four coplanar points. (i) The osculating plane at the point A will pass through the point « provided (pk + pi, PikK+ Ps, PokK+ Ps, Psk+ps0rA, 1)=0; so that three osculating planes of points of the curve, other than the point x, pass through the point «; moreover, the condition that three points whose parameters are roots of (7772730, 1)? should be coplanar with the point «, viz. (Dok + pr) 73 — 3 (Pik + Po) To +3 (Pok + Ps) Ty — (P3k + ps) T = 0, is clearly satisfied by the points of contact of the three osculating planes which pass through the point «. (ii) If A, pw, v be such that both Porgy + pi (uy +N + AW) +P. (A+p+v) +p; = of Pidpy + p, (wy + vd + Ap) + ps(X+ w+v)+p,=0)’ the points A, mw, v, being coplanar with every other point of the curve, are necessarily Vor. XIX. Parr I. 19 146 Mr RICHMOND, RATIONAL SPACE-CURVES OF THE FOURTH ORDER. collinear: since in the equations an assigned value of v leads to a unique pair of values of X and gu, Through each point of the RS may be drawn a line which intersects the curve in two other points. Such a line is termed a trisecant line or triple chord of the R§; a real triple chord must pass through each real point of the curve, but the two further intersections with the curve may either be real, or may have conjugate imaginary parameters, or lastly they may coincide. For the last case we put #=A, and eliminate v: thus H(X)=! (PpipsrA, 1), (prpops0aA, 1)? | =0. (Pipspsid, 1), (papspsi, 1P There are therefore four lines which touch the curve at one point and intersect it at a second potnt; viz. the tangents at the points whose parameters are roots of H(X), the Hessian of the quartic F(A) which gave the parameters of the contacts of the stationary planes. (iv) No line can meet the Rj in three consecutive points unless both (PoPip2Ps 0X, 1)s=0, and (Pip2psps0r, I) =(0) at the point of contact; z7e. unless X is a repeated root of F(A). Conversely, if F(A) has two roots equal to 2, it will be seen that the tangent at the point > must have three points in common with the curve. The necessary and sufficient condition that such a tangent should exist is that the discriminant of F(X) should vanish. (v) Should it be possible to find values of y, v, such that poe tpi (w+ »)+ ps—0) Ppytp(u+v)t+p,=07 Popvt ps(u t+ v)+ P= 0) simultaneously, the points yp, v, being collinear with every point of the curve and coplanar with any two points of it, must be coincident. The condition that such values of » and vy exist is that f= Po Pr Pro | Piss) | Po Ps ps ! should vanish: and, given that J=0, the curve will cross itself if the values of uw and v found from two of the above equations are real; will have a real isolated point if they are imaginary; and will have a cusp if they are equal, the quantity T= pops — 4pips + 3p! vanishing as well as J in the last case. (vi) Should the tangents at the points A, p, intersect, Pop? + 2p we (A+ M) + Po (2+ p+ Aw) + 2s (N+ BH) + ps =O: —————eeeeEEeEEEeEeEeEeEeEeEeEeEeEeEeEeEeEeEeEeEeEeyEeEeEeEeEeEeEeEeEeEeeeeeeeeeeEeEeEeEeEeEeEeEeEe er lee e______CO Mr RICHMOND, RATIONAL SPACE-CURVES OF THE FOURTH ORDER. 147 Two tangents of the curve intersect a given tangent; also the form of this condition shews that the tangents of the curve have a poristic property, viz. that If one skew polygon of n sides can be described whose sides are tangents of the curve, an infinite number of such polygons ewist. It is not necessary here to point out that these properties define certain involu- tions on the curve, nor to state at length the forms of equations determining the parameters of the various sets of points. Il. First standard form of the parametric equations of the curve; connewion with the theory of the binary quartic F(X) and tts concomitants. Unless F(X) has equal roots the four stationary osculating planes of § I. (1) are well adapted for use as coordinate planes, for it may easily be seen that they do not under other conditions coalesce or meet in a point: two or all four may be imaginary, but it will still be true that each real value of 2» determines a real point of the curve; for the present however distinctions between real and imaginary loci are ignored. Let then y%, 1, Yo, Ys, be the roots of F(A), and ~=0, 2,=0, 2=0, 2,=0, the respective osculating planes of the curve at these points: the parametric equations of the curve now take the form px, = A, (A — y;)4; (Ga, il, 2 sy the undefined multiplier p implymg that we are concerned only with ratios of the coefficients w,. For a variety of reasons the most suitable values of A,, dA,, Az, A;, are 2 Ay = po’ (Yo— HY (% — ¥2)? (Yo — Ys); ete, ete., and, these being adopted, it follows that :— When the equation defining the parameters of the points of contact of the stationary planes of the curve, viz. F(\)= Quin Pars: paOr, 1lf=pa- Yo) (A — 1) (A= Ya) (A — Yg)ece ree eceees (3), has no two of its roots yw, 1, Ye, Ys equal, coordinates of points of the curve may be exhibited in the form PX = (A—%)P + {Po (Yo — 1) (Yo — Ya)? (Yo — i) pay = (X= yr)* = { 0? (Ya — Yo)? (V1 — Ya)? (1 — Ys)?f b nee eee eee eee ere eee ees (4). ete. ete. The chief merit of these equations (4) les in the fact that if a new parameter 0’ be introduced in place of X by means of any lineo-linear relation EX’ + FX+ GA’ +H =0, ; 19—2 148 Mr RICHMOND, RATIONAL SPACE-CURVES OF THE FOURTH ORDER. equation (3) is replaced by a quartic (p,'p,'p,'p;'p/ 0’, 1)* in X' whose roots os " Ye> 3 > are such that Eryyy + Frye + Gyr + H=0 ; (r=0, 1, 2, 3), and the equations of the curve (4) become p'%,=(X' — 0) = [Po? (Yo — 1’) (Ye — 2) (Y' — 98')*} 5 ete, ete. ; that is to say :— In the standard form (4) of the equations of the R3, Uf the parameter and equation (3) be transformed by any lineo-linear substitution, the coordinates (a, 2, %, 23) of any point of the curve are the same functions of the new parameter and the roots of the transformed equation, as they were of the old parameter and the roots of the original equation. With each rational quartic curve, then, is associated a single binary quartic F() = (PopspepsPQr, 1) and it will be found that the theory of this quartic and its concomitants is bound to that of the RZ in so intimate a connexion that the points of interest presented by the two are to all intents the same. The classification of the different types of the curves depends solely on the different forms the associated quartic may exhibit; curves for which the absolute invariant of the associated binary quartic has the same value are, as regards projective properties, identical, and may in fact be represented by the same equations: those curves which possess any special projective property are distinguished by a relation between the invariants of F(X); and, on a given curve, the parameters of those points at which any special projective property holds are roots of a covariant of F(A). In order to simplify the equation (4) of the curve yet further we might suppose F(X) reduced to some shortened or canonical form such as (1, 0, m, 0, 1QA, 1)* by a lineo-linear substitution, and on rare occasions such procedure is of advantage; but both this canonical form and the form 44*—JA—J are objectionable for two reasons, (i) they destroy the symmetry among the four roots, (ii) they cannot always be reached by real substitutions. On the other hand, if F(A) be left in its general form, we may confine our attention wholly to forms which possess the property of invariance-—a fact of immense value in investigations. The absolute dependence of the theory of the R, upon that of the single binary quartic F(A) and its concomitants is not confined to the case when the parametric equations of the curve can be reduced or are reduced to the form (4), and may be established, (as indeed it was established long ago by Armenante), without the reduction ; but equations (4) bring the dependence prominently forward, because it will be seen that the expressions for pa, pa, pi, p#;, are themselves covariants of F(A), though not symmetrical ones. It is by taking account of these unsymmetrical invariants and co- variants that I hope to simplify the investigation of properties of the curve, and bring into harmony the analytical and geometrical methods of treatment. Mr RICHMOND, RATIONAL SPACE-CURVES OF THE FOURTH ORDER. 149 III. The concomitants of the binary quartic F(X): second standard form of the equations of the curve. As the simplest way of explaining the notation to be used, I shall consider briefly the invariants and covariants, symmetrical and unsymmetrical, of the binary quartic F()). The complete system of symmetrical concomitants is well known; L = pops— 4p,ps + 3p; J =p.pops + 2p\poP3 — Pops — PY Ps — p»*, the catalecticant ; F =(ppipopsps0A, 1)4, the quartic itself; H =(p.p.—p*)M+..., the Hessian of F; G = (pps — 3p)Pip2 + 2p,*) MB +..., the Jacobian of F and H. Beside these there are certain unsymmetrical invariants and covariants, expressible rationally in terms of the roots (Y,%1,%, 73) of F, though not so expressible by means of the coefficients (p)pip»p;ps), Which are of even greater importance for our purpose. Of unsymmetrical invariants the most convenient are the three ky = Po (Yo — V1) (Y2— Y3)3 2 = Po (Yo — Ye) (Y¥s— M1)3— he = Po (Yo — Ys) (1 — Ye); so that Ky, + key + ks = 0; ke + he + he = 240; (ky — ks) (hs — k,) (hy — ky) = 4382S; and for the discriminant hvk2ke = 432 (13 — 27°). A slight disregard of the perfect symmetry which should obtain between the root Y and the others will be noticed in these equations: it might have been avoided had it seemed necessary or desirable to do so; but, as it is, we may in future interchange the suffixes 1, 2, 3, cyclically in any result we obtain. For unsymmetrical covariants it seems best to choose three quadratic functions of X which are factors of G; ™m OF m(A)= (A=) + [Po (Yo — %a)(Yo = 2) (Yo — Ya)} +O — M1)? = [Po (Ms — Yo) (M1 — Y2)(% — Y)} = — (A= 92)? = {Po (Y2— Yo) (Y2— 11) (Y2 = Ys)} — (A = 5)? + {Po Ys — Yo) (Ys — 11) (Ys — V2) } 5 MOF M(A)=_ (A= Yo)? = {Po (Yo — (Yo = Yo) — Ys)} + A= 2)? + {Po (Ya — Yo) (Ya — V1) (Yo — Ye)} = — (X= 92)? = {Po (Ys — Yo) (Y¥s— M1) (Ys — ¥2)} — (A= M1)? + [Bo (M1 — Yo) CY — ¥2)(% — ¥5)}3 Ms OF s(A)=_ (A= 40)? = [Po (Yo — M1)(Y% — Yo) (Yo — Ys)} + (A — Ya)® = {Bo (Ya — Yo) (Ys — V1) (Ys — Y2)} =— (A=) = {Po (M1 = ¥0)(% — Y2)(V1 = Ys)} — (A= 2)? + [Po (V2 — V0) (Y2— 11) (V2 -Ys)} - It is now not hard to verity the following identities :— plier te alagnel th /bene == (0). nocounsadoososconseopnoadeneoceoee (5), AP = k, (heyy? — k,?n;*) = ke (y?ns? — ken,) = ks (heg?n? — hey?n,?) ; — 48H = hikeghs (ley? + keane + hand); — 324 = kephethemnins- 150 Mr RICHMOND, RATIONAL SPACE-CURVES OF THE FOURTH ORDER. Again, since (X= 0)? = {Po (Yo — 1) (Yo — 2) (Yo — Ys)} SEC m+ M2 + 15); (A=) = {Pol — %0) (1% — 2) (% — ¥s)} = EC Mm — M2 — 7s); (X= 92)? = {Po (Ys = Yo) (2 = Yr) (Y2 — Y5)} = F(— a + M2 — 75) 5 (A — 95)" + {Po (Ys — Yo) (Ys — M1) (Ys — Y2)} = $(— m— 2 + 7); it is clear that the parametric equations of the curve take the form PX =(m+ +13); Pe =(—m+ 2 — MB) | pr, =(m— 2-73) p%s=(—m—m+ 2) J In these equations 7. 7, 7; are definite quadratic functions of X%; equations (4) and (6) are from this point of view identical. But a second method of interpreting (5) and (6) is to consider 7,727; as coordinates of a point in two dimensions, each a quadratic function of a parameter , and therefore determining points of the conic (5); and, by making use of the [1], 1] correspondence between points of the conic (5) and the R,, to interpret the known properties of the former in such a way as to throw light on those of the space-curve: but the two modes of treatment will be found to run on the same lines and may be developed simultaneously. (Lo be continued.) VII. On Divergent Hypergeometric Series. By Professor W. M°F. Orr, Royal College of Science, Dublin. [Received 26 March 1900.] 1. IN a previous paper (Camb. Phil. Trans., Vol. xvit., Part 11.) relations were obtained connecting the convergent hypergeometric series which satisfy the same differential equation as F'(a,...%m3 pi,--- Pn; ), where m—7 [u"H], = — Nal —— (tu) EE een en osee eee (v). The equation (i) becomes div - KV — : [{u” + (K — 1) (u” —u)} #]| = 4p, and the formula div[AB]=BcurlA—Acurl B transforms this into _ KV p+ pw HK ET \(ae su) curl Eh Ara eee AG): The first equation of (iii), when similarly treated, gives {u’V + (K — 1) (u’—u) Vi V+ = {u’V [u”H] + (4K — 1) (u” —u) V [(u” — u) B)}} ” r 1 wu Yr ss +4rpu”+(K—-1)u iy — yu —u) curl H! = | GWAIE. scoosanenee (vil). Now since u, uw” have the same direction, and are constant, we have, if a, B, ¥, 6, ¢, € are scalar constants, (au+ Bu”) {(yu+ du”) V. [(eu+ Cu”) H]} = (yu + ou’) V {(au + Bu’) [(eu + Cu”) H]} = (yu + du”) {0} =0. * Aberration, p. 18. 176 Mr WALKER, SOME PROBLEMS IN ELECTRIC CONVECTION. Hence if we substitute the left-hand side of (vii) for V curl H in (vi), we find KV" + 4p = + fu’ +(K—1) (u”—u)} ({u” +(K-1)(u"—u)} VV + 4rrpu” +(K —1)u {Vy — 7a" —u) curl H}) me! =: fu’ +(K —1)(u"- w))({a" +(K —1)(u"”—u)} = 4crpu"’ + (K — 1) uvey) = = u(u”—u)(4rp + KV), where the last term results from a second use of (vi). Thus p 4 « +m (a 1 COUNT Seri Vern eee ee (viii), where P=K—-K ju'?+(K—-1)(u"-—uj}/V ms KCK usa? I ernceeeteece nr eaeeee (ix). m=l—fu?+(K-1)(u"’-uy}/V=P/K If then we put c=l£, y= mn, z=m€ we have sees dé de? where p’=pn*. Thus wy is the potential in &, 7, € space due to a charge pn*d&dnd& within each element of volume dédndg& Corresponding then to a charge pdadydz in actual space there will be in w a term due to pn*d&dndf; and if the real distribution consist of a point-charge e at (2, y, 2), i will be due to a point-charge e’ at (& 7, &), where _predidnde _ wv. e pdadydz Im?’ + 4c, Tp = 0, *. w=ne/lm’p,, where pP=&+7°+ C=H/P + (y+ 2)/m ............005 (x). Further, by (vi), denoting differentiation by suffixes, [w’ + (K = 1) (w" = w)} (My = My/V = KV + deep = SEV Pree = 2 Yi + Ve) “Viet (My — Mz) = V2(P me) yy + er)/{u" + (= 1) Cu" — w)} = — [w+ (K 1) (ae — w)} (Yay + Yee). The component of (vii) parallel to OY gives V (L.— Nz) = {u" +(K —1)(w" ~ 1} Yay — 7 ("N+ (K —1) (w= 0)? Ne} or V (L,— 1) = {u" +(K 1) (u" —u)} Yay. Similarly V (n?M, — L,) = {u" + (K — 1) (u’ — u)} War. Mr WALKER, SOME PROBLEMS IN ELECTRIC CONVECTION. Wee Hence VL=o,;, WV M =, + \u’ + (K-1) (u’ —u)} ve, WV N =o, — {u" + (K —1)(u" —u)} Wy, where @ is a function as yet undetermined. Now divH=0, .«. V’%=0; and since this is true at all points, »=constant. - L=0, M={u"+(K—-1)(u’—-w)}/Vn?, N= {u’+(K -1)(u’—u)} W,/Vn’...(xi). On substituting in (v) we obtain X=—,, WY =— {14+(K —1)u(u"—u)/V*} Wy, Zi = — {1 + (KH —1) eb (ue —u)/ V7 oy... eeceeseve cence (xil), and X’=— vz, WY’ =—{l—w'/V 4}, n?Z’=— {1 —uw’/V3} Wy.........--- (x11). Hence Ii = 1b = Mi MUA Ve —) MeeVee sce sescncinaweswasateccaes (xiv), N’=N—uY/V =— (u" —u) my, / Vie} and L” =0 M” = (K —1)(u’ =u) (1 — wu’) V?) p./ Vn? | (xv). NY’ =—(K = 1)(w" =u) (1 —wu"/V) ,/ re] Case II. A medium of specific inductive capacity K fills all space and is at rest relative to the ether. Two particles P,, P, whose charges are e, e, are moving along the axis OX with constant velocities wy, U,. To obtain their mutual energy at any time. Here the electric and magnetic forces due to both particles can be obtained by adding those due to the particles separately. Putting u=0 in the analysis of Case L, we find = Kn?, me=K, ne=i —Ka3/V2 and the forces at any point @ due to e, are ls _ldhy oe de? nm? dy ne dz’ Ku Ku, L=0, M= y % N= y Ww, her pe St r= +n +, where Mapes, piabttnt +t and (2%, ji, %)=(Lé&, mm, mo) are the coordinates of @ referred to P as origin. 178 Mr WALKER, SOME PROBLEMS IN ELECTRIC CONVECTION. We shall suppose that ~>u, and that P, lies on the positive side of P,, ie. that x, is positive when Q is at P,. Let the distance between the charges at any time be a. Since the electric force acting on a charged particle moving with velocity u, is E+(u.H]/V, the electric force acting on e, will have a component parallel to OX equal to 7 °e&,/L°m,%p,5, or n,*e,/Ka*. Thus the ponderomotive force on P, due to the presence of P, will be e.m%e,/Ka®?; and it will be noticed that this is not equal and opposite to —n,e,e,/Ka*, which is the ponderomotive force acting on P, owing to the presence aye 12M, If the velocities of the particles are kept constant by the application of force from outside, the rate at which this external agency does work will be Uy . NHC ,€o/ Ka* — uz. n42e,e2/Ka?, Ku C4» or Kat (1 — Us) (1 + re Let us suppose that e, and e, were originally at an infinite distance and were set in motion with velocities uw and u,. The work done in starting them will be an expression of the form 4A,e,+ 4.e.*, not containing any term in ee,. As however the particles are made to move with constant velocities %,, uw. and approach one another at a rate (w%—u.), their mutual influence becomes finite after lapse of time; and the work that has been performed by external agency in keeping the velocities constant is, when the distance P, P, is a, C15 K = i dt ~~ {@+ (w—w) ft}? ” Kk (% — Us) (1 - 2 Thus the mutual energy of the charges is 12> / Kus Bee Ka fe + ) Saaw tie neicoaee we becese sceeew seme (xvi1)*. The case in which the velocities of the particles are equal may be treated as. follows. Let wu and uw, differ by a very small quantity; and when the distance P, P, has diminished from infinity to a, let uw, the velocity of the second particle, be suddenly altered to wu. The work done in this very small change will be ultimately negligible. We thus find that the mutual energy of two point-charges e,, e, moving with the same velocity wu is €,€> Kw ae Gh (1 ~ — aslo Meee sanidstiemeteticeet este veteeees (xvi). * From the mode in which the mutual energy is obtained it is clear that differentiation with reference to a will not give the force between the charges. Mr WALKER, SOME PROBLEMS IN ELECTRIC CONVECTION. 179 The energy of the field in this case. In my book on Aberration and the Electromagnetic Field* I showed that a con- sideration of the forces and stresses leads, if the fundamental ideas of Lorentz be adopted, to the expression = ob Ol @ GES pee = Ce (7a ))p > ic) ane (x1x), as the energy W of the field per unit volume. Let us compare the mutual energy as obtained from this point of view with the formula (xviii). It is clear that as the particles are extremely small, the mutual energy of the charges ¢,, ¢ upon them will be independent of the shape of the particles and of the manner in which the electricity is distributed upon each. We shall accordingly simplify our analysis by assuming that the charges are uniformly distributed through the interiors of small ellipsoids of semi-axes 6/, Am, On. Let us take new variables & y, € given by w=lE, y=mmn, z=m6, as before. In the corresponding (&, 7, €) space there will be two spherical particles p,, p, of radius @, and y will be the potential due to charges n%e,/lm?, n’e,/lm? distributed uniformly through their interiors. The electric and magnetic forces at any point Q =(a, y, z)=(lE, my, mf) in the (x, y, 2) space will be given by yuh 4 ied 1 dw l dé’ mn dn’? ~ mn? dt’ ve 47 ee AG L=0, M=—-, 4, N=, Y. Hence, when Q is within the first particle, oe! a€, | ee oe ea C2 7 € (a e2 X=7,( po as) =a eae reales ase where (€,, 7, €) and (&, », €) are the coordinates of (& 7, ©) referred to the centres of the spheres p,, p, respectively. 1 lm? [ [yf Sars = [[Jaeayae w= Hi dédndt W, and if d&dydg=dr and u=la, the integral through the p, sphere of £,£,/6%.° is EE, _ & ( 1S OCS) Ja Bp. =| dr G3q 1 a + ae DOnOAND } 87? = — —— + smaller terms. 15a? Thus the integral becomes negligible with @; similarly ; dr Te is vanishes in the limit. * gg 3438. 180 Mr WALKER, SOME PROBLEMS IN ELECTRIC CONVECTION. But it may easily be seen that the term involving ee, in the integral of the energy W within the particle is made up of a term in [arB.e/6 52, and a term in lar (a? + £)/8p2. Hence the contribution to the total mutual energy from within this particle is negligible: and we may deal similarly with the space inside the second particle. Thus we need only integrate W through space outside both spheres. For a point Q not within either particle we tind ee (tsar s\n ee De 2 £:) ae isl “:) real fas ae ae > Dear 5s , Ku cS, eK L=0, M =——, 4, ada ae and since the dielectric is at rest a 1 co(K RF: 2 W= mall dxdydz(K¥* + H?). Thus the mutual energy is the integral through external space of I Bag. - FE a ag Ge 8a K pips |KEE, tees ct &) =k Va iat ? and is ee. [ lm@dr { ‘ FON et 6 io8 der | Knipp? " && + (1 - =p) (0 +¢ } > copongausaeqoneoddosose (xx). We now take as new variables the distances p,, p. of any point q in (&, », &) space from the centres of the spheres p,, ps. If p, x be the polar coordinates of g referred to the middle point of p,p, as origin we have pr — ps = 2ap cos x, 2 (p2 + ps) = 4p? +a? && = p? cos? ¥— 4a’, 0 (Pi Ps) _ ap" Sin x C6 La oun | DOR no 8 eee nus nde acosoaAooaoates 1 = 0(p, xX) Pips Se rp ey IN et Hence | dr bs = | | 2crp* sin ydpdx eo = 7 (p27 = pay —ai ~ Qas If dordps Pr ps ; within appropriate limits. Also the contribution from the region in which & is negative is by symmetry equal to the contribution from the positive side of the plane €=0. And that part of the former space which is outside the sphere p,=@ may be regarded as divided by the sphere pi=4a into two regions: in one of these @ =| & BE tf 1) 4 ED (EE) ns oc 0s. cosas (xxiv). The error in this argument lies in the neglect of the inequality of the action and reaction between the infinitesimal charges eén, of velocity u+éu, and the charges ne, whose velocity is u. In the present case the value of the mutual energy given by (xxiv) would be }Zey, or nee, ie C,€» Ima’ ss Ka (1 —-K 7): This is contradicted, as it should be, by (xviii). Case III. Images at a plane surface bounding a semi-infinite drifting dielectric. Let a semi-infinite dielectric of specific inductive capacity K be bounded by a plane face, which we shall take as the plane of reference YOZ: and let the medium be moving with constant velocity u parallel to OX. Let a particle P of charge e be moving along OX with constant velocity u” ( Lm? pr L2=ne=1—u2/V2 me=l, pe=a/l? git a)/m, and 2, %, 2 are the coordinates of S referred to P as origin. * Aberration, (79), p. 47: see also § 33. Mr WALKER, SOME PROBLEMS IN ELECTRIC CONVECTION. 183 We have further by (xiii) as the electric force acting at a point moving with the velocity u of the dielectric, r dy, ; 1 uw’ \ dp, ; 1 uu’ dy 3X6 ——— = — — — = — _ = \\ ete dx’ v n? (2 = dy ” 4 ny? (2 =) dz ~ Due to we at Q we have at S, EB” (= E+ y(u"'H)) =a Ny pe where Wo =5—5 ee : lms? ps I2=ne=1—w"?/V2, me=1, p2= 22/1? + (y2 + 22)/m2, and a, Yo, 2, are the coordinates of S referred to Q as origin : also ,_ dN me eke uu’ dvb, ee al Teens Oars dz’ Y =-,,(1- V) ay pL Pe Oe /V*) de: Due to ve at R we have for a point 7 in the drifting dielectric E* é Eee [u"Ex)) =a where ae = — l?= Kn2, me=K —(K—-1)w/V2, ng=1— {uiv? + (K —1)(w¥ — u)}/V2, ps = @3'/le + (ys + 2:°)/ms?, and «5, y;, 2, are the coordinates of 7 referred to R as origin: further ye _ dy; y= 1 (1 = a) dws ee Ee (1 0D 7) dys dy Ns? WV) dy me 78) de” In order to satisfy the boundary conditions we now choose the quantities b, ¢, uw”, wi, at our disposal in such a way that when S, 7 are taken at some point (0, y, z) on the bounding surface 2=0, we may have p,:p.:p; as ratios which are the same wherever the point may be on that surface. At the time t=0, since y,=y=y;=y and z,.=z2,=2,=z, we must have the ratios a/l? + (y? + 2)/m?2 : 2/12 + (y? + 2)/me? : 2/le + (y? + 2)/m;?, independent of y and z. Therefore fli S Ors Cl Wiehe Wirt Ei BLE Ii scoodococesbocodadoon (xxv), and then Pits Pais Pare tl tks Wi agrenmacnesctseeaeeees sustecyanesne sae (XXv1). In order that the subsequent values of p,, ps, ps; at any time ¢ may have ratios which are the same wherever (0, y, z) may be on the bounding surface, we must have the equations (xxv) holding when a, b, ¢ are replaced by a-—(u—u")t, b-(w”—u)t, c-(u-wu®)t. 184 Mr WALKER, SOME PROBLEMS IN ELECTRIC CONVECTION. Hence, for all values of ¢, fa—(u—w") th/l : (b-—(w’” — u) t}/l, : {o—(u— uv") H/l;= 1:1: 1/m;,. Therefore (w — w’)/ly =(ul” — u)/ Ly = (U — UP) ag Ug. eee eeeereeee eee e neste eens (xxvil). Thus wv”, w are determined: further (u—w fa = (ul” = u)[b = (6 — U") 0.002... saeververercserceses (xxviii), and (xxvi) will now hold for all values of ¢. The conditions which must obtain at the boundary* are that Y’, Z’, L’, M’ shall be continuous. When these restrictions are imposed, the fact that the equations of the field hold in the media on each side of the surface is a guarantee of the satisfaction of the dependent conditions—the continuity of the rate of increase of the total normal electric and magnetic polarisations. At a point outside the dielectric we find in all, Y’=(1 — wu’/V*) ey/hpe + (1 — wu”) V*) pey/lap.'. Inside the dielectric Y' =(1 —ww'*/ V*) vey/lsms‘ps*. Hence for continuity, making use of (xxvi), (1 — uu" /V2)/l, + (1 — we!” V2) pa[ly = (1 — uae /V*) v/dgaiig..sseeeeereereeees (xxix). The condition of continuity of Z’ leads to the same equation. In a similar manner we have, from the consideration of WM’, —(u’ —u) ez/V1.p3 —(u” — wv) wez/ V1i.p.5 = — (w¥ — u) vez/Vismgp;', therefore (u” —u)fly + (w” — w) pa/ly = (UY — U) veig/lg.. eee reneeeeceeeeeeoees (xxx) which also provides for the continuity of N’. By (xxvii) we may put (xxx) into the form TLE 7), =) DreerqaceocepooencacqdopdeonboscongasHodaKGend (xxxl). Since wu”, w¥ are known, (xxix) and (xxxi) give m# and y, and the solution is complete. It may be inferred from the dependent boundary conditions that the surface charge is constant; and since its amount is obviously zero when the charge e is at an infinite distance, it is zero throughout. Or we may easily verify that 470 = Xonteide a {xX of (Kk * 1) X'} inside =(—l+ptyv)e{a—(u—w’) th/Lps =0, by (xxxi). * Aberration, § 17. Mr WALKER, SOME PROBLEMS IN ELECTRIC CONVECTION. 185 The special case in which u” = u. From (xxvii) it follows that if uw” =w(1—e), where ¢ is small, and if squares of e be neglected, w”=u(1+e), uw =u(1 —eKt/m,). Hence, in the limit when w’ =u, we have by (xxviii), a:b:e¢=1:1: {1 —(K —1) wv/KV34, and this equation gives the ratios of the constant distances of P, Q, R from the bounding plane. When all the velocities of translation vanish the solution reduces to that with which we are familiar*. The energy of the system. We shall simplify our analysis by considering the case in which the semi-infinite dielectric is at rest and the particle is moving towards it with velocity w’. We then have w=0, while from (xxv) it follows at once that w’=—w and that 7 = us Sect chine «Sac cn ee MORO e osname shea (xxxil). Thus L2=n2=1 —w?/V2=l2=n2, m2=l=mZ, meZ=K, ng? = 1 — Kui?/V2, 1; = mgn3. Let a=, b=1,8, c=I,y: then (xxv), (xxvi) become Cer Satya Myles th. acti dost ee dscekeececeee se (xxxili) SMUG S Soovdoonoencqaanacadedoosonabocad (XXxiy). For yp, v we find from (xxix), (xxxi) l+yp be a ny GN sfc emet nee oak toauet aeons te ts Sac (xxxv). l—-p=v We shall, as in Case II, assume that the charges e and ye are distributed uniformly through small spheroids of axes (1,0, m,0@, m,@) and (1,0, m.0, m0), so that here each has axes (7,9, 6, 0); and the charge ve will be regarded as distributed uniformly through a spheroid of axes (J,9, m,0, m,0). * See J. J. Thomson’s Elements of Electricity and Magnetism, (1895), p- 164. 186 ‘Mr WALKER, SOME PROBLEMS IN ELECTRIC CONVECTION. Direct substitution then gives for space within the particle of charge e, whose specific inductive capacity will be taken as unity, Sena *). Vn, ie ps In order to obtain the forces outside the dielectric and the particle of charge e, we replace @° in these six equations by p;*. Within the dielectric Y= ve€; _ ven _ vet Sean non.. © Kens K*ngp3 | ( rae cen were XXXVI ~ she. eivels ui*yen LO eT kage VErap3 We shall denote elements of volume in the three regions just mentioned (defined algebraically by pi< 9, p:>p,>98, and «<0) by dv,, dr, dv; respectively or md7,, mdt», lym:dt;. Then the total energy of the system, as given by (xix), is 1 See eae eae = | dv, (BP +H) + i dv, (E+ HP) + * dv, (KE? +H), gmt (Get ee tee tase + ye) (et gs) * (Lm ve) opal) + ge[nen (a Se te te (a+ +(1- ps) arpa) + me | lamzdr, (ae + ae a a : Now Jane a =[an% a= dr, 5 = as Finlowanchemhe, | dr, Oe ard [a Seem , ? 1 G35,3 7 —, Vvanis wit A Pe P2 Mr WALKER, SOME PROBLEMS IN ELECTRIC CONVECTION. 187 Further it may be proved by our former method of integration, taking p,, p: as independent variables, that fant fan tt hors BeaO maeoe ua) pee Pe os ee | dr +e 2 ps Qa, 2 ps , far, fk: _ 9 pei a = pr'ps" 2) - pi'ps a Hence too [ de, B= F = far, 7H a Psy, Pp. On substituting we obtain as the total energy of the system vel, aig 2 Sine = ees 5n,0 “4 TEE eee {Qun,?—(1—p?)} + 5Kmn,2y But from equations (xxxv) we have, on multiplying them, and by (xxxill) y=a/nm;, while na=a. Hence the energy reduces to a (4—n,?) + io Pr (XXXvVil). The first term in this result, corresponding to an infinite value of a, is the energy when the effect of the dielectric is negligible. If then a spheroidal particle of semi-axes (7,0, 0, @) has a charge e uniformly distributed through its interior and is moving along its axis of figure with constant velocity wu’, the energy is e4u/V) 56 (1 —w?/V*)* The second term of (xxxvii) informs us that the mutual energy of the particle and the dielectric is pe’n,’/4a, or Knz — 1% n7e ( i) dann lets since teste aeiieOe erable od caw ee XXXVI Kn,+7, 4a : where n?=1— Kw’?/V? and from (xxxii) it follows that n? as UPB NES 188 Mr WALKER, SOME PROBLEMS IN ELECTRIC CONVECTION. When everything is at rest the energy becomes sé) (KK —)eé 56 (K+1) 4a’ a result in electrostatics which is easily found by direct means. The correctness of (xxxviii) may be verified at once. The electric force along OX acting on e due to the dielectric is pef,/p,*; hence the work that must be done against the electromagnetic field by externally applied forces if the particle is moved up from infinity with constant velocity is 0 pe 7 J" @e=wompee or pe*n,?/4a. SUMMARY. Case I. The fundamental principles of Lorentz are applied to the case of a particle of charge e moving with constant velocity w” (relative to the ether) through a dielectric of inductive capacity K, whose velocity u is constant and parallel to vw’. Let m= K—(K-1)w/V*, ne=l—{w?+(K—-1)(u’—uy}/V, w=en/K*mp,, pr = x / Kn’ + (y?+2*)/m®, where (x, y, 2) are coordinates referred to axes at the charged particle, OX being parallel to u, w’. The electric and magnetic forces at any point fixed in the ether are shown to have components given by , d ‘ uM d x - » WY =— {1+(K—-1)u(u’—u)/V*} a wZ=—{1+(K—1)u(u’ —u)/V*} Be D=0: VneM = ju" +(K —1)(u’ —u)} = : Vr? N=— {u" +(K-1)(u"- ge where V is the velocity of light in vacuo. CasE II. Two point-charges e,, e, are moving with constant velocities %, wm, along the axis OX, through a stationary dielectric. The numerical values of the forces acting on the particles, due in each case to the presence of the other, are found to be €,€o ( Ku Kali 7) on e,, and Cle (4 _ Ku? y ( on és, Ka ve ) i where a is the distance between them. These forces are not equal*. If the particles are originally at an infinite distance apart and their velocities w, u are kept constant by an external agency, the work that has been so expended by the : : 3 Kus time that the distance apart has been reduced to a is Hee aF 2 ) * For a discussion of the principle of the equality of action and reaction reference may be made to a paper by Poincaré, Archives Néerlandaises, (2), 5, pp. 252—278, (1900). Mr WALKER, SOME PROBLEMS IN ELECTRIC CONVECTION. 189 The corresponding expression for the mutual energy when u,=% is also obtained by integrating the expression (AE? + H’)/87 through all space. Case III. A semi-infinite dielectric occupies all space on the negative side of the plane YOZ, and moves with constant velocity uw parallel to OX. A point P with charge e, distant a from the dielectric at time t=0, is moving along OX -at a rate w’( Ay; 2; G= ll. Dien 00 ), j=l or say that Mx=a,j;«;, the caret shewing where the missing suffix is to be supplied. 1B Mr DIXON, ON A CLASS OF MATRICES OF INFINITE ORDER, etc. 191 Let |M be the matrix formed with the absolute values of the coefficients a. We shall assume that the quantities |a,;|, in the jth column of | M|, have a superior limit a; and that a@,+a,+4,+... is a convergent series. When this is the case the matrix M will be called “regular.” Then for all values of ¢ =|a;| is a uniformly convergent series U and its sum is less than some finite quantity, g, which is independent of 7. Suppose in the first place that q<1. Then the system of equations (1 — M) «=a, that is, i COS (CS 1 OA Bootes) haodoone secre aap aracecboradsercne (1), j=l can be solved for 2,, 2... as follows, when a is finite for all suffixes. Suppose @ to be the greatest of the quantities |@. Then we have, for each suffix, |Ma|\+)| M\[a|\>qa; hence | M?a\ +|M|| Ma| + qa and so on; in general | Ma | > q’a@. Hence the series a+Ma+ M?a+M*a+t... are absolutely convergent. Let the sums be denoted by 8, so that b; =a;+ Ma; + M*a;+... G=1, 2... 0). The quantities 6 are all finite for they are less in absolute value than @/(1 — q)*. Hence the series Mb are all absolutely convergent. 2 x a \ Also Mb=M>Ma=%4,; (= Ma; ) , a doubly infinite series which is absolutely con- : r=0 1=1 r=0 vergent and may therefore be summed with respect to 7@ first or with respect to 7 first without the result being affected. We thus have Mb=Ma+ Ma+ Ma+..., and b—Mb=a. Thus «=b is a solution of the system (1). The sets of quantities a, b such that b—Mb=a determine each other uniquely if it is understood that each set is to have a finite superior limit. It is clear that @ is determined by b. Suppose 0’ to denote a set of quantities having a superior limit b and put a=(1—M)0’. Thus a|+6(1+q). Then since |@| has this superior limit b=a+ Ma+ M*a+... =)’ — Mb’ + Mb’ — Mb’ +... =0’, * A superior limit to |b)—a| is qa/(1—q). 25—2 192 Mr DIXON, ON A CLASS OF MATRICES OF INFINITE ORDER, ete. that is, 6 is determined uniquely by a arbitrary. Either of the two sets may be considered as § 2. The value found for 6 may be written x Sa:.Mr- : | Soar ans], 1 | j=1 the summation with respect to j being taken first. Since the general term here is less in absolute magnitude than @q’—'a; the double series is absolutely convergent, and changing the order of summation we may write or still more compactly for M~a,;=1 or 0 according as the missing suffix is or is not j Let A,; stand for = M"2,;, then we have r=0 b;=a;+ S; Aja; @=d2 iy. 20 ). j=l Since A,;==M'a,; it follows that |A,;|$a/(l—gq). Also by taking |a;|=a@ for r=0 all values of j, and making the argument of a; equal and opposite to that of Aj for a particular value of 7, we find that @ S Ajj! 1s a possible value for 6;— qj. j=l Hence = |Aj;| cannot exceed g/(1 —q). al § 3. Now take g to be greater than 1. Since WM is a regular matrix we may take a finite integer m and a positive quantity g such that = |ag|<@n @,/(1 => Ym). Let the values (3) be substituted for 2,4;, @m4.... in the first m of the equations (1), that is, in the system an a Ds AGL; = A; = ils 2 500 nv). The resulting system is ™m x L— > (ain + > ayn) 2,= A+ > Aj C; C= TRAD ese NUV ROS snaeticlee aes (4). h=1 j=m+1 Pe j=m+1 — The coefficients and absolute terms in (4) are all definite finite quantities; for there are finite superior limits to | yj |, |c;| and the series || are convergent; hence j za na the series = ajyjy, and = ajc; are absolutely convergent. j=m71 j=m+1 Hence in (4) we have a system of m linear equations in the m unknowns «7, 22, ... Um and the equations (3) give the other unknowns in terms of these. Thus the solution can be found unless the determinant of the expressions on the left in the system (4) vanishes. If it does, then there is no finite solution of (1) unless the quantities a satisfy one or more, but not more than m, homogeneous linear relations; when these relations are satisfied the solution will not be determinate but will contain linearly arbitrary parameters, one for each of the relations. § 4. Let A be the determinant of the left sides im the system (4), and A’ the greatest in absolute value of its first minors, and let g=|A’/A), it being supposed that A does not vanish so that g is finite. Then all the quantities x will be finite and in fact a superior limit to them may be found as follows on the supposition always that @ is a superior limit to |a). Let p be the absolute value of the numerically greatest of the right sides in (4), then we must have | a; | mgp (1=1, 2... m), and so from (3) ™ | x;| > (a +mgp = a) la = Gm), (=m-+1, ... ). a=1 /'! Let o be the greatest of the quantities 194. Mr DIXON, ON A CLASS OF MATRICES OF INFINITE ORDER, etc. then p}@+o0(1—q,), and the greater of the expressions = Co mga (1 + = B a Co m ——— |14+ (a + ) = a 1 =| pet 1 — dm h=1 = is a superior limit to |z|. Let @K denote this limit. § 5. It may again be proved that the solution is unique, that is, that there is no other set of quantities 6’, having a finite superior limit and such that (1 —W/)b’=a, unless A=0. Suppose A not to vanish and the set b’ to exist, the superior limit of |b’| being b. Then |a|+6(1+4q), so that a superior limit for |a| is implied by that for |b’). Let M’ denote for the moment the matrix formed from M by striking out the first m rows and columns. Then putting (1 — /)b’ for a in (2) we have eae =(1—M’)bi + & ain (on — by) @=m+1,m+2... 0). Hence the process of §1 gives (3) in the form x; = b; + 3 ya (a, —b,) ((=m+1,m42...2). Thus (4) becomes ™m na nm 2,— > (an + & Qj; mn) t= qd - M) bi + > a; (6/ > Yih by’) j=m+1 j=m+1 h=1 h=1 m™ | xo m™m , , =b; > abn, = > > a5ry5n. Dy 5 h=1 j=m+1h=1 Unless A=0, 2;=6, is the unique solution of this set of m equations. Hence again, unless A=0, the sets of quantities a, 6 such that (1—M)b=a deter- mine each other uniquely and either set may be considered as arbitrary, provided always that each set has a finite superior limit. It is conceivable that the system (1—M)«#=a could be satisfied by values of 2, increasing indefinitely with the suffix, but such that the series Mz are nevertheless convergent. We may shew however that this is impossible if the series |M||«! are con- vergent and their sums finite. Let A be a superior limit to these sums, then | Ma| > A. But c=a+ Mz, and therefore [z|pa+A, so that there is a superior limit to {#| and this case falls under that just discussed. Mr DIXON, ON A CLASS OF MATRICES OF INFINITE ORDER, Eve. 195 § 6. In (3) we'may write c; in the form a;+ % Aja; The series 2A,’ is j=mr1 j a; See “ : ; absolutely convergent and in fact T3 F ! ie are (§ 2) superior limits to = |Ajj', |Aj|. —~ Gm — Ym j=m+1 Make this substitution for c in (4). The mght-hand side thus becomes a) ao at D a (ain +. Qi; An) 5 h=m+1 j=m+1 the order of summation in the double series being indifferent because it is absolutely convergent. Hence the final solution is of the form va) x=b=a;+ > Aya; (@=1, 2...0) sc cciccevcceccccecccceescces (5). j=1 q = rl 5 — ij is c 1 te te i € 1 € 1 fr f f € solute] y conver: nt ij ’ S e j series. It is therefore absolutely convergent. Moreover we may take | a;|=@, and suppose the argument of a; equal and opposite to that of Aj; for a particular value of 7 and for all values of 7. In this case b;-—a;=|Aj;|@; this being a possible value of b;— a; j=1 cannot exceed (1+ 4’)@ and therefore 1+ is a superior limit to = | Ax |. j=1 A superior limit to |A,;| may be found as follows. Putting 6;=1, b; =0 («+j7) we have a;=—ajz, aj=1—a;. Again, putting aj=1, a;=0 we have bj=A;z+1, b;=Ay. Thus, by subtraction, when GUN; b=A Aj Now im this case we have a=G;, and therefore [A ,j| + Kay. Thus, denoting by esl the matrix whose coefficients are the quantities 4,, we have the result that 1 1-M is a regular matrix. The expression for b in terms of a@ is then of the same nature as that for a in terms of b. 1 We may thus write b=(1— {#l)a, {#1 being a regular matrix such that 1 am sl ie 1 — M = 1 or (1— #M)(1 — MH) =1. 196 Mr DIXON, ON A CLASS OF MATRICES OF INFINITE ORDER, Etc. § 7. Let M, {¥\, |\M) be the matrices formed from M, {#1, |M| by turning rows into columns. Take any absolutely convergent series e, + @+é;+ ..-. 7 wn Then {#le; => Aje;, an absolutely convergent series since | A;| > Ka;. Moreover j=1 fy zx a = Me; => > Ajie;, i=1 i=1j= an absolutely converging double series. Let f=e—f¥le, then Sf is an absolutely convergent series. Hence with the former suppositions as to a, b La;f; or Laf= ae—- LafPle i=1 = Xae — LefMa = eb; the order of summation in Safle may be changed because the convergency is absolute. Again Mf; and 2M [f; are absolutely convergent series, single and double respectively. i=l Thus Sb(1— M) f= >f(1—- M)b = af = deb. Since all but one of the quantities b may be taken to vanish we have (1-M)f=e. Thus y=/ is a solution of the system of equations Conversely if e=(1—M)f, =f being absolutely convergent, we have f=(1— {#)e. Hence there is no other solution of the system (1—M)y=e such that =|y| is convergent. § 8. Suppose that the solution of the equations (1) fails through the vanishing of A, the determinant of the expressions on the left in (4). To fix the ideas let this determinant vanish with all its first minors, but let the second minor resulting when the first two rows and columns are struck out not vanish. Then two conditions must be satisfied by a, a, a;... if the solution is to be possible, and two independent solutions, say =€, c=’, exist for the system (1—M)#=0. The method of solution will not fail for the matrix derived from M by striking out the first two rows and columns, and we may thus find & £& by leaving out the first two equations of (1) and solving the rest for «;, #,, ... m terms of «, a, which may be arbitrarily chosen. Mr DIXON, ON A CLASS OF MATRICES OF INFINITE ORDER, Etc. 197 Since the coefticients in the first two rows of M form absolutely convergent series we may also (§7) choose 7, 7’ so that y= or y=7/ will satisfy all but the first two equations of the system (1—M)y=0. Then 2 6(1-M) n= S(1-M)&= (Ng and (1—M)n;=0 unless i=1 or 2. Thus £,(1— M) + &(1—M) 9, =0. In like manner &/(1 — M)»,+ &' (1 — M) = 0. Thus (1— M);=0 for all values of 7. In the same way (1—M)n; =0. The conditions necessary for the solubility of the system (1) are For leaving out the first two equations we may find 2, a, ... in terms of 2, a, 3, @,,-.. by means of the other equations, so that we shall have (USM Nee Gs (G8) CH oo Cal) Then m (a —(1 — M) ay} + 9, fa. —(1 — M) 2} a, :{a;— (1 — M) a;} mh Similarly an {ay — (1 — Mya} + ay {as — (= BM) a} = 3 ans. i If then an=0=ay’ the first two equations in (1) will be satisfied also, and there is a solution of (1) of the form aj=b;+r€;+ NE, dX, ’ being two arbitrary coefficients. If Yay, Sam’ do not both vanish the system (1) can have no finite solution. Similarly the conditions for solubility of the system (la) are EIS = Zaki. If these are satisfied, then the solution is of the form Yi=Sit wnt WD: , * To avoid repetition I have not discussed the important question of convergency here and in some other cases. The investigation can easily be supplied by the reader. Vor, XxX Parr I 26 198 Mr DIXON, ON A CLASS OF MATRICES OF INFINITE ORDER, etc, where yw, # are two arbitrary coefficients; if they are not satisfied the system has no finite solution. The method used is quite general, the number two having been taken here for convenience. If finite values have been chosen for &, &, &, & there will be superior limits to &|, &|. Since only the ratios, and not the actual values of &, &, &,... matter, we may in fact suppose these superior limits to be any we please. § 9. A case of special importance is that in which | aj |< eat, | a;|< 4rd}, where e, «, X are real positive quantities of which X< 1. The matrices M, M are thus both regular. xz The equation 2;— aj; =a; may then be written ja - iz . - : . XN a; = > ai; M.A z= an % v= and the system solved as above for the quantities X~'#;. Thus when A does not vanish there will be a superior limit, say 7, to 2 ~'x;| and we shall have | a; |< TA. Suppose that A=0 and take in fact the case of § 8. To find & & we ignore the first two equations of the set (1—W/)z=0 and in the 7th equation take the terms M2,+4 7 over to the right. Since |a,|g, we have @a=2 Ma Ma i é se C 3= @ ae and it follows that the coefficients in © are functions of @ having no singularity, essential or accidental, in the part of the 6@-plane which lies outside the circle whose =) UO 500), centre is the origin and radius q. If another real positive quantity qg,, is so chosen that = |a,;|@m, the process of solution (§ 3) consists of two parts. In the first of these #41, myo --. are found in terms of 2%, 2, ... %m, Amiri, mis, ---- The resulting expressions are functions of @, without singularity essential or accidental in the region outside the circle whose centre is the origm and radius q,. The same is therefore true of the coefficients in the m linear equations for 2,, a, ... 2, Whose solution forms the second part of the process. Since the operations performed in this second part are purely algebraical, finite in number and unambiguous, the resulting functions can have no essential singularities im the region considered, and therefore only a finite number of accidental singularities. Hence in the part of the @-plane outside a circle whose centre is the origm and radius q,, the coefticients in © have no essential singularity and only a finite number of accidental singularities. Moreover q,, may be taken as small as we please. § 11. Since the solution of the system (1) was unique, the functions of @ here considered are uniform. Their expansions in the domain of any point in the 6-plane may be found as follows. Let 7 be the greatest of the quantities | Oa, when a=1 for all suffixes, then for any finite values of a the series @a—(0’—0)O%a+(@ —6)@%a... will be con- vergent so long as |@’—@|<1/r. Let 2 denote the sums of these series. Thus Oa’ = Oa — (6 — 0) Oa + ..., and uv + (0 — 0) @z' = Oa. But now (@— M)® =1, so that (0—M)«' +(0 — 0) 2’ =a, or (0 —M)a# =a. Hence 2’ is what x becomes when @ is changed to @ and if ©’ is the matrix corresponding to © we may write @' = © — (6'— 0) @ + (0 — 6) @_... to w, the expansion holding good when |@’—@| is less than some real positive quantity which is not infinitesimal. Hence the process of analytical continuation may be used to give © when |@| has a value En; = 0, .—) the modified system is insoluble, and in general not otherwise. Thus unless S£ vanishes this value of 6 is a simple pole for the coefficients in @. The residue of z; at this pole, that is, the limit when 6’=@ of (@’— @)a;, is readily found to be 2 a ED an+ & Ein. i=1 i=1 Mr DIXON, ON A CLASS OF MATRICES OF INFINITE ORDER, etc 201 In the same way y; will have a pole, the residue being nj & ev: + & Eni. i=1 i=1 By analogy with the case of a fimte matrix we may call @ a multiplier when it is a pole of @; so far the multiplier has been supposed simple. It may however be double, triple, or repeated any number of times; in such a case there will be 2, 3, ... independent sets of quantities satisfying the equations (@—M)a=0. Another possibility is that S& may vanish; we shall then say that the multiplier is of a higher order than the first. Thus a double multiplier is not the same thing as one of the second order. Then there are sets of quantities £”, § 13. Let @”, 6% be two unequal multipliers. ME® =0%E®, My =O%n"............ n®, £2, »® such that ME” = 0% E, Mn” = 9% yo, wo al es Now S n° ME = €, Mn,”, i=1 iat : ra) 2 that is OYE £0 2 = 0% S Ey, i=1 i=1 oo Dn Hence & &;"n;* vanishes and in like manner so does & €;°n;" i=1 i=1 If @% =@", a double multipher, there are two sets of quantities €%, €% and two But 7”, 7® may be replaced sets 94", 7”, so that we still have the four equations (6). We can therefore take 7” so that by any distinct linear combinations of them. an E E40 =0, i= «2 YE; =0 also, we can and then, unless the values so chosen for 7" are such that i=1 take 7 so that S E,% ni) =(), a= In the excepted case when 2&7") =0 we can take linear combinations of &", £&% 4= ao i) in the place of —", E®, and so secure that ¥ &;%n;°, &;@;" vanish. The same kind i= =I of method apples when the multiplier is repeated oftener; thus in general S& and L&%m” vanish if 8%, 8% are any two multipliers of the first order, equal or unequal. = a repeated multiplier, the residue for z; is ienowald“)i—104)—reee 2) a =z ani” = ani” {) a .@) t= 1 ge a pig}t) = +.. > En SS E,2) ni”) i=1 i=1 202 Mr DIXON, ON A CLASS OF MATRICES OF INFINITE ORDER, etc. § 14 Again, let @ be a multiplier not of the first order so that SE Then, i=1 since this condition is fulfilled, sets of quantities &', 7 can be found such that (@-M)E&=-E£, (@-M)y =-1, or ME = OE + &, Mn =6n\ +>. The quantities » will form an absolutely convergent series. We also have since Ss ni (M— 0) & = S E> (M—@)n;. i] n Dn or (0 = 0)? > ane — ani + (0 = @) > Qn; . i i=1 t=1 t=1 t= Hence it follows as before that (@’—@)«’ tends to a finite limit as 6 approaches 2 6, unless = &;y; =0. Thus the coefticients in © will have a pole of the second order i=1 and we shall say that @ is a multiplier of the second order, unless Lé&y (or LE‘) vanishes, in which case the order will be higher. Multipliers of higher orders can be discussed in the same way, the order being always that of the corresponding pole for ®, which must be finite since @ has no essential singularities away from the origin in the @-plane. § 15. To illustrate the working of the method in general take 6, 0” to be equal multipliers of orders 4, 3 respectively. Then we have sets of quantities* & &, &, &, &0), BM), &20) such that Me= 0g, ME=06' +6 Me*=08+8, ME= 08+ ME =: OE”, Meu = 020) ah £0), Me”) = 6e24) + £10), We also have two sets of quantities », 7" such that Mn = @n, Mn” = An", * No confusion is caused by this notation as the powers of — do not occur in the work. Mr DIXON, ON A CLASS OF MATRICES OF INFINITE ORDER, etc. 203 and these may be so taken that an a >> ni Es = 0, S Ni &24) =0*, =1 i=1 while > 7&2 +0, ne E29 £0; i=1 bas zt otherwise one of the poles has a higher order than we have said. Also Sn&, =n&, SF, EE", Sn all vanish and so do SHE, TV, WME, =n E%, Dy £0), The set 7! is next determined from the equations (0—M) 9 =—7; it is not unique, but may be altered by the addition of multiples of 7», »” and these may be so chosen that =7'&, =y'£*" shall vanish. In the same way 7%, 7°, 7%, 2° may be determined in succession so that (@—M) x =—7, (0—M)y=-77, (0-M)y =, (0-M) -=— po, and so that also >77é, Sn? E20, SE, pew, > VES, Snr E20), 37? MES, >?) E20) shall all vanish. Now if (@— M)z=a, (@—M) y=e, we have, under the usual conditions as to absolute convergency, Ms ayi= 2 y(O—M) a= 3 x (0— My. Sem. =1 =1 c i=1 i Hence in the present case all the sums of the form &;n; that can be formed i=l with the sets BBB Em, £0, pam, m7, Wr, 7,7", 7%, 7, vanish except L&q*, LE'm*, TE*y', LE*y which are all equal and LEV, TEWypw, SE) _w which are also all equal. There is no loss of generality in supposing each of these seven equal to unity. Then it will be found that the part of «/ which becomes infinite when & =@ is Yan Yan Yar? Lan? g) le —8¥ 0)" (oor * g| Yan Xan! Zan? . Lan _, aan dee | et (e—0"* @ =| ne ote a 7 +E gy o Yan San San San San San Ss ean wot aa tee lena opt oo |t 8" 9—o: * It cannot be the same set 7 that satisfies both these while we have conditions. If Sy=0, Syi0)=0, then \, u can be so Dn (AE9 + pe?) =0. chosen that Hence @ is of the 5th order or 4") of the 4th according En (AZ + wz?) =0, as \ is or is not different from 0. 204 Mr DIXON, ON A CLASS OF MATRICES OF INFINITE ORDER, Etc. The infinite part of y may be written down similarly. Other cases of repeated multipliers may be treated in the same way. If 6% is a multiplier unequal to @ and if &*, » have their former meanings, then LEO My = OLE! + ZEN = OLE ; but Dp ME = 2 TEC y}. Hence S£* y'=0 and it may be proved in the same way that x Ss Emo) nin?) =0 —_ v v > i=1 if @”, @® are unequal multipliers of orders at least equal to m+1, n+1 respectively. The results are like those in the theory of finite matrices, which could in fact be established in the same way. Lemmas relating to expansions in sines and cosines and in spherical harmonies. § 16. With a view to some applications of the above theory we shall now prove certain lemmas establishing a connexion between series convergent by ratio and analytical functions. I If in a series a+, sin @ + a, cos 0+... + Gem Sin MO + Ay,cosmO +... the coefti- cients a), @;, d.... form a series converging by ratio, then the sum of the trigonometrical series is an analytical function of @ for all real values of @. For we have |@m'\<«X”. where «, X% are constants and X<1 and the sum of the series is F'(e®), where F(z) = a) +4 (Qo —0d)) 2+ -.. +4 (om — Uama) 2" + «.- + 4 (as + ,) tite ee ee 4 (Ge SF Lem) GI 3.5 which is an analytical function of z in the ring between the circles 2 =X*, |z/=r~ and therefore on the cireumference of the circle |z|=1. Il. Conversely if (@), a function analytical for all real values of @ and having a period 27, is expanded in the form lp + dy Sin 8+ a, cos O+..., then the series Utd, + d,4+... is convergent by ratio. For the function f(—clog z) is analytical on the circle |z|=1 and therefore through- out a ring R,—d<|\z 1>R, and such that for real values of 7 from R, to R, inclusive this solution of Laplace’s equation is analytical for all real values of the angular coordinates @, ¢. Take the formula °V_VwU) de = (Ngee oo I[ (OW —VV U} de dy de = [| {U oY ant in relation to the shell bounded by concentric spheres of radii R,, R,. Let U be the reciprocal of the distance from a fixed point (7”, @’, ¢’) in the thickness of the shell to the current point (7, 0, ¢). The left-hand side of the equation is the value of 47V at the fixed point (7, 6, ¢’). Over the outer sphere, r= R,, we have ay eal : on Or U =(R24+ 72 — 2R,r'p)+ mip (w)+. as P,(u)+..., where « =cos @ cos @ + sin 6 sin & cos (¢ — @’), 4) =F, +3 OU =D at Cones on Re Re 4) Rr P, (4) -.. On the inner sphere, in like manner, av ava. ons’ Th Jie Roe Ui chia 1(H) +-- toma Pa (H) +++ oU nk ra +P, i) ees a Pr(e) — Thus ve = s [aA Y. . ye = e es n 77H | 2 ; i ln fr oV 4 where ae Vl i | Pa |B ei) vi sin 6.6 dé, R; 0 0 or the values of V, = being those on the outer sphere, and Act ae Ree | ze ie OS nv - R, = fin 6 d0 dé, /0 7 the values of V, 2 being those on the inner sphere. Mr DIXON, ON A CLASS OF MATRICES OF INFINITE ORDER, ere. 207 This expansion of V’ is convergent by ratio so long as r’ lies between R,, R,, E V ah. since V, = are finite over the whole boundary of the shell. Hence over the unit sphere we have, leaving out the dashes, V= DS in ar ye ealt n=0 a series uniformly convergent by ratio, since (Y,+ VY») tends to the limit zero as m increases if X is a quantity <1 but > R, and > R,. Since P,(%) as a function of 6’, ¢’ is a linear combination of the 2n+1 funda- mental harmonics of degree n, it follows that the same is true of Y,, V_,, and that these are complete surface harmonics. The sum of the series = r”(Y,+Y¥_,-) is an analytical function of the coordinates 0 n= : fant Sill L : throughout a concentric sphere of radius ,> it satisfies Laplace’s equation and has the values assigned to V over the unit sphere. Hence a function which is analytical over the surface of the unit sphere coincides on that surface with a potential existing and without singularity throughout a greater concentric sphere. A function depending upon two points on the surface of the unit sphere, if analytical everywhere, will thus coincide with a function W, satisfying Laplace’s equation in regard to each point and analytical for each point throughout a concentric sphere of greater radius. Take then radii R,, R, both >1 but such that for values of 7, 7, up to R, R, inclusive respectively W(1,, @,, ¢:, 7, 4, $) is an analytical function of the Cartesian coordinates. Then by the usual formula if 7,< R,, 7,< R,, 2r [a (20 [rx 16m? W (rs By $s Por On b= ff RR, (R?2- 7,2) (R2-72) W (fy » y's his Ry, 00', py’) Sin 6,’ sin 6,’ dd,' dy’ dO,' dy! {R,2+72—2Ryr, (cos 6, cos 6,’ +sin 4, sin 6, cos d, — $y)}3 {Rye +7y2—2Ryry (cos A, cos Oy +8in A, sin 4,/ cos o—y)}* By expanding this in ascending powers of 7, 7, we find Wn, 6, dr, iior, (oly gs) = SS TNE (he $;) Zn (82, py), m=0n=0 where Y,,, Z, are surface harmonics of the degrees m, n. The expansion converges absolutely and uniformly so long as 7,<,, 7,< R,. Hence taking >> = but <1; 1 m oe but <1, we have that in all positions the numerical value of Ymn(@,, b:) Zn (2, b2) 2 is less than «A™u”, where «, X, w are positive constants and XA, pw <1. 27—2 208 Mr DIXON, ON A CLASS OF MATRICES OF INFINITE ORDER, etc. § 18. V. Let F(t) be an analytical function of ¢ for real values between +1 in- clusive. Suppose in IV. that the function V is equal to F(cos@) over the unit sphere and let the arbitrary values of 2 on that sphere be also chosen so as to be only dependent on cos@. Then Y,, Y_, will be independent of ¢ and thus V is expanded in zonal harmonics. Hence we have the expansion of F(t) in a series La, P,(t) and since the value of P,(1) is 1 it follows from IV. that the series =|a,| is convergent by ratio when F(t) is analytical between +1 inclusive. Also a function of z, y, analytical for values of each between +1 inclusive, may be expanded in a series YLaun P(e) Pn(y) and Gm, will be numerically less than «d™u", where «x, A, mw are positive constants and X, pu < 1. io] Again if =/a,| is convergent by ratio { a, P,(t) will be an analytical function n=0 a of t for real values between +1 inclusive. For as in III. = a,r" P,, (cos @) is an analytical n=0 function of the Cartesian coordinates throughout a sphere of radius R>1. This function is independent of @ and so must be an analytical function of z and a*+y*, or of 2 and #+y?+2. Thus when 2?+7°+2=1 it is an analytical function of z for real values between +1 inclusive, which was to be proved. Integration-Matrices, § 19. Let a(z, y) be a function of two variables x, y uniform and finite for all real values of x, y between certain limits, say a, b. Let (x) denote a function of « uniform between a and 6, and consider a transformation N applied to this function and defined by the equation 5 Nes y= | (2, y) & (y) dy. This is a limiting form of the homogeneous linear transformation, the variables, infinite in number, being the values taken by ¢ (a) for values of « between a and b and the coefficients being the infinitesimals a (a, y) dy. The problem corresponding to that on which the discussion of §§ 1—15 was founded is to find a function (zx) such that (x) — No(x) =f («), a given function. If there is a positive quantity g, less than 1, and such that when | ¢(«x)| $1 for all values of « between a and 6, then | N(x) +q for such values of a, a Mr DIXON, ON A CLASS OF MATRICES OF INFINITE ORDER, etc. 209 b or in other words such that g is the greatest value of | a(«, y)\|dy for such values of «, then the series F(a) + Nf (a) + N2f(2)+ N%f (x)4+... is absolutely and uniformly convergent, if f(«) is always finite. The sum of this series is the required function ¢(«) and there is no other solution. The proof is exactly as in § 1. § 20. When there is no such quantity as q this method fails. The solution may however when a(z,y) is an analytical function be made to depend on that of the problem of § 3. We may without loss of generality take a, b to be +1 and then suppose a(w, y) expanded in a doubly infinite series of zonal harmonics. Suppose that a(z, y)= = = a;; P; (x) P; (y), =07=0 where P;(x) is the zonal harmonic of order 7. Take for $(#) the expression = «;P;(2) ; i=0 = S ss Be 7 a; P; aL then No (a) i=0 joo Qj+ 1 Kj hij (x), and thus the coefficients «, , 2, ... are transformed by such an infinite matrix as was taken for M in § 1—15. In this matrix, since a(#,¥y) is supposed analytical, the typical coefficient el is less in absolute value than ap’, where a, p are real and positive and p p on or aes Thus zab=— 27 (*) =~ 20 (=) and the value of o at c is that of oy Oa ds)z On/z = or = at Z. Thus when the assigned boundary values are denoted by F(s) the potential without F ae : Wf singularities that has these boundary values is oe | F(s) ods. 27 | Along an analytical part of the boundary o is an analytical function of s. Regarded as a function of the internal point, o is a potential without singularity throughout the area, except at the point of the boundary that is concerned, where it becomes infinite 7, like the real part of —; ae along the rest of the boundary o is zero. 2 z—Zds § 28. Suppose then that there are two areas A, B for each of which the problem, with singularities, has been solved, and that these two areas have a common part C. Can the problem be solved for C, and for the combined area A4+(B—(C)? Take C first. Let dA, denote that part. of the boundary of A which also separates C from B—O, and A, the rest of the boundary of A, B, that part of the boundary of B which separates C from A—C, and B, the rest of the boundary of B; A,, B, may thus have parts in common, but it will be supposed that where A,, B, meet, they meet at a finitesimal angle and at a point where the boundaries of A, B cross each other. 28—2 216 Mr DIXON, ON A CLASS OF MATRICES OF INFINITE ORDER, Etc. Suppose ¢@ to denote a series of values along dA,, and N@ the result of the following operation. A potential y is constructed for A, without singularities, and having the values @ along 4dA,, 0 along A,. Then a potential y is constructed for B, without singularities, having the same values as x along B, and vanishing along B,. The values of w along A, are denoted by V¢. The operation NV belongs to the type that has been discussed in §§ 19—23. We have in fact, taking w, y to be, say, ares of A, measured from a fixed point and to be the independent variables, No (x)= | a(x, y) b(y) dy, Tae where a(za, y= Lim 5 when ¢$(#) is taken to be 1 from y to y+ 6éy and 0 else- éy=0 where. There is uo difficulty in expressing a(z, y) by means of such functions as o of § 27. Again, so long as at least one of the boundaries A,, B, exists, there will be a fractional factor g, real, positive and <1, such that the greatest value of V@ does not exceed the greatest value of gd, and thus the solution of the equation G-WN) =f, for @, when 7 is known, is given by the series ¥+ Nf+ N*7f4+ Nef +... as in § 19. : The boundary of C consists of A,, B, and possibly C, a piece common to Aj, B. Let g denote the assigned values on the boundary of C, b a real constant. Construct a potential w for B, having the given singularities in C, equal to g+b along B,, OQ, and having any values h along the rest of B,. Find a series of values ¢, along A,, such that ¢—-Nd=g-utb. Form a potential v for A, without singularities, equal to @ along dA, and to 0 along A, and also a potential w for B without singularities, equal to v along B, and to 0 along B&,. Then uw+v—w-—b is a potential for C having the given singularities; along B,, C, we have uw=g+b, w=v and along A,, v=¢, w= NQ, v—-w=o-—Ngd=g-—utb, utv—w-—b=gy. Thus u+v—w-—b is the potential sought. § 29. Next, to solve the problem for 4+B—C. Let g again denote the assigned values along A,, Bj, f any values along A,. It may be that C consists of detached parts, 0, C%,... and if so it may be possible ) P vy; P OOOO —n—n—n Mr DIXON, ON A CLASS OF MATRICES OF INFINITE ORDER, eve. 217 to draw an irreducible circuit from a point of A through (™, B, C™ and back to the starting-point in A. When this is so let a, be the increase in value of the required potential due to the passage of 2 round a closed circuit from a point in A through C”, B, (1, and back to the starting-pomt in A. Thus a,=0. There may of course be irreducible closed circuits in A or B or both already. The assigned values g must be consistent with those of the moduli for closed circuits, and so will in some cases not be uniform but dependent on the path by which the point concerned is reached from some fixed origin. Construct a potential u, for A with such of the assigned singularities as fall within A or on its boundary and having the values g, f along A,, A, respectively. Then construct a potential ~ for B having such of the assigned singularities as fall within B or on its boundary and equal to gy, m—a along B,, B, respectively. Here a is to be taken as a), a, @;,... according as the part of B, in question belongs to C, C2, C%,..., and similarly for A, in what follows. Construct a potential uw, for A with the singularities and equal to g, %4+a along A,, A, respectively. Find values ¢ along 4,, such that ¢— Nb=H-m=f—hf,, J, denoting the values of mw, or u,+a, along A). Repeat the above construction, forming three potentials w, v,, v2 aS WU, 2%, Us. Were formed but starting with g, f—@ as the boundary values of y. Then m—%, m—%, U,—% are potentials without singularity for A, B, A respectively and the boundary values are for u—%, 0 along Ay, ¢ along A,; for u—v, 0 along By, uw—w along B,; for %— v2, 0 along Ay, w,—, along A,. Hence along 4,, Uz — Vz = NV (uy — %) = N¢, n=f-Nb=/-$=n. Thus v,—v is a potential for A, vanishing along the boundary and having no singu- larity, or v,=v, throughout A. Again, %,+a,—% is a potential for C” without singularity and vanishing along the boundary, that is 1,=%—a, throughout C7, and v, can be continued throughout B. Hence %, 2, % are really one potential v having the assigned values g along A,, B, and existing throughout A +B—C, with the assigned singularities, including the modulus a, for the passage round AC’BC1A (r=2,3...). § 30. When the combined area A4+5—C has no boundary, the above applies with one point of difference, namely that 1 is a multipler for the transformation JN, so that the equation 6—-N¢=f-fe 218 Mr DIXON, ON A CLASS OF MATRICES OF INFINITE ORDER, ETc. cannot be solved for @ unless a certain condition is fulfilled*. We suppose the curves A,, B, and the assumed values f to be analytical, and also that there is no point common to A,, B,. To see that 1 is a multiplier take ¢=1. Then the potential y for A which has no singularity and is 1 along A, is 1 everywhere and therefore along B,. The potential w for B which is 1 along B, and has no singularity is 1 everywhere and thus V1 = 1. This multiplier is not repeated and is of the first order. For when ¢ is not con- stant along A,, the greatest and least values of yx are to be found on A, and thus the range of yx on B, is less than its range on dA,; in lke manner the range of on A, is less than on B,. But if N@—@ were constant the range of w on A, would be equal to that of ¢, which is absurd; therefore the multiplier 1 is of the first order and is not repeated, and there is only one condition to be satisfied in order that the solution may be possible. When the solution is possible, @ and therefore the resulting potential are not unique, but may be altered by the addition of an arbitrary constant. Thus by successive steps the solution of the problem with singularities is found for the whole Riemann surface. It is desirable that at the last step when the boundary is altogether done away the boundaries 4,, B, should consist of analytical curves and should have no point in common. So long as the combined area has a boundary the method of solution applies unconditionally but a single condition arises when the boundary finally disappears; when this condition is fulfilled the solution is possible. § 31. There can be no doubt what this condition is. Suppose wu to be the potential which satisfies the conditions, v the conjugate potential so that w+.=w is the resulting monogenic function. Then the whole increase in w when z travels round the perimeter of the dissected surface, supposed so drawn as to avoid the singular points of w, is Qu k,, where k, is the coefficient of log (z—c,) in that part of w which becomes infinite at c,. The real part of 2ur=k, is the increase in u, which is assigned before- hand and in fact must have the value zero, being the sum of all the moduli for irreducible circuits, each taken twice with opposite signs. The condition with which we are now concerned is the vanishing of the imaginary part of 2urXk,, that is of the real part of Sk,, or of the sum of the coefficients of the logarithmic infinite terms in the singu- larities assigned to u. (Forsyth, Theory of Functions, pp. 401—2.) The condition may be found in another way, which will further illustrate the general theory. Let @ be a potential without singularity for the space C common to 4, B, the value of @ being 0 along A,, 1 along B,. When C consists of separate parts the function @ in one part will be quite distinct from the function in another part; the method of § 28 will however still apply. The curves A,, B, have been supposed analytical so that @ can be continued across them. * This difficulty seems also to affect the solution given for C, but it may be removed either by a proper choice of the constant b or by cutting out a circular piece from 4 or B. Mr DIXON, ON A CLASS OF MATRICES OF INFINITE ORDER, eEvc. 219 Let C,, C, be two curves along which @=6,, 6, respectively, and take 0<6,< 6,<1, so that the space between C,, C, les wholly in C. Take the equation fie = —u = ds =|) {OV2u — uV0} dady, for the space between C,, C,, w being a potential without singularity for this space. It follows that r ou 00 ih (pee [18 on ~ "oat @=!, Jor — ust as, n denoting the normal drawn towards B,, so that ue is positive. We have also on Cu ou ae = ae ds =U, say. j 08 p ae Thus (0, — 02) U=| % an ds =i Ua ds. Now if w is a potential for B or A without singularities U=0; this is the case for both y and w of § 28. Hence going to the limit and putting 6,=0, 6.=1 we have 00 i ed SS ed va, “=|, ve ant But along B, y=wW and Ga A, x=¢, ~=N¢. Hence it follows that for any assigned values } 00 ; eee |? on i=] No.5 00 Thus a is the second associated function for the multiplier 1, the first being a constant. In order that the equation é $—Nb=f—f, of § 30 may be soluble, it is necessary that 5 om ae [UM 5, @=0 or i. (UW — hy — 0 as=0. Now ™—%-—4, is a potential without singularity for C’, so that we have : 00 cds 0 (UM — %) [em 5, | (tom — a) 5, do = — eae ds. 220 Mr DIXON, ON A CLASS OF MATRICES OF INFINITE ORDER, Etc. We may suppose that C, does not pass through any point where there is an assigned singularity. Then since uw is a potential with singularities throughout A we may replace ou, : - Sake ; é —°ds by the same integral taken round small circles containing the respective assigned c, On S =| : singular points which lie on the same side of C, as B,; similarly for -{ a we may C, put the same integral taken round small circles containing the respective singular points on the same side of C, as d,. Thus | Sah is numerically equal to the real part J¢, xs F 30 : of 27Sk,. Also along B,, y%.=%—a and thus (Uy — Uy — a) ds=real part of 2772k,. Jia Ne The condition is therefore again found to be the vanishing of the real part of Dh. Matricial Functions and Integrals. § 32. I propose next to apply the method to prove the existence of factorial fune- tions and integrals, and more generally of the functions which Poincaré has called zeta- fuchsian and zetakleinian. These may be called matricial functions on the surface, for they occur in sets and are transformed by means of a matrix when the argument place passes round a closed circuit. Similarly a set of matricial integrals is a set of m functions w,,W2,-..Wm such that by passage round a circuit w is changed into Mw+y, where 4;, %, +.» Ym are any complex constants, and JJ, is a matrix. A general account of the process will first be given (§§ 32—36) and afterwards some points that arise will be discussed more carefully. Let the surface be dissected in the usual way by cuts drawn from any point QO, so as to become a curvilinear polygon of 4p sides, simply connected, the (4¢+1)th and (41+ 3)th sides being opposite sides of the same cut, as also the (4¢+2)th and (4¢+4)th. Let cuts also be drawn from O to those points within the polygon passage round which is to produce a linear transformation of the functions; simple logarithmic singularities need not be thus treated. Let J be the whole system of cuts as now drawn. Take another system of cuts J, formed from J by slightly varying each of the cuts in J without changing the terminal points, and such that J, 7 have no points in common but the terminal points. Let R be the region bounded by J and J, in such a way that R contains every point of the surface and covers twice the lunes L between I and J. From R form a surface R’ by describing circles round O and the other terminals so as to exclude these points. Let ZL’ denote the space common to L, R’. Then R’ is a simply connected surface whose boundary consists of circular ares round the terminals, which ares we shall call in general &,’, and of parts of the lines Mr DIXON, ON A CLASS OF MATRICES OF INFINITE ORDER, etc. 221 I, J which we shall call R,’. The lines R, occur again within R’ and will then be called R,’. Let the original surface be S and let the part outside the circles be S’. Suppose the ares J, J so drawn that the boundary value problem can be solved for R’. Then we wish to find for S’ a set of potentials wu, uv... uw” having given boundary values and given singularities within the polygon, and such that passage round closed circuits produces given linear real transformations of the set; the transformations need not be homogeneous: they will be the same for all equivalent circuits. Let WM denote these transformations in general, so that the passage from a point of R,’ to the corresponding point of #, through R’ changes u into Mu. M thus denotes a set of matrices of order m+1, affecting w, wv... w”, 1; the last row in M will be 0,0... 0,1. Let M, denote the corresponding homogeneous transformations, that is the matrices formed from M by striking out the last row and the last column. The assigned matrices M must satisfy such conditions that the effect of passing round the whole boundary of the polygon shall be to restore the original values. These conditions will be discussed later. § 33. Let a set of potentials w,, u%?... um" be formed for R’, having the given boundary values g on R,’, the given local singularities within R’ and on its boundary, and any assigned values /f, on &,'. Singularities and boundary values occurring within ZL must be taken twice, and at the second time must be transformed by the proper matrix. Denote by f, the values of these potentials on Ry. Suppose a set of m potentials to be formed for A’, without singularities, vanishing along A, and having values ¢@ along R,. Let x denote the values of these along R,’ and let WV’ be such an operation applied to @ that V’6=M,"y. Let ¢@ be so determined that 6-N’¢=f,-— Mf, and let a set of potentials w!, wv... w" be formed for R’ having the given singularities and boundary values on £,, as before, and having the values fo—@ along R. Then the values along &, will clearly be fA—y, that is A-M.N’¢=f,-M[¢-fi+ MA] =M(fo— >) — (M- My) (fo— $) + ME - Mo) Mf, =M(f.- >); since the effect of the operation M— WM, is independent of the quantities operated on. Now let us take the values of w at. any point of LZ’ and compare them with 1 the values at the same point in the overlap. The boundary of L’ consists of parts of Rj, Ry, R.’ and along all these u,—Mw vanishes while it has no singularity inside L’ or on its boundary. Hence throughout L’, w= Mw and therefore throughout S’ the potentials w are as desired. It should be borne in mind that M,, M,~ are distributive but M, M~ are not. Thus if %=Mu, we have M1u,=u, although M~™(u,— Mu) is not zero. For different parts of R,’ M has different meanings and the substitution denoted by M for one side of any of the lunes Z will be denoted by M™ in respect to the other side of the same lune. Wo, XID IPA JOM 29 222 Mr DIXON, ON A CLASS OF MATRICES OF INFINITE ORDER, eEvc. § 34. Let us consider next the question of finding a set of potentials w}, u?... u™ for the area of a circle, having given boundary values and local singularities and such that passage round a circuit including the centre causes them to undergo a real linear transformation which we shall again call M. WM, being a matrix of order m+1 operating on uw, u?... uw", 1, can be reduced to its canonical form. This simplifies the problem and we have to consider four cases, according as the multipliers are real or imaginary, of the first or of a higher order. I. Take a real multiplier of the first order. Let it be e. We are to seek a potential w having given local singularities and boundary values and such that passage round a circuit enclosing the centre multiples it by e**. Let w be the resulting mono- genic function, of which w is the real part. Take the centre as origin, the radius being unity. Then wz is unaffected by passage round a circuit enclosing the centre; its singu- larities and the boundary values of its real part are known, so that it can be found by the known method. It is unique save for an additive imaginary constant when the accidental singularity, if any, at the centre* is specified. If there is no pole and no local singularity and if the boundary values of w are denoted by /(¢), then the value of w is — [reer S* ag. = II. Take a real multiplier of the second order. Then having found wu as above, we are to find w so that passage round the centre changes it into we™*+u, boundary values and local singularities being again assigned. Let w' be the resulting monogenic ; logz . emah TES function. Then w! — e = w — 8 is changed by the circuit into Quer 8 , log z+ 2uar wie =e a aes 5 that is, it is multiplied by e. Also its: singularities and the boundary values of its real part are known and thus it can be found by L; thus w! and w are determined, since w is known already. For real multipliers of higher order the same kind of treatment will avail. If w, wi, w?... w™... are monogenic functions changed by passage round the centre into ew, ewi+w,..., er w*+w" ..., then the functions log z ek w, gr es amd ay °s eA... \ Qur # _, log z orn, W"” log 2 (log z log z ZA) wy — ea +(-1) e™ (ee 37k )\s “56 . to n+1 terms }... Quer” r! em \ Qear * Note that such will generally be introduced by differentiating. If wz‘ does not vanish at the centre gta = will have a pole there, / Mr DIXON, ON A CLASS OF MATRICES OF INFINITE ORDER, ere. 223 are unaffected by the passage. Their singularities and the boundary values of their real parts are known in the case supposed. Hence they can be found and the functions w, w'... w” ... determined successively. The multipher may be 1, in which case X=0. If this multiplier is of the rth order in M,, it is of the (r+1)th in. M, the first function w being a constant. There is no further speciality in the case. III. Imaginary multipliers occur in conjugate pairs since WM is real. Let such a pair be e+“. Then the problem is to find two monogenic functions w,, w., whose local singularities are given and the boundary values of their real parts and which are changed by passage round the origin into e™ (w, cos 2um — wW.Sin 2wm), e™* (w, sin Zum + w, cos 27°). It readily follows that ZA (Wy + We) + 2'A>t# (w, — wwe), and ZF (wy — Wy) + 2% (w, + LW), are unaffected by passage round the centre. They are monogenic functions; their singu- larities are known and so are the boundary values of their real parts. Hence they can be found and from them w,, w,. A pole of any order at the origin for either of the functions 2‘ (w, + w,), 2'***(w,—ww,) may be in the list of assigned singularities. IV. In discussing the case of imaginary multipliers of higher order a symbolical notation will be useful. Let Ew,, Ew, denote terms in two series of monogenic functions, such that by passage round the centre H”(w,+.w,) is turned for any value of n into er Qe) Bn (w, + ww.) + LH (w, + w,). The functions w,, w, or H°w,, Hw, are to be those discussed in III. Then the expressions ge {1 +e" (A+tu) [SES kn (w, + We) 4+ gid+e {1 +e" (Acne) Eig | ICE Aiaerg En (wy, = iW), and 1z'A-# {1 + ener (Aue) 1D En (w, + WW) —ygihte {1 + ene (A-um) EY logzizm in (w, = We), when expanded in descending powers of # by the Binomial Theorem, represent functions unaffected by the passage whose singularities are supposed known and also the boundary values of their real parts. (Negative powers of # may be left out, as their effect is to reduce to zero the functions on which they operate.) The solution can therefore be found. § 35. The imaginary additive arbitrary constant in wz (Case I.) must not be for- gotten. It leads to a want of uniqueness in the determination of w. In fact w and w” may both be increased by any real multiple of oz. In the same way w,, w, or H"w,, EH"w, may be increased by any quantities of the form (+ tks) LZ~ ATH + (4, — Ky) 0Z~4-#, (K, + tk.) 27TH — (kK, — bk) 2-H, where «,, & are real. 29—2 224 Mr DIXON, ON A CLASS OF MATRICES OF INFINITE ORDER, Etc. When A=0 we still have the want of uniqueness. Any real multiple of log z may then be added to w. For the sake of definiteness I shall call the solution expressed as a definite integral the primary solution, the function that may be added to this the complementary function. The complementary function is purely imaginary along the boundary. When the assigned values of the real part on the boundary are all zero the primary solution vanishes every- where in the area. § 36. On the surface S of §32 let circles be described concentric with those that form &,’ and outside them but not meeting each other. Let YZ denote the aggregate of the areas of these circles, 7, that of their circumferences. Then the boundary value problem with singularities, matricial and other, has been solved for S' and also for the several parts of 7. We are to deduce the solution for 8S. Let f denote a series of values along 7,. Form a set of potentials u, for TY’ satis- fying the general conditions of the problem and having the assigned singularities so far as they fall within 7’ or on 7, and the values f along 7,. Form a set of potentials uw for S’, agreeing with uw, along &, and having the assigned singularities, local and matricial. Let 7, denote the values of u along 7). Again let @ denote a series of values along 7,. With these form the primary (§ 35) set of potentials y for 7 without local singularities and with the matricial singularities assigned except that they are made homogeneous, M, taking the place of M. Form a series of potentials wy for S’ without local singularities and with the matricial singularities assigned, except that again M, is to take the place of MW; also along R,’ W is to agree with y. Denote by N¢ the values of Y along 7. Then let @ be found so that o- No=f—p. Let w%, u, be formed as were uw, wm, except that the values of w along TZ, are f— ¢. Take the same complementary functions in uw, as were taken in w%. Then %—%&,, %— us are sets of potentials for 7, S’ respectively, without local ordinary singularities and with the assigned homogeneous matricial singularities. They agree along R, and thus on 7), since u%)—%, is the primary set of potentials with the boundary values ¢, Uy — Us = NV (uw — Us) = INfor, Us =u — Nop =fr:—-N¢=f- $=. Hence uw, us agree along #,’ and J, and have the same singularities. They must then agree throughout the space between the two circles described with common centre Mr DIXON, ON A CLASS OF MATRICES OF INFINITE ORDER, erc. 225 0, since they are uniform in this space. Hence they must agree throughout S. Thus a set of potentials is found for the whole surface S having the given ordinary and matricial singularities. Some conditions are necessary for the success of the method; these will now be discussed. § 37. In the first place the assigned matrices must satisfy a condition as was said above (§ 32). It is advisable to consider separately the homogeneous matrices M, and the absolute terms in M, which may be denoted by M0; I shall call the latter the addenda. Functions or potentials for which the matrices M, are the same will be said to form a family and the different members of a family will be distinguished in part at least by their singularities and addenda. The boundary J consists of say 4p+27 pieces; among the first 4p sides the (4¢+1)th and (47+3)th are opposite sides of one cut, so are the (4¢+2)th and (4¢+4)th, and among the last 2r the (49+ 22+1)th and (49+ 27 +2)th, 7 being any admissible integer. Let A, A,“ denote M, M, for a circuit equivalent to the (4¢—2)th side (i=1, 2...p), Be, B® » ” ”» (40 as, 1)th » ” oo, ¢,0 - ; from the (49+2i—1)th to the same point on the (4p + 22)th (Gal, 24 S00 79): Then we must have for any family Y Y p)— _ _ = CG" Coe ... Go A,” 1B) Ay?) Bow... A,® 1B, A,® Boe =1. This is the only condition to be satisfied by the homogeneous matrices and thus all may be chosen arbitrarily except one, say C,; this is then determinate. If there are no matrices C,, or as we may say no local matricial singularities, the question is rather more complicated. We may then write the condition A,'?) B,?) A,” = B,® A, BY A,® B® ... Ap? . By). Suppose 4,”, B,” ((=1, 2...p—1) to have been arbitrarily chosen and let H, stand for B,! A, Bw A,” ... A,2™ so that A,B,” A,” = £,B,®. The multipliers of A,'?)7B,'”) A, are the same as those of B,'?. Hence B,'”) must be so chosen that its multipliers are the same as those of #,B,'”). The determinant of B,'?}—@ must then be identically equal to that of #,B,”)—@ or of B,”—-0F 7 since the determinant of £, is clearly unity. The order of the determinant being m we have here m+1 equations to be satisfied two of which are identities, namely those which arise from the term in 6” and the absolute term. The other m—1 are linear in the constituents of any one row or column in B,"”), Thus we may take arbitrarily the constituents in all the rows of B,'”) but one, and one even in that row, the others being then determinate. Let the multipliers be @,, 0,...0m. Then we may suppose B,'?) in its canonical form to be a,=0;0; (¢=1, 2...'m); 226 Mr DIXON, ON A CLASS OF MATRICES OF INFINITE ORDER, ete. and ZB, in its canonical form X;/=6;X; ((=1,2...m). Then A,'?) must be X=)j,2; ((=1,2...m), where 4,... %» are arbitrary quantities. This investigation will need to. be modified if the multipliers of B,'”) are not all simple and of the first order, but in any case the number of arbitrary quantities is the same, since there are 2pm? coefficients in the matrices A,, B, and these must satisfy m? conditions of which one is illusory since the determinant of B,” A,“ B,%— A,” B,® ... Ay'”) is necessarily unity. § 38. Let us now consider the addenda, of which there are m(2p+r). These are restricted by m linear conditions in any given family. For the transformation OM Or) ... CV AMB® Aly) Bw... AVABO AW Bo. is that which corresponds to a passage round the whole boundary 7. The homogeneous part of this transformation is unity; the m addenda are determined by the local logarith- mic singularities, which are assigned beforehand. Thus there are m conditions to be satisfied by m(2p+7r) quantities and therefore only m(2p+r—1) of them are arbitrary. This number is effectively reduced by m more, since in any family all the functions may be constants. There being m functions, each of which may have any constant value, we have m independent sets of addenda arising in this way and so for functions properly so called there are only m(2p+r— 2). A family in which these reductions are not made is that in which M, is unity throughout, in fact that of the Abelian integrals and functions. In this case the con- ditions ordinarily connecting the addenda with the coefficients of the logarithmic singu- larities only involve the latter so that the addenda are unrestricted; also when the functions are constants the addenda are all zero. To find whether the reductions must be made in all other families we may first find the relations connecting 170 and M0. Now 0=M—, M0=M,". M0 + M0. Hence M0=—-—M,7 M0. Thus C0” Or> ...0% AM BH A BH ,,, A” BY0 : es Gm ¢,">) md Cc," Or—+) 0 a s O06,» 0. CO, Ay?) B,'?) A,'?) Bias A,@4 (B® = 1) A” 0 ~. P . 5 . r +50," 0")... 0,0 A,?... 49 (1 — B,” A, By) Bo 0. This stands for m expressions no linear combination of which can vanish for arbitrary values of AO, BO, CO except when there are no matrices C, and when as applied to a certain expression By = eA) sill (ley cs) = That is 1 must be a multiplier for each of the matrices and the expression multiplied by this multiplier must be the same throughout. If this multiplier is of the first order throughout, the family may be divided into two independent families, Mr DIXON, ON A CLASS OF MATRICES OF INFINITE ORDER, etc. 227 one the Abelian family of single functions, the other a family of sets of m—1 functions. If the multiplier 1 is not of the first order throughout, such a division will not generally be possible. Let s be the number of linear combinations of functions of a set which are Abelian, that is which are unaltered by all the operations M,. Then the number of conditions to be satisfied by the addenda, when there are no matrices C, and no local singularities, is reduced by s. Whether there are matrices (, or not, the same reduction must be made in the number of sets of addenda which correspond to constant values of the functions. Thus we may write as the effective number of independent sets of addenda m(2p+r—2)+s+s, where s’=s when there are no matrices C,, but is zero when there are. § 39. There are two cases of the process of combination of areas in the method; one is the change from R’ to S’, the other the combination of S’ and Z. In each case the solution depends on finding ¢ from an equation of the form ¢—Nd=f where WN isan operation like © of § 23. If 1 isa multipher for WV in either case one or more conditions will have to be satisfied. This may be avoided in the change from R’ to S’, for we shall now prove that if the circles R, are made great enough all the multipliers will be less than 1 in absolute value. Let ~ be a real positive quantity such that when M, is applied to any set of quantities not numerically greater than 1 the resulting set cannot be greater than pw. Let qg be the greatest value on R, of a potential for A’, without singularity, which is equal to 1 along R, and to 0 along Ry’. Then the greatest numerical value of N’$ does not exceed qu times that of and so if qu<1 the multipliers of N’ are all less than 1 numerically and the solution of the equation ¢—N'¢=f is g=f+N'f+ N°f+... to infinity. We are to shew then that g can be made <1/p. Now let R be conformally represented on the unit-circle. Let K,’, Ky, K,) be the curves corresponding to R,, R,', R,’. Then Ky’ consists of parts of the circumference, K, of ares with both ends on the circumference, K,/ of arcs wholly within the circum- ference. Let a concentric circle of radius p be drawn so as to enclose the ares Ky’. We may treat p as a constant if we suppose the radii of the circles R,/ not to fall below certain quantities. Now of two potentials without singularity, the one of which is unity along K,’ and zero along K,’, the other unity along Ky’ and zero along the rest of the circumference, the latter will have the greater value at any point of K,. But the greatest possible value of the latter, within or on the concentric circle of radius p, is 2 iota ( Ota 2", T l—p 4 * This is the value at the point whose polar coordinates this potential has the value unity, its value at any point are p, 0 of the potential which is equal tol on the circum- P is ference between the vectorial angles +4w and to 0 over 1 the rest of the cireumference, If AB is the are over which Qa (22 APB-w). 228 Mr DIXON, ON A CLASS OF MATRICES OF INFINITE ORDER, Etc. where is the whole angle subtended at the centre by K,’. This value can be made less than 1/u by taking a small enough value for @, that is, by increasing the circles Rj. We need not suppose the curves #, necessarily to be circular on the original surface; it will be enough if each is circular in some conformal representation and thus the possibility of their cutting each other before they are great enough for our purpose may be avoided*. Hence it is possible to take the curves R,’ in such a way that qu<1 and that therefore the construction given for S’ shall apply unconditionally. § 40. The nature of the solution for S’, when there are no addenda or ordinary singularities, can be discussed as in § 26 by help of a set of potentials X vanishing along the boundary, having addenda zero and no ordinary local singularity except a logarithmic infinity at a particular point c, and belonging to the family contragredient to the given one, that is the family in which M, is replaced by M,~. Let 8 denote the coefticients of the logarithm so that, near c, the functions X—Blog|z—c| have no singularity. The expression SAdu is uniform on 8S’, for it is changed into TM 2. Mu or MM, 2r.du. If the addenda of uw are also zero, then Suwdd is also uniform. Hence we have the equation i(enen: du Lp ip eee = YuV2r~ — SrV” ra [is a 7a ds I {SuV2rA — TAVu} dady, for the area of S’, less a small area containing c. Thus 27>8u(c)= | Sudy, taken round the boundary of S’, ~ being the potential con- jugate to A». Since the quantities @ are arbitrary this gives an expression for each of the potentials w at c, in terms of the boundary values. When the integral is taken with respect to the are of the boundary the subject is sult, and 2 is an analytical function of s since the boundary curves are analytical. Writing in the form =A;4;“ we have ry Qarui(c) = | Suidyy', taken round the boundary of S’. Again let o denote a set of potentials cogredient with w having zero addenda and vanishing along the boundary and with no additional singularity except that near the point Z of the boundary they become infinite like , y being a set of real z—Zds constants. Then it is found as before (§ 27) that / On do © — = Ss —— — 2m Bo (c) = Rice Ne ) ds, * For instance we may fix all the circles R,’ but one, say that about O, and take the conformal representation of the whole surface, with these circles cut out and other suitable cuts drawn, upon a circle whose centre shall corre- spond with O. Then a concentric circle of suitable radius can be drawn within this one and the curve it represents taken to complete the set Ry’. Mr DIXON, ON A CLASS OF MATRICES OF INFINITE ORDER, eve. 229 A being a small are drawn to exclude the point Z By the method of § 27 this is : i) : aie 4 : again found to be 2aSy(“% and so we have 27wi(c)= | Su'o/(c)ds, where o/(c) is 5 yi a5 : z=Z the real part of an analytical function of c¢, and is such that o = Xy;0;". i § 41. To shew the dependence of co on two points we may write it o(c, Z) or a(c, s) It is not a periodic function of s or a uniform function of c, nor are the assigned values of u periodic in s. For the sake of convenience in applying the results of the infinite matrix theory it will be well to change the form so that the functions involved shall be periodic. This is substantially what was done in the solution of the problem for the circle (§ 34). For the circle round O no change need be made as o and w are both periodic on this circle. On each of the other circles included in R, a transformation of the assigned values must be made according to the corresponding homogeneous matrix. Thus the operation WV that occurs (§ 36) in the combination of the areas S’, 7’, is of the same type as © (§ 23); all the functions involved are analytical and may be taken as periodic after the transformation just referred to; the range is the period in each case. Hence the solution is possible unconditionally unless 1 is a multiplier for 1. Suppose for the present that this is not so, and consider functions and potentials of the first kind, that is, without local singularities other than those common to all matricial integrals of the family. The family of Abelian integrals cannot be included since 1 is not a multiplier for V. Thus s, s’ vanish and the number of independent sets of addenda is effectively m(2p+r—2) (§ 38). The number of sets of potentials of the family of the first kind is greater than this on account of the arbitrary constants that enter into the solution of the problem for a circle (§ 35). In the case of a single circle the complementary function contains an arbitrary coefficient for each multiplier and so in %, %, fz (§ 36) there are complementary functions involving mr arbitrary coefficients. The number of sets of potentials of the first kind is thus increased to 2m(p+r—1). These may be arranged in m(p+r—1) conjugate pairs, and thus the number of monogenic functions of the family of the first kind is m(p+r—1). § 42. Now let 1 be a multiplier for NV. Then the equation ¢—N¢=f—/, will not be soluble for ¢@ unless a certain condition is satisfied by f—jf;. This condition will be a linear homogeneous relation among the addenda, the coefficients of the local ordinary singularities and those of the complementary functions in f,. The condition cannot have anything to do with the assigned values f, since a change of these into, say, f+ would only cause f, to become 7,+ Vy and ¢, ¢+7. In a potential of the first kind there are m(2p+r—2)+s+s' addenda and mr coefficients of complementary functions, that is 2m(p+r—1)+s+s' constants in all. These are now restricted by a condition, but on the other hand when the condition is satisfied there is a new arbitrary coefficient in the solution of the equation for ¢. Thus the number of arbitrary constants is undiminished on the whole in general. The case Wor, JID, 1B UE 30 230 Mr DIXON, ON A CLASS OF MATRICES OF INFINITE ORDER, Ete. when the Abelian family is included needs special examination. The multiplier 1 will certainly occur s times but the corresponding arbitrary coefficients in ¢ and in the final result are only additive constants, and do not therefore compensate for the loss due to the s conditions that have to be satisfied. On the other hand when there are no local matricial singularities these s conditions do not affect the addenda but only the coefficients of the local logarithmic singularities and so are illusory when the potentials are of the first kind. The number of independent sets of potentials of the first kind is thus 2m(p+r—1)+2s' at least, and of sets of monogenic functions of the first kind m(p+r—1)+s’ at least. In general there will be mr sets of potentials of the first kind with zero addenda; these arise from the complementary functions and when 1 is a multiplier for NV there may be more of such sets. It does not follow however that there are monogenic func- tions of the first kind with zero addenda, since the addenda of the conjugate potentials would then have to vanish also. This is known to happen sometimes with the factorial functions and it may happen in the more general case also, but it is not ensured by the mere existence of potentials of the first kind with zero addenda. Families with Complex Matrices. § 43. So far all the homogeneous matrices involyed have been real. Let them now be complex and be denoted in general by M,+cM,. Then the monogenic functions w are to be turned into (+ cM')w by passage round a circuit, and the functions ww will become (—M’+cM)w so that the 2m functions w, cw are transformed by real matrices with complex addenda and belong in fact to the family characterized by the matrices ( M,, M,’). Let F, G, H be the families characterized by the matrices M,+ cM, |- My, M,. M,—.M,/,( M,, My). Let w, w' be a system of monogenic functions of the family H |— M,’, M, of the first kind, found by the foregoing methods, so that w, w’ become after passage round a circuit Mw+Mw’ and —M’w+ Mw’ respectively. Then w’, —w form another set of the same family and these two sets give the combinations w+ ww’, w’ —w and w—w, w'+w, also belonging to H. The set w—ww’ belongs to the family F and w+ww’ to the family G, but unless the sets w, —w and w, w’ are linearly independent one of these will vanish identically. Hence the number of sets in F, G together is exactly equal to the number of linearly independent sets in H, which is 2m(p+7r—1)+ 2s’ at least, since 2m, 2s’ take the place of m, s’. There is not yet anythig to shew how this number is divided between the families F, G, but it will be proved that each has m(p+r—1)+s’ functions of the first kind at least. The difficulty does not arise when there is a local ordinary singularity; a function having such cannot vanish identically. Hence matricial integrals, not of the first kind, exist in both families #7, G; on differentiation these yield matricial functions. Mr DIXON, ON A CLASS OF MATRICES OF INFINITE ORDER, etc. 231 § 44. Take any m+1 sets of matricial functions w,, W....Wm, belonging to the same family and having only algebraic singularities in addition to those that characterize the family. The m-+1 determinants | GUAM: tod oes te | Ci Ui ancas Wms ae aecsents | TO, WE Sena Wren || are factorial functions on the surface and their ratios are uniform functions. Thus Wy), We +.» Wms, are connected by a linear relation whose coefficients are uniform functions. In particular the first m+ 1 derivatives of any set of matricial integrals are connected by such a relation, and thus the first derivatives, which may in fact be any set of matricial functions with only algebraic additional singularities, satisfy a linear differential equation of the mth order whose coefficients are uniform functions on the surface. If only m sets w,,wW.... Wm are taken these will generally not be connected by a linear relation with uniform coefficients. For if such were always the case a set of matricial functions would satisfy a linear differential equation of order m—1. The group of such an equation would consist of matrices of order m—1, whereas the matrices of the given family were supposed of order m. The first minors of the determinant of the m sets w,, w.... Wm, divided by the determinant itself will form m sets belonging to the contragredient family, and if Ui aE nag UNE. We, See WS are sets of matricial functions belonging to contragredient families {Ww will be a uniform function. § 45. We can now estimate the number of matricial integrals of the first kind in any family, whether the homogeneous matrices are real or complex. Take m particular sets of matricial functions W,, W.... Wm belonging to the contragredient family and not connected by a linear relation with uniform coefficients. Then if w stands for any set of functions in the given family }W,w, XW.w..., say X,, X,..., are m uniform functions: when these functions are known the functions w are known and conversely. Take w to be the differential coefficient, with respect to an Abelian integral @ of the first kind, of a set of matricial integrals having one additional singularity, namely a logarithmic infinity at c. Then w has 2p+r—1 poles, namely c and the zeros of do, and the matricial singular points (see § 34 n.). The uniform function {Ww may have these poles in addition to those of W, that is k+2p+r—1 poles in all, if « is the number of poles of W. Hence in >Ww we have «+p+r constants at disposal, this number including residues and an additive constant. The m uniform functions X thus involve m(p+r)+« constants. Now when the m equations {_Ww=X are solved for w the determinant of W,, W,... will appear as a common denominator and so the zeros of this determinant will be poles of the resulting values of w unless the functions X are restricted. 30—2 232 Mr DIXON, ON A CLASS OF MATRICES OF INFINITE ORDER, etc. The determinant is a factorial function having =« poles and hence x is also the number of its zeros. At any one of these zeros the equations {Ww=X are incon- sistent, that is, they give infinite values for w, unless the values of X satisfy a single linear homogeneous relation. The number of arbitrary coefficients in X is thus reduced by S« at most and we find m(p+r) as the least number of arbitrary coefficients in w, that is in a set of matricial integrals having one additional logarithmic singularity. Now in the general case the m coefficients of (z—c)* in w are arbitrary and thus there are at least m(p+r-—1) sets of matricial integrals of the first kind. This is the same number as was found above (§ 41) for a family with real matrices. We also found 2m(p+7r—1) as the sum, when s=0, of the numbers in two families with conjugate complex matrices (§ 43). Hence each must have exactly m(p+r—1) sets of integrals of the first kind. § 46. There are exceptional cases in which the number of sets of integrals of the first kind is greater than m(p+r-—1); in these 1 must be a multiplier for V in § 36. (1) Suppose the Abelian family to be included s times in the contragredient family. Then we may take W,, W,... W, as constants. Hence in the original family s linear com- binations in a set of matricial functions are uniform, that is, the original family includes the Abelian s times also. Now if r=0, X,, X,... X, have as their only poles ¢ and the 2p—2 zeros of dw. Thus the residues at c must vanish and only m—s’ of the coefficients of (—c)* im w are therefore arbitrary. Hence there are m(p+r—1)+s"’ sets of matricial integrals of the first kind. (2) More generally suppose the contragredient family to include a set of matricial functions W, without poles. Then, if r=0, the residue of X, at c¢ must again vanish, and the original family will have one extra set of matricial integrals of the first kind for every set of matricial functions without poles in the contragredient family. (3) Suppose it possible to take W,, W.... Wm all without poles. Then by giving constant values to X,, X,... Xm we get m sets of functions without poles in the original family *. In either family when r=0 there are mp sets of matricial integrals of the first kind and matricial integrals with a single additional logarithmic singularity cannot exist. If more than m sets of functions in the family are finite everywhere, let them be denoted by w,, W2...Wmii. The determinants Win Westra * If W,, We... Wm, can be taken to have no poles, and if m—kp is positive, it can be proved in like manner that m—kp sets in the original family have no poles. Mr DIXON, ON A CLASS OF MATRICES OF INFINITE ORDER, ere. 233 are then factorial functions, finite everywhere, so that their ratios are constants. Hence the m+1 sets are not linearly independent, that is, there cannot be more than m sets of matricial functions everywhere finite. § 47. When p=1, logow and wu atford an example of a pair of matricial integrals, u being the elliptic argument and cu Weierstrass’ c-function. This is a case in which the family meludes the Abelian family. In like manner the logarithm of a factor- function, as defined at Proc. Lond. Math. Soc., Vol. xxxm. p. 11, with the Abelian integrals of the first kind or some of them, forms a set of matricial integrals, whatever the value of p. An account of factorial functions is given in Mr Baker’s Abelian Functions. Matricial functions are the subject of M. Poincaré’s memoir (Acta Math. v.) and are also referred to at the end of M. Appell’s memoir on Factorial Integrals (Acta Math. x111.). The functions are of great importance in the solution of homogeneous linear ordinary differential equations with algebraic coefficients. The first steps in the solution, after the Riemann surface on which the coefficients are uniform has been constructed, are to find the critical points of the equation and its group, that is the group of matrices by which the integrals are transformed when difterent closed circuits are described on the surface. Then the family of matricial functions to which the integrals belong is known and we may construct m particular sets of functions (m being the order of the equation) belonging to the family, and such that all other sets belonging to the same family can be expressed linearly in terms of these m, the coefficients being uniform functions on the surface. These uniform functions are then to be identified by their singularities. If all the critical points are regular these singularities must be accidental. If there are irregular critical points, the integrals have essential singularities, but these are now reduced to essential singularities of functions uniform on the surface, combined with the singularities which characterize matricial functions. M. Poincaré gives an analytical expression for zetafuchsian or matricial functions from which it is of course possible to argue their existence. This argument is however limited in its application by the restriction (Acta Math. v. p. 258) that the group is to be of the first species, or in the notation of the present paper that the multipliers of the matrices C, are all to be equal in absolute value to 1. An advantage of the method of this paper is that this limitation is not needed. XI. Seminvariants of Systems of Binary Quantics, the order of each quantic being infinite. By Major P. A. MacManon, D.Sc. F.R.S., Hon. Mem. Camb. Phil. Soc. [Received 8 November 1901.] Introduction. Ir we take a single quantic of infinite order = n\ _ n\ — a, = (GQ), + AsLy)" = Ax} + (;) G23 122+ (3) Tire TS oc the seminvariants, as is well known, satisfy the linear partial differential equation G0, + 20,07, + 34207, +... =0. If we write s!@, for @, the differential equation becomes Og =40¢, +02, + 0c, +... =0 and the solutions are now connected with the symmetric functions of the roots of the equation a” — 2" + Ga" — ... =(“£—a,) (4 — a.) (w@— ay)... = 0. The whole of the linearly independent or asyzygetic solutions for Q,=0 are in fact given by the expressions Daytantsa,’s ... where t,, &, t;, ... are any positive integers except unity. It follows from this that the enumerating function for the asyzygetic forms of degree @ is 1 G7) =f) =)) ae): Moreover it is convenient to denote 2a rae DY (tikats <>), and now (t,ft;...) is any collection of non-unitary integers and any such collection, account not being taken of the order of the numbers in the collection, represents one Mason MACMAHON, SEMINVARIANTS OF SYSTEMS OF BINARY QUANTICS. 235 of the series of asyzygetic seminvariants. From these asyzygetic forms it is possible to select a certain number which possess the property that every seminvariant is expressible in terms of them as a rational integral function. Such forms are termed perpetuants and it was established by the present author and by Stroh that these forms of degree @ are enumerated by the generating function ge ia d—2)(—2)...(—2)’ and by the present author that the collection of integers (GELS) which denotes a perpetuant of degree @ is the number @ occurring at least once . a 6-1 a . once - : 6-2 re SS twice FP a 6-3 Bs = 2° times % a 6—4 a 5 2? times . 4 3 A i: 2°-4 times ; 3 2 3 any number of times; so that, for example, a perpetuant of degree 6 is any collection of non-unitary integers, 6 being the greatest integer, which involves the collection (654233), The object of the present paper is to generalize these known results to the cases of systems of two or more binary quantics of infinite order. § 1. It is first necessary to establish the generating functions and forms of the asyzygetic seminvariants. In the case of two binary quantics a0 ot — 10) we are concerned with the partial differential equation Og + Og = GOz, + H0¢, + G0z, + «.- + bd5, + 6,05, + body, +... =<()) and the symmetric functions of the roots of the two equations a” — ax" + an" — ... = (“4 —%) (vw — &) (@— a)... = 0, a” — b, a" + b, a” — ... = (x— B;) (x@— B.)(a@—f;)...=9, when n=. These symmetric functions will be of degree 6, in the coefficients of the first equation and of degree @, in the coefficients of the second equation. Supposing 0, > 0@, 236 Mason MACMAHON, SEMINVARIANTS OF SYSTEMS OF BINARY QUANTICS, we know from the researches of Cayley that the enumerating generating function of these forms is 1 {i —z)(1—#)... —)} {(Q—2) 0-2)... —2)}’ where the second factor of the denominator is to be omitted when @, <2. Let ©, with or without a suffix, denote any collection of positive integers, subject to the conditions that no integer is >@ or <2 and further let (©),, (©), denote the corresponding symmetric functions of the roots of the two equations above written respectively. If the highest integer in © be s where @>=s, then ay * (O)a will denote a seminvariant of degree @ of the first equation. Noting the well known results o,(@).=0; OFC); —Crs>; we can construct a solution of the equation OF =F QO, = (0). viz. :— (®,)q (Q21*), — (O1)a (O14 )p + --. + (O11%)a (On) of degrees 6,, 6, in the two sets of coefficients; for, operating with 0,+ 0), we obtain (9;)a (O15 )p T= (O,1)a (0.1%), + eee —(O,)q (O.1*), + (O11 )a (0.1), — -.., or zero, since the terms destroy one another in pairs. The solution obtained represents the whole number of asyzygetic solutions which are of degrees @,, @, in the two sets of coefficients. Firstly I say that assuming the forms to be asyzygetic they are obviously enumerated by the generating function of Cayley above given, and secondly I say that the forms are in fact asyzygetic; for suppose, to the contrary, that a number of such forms (®,)a (Q21*), — (OL )a (O.1*)y + (Oya (9,'1"), (Oy1)a (0,'1"~), + eee (0,)a (O."1*")y — (O11) a (921%) + --- ere ere eee eee eee eee eee eee eee eee eee eee eee ee ee ey are connected by a linear relation. It follows that the leading products of each must be of the same weight and degree as regards both the a and the 6 coefficients and that therefore the supposed linear relation must connect the leading products. If then «, x’, «”,... be in descending order of magnitude and we operate a sufficient number of times, upon the linear relation connecting the said leading products, with Q, we must obtain a linear relation connecting a certain number of the forms (O,)a (9.)s, (Oya (O,’)y, tee but this is a linear relation connecting the forms (@,)a; (@,')a; weey THE ORDER OF EACH QUANTIC BEING INFINITE. 237 which we know, from the theory of a single quartic, does not exist, Hence all the forms involved in the constructed solution are asyzygetic and we may take (®1)a (@21")5 — (O,1’)q (O21), + ... + (O11) g (©.), to be the general type of asyzygetic seminvariant, of degrees 6,, 6, in the two sets of coefficients, of the system of two quantics. It is convenient to denote the general seminvariant obtained by F(a, 0; b, x), so that (®,)a = F(a, 0); (O,1"), = F(a, «,), and so on. I also write F (a, Ky b, Kq)'= (O,1"), (0.1%), = ie i ") (O14), (9,1), Ki + ky Ko ) (@18+8), (@,),. ot This expression has been constructed so as to lead to the relation (Qa + 0%) F (a, 13 b, x) = F(a, e,—1 SDKs): for we now find that it is easy to construct a solution of the equation Og + O,+ 02, = 0, appertaining to the seminvariants of a system of three quantics. The form is Hi (a, 0) E'(b, ka; 6, 3) — F (a, 1) F(b, k—1; ©, «) + F(a, 2) F' (6, 2-2; 6, es) —..., which I write F(a, 070s ca C), es)! It is manifestly obliterated by the operation O,+ OF + 0, since, after operation, the terms destroy one another in pairs. The leading term of. this seminyariant is (O,)a (9,1"), (O,1*), of partial degrees 6,, 02, 03. Considerations similar to those brought forward in the case of two quantics establish that these forms are asyzygetic and they are clearly enumerated by Cayley’s generating function 1 {(a-2)Q =)... —2)j (a —2)0-4)...0-#) 1a —2)(1—2)... 0-2)" It is convenient to have the leading term of the seminvariant before us, so that I write F (a, 0) = | (®:)a| = (Oy)a F(a, 0; b, #2) =|(O,)a (Oq1"), | F(a, 0; b, k2; ¢, «3)=|(@,)a (©,1*), (@,1*s),. and so on. Vou. XIX. Parr II. 31 238 Mason MACMAHON, SEMINVARIANTS OF SYSTEMS OF BINARY QUANTICS, And further it is convenient sometimes to write similar expressions which are not seminvariants in a similar notation. Proceeding to the system of four quantics and the solutions of the differential equation 0,+ 04+ 0,4+ 2,=090, we first construct the form PUG 103: 0, Kas) C) Ks) = MONA) AOS Oh 23) = (* +1 1 ) F(a, Kk, +1) F(b, w.—1; ©, ks) 9 + +2) P(a, +2) Fb, Ko — 23 C, Ks) =(* x s) F(a, + «:) Fb, 03 ©, 15); and observe that (Q74- OF ©. IF (Gericke sn Ce ies) 21s ( One — hg Ost Gs): Thence as before we find that the general asyzygetic solution of O,+ + O,+ N4=0 is JE (W)) O(a Cae Ch ON) —F(a,1) F(b, 2.—1; ¢, «3; d, #3) + F(a, 2) F(b, x.— 23 ¢, «3; d, Ks) WB (Gs vice) st (050s ewecais °@,) 14); which we write F(a, 0; b, ko; ¢, #33 d, Xs) =| (Ox)a (Ool™)y (Os1*)- (O41 )a |. All these forms are asyzygetic and manifestly enumerated by the generating function of Cayley. We are now in a position to proceed to a system of n quantics and the solutions of the equation Og + Ny + 1, +... + On = 9. For all integral values of x we write TGR AE URNS Oy 53 Soe iy han) - F (a, k) TON (GRAB (Hs KAR onan an) Es (Fa *) F(a, ey VE (O Ne — Chik) els (Kp) a: Gi ae K, + 2) F'(b, to — 23 C, 33 -.. N, Kn) Ky + Ky Ke )F@ fee ea NE DE wea <<<. make) THE ORDER OF EACH QUANTIC BEING INFINITE. 239 and then the general asyzygetic seminvariant of partial degrees 6,, 6,, ... 0, is JEGEOS Oh 0a8 GQTES Soon ke) or | (O.)a (O.1%), (O,1's), ee (9,1*"), in agreement with the generating function of Cayley. § 2. The method adopted by Stroh for the investigation of the enumerating generat- ing functions for perpetuants may be presented in the following manner. Let pi, ps, ps, --- be a number of numerical magnitudes whose elementary symmetric A : a? Oe a,§ functions are a, a, a3, ... and denote a, symbolically by — or — or Ale where f i my aio s! ! M%, @, M,... are umbre. We may write 1l+a,r7+ aa*+...=(1+p,2)(1+ pov) (1 + p32)... = ENT = E%st — ast — Further let 14+ A,7+ A,av?+ ... +A," =(1+o,2)(1+ 022)... (1+ on). Substituting o,, o:, 7, ... Om» for # in succession in the former identity we obtain 1+aqo, +a.0) +...=(1+pi0,) (1 + poo) (1 + po.) ... =e", 1+ao, + a0. +...=(1+ pio.) (1 + poor) (1 + psn) ... = €%%, 1+ Om + QOm +... = dd a PiFm) d ar P2Fm) a se PsOm) oo. = ETI ¢ and by multiplication T=om P=Py I] (l+a,o+a.o0?+...)= Il (1+A,p+ A.p?+...+ Amp”) o=0c, P=P = em +o ,a,+...+o7mam . If the product Il(1+ Ajp+ A.p?+...+ Anp™) p be multiplied out we find that the coefficient of any product Aly At Aiea is that symmetric function of the quantities p,, ps, p3,... which is denoted by (pqr-».); hence 1f we expand EMA FF 209+... Fomam by the exponential theorem and express the symmetric functions of the quantities Oy, O,... Gm mM terms of Ay, A,,... Am we shall obtain, as coefficient of any product A,A,A,..., the symmetric function (pqr...) of the quantities p,, pz, p3,... expressed in symbolic form. To obtain particular theorems the quantities o,, o:,... om are at disposal. 3) 240 Mason MACMAHON, SEMINVARIANTS OF SYSTEMS OF BINARY QUANTICS, If we wish to restrict consideration to non-unitary symmetric functions we have merely to suppose O, + G2+... + om=), for then A,=0 and only non-unitary symmetric functions of py, ps, ps, -.. cam present them- selves. In this case we have seminvariants expressed as functions of differences of the symbols a, Qs, Gs, --.- Ex. gr. erede (2)+ A. (2?) + oo, = emit osme = gt (4, —22) Le sate: 4 4 = Al sera) re at — MP +45 % (a,—@)i+... where Ae 1070 — one Hence ()=-5,(4- a) 2) — 1 4 (2 ) ars g(a = a>) and generally 1 2) =(—) (gy (a — 2 Ex. gr. If oj, o2, o3,-.. Gm be the m, mth roots of unity, the l= 5 Sl SO ama! sl I leading to ]— (m) + (m?*) = (m*) +..0= ENUF Tat --- EomIm Si ily : so that (—}* (m*) = (ms) ! (O10, + Golly + ... + Om%m)™. Numerous results of a similar nature may be obtained by considering special relations between oj, 2, ... Gm or by giving them as above special values. The case of seminvariants of a single quantic is obtained by impressing the condition O, t+ G.+ 2... + 5n=0. The general seminvariant of degree m and weight w is then represented by ul ail (71% + G2 + O3%3 +... + Fm&n)” 5 where, when m>w, we must multiply by aj”. Ex. gr. for degree-weight (4, 2) E Ay" 2 om (oy) + Fy + O30; + O40), > (Gr : =a? & Yo2+ aa.Sov0s) ; = de? {a.(— 2A.) + a,°A3}, =a, (2) As. THE ORDER OF EACH QUANTIC BEING INFINITE. 241 Stroh assumes that the seminvariani is irreducible unless it be expressible in the form 1 w! (P+ Q)" where the auxiliaries which occur in P satisfy the seminvariant condition; 1e. their sum is zero. This being so expansion by the binomial theorem exhibits the reduction. Stroh expresses the condition FP (oy, 62, 03; «+» Gm) =0 that the sum of fewer than m of the auxiliaries oj, o2, ... @» may vanish, and finds that the condition is of weight Qm-1 = 1 in the quantities A,, A;, ... Am. Hence the conclusion that the generating function of irreducible seminvariants, or so-called perpetuants, of degree m is ol at (-2)G-—2)...q-2)' (m > 2.) The present author further established that the function of A, A;,...A,, that vanishes in consequence of the condition of reducibility contains a term Aen Aen Aaa Are and that therefore the associated seminvariant which contains in its partition representation 1 number equal to m 1 . _ m—1 2 numbers _,, m— 2 22 x - m—3 ” ” Qm—4 ‘ Bw 3 is for m>4 the simplest exemplar form of perpetuant of degree m. The forms for m=2 and m=83 are (2) and (3) respectively. The general form of perpetuant of degree m is thus ¢ git=4 .3 where Kin, Km—1, Km—2) ++» Ks, Ko are arbitrary positive integers. Gn m2 +3 DK) I seek to obtain the corresponding results in the case of perpetuants of a system of n quantiecs. The Strohian form of simultaneous seminvariant of two binary quantics is I 1 , , ; a (G10, + O00 +... + O4,%, + 01/8: + Or Bo + ... +o, Bo,)”, 242 Mason MACMAHON, SEMINVARIANTS OF SYSTEMS OF BINARY QUANTICS, the weight being w and the partial degrees 6,, @., where Oy t+ o.+... +09, +0) to.+...4+ a =0, and 8,, 82, 8s, .-. constitute a second set of umbre associated with the second quantic in such manner that Bt_ At 8t a, Sea sls si To see how this arises write l+ae+qet+...=(1+pi2)(1+ pz)... =er*=e7=... 14+),¢4+b.0°+...=(1+ x) (1+p.xr)...=87=87%=... 1+ A,v+ A,a+ ...+ Ae 2 =(1L+o,2)(1 +22) ...(1+ 2,7) 14+ Bc + Bia? +... + Bo,v® = (1 +0)2)(1+ 0,2)... (1+ 09,2), identities which lead to TI 11 (1+ Ap + Asp? +... + Ag,p%) (1 + Bip’ + Bop? + ... + By,p™) p p' = Mitt Foy +... +9.A9, + 61/8, +02B.+... +09 Bg To find the condition that this function may be a seminvariant observe that the sinister may be written {3 A,A,A,... Ay (par ... 1%)q} {2 By By By ... By (p'qr’ ... 1%)o} and that operation with Q,+ 9, gives {SA,AqAy... Ar (pgr ... 187)g} (2 By By By ... Be (p'q'r’ -.. 1%)s} +{2A,A,A,... A® (par... 1%)q} {2 By By By ... By (p'q'r’ ... 1=™)s}, an expression of which A,+ B, is a factor. Hence the seminvariant condition is A, +B, =O, tOn+...+09, +0) +o. +...+06,/ =0, as was to be proved *. Before proceeding to the general theory of the expression of the perpetuant forms we can observe at once that there is no difficulty connected with the enumerating generating function of perpetuants of any given partial degrees in the coefficients of any number of quantics. The investigation of Stroh is sufficient to shew that if @ be the whole degree of a system of perpetuants of partial degrees @,, @., 03,... the generating function is part Ort Ost. -1 4 (1—z)(1—z)...1—2%)} (A —2)(1— 2)... 1-2}... {1 —2)(1— 2)... 1—2=)} (1 2") (124). .(1—2%)} a fraction which if its numerator were unity would enumerate the asyzygetic forms. * This theorem is obviously generalizable to the case of three or more quantics. THE ORDER OF EACH QUANTIC BEING INFINITE. 243 For the actual expression of perpetuants :— Take =i, (k=l. For reducibility either ¢, or oc,’ must be zero, Hence the condition of reducibility is oc, =0, where o,= A,, o,'= B, and 4,+B,=0. We must therefore have A,=0 and B,=0 so that no form is reducible in this manner; every form is a perpetuant. The generating function for the asyzygetic forms is 2 l-—z which yields the series 1 ayb, — a,b, Ab. — a,b, + ab, ybs — dbs + dab, — ash, &e. and is represented by the general formula | io (1*)p |, which should be compared with | (P1)a (9,1*), I, wherein (@,), has been put equal to a, and ©, is absent. Take (h=2, (ht For reducibility either o,, c, or a, must be zero. Hence 010.0, =0, or A,B, = A,A, = 0. All forms associated with products containing A,B, are consequently irreducible, The simplest of these is associated with A,B, itself and is of degree 2 in the a coefficients and of degree 1 in the b coefficients. We take it to be (2)a (1), aa (21), bo, or |(2)a (1) |- The general form of the perpetuants we may take to be | (at), (l=), |, a enumerated by d—-2z)a-2) és 244 Mason MACMAHON, SEMINVARIANTS OF SYSTEMS OF BINARY QUANTICS, Take 6,=2, @,= 2. The condition is O00; oo (a, + o)(o,+ a; ) (a, +o.)=0, or A2B,B,+ A.A,B2=0. This relation shews that there are two separately irreducible seminvariants of — weight 7 but that they are connected linearly with reducible forms and that thus we must regard only one form as irreducible for the weight 7. Selecting A,°B,B, as the base product we are led to the simplest form (2*)a (21), = (2*V)a (2)s, and to the general type (2842), (Qeet1 1X41), | corresponding to the generating function d@—20-2)" We may of course replace B, by A, and then corresponding to A,*A,B, we get the simplest form (2), (2*1)a a (21)p(2°)a, which is merely the former with changed sign in agreement with the identity ASB,B, + ASA,B, =0. So much for some simple cases; we may now consider the general case of any number of quantics and given partial degrees @,, 02, 03, . First consider Stroh’s method for one quantic and forms of degree 6: 1 sl (4%, + ole + O9hs + Tsay + T3A5 + Tots)”. The condition of reducibility is 010030 0505 (0, + G2) (0; + G3) (0, + 04) (G+ G5) (G1 + os) X (2+ 03) (G2 + 5) (G2 + Gs) (G2 + Oe) (Fs + O4) (G3 + 5) (Fs + Os) X (64+ 05) (0+ O56) (65 + Os) (0, + 2 + Gs) (, + 2+ G4) (a, + G2 +65) (0, + 72+ O%) (G) + Os +04) (G1 + O53 + G5) (OL + Ts + Gs) (a, + o,4+05) (a, + 0,+ 0%) (a, +05 +o) =((). In this one term is > o{oso$oso5o;5, associated with the product A,A,AJA¥. Thence arises of course the simplest sextant perpetuant (654239). THE ORDER OF EACH QUANTIC BRING INFINITE. 245 We may regard a, a2, 03, 04, 05, o aS belonging to different sets of auxiliaries (the number of sets being equal to the number of different quantics) in a number of ways equal to the number of partitions of 6. Thus we obtain Partition Symmetric Function (6) > ofososo3o; 05 (51) S o8o.{0,%o30,"|o5 (42) > o{o%0°o,4\0205 (41°) > ofe,2o;%o,'|0;7|o5 (3?) > o,'o.80;°|o 40205 (321) > o8080;°|o,'0;7|o6 (31°) > ofo,%o35| o,4|o;| a6 (2°) > o{o.8|o%o3|o205 (271?) > o80.3|08o,|o2|o¢ (214) > ofo.8|c5°|o.|o;2|o5 @ Lof|o,5\o5|o3|03| 05 The different sets of auxiliaries are to be connected with elementary symmetric functions denoted by letters A, B, C,... with suffixes. Thus for three quantics and seminvariants of degree (2, 2, 2) we associate the function o,c,'\o;50,'\0,0, with the lettered product ASBAB3C.C,, where observe 2%, 2414, 21 are the partitions conjugate to 8, 84, 21 respectively. We thus get the simplest seminvariant perpetuant of the type, viz. :-— |(2°)a (2414), (21)<|; and the general perpetuant derived from it by increasing each of the five exponents by arbitrary numbers. In the case of the partition (1°) we are concerned with seminvariants, of six quantics, which are linear in the coefficients of each quantic; we are led from a \o8\os\os\o7\o¢, to AYBYCSDALYF, where A,+B,+0,+ D,+ #,+ F,=0, and thence to BYCSDsEZF,, indicating that we may take the simplest perpetuant to be | dy (1**)y (1*)c (14)a (1?) (1 )y| - We can therefore form the following table of results for perpetuants of degree 6 in respect of one, two, three, four, five or six binary quantics. No. of Quanties Degree Simplest Perpetuant 1 6 (654°3*)q 2 5 I |(574234), (1)p| 2 4,2 |(434)q, (21),| 3 Apia |(4434)a (1?)p (1)e| 2 3,3 |(3*)q (321*)p| 3 Buel |(3°)a (271*)p )e| 4 Sy il, a5 1 |(38)a (14)5 (1?)e (1)al 3 2; 74) re, 1(2°)a (2414), (21),| 4 BOTT |(2°)a (2414), (1*)e (1)a!| 5 Deieat Tad |(28)a (18)o (14)e (12)a 1)e| 6 rigcl 3 on in | | Qy (1p (1*)e (14)a (1*)e (Ly| Vou. XIX. Part II. 32 246 Mason MACMAHON, SEMINVARIANTS OF SYSTEMS OF BINARY QUANTICS, The general perpetuants are formed by adding arbitrary numbers to the exponents which present themselves. As the general formulae to be presently given require modification in the case of the low degrees, it is convenient to give here the results for the degrees 2, 3, 4, and 5. No. of Quantics Degree Simplest Perpetuant 1 2 (2)a el |qto (1?),| 1 3 (3)a 2 2,1 \(2)a (1p) 3 Lb all ao (1*)p (1)e| 1 4 (43)a 2 34 (3*)a 1)o| 2 2, 2 |(2*)a (21), 3 2,1,1 (2?)a (1*)o 1)e| 4 ite wale |@o(1*)y (1%). (1 al 1 5 (543*)q 2 4,1 |(4°3*)q (1) ; 3,2 (35a (21) 3 3. le (3*)a (1) (1)e| 3 2, 2,1 |(2*)a (271*)y (1)e| 4 rl ha 1(2*)a (14), (1%) (1a 5 1,1,1,1,1 Go (1%), (1*)e (1*)a 1e| Of the degree 6=0,+6,+0;+...+6, the selected term of the o product which expresses the condition of reducibility is 26-3 56-4 96-8 (a, 0205) Gay etre. Cae o, AM Be yl S24 Wiig pone ES Mi \clejcinlalsiete cleinieia metnta aiaietacivieielelsiele b ar eee ra Te 3 Grin and the corresponding literal product 99-8, 499-8, 409—-@,41 99-4 AG AS Meee os 298-8, Jy8~8,—8, p99-8,- 8, +1 28—61-¥ Br. Dey Bae sone 99-8, — 8-83 7-99 )— 82-05 98 — 81— 0,—8s+1 98 -6,-8,-2 oy On tase Grae sO eee eee eee eee eee eee eee eee eee ee eee eee eee Ne We _, W2 8,-1 THE ORDER OF EACH QUANTIC BEING INFINITE. 247 Whence the corresponding simplest perpetuant of the type is O42 ae ee 0-4 \(@ °@-1 6-2 BACT aay 6-6-6, 0 —0y—8+1 (Cs ET 6, —2 Sh sleet) 29—0:—0.—05 20-0, —0,—0,+1 Osa} = a= 6—8,—0,—2 (CAS am 6, — 2 Seger ys PRO ewe e eee ee eee e ewer eases sees ee eee eee ee eess setter eensessseee 2 2 fs oe (CARA? SS: CEE Wore ia op This expression is valid as it stands when @, is not less than 4, 6,, 63, 0;, ... On not less than 2. If @, is equal to 3, the first term er % in the first bracket must be taken and not the last one 3204 Similarly if 6@,, @;,...0, be unity the first and not the last terms in the corre- sponding brackets must be taken. When @, is equal to 2 the first bracket is simply (Came? when 0>3. When 0@,, @,, 0;, ... Qn are each equal to unity the form is Veneta en lsraa) eee Cn : Ex. gr. The simplest perpetuant of four binary quantics of partial degrees (7, 4, 2, 1) is (f= 618 4256 4512 SHE) (48 38 236 1°), (2? 1*), ()a | of the total weight 2" —1=8191. The general problem of the perpetuants of binary quantics has therefore been com- pletely solved; they have been enumerated and every single perpetuant has been identified. It will be useful in conclusion to give a few illustrations of the calculation of the new symmetric functions that have been brought to light. Ex. gr. the perpetuant | ao (1?)5 (1)o | = | (1*)y (Le | — | (1)s 1c | + a2] Bo (1)e | = ay (bc, — 3b,6)) — ad, (010, — 2b.€)) + Ay (bye, — be). The five other seminvariants obtained by permutation of the letters a, b, ¢ are all reducible by means of the above form and the three compound forms dy (by €3 — b, C2 + by, — b3¢y) Bo (CAs — C12 + CoM, — C3 My) Cy (gb; — a,b + Ab, — a3b)). 32—2 248 Mason MACMAHON, SEMINVARIANTS OF SYSTEMS OF BINARY QUANTICS. The number of distinct terms in any form | (Oya (O21), (O51), ... (0, 1%), | is («2+ 1) («3 +1)... (kn +1), as may be easily gathered from its definition. We may therefore expect 30 terms in the expression of | a, (1*), (12).(1)a|. It is de | (1*)5(1%)e a | = as | (14)p (12)e La | + a | (12) (12). | — ds | (1)p (12). (L)a| + as | bo (1*)e (1a, wherein (1*), (1?) (1)a | = (199 |()-(Dal= (“TF ")A%| De Mal+ ("$ 7) Glo De! = Be(cat, = Beate) = (“T*) bess (Cid — Beads) + ("3 *) basacads — xd) Thence | a, (1*), (1). (1)a | = dy {by (Cod, — 3e,dy) — 5b; (c,d, — 2c,d,) + 15d, (od, — c,d,)} — a, {b; (c,d, — 3e;d,) — 4b, (e,d, — 2c.d,) + 10b; (ed, — e,dy)} + dz {b, (c,d, — 3¢,d,) — 3b; (¢,d, — 2c,d,) + 6b, (cod, — .d,)} — dz {b, (Cod, — 3c3d,) — 2b, (c;d, — 2czd,) + 3b (cod, — c,dy)} +a, {by (cod, — 8e,d,) — 6, (ed, — 22d.) + by (ed, — ce, d,)}. Ex. gr. Consider the form F =| (@,)a (O21*)p (31%). (O.1**)a |; we have 0,F= — | (@y)a (O21); (O31), (O,1™)a! Oy F = | (Qy)a (O.1* Yo (Os 1*)e (Os1)a | — | (Or)a (O21) (O21). (O41")a | O.F = | (@;)a (921%), (O31). (O,1**)a | — | (O1)a (O21) (Os1*)- (@s1™)a | Og F = | (O,)a (O21) (Os1%)e (O19 ™)a |, so that the separate operations produce seminvariants of the same type and the relation (Q¢+2+0,4+ Oa) F=0 is verified. Expressed as a sum F=> (-)P (0, 1?)q | (O,1*?), (O;1*), (O,1*)a P = SE (ra (SPF 4) 9,17), (O,19-P*2),| (15-2). (819)a Pg q = DED (jeter sé = te ") ( a + ") (O,1”), (O,1%-?+4), (Q,1#-2+"), (@,1*-7),, par in which the upper limits of the values of p, g, rT are Ky, Ks, «, respectively, so that the series involves (x,+1)(«,;+1)(«;+1) terms, as should be the case. If wa, Wy, We, Wa be the weights of @,, ©,, @,, @, respectively, the maximum weights of the seminvariant in the coefficients of the separate quantics are Wat Ko, Wy +KotKs, We+Ks +k, Wat Ks respectively. XII. On certain Quintic Surfaces which admit of Integrals of the First Kind of Total Differentials. By Arruur Berry, M.A., King’s College, Cambridge. [Received 4 January 1902.] CONTENTS. PAGE Sols eintroduction ep 2) : SMe aides SesCAN bate PM oct de AINE cet Wd Ghee L250 PART I. §§ 2— Preliminary classification of quintic surfaces . 5 - : A ¢ : 5 9 . 252 § 3. The case of a double quintic curve . : : : c : 4 ; . : : . 253 § 4. The case of a double quadriquadric . C . 4 3 : 3 5 j : - 254 § 5. The case of three or more intersecting double straight ines). 6 ; : j : . 257 § 6. The case of three intersecting double lines, two of which coincide : : : . - 263 § 7. The case of a curve of multiplicity greater than 2 5 0 ¢ : : : 5 - 264 PART II. §§ 8—11. QUINTICS WITH TWO LINEARLY INDEPENDENT INTEGRALS OF THE FIRST KIND, WHICH ARE FUNCTIONS OF ONE ANOTHER, § 8. Preliminary : : : ¢ : : 2 . C 0 : a 5 eel 9. Reduction of the differential equation . ‘ : F : 5 7 % 5 5 pail § q 10. Integration of the differential equation . 5 5 : i 3 5 4 5 5 a ae § g q § 11. Summary of Part IT. : 3 . o : : : . ; 6 : ‘ : - 282 PART III. §§ 12—18. THE CASE OF TWO FUNCTIONALLY INDEPENDENT INTEGRALS OF THE FIRST KIND. § 12. Preliminary 5 5 : : : : : : : : c . 283 § 13. Discussion of possible en laritics : ; d C : : 3 : , : . 284 § 14. The case of an ordinary conical double point . ‘ é : 0 é 3 é : . . 285 § 15. The case of a biplanar double point 3 : : : : Pie as 5 : yi 5 | AS 16. The case of a uniplanar double point. : j : : : : E : é 5 PASy/ 17. The case of a double straight line . ; : A ‘ 5 PAB § 18. The case of a tacnodal point having the adjoint as , tangetit Slane’ : 5 : . 291 250 Mr BERRY, ON CERTAIN QUINTIC SURFACES WHICH ADMIT OF § 1. IntTRODUCTION. THIS paper may be regarded as a continuation of one which was presented to the Cambridge Philosophical Society rather more than two years ago*, which I shall refer to as “Quartics.” In the introduction to that paper I gave a short historical account of the subject, which it is unnecessary to repeat. some of the chief points in the theory. It is however convenient to restate If f=(a, y, 2, 1)”=0 is an algebraic surface, and if P, Q are rational functions of x, y, 2, Which satisfy, in virtue of f=0, the condition of integrability oP 2aQ ay ae” then [Pdx+Qdy is called an integral of a total differential. If the integral is finite all over the surface it is said to be of the first kind. It is known that on a non- singular surface of any order no such integral exists, and the determination of surfaces which admit of such integrals is a problem of some interest, which has so far received few solutions. If the surface is a cone the integral reduces to an ordinary Abelian integral associated with a section of the cone; such cases can therefore be omitted as belonging to a familiar theory in plane analytical geometry. Further no integral of the first kind can exist on a rational (unicursal) surface; and therefore none on a quadric or non-conical cubic. Of quartic surfaces admitting of such integrals two were discovered by Poincaré, four more by De Franchis+ and three of the latter independently by myself}. The present paper is a contribution towards the solution of the corresponding problem for quintic surfaces. The problem has been shewn by Picard to depend on the solution of the differ- ential equation oF 6 sees ide Chere == 10) srevesseseccseacseeeesens Bapacace (1); where the equation, 7=0, of the surface is written in homogeneous coordinates, and * «On Quartie Surfaces which admit of Integrals of d 9 d 3 d oa the first kind of Total Differentials”: Camb. Phil. Trans. * ay +29 get apt” Vol. xvii. (1900), PP- 388347. ’ ; ; which is identical in form with the familiar equation satis- + “Le superficie irrazionali di 4° ordine di genere fied by a seminvariant of the binary cubic (x, y, z, w¥£, 7)°. geometrico-superficiale nullo”: Rend. del Circolo Mate- ]t admits of the three independent solutions, x, u=z—y’, matico di Palermo, t. xtv. (1900), pp. 1—13. Read on Aug. y,H42)_ 3cyz+2y8, I omitted to notice that from these 16, 1899. . could be constructed the rational integral solution + Comptes Rendus, 2 Sep. 1899, and “ Quartics.” I take 3 cree o= (v? + 4u) a? = 22h? - — 3222 33 3, this opportunity of correcting the error of omission to Cae aR a which my attention was called by De Franchis’s paper. In my case I. (i) (‘‘Quartics,” p. 337) I get the differential equation This may be rather more conveniently written in the form We thus get the surface ap+brv+cu?+dzi=0, (F), equivalent to the surface called F,“*” by De Franchis. He points out also that the surfaces which I call (C) and (D) (‘*Quartics,” p, 337) can be treated as varieties of a single ease. It seems to me however more convenient, from my point of view, to treat them as distinct. INTEGRALS OF THE FIRST KIND OF TOTAL DIFFERENTIALS. 251 the quantities @ are polynomials of order n—3, which satisfy the subsidiary differential equation dé, d6, _d@, dO, __ dx t dy * dz * dw iefeletoralejeveletelaleraisiclalstevelaiatefereinielvtetelotetelteteletere (2). The six polynomials of order n—2 chosen from the array || 1, A, Os, % | | a, y, 2% wil have further to satisfy certain conditions, not yet fully known, at singular points of the surface. These conditions being satisfied, the most general form of the required differential is * | dx, dy, dz, dw | Toy ey ai Of Opler |e AW al cca cane ae Wis Wy 90 where J, m, n, p are arbitrary parameters. By making any three of these parameters vanish we get four equivalent forms, which are simpler but less symmetrical. In the case of n=5 the @’s are quadrics and I have not been able to discover any general method of integrating the fundamental differential equation, so that the problem has had to be attacked by quite different methods. I have begun by considering the various singular lines which a quintic can possess, those of highest order being taken first. A certain number of quintics are known to be or can be proved rational, so that they can be rejected. Others can be birationally transformed into quartics or cubics, so that in these cases the problem is reduced to one which is already solved. In some cases the conditions imposed by the singularities on the surface and on the quadrics @ produce simplifications in the differential equation, which render it integrable. But these processes become rapidly more laborious .as the number of singularities diminishes. In the cases when the quintic has no double or multiple curve, or only a double conic or a double straight line, I have not been able to solve the problem completely, and, though I have met with various surfaces of these types which admit of the first kind (some of which occur in Part II. of this paper), I make no attempt to deal with them systematically. The remaining cases, viz. when the quintic has a double curve of order greater than 2, two non-intersecting double straight lines or a triple (or multiple) curve are dealt with in Part I. of this paper (§§ 2—7). Among the surfaces of these types those which I have found should be the only ones admitting of integrals of the first kind, but some of the algebraic work is so long and complicated that I can hardly hope to have escaped all errors. In Parts II, and II. I deal with quintics which admit of two or more linearly independent integrals of the first kind. If these are functionally dependent on one * Cayley: ‘‘Note sur le mémoire de M. Picard ‘Sur _premiére espéce’.” Bull. des Sciences Math., Sér. 11. t. x. les intégrales de différentielles totales algébriques de (1886); Coll. Math. Papers, t. xm. no. 852. 252 Mr BERRY, ON CERTAIN QUINTIC SURFACES WHICH ADMIT OF another (Part IIL), I find (§§ 8, 9) that the quadrics 6 necessarily have a common linear factor, so that the fundamental differential equation reduces to an integrable form. On integrating it by processes exactly analogous to those used in “Quartics” I arrive at certain surfaces admitting of two integrals, as well as at certain others which admit ? of only one. Lastly it is shewn in Part III. that in no case can a quintic have two functionally independent integrals of the first kind, a result established by Picard for certain kinds of quintics. PART I. PRELIMINARY CLASSIFICATION OF QUINTIC SURFACES. § 2. I have not been able to find any systematic classification of quintic surfaces. One of Schwarz’s earliest papers * enumerates all possible ruled quintics, and a paper by J. E. Hill+ gives a useful bibhography of the subject, and briefly discusses a number of cases. Most of the other papers on quintics that I have been able to find deal chiefly with the properties of special surfaces and in particular with rational ones, and I have not been able to make much use of them. I take first the case when the quintic has a double curve, postponing (§ 7) the case of a curve of higher multiplicity. An irreducible plane curve of order 5 has at most 6 dps (double points). Hence if the double curve is of order greater than 6 every plane section of the surface is reducible, and so is the surface itself, that is it breaks up into two or more surfaces of lower order. It is clearly unnecessary to consider such surfaces. If the double curve is of order 6 every section is rational (or reducible) and by a theorem of Noether’s§ the surface (if irreducible) is rational. Again, if the double curve consists in whole or in part of a plane curve of order greater than two, any line in the plane meets the surface in six or more points, and therefore lies wholly on it. Thus the plane is part of the surface, which accordingly is reducible. Moreover, if the double curve consists of one or more rational curves, which collectively have at least one adp (apparent double point), we can draw a chord through any two points of the curve which meets the surface in four points on the curve and therefore in one point elsewhere. The coordinates of this point are therefore expressible as rational functions of the two parameters which determine the positions on the double curve of the ends of the chord. As there is at least one adp, such a chord can be drawn through any general point of the surface, and the coordinates of any general point are * “Ueber die geradlinigen Flichen fiinften Grades” : Crelle’s Journal, t. 67 (1866); reprinted in Gesammelte Abhandlungen, t. u. pp. 25—64. My references are to the latter. + “On Quintic Surfaces’: Mathematical Review, t. 1. no. 1 (1896). + It seems to be tacitly assumed by writers on solid geometry that if every section of an algebraic surface is reducible, so is the surface itself ; but I have not been able to find any published proof of the theorem. I have obtained what I believe to be a rigorous proof, though a clumsy one. § ‘*Ueber Flaichen welche Schaaren rationaler Curven besitzen’’: Math. Annalen, t. 111. (1870). INTEGRALS OF THE FIRST KIND OF TOTAL DIFFERENTIALS. 253 accordingly expressible as rational functions of two parameters; that is, the surface is rational. Thus we see that if a quintic exists, which has on it a double curve consisting of a twisted cubic, a rational twisted quartic or quintic, two non-intersecting straight lines, a conic and a straight line not in the same plane with it, or two conies not in the same plane, it must be rational: and this is still true if the surface has any further singular curves or singular points. We have already seen that rational surfaces can be rejected for the purposes of our problem. Thus of quintics which have a double curve we need only consider those in which the double curve consists of either: (1) a non-rational irreducible twisted quintic (§ 3), (ii) an elliptic twisted quartic (quadriquadric), with or without a straight line in addition (§ 4), (ii) three, four or five concurrent straight lines, no three of which are coplanar ($ 5, 6), (iv) a conic, or two intersecting straight lines, or (v) one straight line. We shall now discuss in order the first three of these possibilities. The fourth and fifth are not dealt with in this paper for reasons given in the Introduction. “4 § 3. THE CASE OF A DOUBLE QUINTIC CURVE. If the double line is a non-degenerate twisted quintic, then, the general section of the surface being elliptic, we know by a theorem due to Castelnuovo* that the surface is either rational or ruled; and Schwarz’s list shews that a non-rational ruled surface of this kind actually exists f. This can be verified independently from a consideration of the possible twisted quintic curves. According to Cayley’s classification}, the number of adps is either 4, 5 or 6; and of the species with 5 adps one type has an actual dp as well. If there are 6 actual or apparent dps, the curve can be projected into a plane quintic with 6 dps, and is therefore rational. The two cases left for consideration are the quadri-cubic represented by the symbol 6—1 (the partial intersection of a cubic surface and a quadric having a common generator), and cubi-cubic of the second species without an actual dp, represented by the formula 9 — 6 +2 (the partial intersection of two cubic surfaces which meet also in an excuboquartic); the former has 4 and the latter 5 adps. In the former case any generator of the quadric belonging to the same family as the given generator meets the cubic in 3 points which necessarily lie on the quintic curve. It therefore meets the quintic * Cf. Castelnuovo et Enriques: ‘‘Sur quelques récents + ‘‘Considérations générales sur les courbes en espace”: resultats dans la théorie des surfaces algébriques”: Math. Comptes Rendus, t. 54 (1862) and Coll. Math. Papers, t. v. Annalen, t. 48, § 39, and the references given there. no. 302, p. 15. Cf. also Noether’s paper, ‘‘Theorie der + The surface B 1., loc. cit. p. 63. algebraischen Raumcurven,” Berl. Abh. 1882. Won, SOD IAs INE 33 254 Mr BERRY, ON CERTAIN QUINTIC SURFACES WHICH ADMIT OF surface in 6 points, ie. lies on it. Thus the quadric is part of the quintic surface, which accordingly degenerates. The argument fails if the quadric is a cone; in that case a generator meets the cubic once at the vertex of the cone and in general twice elsewhere, so that as before the cone is part of the quintic surface; but if the cubic has a double point at the vertex of the cone a generator only meets the cubic once elsewhere; in this case however the quintic curve has a triple point at the vertex of the cone, and can accordingly be projected from it into a conic, so that it is rational. Thus the only possible case is that of the quintic represented by 9—6+ 2, which is of deficiency 1. This can be projected from a point on it into an elliptic plane quartic; there are thus two adps as seen from any point on the curve: so that through any point on the curve can be drawn two straight lines meeting the curve twice elsewhere; there are thus oo? straight lines which meet the quintic curve three times, and therefore the quintic surface six times, so that they lie on it, and the surface is ruled. We thus arrive at a single ruled surface, that given by Schwarz. His investigation shews that there is a one-one correspondence between the generators and the points on either of two plane non-singular cubic curves which the generators meet, and that to an arbiérary point on the surface (not lying on the double curve) corresponds only one point on either curve. Therefore the coordinates of a point on a cubic are rational functions of the coordinates of any point on the surface; the Abelian integral of the first kind associated with the cubic is therefore an integral of a total differential of the first kind on the surface. § 4. THE CASE OF A DOUBLE QUADRIQUADRIC. Let the quintic have as a double curve a quadriquadric or elliptic twisted quartic; this is the complete intersection of any two quadrics of a cluster. It is known that such a quintic is in general rational*, but it may cease to be rational if it has other singularities. This possibility has therefore to be investigated. There must be one or more points on the double curve at which the tangent planes coincide. Let one of these be taken to be c=y=z=0, the tangent plane there being «=0. Let S,=0 be the quadric of the cluster which has this tangent plane. Then since no two quadrics of the cluster touch, the tangent plane to any second quadric S,=0 of the cluster can be taken to be y=0. The line z=x=0 cannot meet S, again and therefore does not meet the quadriquadric again; it must therefore meet S, in a point not lying on the quadriquadric, which we can take to be z=w=x2=0; let w=0 be the tangent plane to S, there. The equation of the quadrics are now Sy=cwst w=10, where: = Ch wyy Ss) ccs. cotwacorectencestcnmcere (3), and SoS gui On wheres o)= (a) 2) "haere. -ceeese snot * Tn Picard’s fundamental memoir in Liouville, Sér. rv. t. 1 (1885) it is apparently proved (pp. 308—312) that no quintic surface can have two functionally independent integrals of the first kind. Iam however unable to see any justification for the statements (§ 6, p. 311), that such a surface must be of genus 1, and that if so it must have a double conic. It is evident that the genus must be at least 1, but the cases of genus greater than 1 seem to be over- looked. Moreover a surface with multiple points but no multiple curves can be of genus 1; a quintic with three ordinary triple points is an obyious case. In Picard and Simart’s book, written 12 years later, the same problem is dealt with (ch. v. § 17), but the existence of a double curve of order >1 is now explicitly introduced as a hypothesis, ¥, 2 |= Qf ef Ts VST w; 0, ’ @; Ose 0, not “proved” as before. At the end of the section a proof is promised of the more general result that no quintic of genus 1 has two independent integrals. I assume there- fore that in the early memoir, when the author was sketeh- ing the outlines of a large new subject, he accidentally overstated the generality of his result, and that he corrected the overstatement in the book. I have not been able to find the proof promised by Picard and Simart for the general quintie of genus 1, but it appears to be an almost immediate inference from certain general properties of surfaces possessing two integrals, which haye been established by Picard, Noether (Jath. Annalen, t. 29, § 8) and Painlevé (Théorie Analytique des Equations Différentielles, Legon 16). 284 Mr BERRY, ON CERTAIN QUINTIC SURFACES WHICH ADMIT OF or 5 f {(0,0. — 0,0.) -—tvfwl=fe(|t, y, 2 ies Q to 41, %, 6 | | Oy, 0, 0, But by the same considerations as before we see that the determinant on the right must be of the form — Q.f,,— af, where Q’ is an adjoint and @ a constant; whence 5 f {(0,02 — 0,0.) — 3 vfw + $ Oe} =(Q—Q) fr fe- But f cannot be a factor of Q—Q', fz, or fw, so that O= Gand) 60s — On Cs—-b (Waa fiz) ease eee ee ee eeee eee eee (25). We have corresponding expressions for 6,0,'—@,0,' &c.; in fact, paying attention to signs, we see that the non-corresponding minors of the two arrays 6:; @., @;, A, tee, exon “outer , | 4, & Y || Ve ee | are equal save for a numerical factor. Substituting in (24) we obtain £{-@Of2— fz) — y (Mfe— vfy) + Ufo — Bfz)} =Q-fet vs, whence = — L(A LY VS A DW) ™ onc o noes cco cscecedecrcensieenen (26). Let us now choose the plane of the adjoint to be one of the planes of reference, say z=0; then we take XN=~4=a@=0, and v=5x, where « is a non-vanishing constant which we can evidently put equal to 1 if convenient. The equations of the type (25) then reduce to HRS Chihes rapa, CH SINR, CH SORES, - sccsenasoses+e" (27), and 6:0 '05G5 = 0 (a= 1h DAN ac ncte ot esac ete core eee (28). From the last set of equations we deduce § 13. DISCUSSION OF POSSIBLE SINGULARITIES. We know that the adjomt Q=0 must pass through the multiple curves, if any, of the surface, and through isolated triple and higher points, as well as through certain double points of a complicated character. Moreover it has been shewn by Picard} that in the case that we are considering, when there are two integrals of the first kind, Q=0 must also pass through the double points of all kinds. * These results are in substance contained in Picard et Simart, ch. v. §17. I have given a somewhat different proof in order to get the results in a general and symmetrical form, + Memoir just cited, p. 306. INTEGRALS OF THE FIRST KIND OF TOTAL DIFFERENTIALS. 285 Thus all the singularities of our surface must lie in the plane Q=0. We can at once shew that there can be no triple point. For, if we take the triple point as origin (c=y=z=0), the section of XO, + pO,=0 by Ax+ py +vz=0, must be adjoint and must therefore have a double point at the origin; hence «=0 must touch 6,=0 at the origin, so that f) 6,=ax+®,, where ©, is quadratic in a, y, 2 and similarly 0, = by +O). Then Ax+py+vz=0 touches raw + pyb+rO,+40,=0, for all values of X:p:»y, and this is only possible if a=b=0. Thus 6,, 6, and similarly @,’, 6,’ are quadratic in , y, 2, so that Kf = 0,0, —0,'@, is at least quartic in w, y, z, which is impossible at a triple point. We need therefore only consider the cases when the quintic has one or more double points, or an infinite number forming a double curve. The origm being a double point of any kind, @,, 6, must pass through it, and - we can therefore always start by assuming A,=ac+by+o2+9,, 0, = aye + boy + oz + O,, Gk= Oy, MEPS ik S Riesasceuctvasanstoscmeenmesaeacis (30), 6,=d+ ae + bytez+O,, | Sf Set Us + Uy + Us, where ©;(1=1, 2, 4) is quadratic in a, y, z, and u(j=2, 3, 4, 5) is of degree j in x, y, z Also u,+0, and from the equation 00, , 00, | 00, | 00, _ Gaey carl dom we have, among other relations, a,+b6,+2d=0 We now consider in turn the different kinds of double point, beginning with the simplest. $14. THE CASE OF AN ORDINARY CONICAL DOUBLE POINT. We can take m=2?+7'+2*, or u=2*—yz, according as the adjoint z=0 is not or is a tangent plane to the tangent cone. In the former case the only term in the fundamental differential equation involving 2? alone is d. 32%, so that d=0. Wo, IDS Terese IE 37 286 Mr BERRY, ON CERTAIN QUINTIC SURFACES WHICH ADMIT OF In the latter case the terms in the differential equation involving 2°, yz are respectively (2a,+3d) a, —(b,+3d) yz; so that 2a,+3d=0, 6,+3d=0, and therefore from (31), d =0. Thus in either case d=0, and similarly d’=0, therefore 6,0,’ — 0,0,= «xf, is quadratic in z, y, 2, contrary to hypothesis. § 15. THE CASE OF A BIPLANAR POINT. We can assume u, to be of one of the forms y2, Yrs, wry, according to the relation which the adjoint z=0 bears to the tangent planes. In the first case, since y=0, 6,=0 must be an adjoint curve, a,=c,=0. The quadratic terms in the differential equation then give b, + 3d =0, so that, from (31), aq=d, b,=— 3d. The cubic terms in the differential equation all divide by 2z, save for terms contained in d d 3 age {(de+by) AE 3dy oa us + 2dus, therefore if w, =z’ +v;, where v;=(z, y)’, we have (a! d d fet SEY ae Re d (ae cae 2) 0%; + by Fes 0. Now, if z* be the highest power of # occurring in v;, we have d {a—3(3 —a)+2}=0, whence 4a=7, which is impossible, or d=0 which leads to a contradiction as before; thus the only alternative is v= 0. Repeating the same reasoning with the quartic and quintic terms we find that z must be a factor of uw, ws as well as of w and us, so that the quintic degenerates. In the second case, the quadratic terms give (av + boy + 2). 2y + 3d (y?+ 2) =0, whence d=0, leading to an impossibility as before. In the third case, the quadratic terms give (aye + by + 2) 2a + (aor + bay + C22) 2y + 3d (y? + 2*) =0, whence 2a,+3d=0, 2b,+3d=0, so that, from (31), d=0, as before. Thus there can be no biplanar point on the surface. INTEGRALS OF THE FIRST KIND OF TOTAL DIFFERENTIALS. 287 § 16. THE CASE OF A UNIPLANAR POINT. We have to distinguish two cases according as the tangent plane is not or is the adjoint z= 0. In the former case we can take the tangent plane to be y=0, so that u.= y* The section of the surface by y=O has a triple point at the origin, and therefore the section of @,=0 has a double point there; therefore a3— ce 0: Also from the quadratic terms in the differential equation 2b, + 3d=0, and as before a,+b,+2d=0, whence Oe SO ee a — DO eee shoo iee nace a. Seca ekene (82). The cubic terms in the differential equation now divide by y save for terms in d (a,x + 2) ee 4a, Us, therefore if, as before, u;= 7» +03, Where v;= (a, 2)’, we have dv (ax + (2) ag — 4a,v; = 0, therefore, if «* be the highest power of x occurring in 2s, a,.a—4a,=0, but a+ 4, therefore either a,=0 and d=0, leading to an impossibility, since xf, = 0,0, — 0,0, has then no linear terms, or v;=0, so that u, divides by y. The origin is now not merely an ordinary uniplanar point but a tacnodal point ; ie. any section of the quintic by a plane through the point has a tacnode (equivalent to two ordinary double points) in the ordinary sense of plane geometry; it follows that the section of rO, + 1A, = 0, by Aa + wy +vz=0, must touch y=0 at the origin, therefore aye + (AD, + Bua,) y + AQz = 0, e+ py + vz =0 meet on y=0, and c,=0, a,=0, and as just shewn this is impossible. In the latter case, we have u,=z*. Further by a linear transformation from w to w+la+my+nz we can remove the terms in 2 and z in us, so that f= + 22+ +U,+U;s, where v;=(a, y)!. 37—2 288 Mr BERRY, ON CERTAIN QUINTIC SURFACES WHICH ADMIT OF From the quadratic terms in the differential equation we have d = 0, and (31) reduces to a4,+b,=0. Let us first assume that v,#0, so that the origin is not a tacnodal point on the surface. Let « be taken to be one of the factors of v,; therefore the section of f=0 by «=O has a tacnode at the origin, and therefore the section of 6.=0 by x=0 must touch z=0 at the origin, therefore b, = 0. (i) Let v, have a second distinct factor which can be taken to be y, therefore a, = 0 similarly. The cubic terms in the differential equation now divide by z save for d d (ae ai + boy a) Vs, or a (2 f-9e)o y therefore a, = b, = 0. Therefore the quadratic terms in f,, f, divide by z, which is impossible, since 2; = 0. (ii) Let v; be a perfect cube, Xa*, say, where \+0. The cubic terms in the differential equation divide by z save for ae = Ae, therefore a, = b, = 0, : so that we now have 6, = 42+ 0,, 0, = at + co + ©,, 0, = az + by + cz + O,. If a,=0, then as in (i) we arrive at an impossible result; we may therefore assume that in this case a,+ 0. The cubic terms in the differential equation after division by z reduce to d d Ger (ZU, + Av*) + (Ax + C22) a Va + (asx + by + C42) . 82. OE INTEGRALS OF THE FIRST KIND OF TOTAL DIFFERENTIALS. 289 Equating to zero the coefficients of 2’, z, 2°, we have, therefore, Gy SUL d d - ss (a s+ egy) ™ +3 (aw + by) =0 ai acees Senter easeeameasese (33), dv, _ 3AC,@ + as a Qaes esecedans sides ticaceneoaceccedeces (34). From the last equation we see that 2 is a factor of v,; if the other factor is 2, or if v,=0, c,=0, and xf,= 0,0, —6, 6, has no quadratic terms, which is impossible. We can therefore assume that v, has a second factor distinct from 2, which can without loss of generality be taken to be — 3y, so that v,=— 3xy; and we then have Co = Ay, CG = b,, AC, = Ap. Further by a modification of the coordinate # we can arrange that X=1, so that we now have f =2— 3xyz + B+ Ut Us, A oz + 9,, 0, = C2 + Coz + Oy, 0,= oo + yy + O,. Further by working with suitable linear combinations of our two integrals we can arrange so that in the corresponding quadrics 6 and @’ we have respectively c,=1, c,=0, and ¢, = 0, ¢.=1, so that 0#,=2+90,, 0/ = OY, 6,= 2+ ©, 0,, =z + Oy, 6,= 7+ 9, 6, = a+ OY, where ©; = (Ai, B:, C;, Fi, Gi, HiGa, y, zy, and @; = (4; Bi, Cy, Fi, Gi, Hi Ve, y, 2F (i = 1, 2, 4). We now have the identities 6.0, — 0,0, = + fhe, 0,0, — 0,0, = 4 fy, 6,0, — 0,0, = 4 fw: By this transformation we have simplified the expressions for 6; and @,, but have lost symmetry as between 6; and @,.. The quartic terms in the fundamental differential equation formed with @,' are By (— ye a8) +30 (20) +2 7 ugt 20 (~ Saye to!) + 30, . 2 These all divide by z save for terms contained in 30,2? + 2a; 290 Mr BERRY, ON CERTAIN QUINTIC SURFACES WHICH ADMIT OF therefore these terms must also divide by z, so that 3 (Ae + 2H,'cy + By’) + 2a = 0, therefore Ante eget and, from the subsidiary differential equation dé,’ i dé,’ dé,’ , dO, _ 0 dz ay dz deol = OF we get 2A,'+2H,'+1=0, H+ By =; whence Et DO: Again, the coefficient of a2*y? in f, is the same as that of 2a*y in f,; therefore A,B,’ — A,'B, + 4(H,H,’ — H,'H.) =4(A,' — 2M), therefore 2B,+3H,=—-1-4,. And, from the subsidiary differential equation in 6;, 2H, +2B,H =0, therefore H,=0, B,=—-4. Also the coefficient of y° in f; is equal to that of yx in fo, therefore — Bs = 2(B,H,' — BY H,) + 2 (A,B, — H,B.). But By, B,’, H,, H,’ vanish, H,’+0, therefore Br=0: Similarly, equating the coefficient of y in f, to that of 4cy° in f,, we have 4(B,B/ — By B,) = 2 (B,Hy — B/H,) +2 (H,B/ — H,B,), whence B,=0. Also the coefficient of 3y*z in f, is equal to that of y’z in fo, therefore 4(2F/ — BY) = 2 (BF, — BYF,) + 2 (BF — BY F,), whence 2F/=—-—3F,/, F/=0. Lastly, equating the coefficients of «yz in the equivalent expressions thw, 0,' — 0,'0, we obtain —2=2H,' + 2F/, which is impossible since F,)=0, 4H,’ =}. We have thus shewn that a uniplanar point is impossible, with the possible exception of the case when it is a tacnodal point having the adjoint plane as tangent plane, 1e. the case when v,=0, with the notation of this paragraph. This case we shall deal with in § 18. INTEGRALS OF THE FIRST KIND OF TOTAL DIFFERENTIALS. 291 § 17. THE CASE OF A DOUBLE LINE. It has not been assumed in the discussion hitherto carried out that the double points considered are isolated. Hence if there is a double line every point of it must be a tacnodal point with z=0 as the tangent plane, since we have proved that no other kind of double point can exist. The line must accordingly be a tacnodal line, with a fixed tangent plane; such a line being equivalent to two intersecting double straight lines or a double conic, the surface has py=1, so that im accordance with Picard’s results two independent integrals cannot exist. § 18. THE CASE OF A TACNODAL POINT HAVING THE ADJOINT AS TANGENT PLANE. As in § 16 we have fH=F+%2m+U%+U;, Where v,= (a, y)?, and 0,= Qr+by+oaz+9,, 0, = age — ayy + 02+ Oo, 6;= 0: 0,= ae t by +0,2+0,, where ©;=(4;, B;, C;, Fi, Gi, HiGa, y, 2), with similar expressions for the quadrics @,’. A section of the surface by any plane Ax+ py+vz=0 has a tacnode at the origin; hence the section of X6,+46,=0, by this plane, being adjoint, must pass through the origin and touch z=0 there. Therefore (Aa, + fay) & + (AD, — way) y= 0, whenever Aw + py = 0, therefore # (AG, + fd») — A (AD, — pay) = O, for all values of X: p, therefore a, = b, =a,= 0, and we have =62+9,, 6,= Coz + @,, A, = av + by + 42 + O,. The cubic terms in the fundamental differential equation are now #(oEtad) ut 3A (ant) + ¢z)=0 ala dy 2 4) sy 4 , whence C= 0, d and age + by =—4 (c ae + Ce aa) (Oy Sa pdaoeaGanbond nebobOSRUCoReneSe (35). 292 Mr BERRY, ON CERTAIN QUINTIC SURFACES WHICH ADMIT OF It is now convenient to treat separately the three cases which present themselves according as (a) v,.=0, (8) v, consists of two distinct factors, (y) v. is a perfect square. Case (a): v2,=0. Then from (35) a,=b,=0, so that f H=2 ++ Us, 6, = G2 of @,, 6, = Co + @., 1= e,, with similar expressions for 0;. If we arrange the constants so that the fundamental identities (27) assume the form (CONS ii (CHS (COOH pn cocnsacosc0bsdosasnacocdes: (36), where we have introduced for brevity the notation (xy’) for wy’ —a’y, we have (qc. ) = 3. We now see that the cubic terms in f;, f, divide by z, so that u,=zus; hence the section of the surface by z=0 consists of 5 straight lines passing through the origin. If these straight lines are not all coincident, we can take two of them to be respectively «= 0, y=0; and then @, vanishes along the first, @, along the second, therefore An — bur Also from the subsidiary differential equation A,+H,=0, H,+B,=0. Similarly A,', By, A; + H,’, H,’ +B, vanish. Now the cubic terms in f or f, vanish identically, and the quartic terms divide by z; therefore (c,9,’) — (¢.0,') = 0, and (@,@,’) divides by 2; therefore (c,4,)=0, —(c¢Ay)—(@HY)=0, —(¢,H’)=0, and (A,H7)= 0. From the first and last of these equations we get that ether (c,H;))=0; or A,=Ay —0: If (c,H,’)=0, then from (c,H,’)=0, we deduce H, = H,' =0, whence (¢,Ay)=0) and A,=A,'=0. INTEGRALS OF THE FIRST KIND OF TOTAL DIFFERENTIALS. 293 Thus in either case A —pAna—() and then also Hien): Thus ©,, ©,, @,, @,, all divide by z; therefore so do f,, f,, and therefore also f, so that the surface degenerates. But if the five lines, f= 0, z= 0, all coincide, say with «=z=0, we can take fH=2 + 2s + Zu) + Aa’, where X+ 0. Everything in the differential equation divides by z save for some terms in we €\. ©, . da 3 therefore Av Hi B= 0: whence also H,=B,=0, smce A,+ H,=0, H,4+ B,=0. We now see that the quartic terms in fy, all divide by 2 save for a term with zw as a factor; therefore u,; has no term in y* or «xy; therefore neither f, nor f, has a term in yz. Therefore (4, By) = 0, (¢,B,) = 0, whence BB 0: therefore 7, has no term in zy’, and therefore wu, has no term in y*. Therefore f is quadratic in z, z conjointly, so that 2=z=0 is a double line, a case already dealt with in § 17. Case (8): v2 = xy. Then from (35) a,=—4¢e,, b,=—4¢,, so that f =H H+ aye + Uy + Us, 6,=¢2+ O,, 6, = coz + Oo, 6,= — Fee + Gy) + O%, where (from the subsidiary differential equation) AM eG FO; Oy edt aC) OMamesnnscaccissceesisns seers (37). There are similar relations between the dashed letters, and we have (c,c,’)=3, as before. From the cubic terms in the identity f, = (@,@.) we have (PENN ER CSG S(t 25 GONE, ea ee (38), (GZ) = (GUMS 1, Gr GAINS (G@LANSO seosoescnocasassadcaoso: (39), and (CEB (CRB NO gente vas cece vatclessleeieis ie sciseiele stasis sei (40). Vout. XIX. Parr II. 38 294 Mr BERRY, ON CERTAIN QUINTIC SURFACES WHICH ADMIT OF Also, equating the coefficients of 32*y, xy’, 2* in f, to those of a’, a*y, 3zy* respectively in f;, we have 2 (c.H.’) + (c, Ay’) =—3 (¢,A/’), (c.B,’) + 2(c,H,’) = — 2 (c.Hy’) — (¢, Ay’), and 2 (c, Hy’) + (¢,B,') = — 3 (¢,B,’), whence, eliminating the coefficients H by (37), (CRAY (GAN =O rec sccceecnsseebacenewecseres sccseencer (41), (GrAG EA (Ga BD) =O Fre senccaes- ten cans os cost aocneseaccs (42), (CeB SV SEUCBR i= Ole cacecs chiclescnaees dceceeeumacecute ces (43). From (38) and (41) we deduce (ay: A eal (evs Val snp piece cceoceasuscenesee sconce (44). From (40) and (48), (Gai) = (5/2) SO sasonaonnacconcconcesdnoossosonc (45). “From (39) and (42) (eal) SS (Gue)) SO) sencorosanecancodcdonsonosecnceec (46), whence A,, A,’, Bo, B, all vanish, and Cea 2 Cae The coefficient of y* in fp =(B,B,’)=0, therefore also the coefficient of y° in f, vanishes, ie. (¢,B) = 0, therefore from (45) B,=B, =0. The coefficient of ay? in fy is now 4 (H,H,)=4(c¢))=4; but this is also the coefficient of 2vy? in f, or 4(c,H.’) = 74 (qe)=4. These results are inconsistent, and this ~ case is accordingly impossible. Case (vy): v is a perfect square, say v. = 2°. Then from (35) a,=— 3c, b,=0, so that f =Ft+ BE+Uy + Us; 6,=¢,2+9,, 6, = Coz + Op, 6,=—feo,27+0,, Ay + Ay — 4.6, 0) Filet By 0 iegun nae 'ccepeseencle esac vn | e*— dx=| e*>u,—, da, 0 0 n 0 0 Bo ea Le. (3) Stu=| eu (a) dz, 0 0 Da na where u(x) =z Un —. 0 n: When therefore Sw, is divergent we may define its ‘sum’ as being equal to the integral | e* u(x) da, 0 whenever this integral is convergent. This is the definition given by Borel, whose point of view is however ditferent. CONDITION OF CONSISTENCY. § 3. It is obvious that our definition will involve us in contradictions unless equation (3) is true whenever Su, is convergent. It is therefore essential to prove that this is the case, ppt 2 ° aL . : . Now, if Yu, is convergent, Le-* wp arta uniformly convergent in (0, X), however great be X. For, however small be the positive quantity «, we can choose N so that |Un| 6" Uy, | < ce F— N’ nh. o nh. <<, for all values of z in (0, X), all values of N’ 2 N, and all positive values of p. w.4 2 an C7 > Un — dar. U 0 0 v! «© 4 an Hence (aly 2 un | GE ex dx =| 0 nN: 0 OF DIVERGENT SERIES. 299 We shall be justified in replacing X by o if only i x ao a lim = uy, | e* — dx=0. X=m 0 DG n: a But | e* a” da = e-* {X" 4+ nX" + ..... +ntj, ie so that this series is J n i _ 2% X” D eS u, > = He 7d — = uw. 0 o tb: i) Oh A Da For if Ss, = Du; n = X2\ a XxX? ty + (+ X) + ug (1 + X tes) seseine +un(14+X+ aude +=) ay ut @ , Sie oti. Caer aone + Sp X” Tiwi (1+ X+ ecccee ae ji and the limit of the last term for n=o is zero. Now let o be any assigned positive quantity, and determine WV so that if xn 2N lsnl 1 | < 20 | 0 nm: mi Olly WCNC AM ATE . x“ Noel a lim = uy | e-* — dx = 0. X=xn0 xX nN: Therefore we may replace X by « in (1), and so o }A8 pice Ln gn Un =| e* uy, — dz, 0 n: it} whenever the series on the left is convergent*. * Since writing this I have proved a more general theorem, viz. that if ¢,=uy)+u,+ an e7* 2 uz, — dz, 0 oo mt Resies +uU,, lim (F714 + + 9n [: 2 n ~ if the left hand is determinate, and a certain other condition is satisfied. 39—2 300 Mr HARDY, ON DIFFERENTIATION AND INTEGRATION It is to be observed that we have not proved that | e*u(x)dx will diverge to a 0 definite infinity whenever Sw, does so, This is true, however, when all the terms of the series are positive. For then, however great be G, we can determine N so that N > un> G. 0 4 0 N a That is I e* Su, —da>G; 0 0 n. «2 and a fortiori [ e*u (x) da > G. 70 But this can only be the case if the last integral diverges to + 2. § 4. When the integral / e* u(x) dx “0 is convergent, I shall say, with Borel, that the series Lu, is summable; and I shall denote its sum by Pum: 0 It is not obvious that the sum of the series O+ Ut my +... is equal to that of Uy Uy Ug Fo ve ceeee The sum of the first series (if it is summable) is [" e*u(a) da where V(L) = We + UY cane SAdade ra op =|. u(x) da, provided w(x) be uniformly convergent. And x a x Se (pe | e* | u(x) da =— l= / u(x) ar ao I e-* w(x) da. JO 0 J0 0 0 Hence 0+%m+%...... will be summable, and its sum equal to w%+%+U...... if u(x) is uniformly convergent and lim eae u (a) da = 0. X=0 0 And under these circumstances A+ Uy + Uy eeeeee = A+ (Up th «we. ). We can deal similarly with the series at+b+...... +k +p + Uy «20... But I shall not enter in detail into these points at present*. * See Borel’s Legons sur les Séries Divergentes. Borel confines himself to absolutely summable series. I may remark that there is no difficulty in seeing that we may prefix any number of zero terms to the trigonometrical series considered later, OF DIVERGENT SERIES. 301 UNIFORM SUMMABILITY. § 5. Now let us suppose that the terms of the series are functions of a variable a, a) a” Let Uw (x, a) => Uy, (a) 0 n! We shall say that Pup (a) is uniformly summable in (B, y) if 0 | e* u(x, a) da 0 is uniformly convergent in (8, 7). CONTINUITY OF THE SUM OF A DIVERGENT SERIES. § 6. Theorem I. If all the terms u,(a) are continuous functions of a, and g Uy, (a) is uniformly summable, and = a” = Uy, (a) — 0 nm Va uniformly convergent for any finite value of «x, in an interval (B, x), the sum of the first series is a continuous function of « throughout the interval. In the first place, e*w(x,a) is a uniformly continuous function of a throughout the domain (0, XG B, 1), however great be X. : : XG : : To prove this we observe that, since > wu, (a) nt uniformly convergent in (8, ¥), we can determine a value of NV, corresponding to any assigned positive quantity o, so that n)\ | tn (a) al | u, (a) is convergent and equal to 1—acosé 1 —2acos@+ a2" If Ya" cosn@ is continuous at the extremities of (—1, 1) we may make a=—1 0 and 1, and so obtain eS 1 (1) ¥(—1)" cos nb = 1 — cos 6 + cos 26...... =5> 0 a 2 1 (2) FcosnO@=1 +4 cos 6 + cos 26...... =5- 0 Let us see whether the conditions of I are satisfied. All the terms are continuous, LD : : - and >a" cosn@— is evidently uniformly convergent for any finite value of « It only = J 5 remains to show that a a | e*u(x, a) dx =f e*0—ac0s 8) cos (ax sin 0) dx 0 0 is uniformly convergent in (—1, 1). This is so provided @ is not a multiple of 7, in which case uniform convergence ceases at one or other of the ends of the interval. If for instance 0<@0<7 oe | e- 72.0088) cos ax sin 86 da X: x < | e7%(1—a cos 6) dz 1X 1 : << pee ex 1—acos@) | l—acos 6 for all values of a in (—1, 1), and this can be made as small as we please by choice of X alone. But if @=0 uniform convergence ceases for a=1; if @=7, for a=—1. And in fact in these cases (1) and (2) each give 1 er le el obec == a result in contradiction with § 3. OF DIVERGENT SERIES. 303 ELEMENTARY TRIGONOMETRICAL SERIES. § 8 If 6+(2n+ 1) 7, ¥(-)" cos nO = [ | e-7(1+e08® eos (x sin 6) dx = 2 ; 0 ~ 0 - Similarly we deduce all the following series: (1) Y¥(-)" cos n8 = 1 — cos 6 + cos 26...... = ey 0 = a 1 (2) ¥cos nO =1 + cos @ + cos 20...... =5> 0 — z : F : 1 1 (3) Y(—)""sin nf = sin 0 —sin 26...... = 5 tan; 0, 1 2 2 core : 3 1 1 (4) Ysin nO =sin 6+ sin 26...... =-— cot — 0, i 2 2 (5) y(-)" cos (2n + 1) @= cos 0 — cos 36...... =} sce 6, 0 (6) cos (2n + 1) 6=cos 6 + cos 36 ...... =1()) 0 (7) Y(-)" sin (2n + 1) @=sin 6 — sin 36...... =()), 0 aaa ; 2 1 (8) &sin (2n+1)0=sin@+4+sin 36...... = 5 cosec é. 0 - In these series @ may have any value except those for which the series takes the form in the case of (3), (4), (5) and (8) these are the values for which the function which represents the sum of the series becomes infinite. And it is easy to see that any one of the series is uniformly summable in any interval of values of @ which does not include any of these exceptional values. By writing 6+ 94, 0—¢ instead of @, and adding or subtracting the results we can obtain a number of more general formulae. It will be sufficient to give the following: (9) 1+ cos @ cos g + cos 20 cos 2¢...... = Es (10) sin @sin ¢ + sin 20 sin 2¢...... = (0), : : 1 sin b saaidha = GY!) Bem 15) Se : (11) cos@sin ¢ + cos 26 sin 2¢...... 5 acre 1 sin (12) cos @sin ¢ — cos 26 sin 2¢...... a ci ee s os These hold so long as 0+ and @—@ have not certain particular values easy to specity. 304 Mr HARDY, ON DIFFERENTIATION AND INTEGRATION DIFFERENTIATION OF A DIVERGENT SERIES TERM BY TERM. § 9. Suppose that un (a), whatever be n, has a derivate u,’(a) continuous throughout ~ (a, —&, a + &), and that F Un (a) is summable throughout this interval, w(a«) being its sum. Theorem If. IF x Fu, (a) 0 is uniformly summable in (a —&, % + &), and a S a” > un’ (a) — 0 nm. eo) uniformly convergent for any finite value of «; the series Fun(a) may be differentiated 0 term by term for «=a. du (x, a) 0a 5 ; oe : 5 2 For since > w,’ (a) ai 8 uniformly convergent, its sum 1s = u' (a, a). Also we can prove, as in § 6, that w’(«, a) is a uniformly continuous function of a throughout (0, X, a — &, a +€&), however great be X. Finally, since Fu,’ (a) is uniformly 0 summable, ea | ew (a, a) da 0 is uniformly convergent. Therefore w (a)= sf e*u(a, a) dx ) 2 = / e-*w' (x, a) dx /0 au 20 a = | e* > uy’ (a) — dx /0 0 n. = ¥ Un’ (a), 0 for a=4). § 10. Consider for instance the series ji ae Uae em + ere + one ito COU a The result of differentiating the left hand p times is wo PF nP era, 0 OF DIVERGENT SERIES. 305 and the sum of this series, so long as a+2mr, is 2) iv | €™ Up (x, a) dex, 0 where nr = - & Up (&, a) = > nP era = 1 This series is uniformly convergent in any interval of values of a, whatever be a. And in order to prove that we may differentiate Y™* term by term any number of times, for a=a,, we only need to be assured that, whatever be p, we can make 2 | \/ e Uy (x, @) da | lJ x | assignedly small by choice of X, for all values of @ in a finite interval (a,—£, a+). = 7 a d po gh enia d\P Now 1? Up (x, a) = ( ) = “al =( da = ae If we differentiate out and replace each term by its modulus we obtain a finite sum of the form Pp ercosa SF a,x, 0 Hence we need only prove that we can make fie (l—cosa) ov dp 5 assignedly small for all values of @ in an interval (¢,—£& a,+é&). Now if a+2nmr we ean find an interval of this kind throughout which l—cosa>y, a positive constant. And then | (FMEA ihes 0. That is to say are __1—peosé (1) Su) Con) — aor eenU LR Ch aaa psiné (2) Sp sin n0 Voc uae if 1—peos@>0. If —10. If —1 1) “we : da\ @7% (1—-0082e) Gog (x sin 2e) 7 2 DL < | et (I— COS 2e) ant dz 0 Wee EG) ~ (1 —cos 2e)?’ and therefore becomes infinite for e=0 at the most like e7?; so that : pe : lax: lim !$ (47 + €) — $ (47 —€)} | Gat e822 COS (asin) 26) = —1()) e=0 ».G ce And we can also choose € so small that | e“® (a, e)dx is as small as we please. 0 Consequently lim [ e*® (a, e)dx=0. e=0 J0 312 Mr HARDY, ON DIFFERENTIATION AND INTEGRATION Hence* : iz [tan ad (a2) da = g (-)> [sin 2Qnad (a) da. /0 1 /0 We can prove in the same way that we may multiply any of the series (3), (4), (5), (8), (11), (12) of § 8 by $(@) and integrate term by term, provided that neither limit is a point at which the sum of the series becomes infinite, and that we insert the sign of the principal value whenever any such point is included in the range of integration. §18. I shall give a few examples of the use of this theorem. Since ; tan (0 — 6) = ¥ (—)" sin 2n (0 — 4), _ 9 = 5 cot (9-— $)= Psin 2n (8 — >), 0 iP [tan (@— d) cos (2n + 1)6dé=0, P{ “tan (@—¢)sin (2n +1) 0d0 =0, 0 and the two corresponding principal values containing cot(@—q) are also zero, And Ie: | ~ tan (6 — ) cos 2n6dé = (—)” 2m sin 2ng, 10 P is tan (@ — d) sin 2n6dé = (—)" 277 cos 2nd, /0 Te le cot (@ — d) cos 2nO dé = 27 sin 2nd, 70 Qa Pp] cot (@ — d) sin 2n@dé@ = 27 cos 2nd. /0 These formulae are true so long as the subject of integration does not become infinite at either of the limits. Similarly if cos n6dé sin nd » cosO—cosP ” sind’ if 0< <7; and so on. $19. The question of the integration of (1), (2), (6), (7), (9), and (10) of § 8, is however of much greater importance; and it is plain that integration term by term is not always legitimate. If for instance we integrate cos 6 — cos 26 +... =4 from @=0 to 6=7, we obtain the obviously false result 0 = 47. Let us consider then whether the equation 1 +cos (a—@)+cos2(a—@)...=4 * I proved this formula by an entirely different method in the case in which the series on the right hand is convergent, in the Proc. Lond. Math. Soc., xxxtv., p. 80, OF DIVERGENT SERIES. 313 may be multiplied by $(a) and integrated from 8 to y, (a) being a function whose first and second derivates are continuous, In the first place, integration is permissible if (8, y) does not contain any of the points a=2n7+0. Hence if 0—27< 8 <8, and é is a small positive quantity O@-e mn wo fé—-e | Fcosn(a—O) db (a) da= | cos n (a — 0) d (a) da, B 0 0/8 and therefore [cos m(a— 6) 6(a) da B 0 o £é ao é =9 | cos n (a — 0) h (a) da — lim #{ cos n(a—@) d (a) da, 0 O-—e 8 e=0 0 /6- provided that any two of these three terms be determinate. The left hand is simply 2 |b (ada while the second term on the right is the limit of nr bd 2 6 i; ceded = | cos n (a — @) d (a) da 0 C SG a2) () 2 ml =| e*dax $(a)da>~ cos n(a—@) 0 y @-e 0 a 8 ; =R if eda | er) & (q) da| 0 J O-e = | ‘€*@ (a, eda, /0 @ : where © (a, «)= R| ev? =" 4 (a) da. 6-e Now this integral is [ery (wu) du=— L eiutae—M al (y) ot =| en oe {ep (w)) du 0 1 0 two du ‘ a where y(u)=¢(@—wu). The second term is 1 i rei d iu F WG aze—tu d iu F a es du {e vin - 2 € (Gu? ) ab (u) du. Hence O(a, «)= -= eFOS© sin (e— x Sine) W(e) + X{#, €) ; i 2 1x (#, €) | < He; H being a constant. Now if & be any positive quantity firm | == a eyo lie | CNG OES e=0 Jé e=0/0 VOT OX ART MELT 41 314 Mr HARDY, ON DIFFERENTIATION AND INTEGRATION and it follows as in the last paragraph that we may neglect the term y/z? in the expression of ©(, e) found above. Hence our limit is the same as hes) dz — lim ¥ (e) e-* 0 —cose) sin (e — # Sine) —. <= J¢ x =0 ba da Now lim sin | e *4-cose) eos (x sine) —|=0; e=0 é & for we may suppose &>1, and then -® *D | < e-Z(1-c0se) »—3 dx Rt /0 lr) (1 —cos €)*” apo If ie et: = tthe And lim | cos € | e-7(—cose) sin (a sin e) «=0 vé “ & sf weiirs : : dx: = lim | e-7(—cose) sin (# sin €) = e=0 /0 L 1 => 5 Ths “a 1 Hence finally lim | e* © (a, e)de=5 my (0) e=0- 0 = = : To (0). ré o 6 Consequently | (a) da= x | cos n (a — 0) d(a) da — Bs (@). =-J8B 0 JB a That is to say 6 - y il [" yl ( r= 15 AMIEL ee: n(“2— 0) (a) dat. Similarly, if 0< y<27+60, 2 (0) =— {5 |’ (a) da+ 7 [cos n(a— 0) h(a) dat : mw |2J6 1/06 And generally /,¢ (a) da + 9 |" c0s n(a— 60) b(a)da=7 enh (2n7 + 4), the summation extending to all values of n such that 2n7+@ lies im (8, y), and e¢ being =1 in general, but =4 if 2n7+0=B8 or =y. This is a form of Fourier’s Theorem. Of course the conditions which we have imposed upon ¢ are much narrower than they need be*. I shall not, however, attempt to generalise them, beyond * The ordinary proofs of Fourier’s theorem show that legitimate in the case of the series cos @—cos 26+..., when as a matter of fact the series on the left is not only S=0, y=7. summable but convergent. I need hardly say that my When we integrate term by term we assume that object is not to give a proof of Fourier’s theorem equal in 0 = = re) generality to the accepted proofs, but to show how naturally | ; dx - F (x, a) da= | ‘ da | F («, a) dx, one is led to it from a point of view quite different from those usually adopted when F (x, a)=e7*@+*) cos (a—x sina), and this is un- true, the left hand being =0 and the right hand =47, as is It is the inversion of integrations in ¢13 whichis not easily seen on working out the integrations. OF DIVERGENT SERIES. 315 remarking that @ may have a finite number of ordinary discontinuities in (8, y). In this case, if any one of the points 2n7+ 6 be one of them, we must substitute 3 {h(Qn7 + 0—0)+ $ (Qn7+6+0)} for @ (2n7 + @). INTEGRATION OVER AN INFINITE RANGE. § 20. I shall now suppose that the range of integration is infinite. Let us assume that, however great be y, Dd — a Y i Fu, (a) da= S| Uy, (a) da. ~B 0 o/s eam Ee 5 5 Then if lim Y | Un(a)da is determinate and equal to y=2 0/8 x | Un (a) da, o/s this equation passes over in the limit into [ S Un (a)da= | “Un (a) da. “8B 0/8 0 The additional condition which must be satistied is therefore that lim gf : Un (a) da=0, y=u Oy lim | eeaes 2 a@ da =0. or y=x2/0 0 Y Now let us assume that nal an a Ss “| Un (a) da o NeJB converges uniformly up to and including a=. Then it may be integrated term by term over (8, 2), and our condition becomes lim | cade | u(a2, a)da=0. ( Y y== ) This will certainly be satisfied if | ede | "u (@, a) da 0 ry; is unifurmly convergent up to and including a=; Le. if g “un (a) da Oy is uniformly summable up to and including a=a. Theorem VI. If the conditions of ILI. are satisfied for any finite value of y, however great, and Y “un (a) da 0 B 41—2 316 Mr HARDY, ON DIFFERENTIATION AND INTEGRATION is uniformly summable in (B, & ), and ner a yh a : > —] un(a) da 0 els uniformly convergent in (8, 2), for any finite value of x, the series Pf Un (a) 0 may be integrated term by term over (8, x ). § 21. Theorem IV. was designed to meet the case in which integration term by term is permissible, although the original series ceases to be uniformly summable at a number of isolated points. In the corresponding case in which the range of integration is infinite we need the following theorem. Theorem VII. If the series Sun(a) may be integrated term by term over (8, y), for any finite value of y, however great; and the integral series al Un (a) da converges and represents a continuous function of a for a=; the series Su,(a) may be integrated term by term over (8, ~ ). I need not delay over a formal proof of this proposition. § 22. We have next to consider how to extend V. in a similar way. Let us suppose that Yu, (a) is uniformly summable over any finite interval which does not include any one of a set of isolated points y;, near which it behaves as in V. No new point arises if the number of these points is finite, so we shall suppose it infinite; also yi< yin, and lim y;=2. Then for any finite value of y, distinct from any , 1=0 (1) P'S un(a)da=VP "un (a) dat JB 0 ols Now let us suppose that ¥| Un (a) da 8 0 J is summable, and that when y tends to «, in a manner subject to certain restrictions (one of which must obviously be that of never taking any of the values y;), the right hand of (1) tends to a limit equal to the sum of this series. Then lim 2) g Un (a) da = a (a) da. B 0 Oo/Bp I shall write this in the form P| ‘ Prin (a)da= ai (a) da, B 0 o Jp For a detailed discussion of the definition of principal values such as that on the left I must refer to the papers on “The Theory of Cauchy’s Principal Values” already mentioned. OF DIVERGENT SERIES. 317 Theorem VIII. If P ly P Un (a)da= g [om (a) da, Jp 0 o/s for every finite value of y distinct from any of a certain set of values y;; and if, when y tends to © in a manner subject to certain restrictions, y [’ Un(a)da tends to a finite 04 linut equal to 248 | Un (a) da ; 8 then P ‘ Prin (a) da= 2 | Un (a) da. Jp 0 oJ/B § 23. We saw in § 17 that if $(a) satisfies certain conditions bole P| tan ad (a) da= ¥(-y"2 ie sin 2na (a) da. J0 0 1 And if these conditions are satisfied throughout any finite interval of values of a 1 Nr oO pNr 5 Ei tan ad (a)da= Oe i sin 2na ¢ (a) da, _ for any finite value of N. Can we replace the upper limit by x ? Let us suppose, in the first place, that the series (-)r> i sin 2n« db (a) da “0 #8 is summable; and let us consider the series Y(-y | sin 2na (a) da 1 u Nr (—)=s =| sin 2na b (a) da /J0 7 1. J Nr 7D =| e*(@, V)de, - 0 where O(2, N)=R E | eria—zera dp (a) da | : J Nr The last integral is 1 seria - yr ia 7 . ons E to) (@)| - lh ee d' (a) da; and the second term of this is 1 ee rr ey i oA el : pea —2ia—xer'e py ae Se —rertia Y (nig 4! 4a? E $ @ |. | 4 da ©) ae, Nx 318 Mr HARDY, ON DIFFERENTIATION AND INTEGRATION provided this double integration by parts be justified. This will certainly be the case if, from some finite value of a, (a), ¢ (a) tend steadily to zero, as then ft $’ (a) da, ie ” (a) da are absolutely convergent. —% f N Hence ® (x, N)=— S- ob (Nar) + Ss where lap (a, NV) | < He*, H being a constant. o te Dn Now [ ee@(a,Nyde=[ 4]. /0 “0 IX We can choose X so that | oY de, which is numerically less than x is as small as we please; and the same is evidently true of Wn) [ b. | sinaa da 9 sin bal+¢e¢é 1 =97 § 25. In these examples the series of integrals is convergent, but all that is essential to our theorems is that it should be summable. Suppose, for stance, that we multiply by a cos aa ———.. ao Then, so long as y is not an odd multiple of 47, = fell cos aa tan 4 Sede = 9(-) fa BBE sua aire 2 9 AD fe 1 0 ty To prove this we have only to observe that, by § 23, d = | cos aatan a i a= | = Gay 4 hed 1 ae J Nr Nar if Na is any multiple of 7>vy; and, by § 16, 1 pe ada z Be al cos aa tan a———. = ¥(—)" | ¢ < 0 Oy 1 J0 * Two other proofs of these formulae will be found in the Proc. Lond. Math. Soc. xxxtv. pp. 61—65 and 83—84, and a fourth in the Quarterly Journal, 1900, p. 120. 320 Mr HARDY, ON DIFFERENTIATION AND INTEGRATION ra Ba 0: a cos aa sin 2na Now P [ —————— 0 ay = ada Br anaes 5 1 and so P} cosaatana——,=7 cos ay ¥(—)" cos Iny =5 7 COs ay. /0 ay; 1 2 ie) ae ada 1 Similarly P | cos aa cot a aia = GT O08 OY; 0 aly a = so long as vy is not a multiple of 7. In the same way we may prove that “= cosaa da a Jo cos bal—@& *sinaa da = —_. = 0, o sin bal —e if 0 €nh (2n7 + 8). Let us assume that val cos n (a — 0) db (a) da Ovy is summable, and seek its limit for y=. The sum of the series is as before proved to be * | e* O(z, y) dz, “0 where O(a, y)=R i ec) & (x) da. vy OF DIVERGENT SERIES. 321 We may suppose that y=(2N+1)7+0, N being a positive integer which tends to «2. Integrating by parts we obtain, instead of the last integral —(2N+1)7 ; = (2N-+1) sy a[ee na | fe = lee u a {e enw) | i —(2N+1) 7 Sere eons a vv (u) du, where y(w)=$(@—u). We suppose as in § 23 that ¢’(a) as well as f(a) tends steadily to 0 for a= The first term vanishes, and to the other two we may apply the same argument as before, which shows that lim | é)7O @-y) da = 0; 0 y¥=co Thus All (a4) da+ gf cos n (a — 0) 6(a)da=a7 S Enp(2n7 + @). B 1Jp The applications of this formula which we obtain by making (4) = are so well known that I need hardly give any. Wii, SOD IPA INE 42 XIV. On the Classification of Integral Functions. By E. W. Barnes, M.A., Fellow of Trinity College, Cambridge. [Received 9 January 1903.] INDEX. SECTION PAGE Introduction 323 Part I. The Asymptotic Expansion of Integral Functions of Double Linear Sequence. 1. Definition of double linear sequence. The ratio ,/, must not be real and negative . c 323 2. Order defined. The standard functions. : 324 3. The general form of the Maclaurin sum formula . 325 4. Its extension to the case of two parameters. 326 5. The extent of its validity 327 6. The asymptotic expansion of the eiaaiand oubie re 328 7, 8. The asymptotic expansion of log Pp (z|@, 2) - 328 9. The region within which the expansion exists : 330 10. The general function of finite order less than unity 331 11. The standard function of zero order 2 333 12. Its transformation into a function of triply- infinite sequence 335 13. The asymptotic expansion of log Qp(z| @,, @,) 2 : 336 14. The general function of finite non-integral order greater than unity . 3 5 2 : 5 336 15. The standard function of finite integral order not less than unity 337 16. The asymptotic expansion when p is odd 338 17. The asymptotic expansion when p is even 338 18. The general functions of finite integral order ; 339 19. Deduction of the asymptotic expansion of the double gamma Pace : ° = : 2 339 20. Conclusion Mr BARNES, ON THE CLASSIFICATION OF INTEGRAL FUNCTIONS. 323 Part Il]. The nature of the Taylors series expansions of Integral Functions. SECTION PAGE 21. Nature of the problem. Poincaré’s method of approximating to a, . : ‘ s 2 : 341 22. The case of P,(z) . , : ; ‘ 5 : ; : é : 3 i s : : 341 23. A transformation of P,,(z) . 0 C : : ¢ : : : : p : 6 342 24, The complete asymptotic equality when p=2 4 : : : : ¢ : : : 343 25. The complete asymptotic equality when p=3 : F : : . : 2 6 é : 343 26. The complete asymptotic equality when p=4 c : : . : ‘ : : : 343 27. The complete asymptotic equality when p=5. General conclusion . ; : c 3 : 344 28. The attempt to apply Poincaré’s process to the gamma function ¢ : 344 29. Approximation to a,, for the standard function of ae sequence of He, ibs fan Hates : 345 30. The case when p is a multiple of 2. . : 0 ¢ ; 346 31-33. The approximation to a,,~!’" for various types of zeros of Facing of onder less than unity . : z 5 : : : : : 6 0 5 : . : 347 34. The corresponding functions of double iene sequence : : : c ° : 2 : 348 35. General conclusions : c : 5 s : : : a 16 : = 5 6 : 349 36. The converse problem . : : : c : o . ¢ ¢ é : 0 : 350 37. Suggestions towards a general Hipory c : : : . : : ¢ : ‘ é 0 350 38. The function @p(z) of simple sequence . : : 2 : : : P 0 > 0 : 351 39. The corresponding function of double sequence . : : : : : ¢ ‘ : : 352 40. Application of Hadamard’s process : 0 : : : : 5 c : . : : 352 41. The case of functions of finite integral order : : : 5 : : 3 . 4 5 353 42. The case of the standard function of zero order and simple sequence : : ; : 353 43. The case of the standard function of zero order and double sequence c 2 a : : 354 44, Conclusion : . 2 : 5 : 5 5 . : - c : 6 o : 355 INTRODUCTION. Ixy a Memoir on Integral Functions* which was recently published, it was shewn that such functions could to a great extent be classified by their behaviour near infinity, and asymptotic expansions were obtained for simple and repeated functions with a single sequence of zeros of finite or zero order. In the present Memoir I continue these researches. Again I thank Professor Forsyth for his kind criticism. Part I. The Asymptotic Expansion of Integral Functions of Double Linear Sequence. § 1. In LZ. F. § 20f it was pointed out that the typical zero of an integral function may be a function of two or more numbers necessary to define its position in the series to which it belongs. When such zero is a function of but two numbers it is said to belong to an integral function of double sequence. Among such functions the * Phil. Trans. Royal Society (A), Vol. 199, pp. 411—500. + In this way I propose to refer throughout the present investigation to the earlier paper just cited. 42—2 324 Mr BARNES, ON THE CLASSIFICATION OF INTEGRAL FUNCTIONS. most simple are those for which the general zero, a say, is a linear function of the numbers, e.g. a=f (a+ na, + 2,2), nm, and n, being the two numbers and a, @, and @, arbitrary parameters. A function with such a general type of zero, I propose to call a function of double linear sequence. The most simple examples of such functions are the Weierstrassian o function and the double gamma function. If n, and m are positive integers and @,/@, is real and negative, a+n,@,+%@, will by suitable choice of mn, and x, represent a point indefinitely near any assigned point on the line joining a and a+@,: consequently this line will be a line of essential singularity of the function. In this case Eisenstein’s series = 5S Jw. Barat n,=0 n=0 (a +2 @, + NW)” not be convergent if p>2. We shall assume therefore that @,/@, is not real and will negative. We shall assume that the general zero is such that both m and n, range from 0 to o. Thus the Weierstrassian o function is a function with four double linear sequences of zeros and is the product of four functions of the type which we proceed to consider. § 2. We shall put, for brevity, Q=n,o,+,,: and then the general function to be considered will be (=z)? TU 2) emt P at Ly it + F@+0)| e PIf (a+)? | where p is an integer independent of m and n, such that Soar is divergent, il a al CEO a is convergent. It will thus be seen that we do not consider functions for which p is infinite for n, and . Corresponding functions of infinite order have been put on one side in the previous discussion: they present peculiar difficulties (v. 7. F. § 12). When », and n, are large and f(a+) admits an expansion in descending powers of (a+) of which the dominant term is (a+), the function is said to be of algebraic sequence, p being real and finite. If p>2, the integral function may be written i [1+ ero for p is bow zero. Mr BARNES, ON THE CLASSIFICATION OF INTEGRAL FUNCTIONS. 325 The order of a function of linear double algebraic sequence is defined to be 2/p. Thus, if the order is less than unity, there is no exponential factor in the typical term of the Weierstrassian product. We shall consider in turn the three standard functions :-— (1) of order less than unity, (2) of non-integral order greater than unity, (3) of integral order not less than unity. These functions we take to be oa aw Z = est Ne 2. Neng) a= HE i er =|. bere era -lill Nd Se we Q. (Z| ar, 2) = Mt ut (2 ¥ (a+ Oy) rae \(@+ OY bes p(at mall , Mm Ny where p is <2 and 2/p is not integral, and where p is an integer such that p+1>2/p>>p; _= & Zz : —2 (—2z)P B (elo, 0) =H [1+ @ Fy) 7 lezayt + para | Ny, No where 2/p is an integer p. By comparison with these standard functions we obtain the nature of the asymptotic expansions of all functions of double linear algebraic sequence. It should be remembered that repeated functions of simple sequence arise from functions of double linear sequence by coalescence of the parameters (J. F.§ 20). Thus many of the results of J. F. Part IV. arise by equating the parameters in the ensuing investigations. The Extension of the Maclaurin Sum Formula. § 3. The Maclaurin sum formula, which was fundamental in the investigations of m-1 I. F. Parts I{l. and IV., gives an asymptotic value for = (nm) when m is large, under n=1 certain restrictions as to the nature of the function ¢(n). An investigation of this formula, more satisfactory than the usual one based on the calculus of finite differences, was given in J. F. § 41. By the methods there developed it may be easily seen that the formula can be written in the more general form "S 6(@+na)=Z+ S ee ie i oe | =mu n=0 a n! da” =mw where Gas a; §(—s, a, @), if d(#)= 3 Ag2. 8s=0 s=0 326 Mr BARNES, ON THE CLASSIFICATION OF INTEGRAL FUNCTIONS. § 4. We proceed now to shew that, if ,/@, is not real and negative, the asymptotic m,—-i ™,-1 P P expansion of £ = (a+) may be written in the form n,=0 ng=0 SA) ae Gh as S) . Z+ P| 3 vol, Pe a a (1), n=0 where $(«) represents the integral function = a,2* and Z= 3a 7 s=0 r=0 pa (0) = | * dx | “ (a) da. The notation for the double Bernoullian numbers is that given in my paper “On the Double Gamma Function*.” The symbol F, is suggested by Cayley’s notation of matrices and was introduced in § 49 of the paper cited. It is such that F, (f(a + 2)|c-me =f (a + m4; + M02) — f(a + Mo) —f (a+ ms). In § 40 of the paper cited it was proved that ns? —~_~ | e™(-—z) dz= se L where n?=e"" and logn. has a cross-cut along the axis of —1/Z, and is real when n is real and positive. The axis Z coincides with the bisector of the smaller angle between the axes of 1/@, and 1/,, and the axis of m les within the smaller angle between the axes of @, and a. Hence, if a is positive with respect to the o’s, m,—1 M,- m,-1 My=1 © 2 = @(a+Q)= 2 2 ae > a, T (1 +r) e~@mertmen z (— 2) dz n,=0 n,=0 2,=0 n=0 47 J r=0 == ee ah Te zn | i eo? _ Le e- ez Take now 2 GL (t=) e-% fe i ee ! ==> (- zy qd ae Ea) el =s a) dz, it being assumed that $(«) is such that the integral is finite. Then the given sum is +f $aPQ+r) Sieniiay. 2+¥lge[ 3 Co ase) |, equal to z= ig Oe) ee) i =Z+ F, Be =, (- zyrh ao) ( 2) i eae as we see by the employment of the expansion mr in §15 of the paper cited. Thus the given sum is equal to Z+F.| § oS, (a) t [s 2 a,0 (1+7r) edz | n=0 r= z2=mw nt Qa} yao (— 2) =7+ Hi eens a, (1+r) ia Reh DADE Ey none T(r+3—n) eis * Phil. Trans. Royal Society (A), Vol. 196, pp. 265—387. Mr BARNES, ON THE CLASSIFICATION OF INTEGRAL FUNCTIONS. 327 Now 2 a, (1 +7) d” 2a, A+r)a" 1202" T (r+3—n) = or T(r+3) = & w(x), where [rae le (x) dx = (a). Thus the given sum is equal to 2 (2) Hes eS Ons a (a | n=0 da z=mw* We have assumed that ¢(z) is such that ae a,U (1 +7) Pass ie Or} gy (—2Y". (l—e-™) (1 —e-™?) is finite: this will be the case if = a,['(1+r)2" has a finite radius of convergence p, and the bulb of the contour along which the fundamental integral is taken entirely includes a curve of radius 1/p. a Ge : : = 3S,” (a) We cannot, however, expand in the series > — ~~ (—z) (1 = (Fa) qd — es?) n=0 n! : ae Qare| | Qare : unless |z| is less than the smaller of the two quantities | iB i If then p is | a, | Ds | greater than the greatest of = : = , the theorem will hold good. And by taking account of the terms which arise when the bulb of the contour of the integral closes entz (l—e?) 1d —e ™?)’ of the theorem which is true when p has any finite value, and therefore when ¢(z) is a function whose order is greater than or equal to unity. and therefore passes over poles of we may obtain a modified form ~~ When p is greater than the greatest of ls |: ee , we have | Qare | 2are 2 2 a4r (2) Z= > 2(—7, @| a, @,)= > (Ee aa & ( 1 2) 5 1+r Corollary. On making a=0, we find -1 m— : zi eSignal =0 n= @, + @, < B = Z— d (0) slehecars: me [yy («)]-— ma 2P nti d” 7 FLY @leamot & YF, |“e (@)y ’ 20, Oo CEO \ er where .B, is the nth double Bernoullian number [Double Gamma Function, § 7]. § 5. The integrals used in the foregoing formulae are only valid when a is positive with respect to the o's. But it is now evident that the fundamental asymptotic equality [(A) § 38, Double Gamma Function], which (§ 41) is true in general if the 328 Mr BARNES, ON THE CLASSIFICATION OF INTEGRAL FUNCTIONS. many-valued functions with s as index have their principal values with respect to the axis of —(#,+,), may be written m,—1 mg>1 1 ~ oS. (2) (a) d® geste > > ——=464le,,0) 4+ > —— fF |\55>— So (2), n xr=mw n=0 n=0 (a+)? ay oe da™s—1.s—2 and that therefore the fundamental formula (1) holds for all values of a, w, and ao, a2 provided = a,f,(—8, @|@,, @,) is a convergent series. s=0 The formula is true in many cases when ¢ (x) is not an integral function. Such cases are best considered as they arise. § 6. If in the foregoing formula (2) we make s=e, where e€ is very small, and after expanding equate coefficients of first powers of e, we obtain m—-1 m-1 SS og (a+ 0)=-2 6.66, ale, ont 2 © F, lao 9 (8 2 1-»] n,=0 n.=0 : n=0 r=mw a formula which may also be written m,—1 m,-1 ie (a) es ee ae oe (a) log IT Il (a+ 0)=— log m=0 nQ=0 Pe (,, @) n = Sn” (a) xe = n! Py BE 2 esses »| ; (Double Gamma Function, § 49.) On differentiating with regard to a, we have AS Se eee aps (a) + 2 (m + m’) mreS,' (a4) + =? Sn8 we) F, “(@ log « — x) m=0 n=0 &+ 0) i oa n=0 n! Se extn | and m,—1 m-1 1 : wo WS. (2) Sn” (a) > > (2) 2 y (2) => Sek ees: ay = yr," (a) — 2 (m+ m’) wi,” (a) > ae F, E , log = ae These formulae are more convenient than the corresponding forms obtained in §§ 51 and 52 of the paper cited. The logarithms imvolved have their principal values with respect to the axis of —(@, + @.). Functions of Finite (Non-zero) Order less than Unity. §7. We proceed now to find the asymptotic expansion of log P,(z\@,, @,), where p>2. The process adopted is the direct extension of that previously employed (Z. F., § 50). In consequence it will be assumed that the theory of asymptotic series developed in J. F. Part II. is familiar to the reader, so that merely the bare outline of the analysis need be given. Suppose that m, and m, are two values of m, and m, respectively such that |z), |m,o,|, and |m,,| have, when |z|, m, and m, are all large, ultimately a finite ratio. Then ; m,-1 My- m,-1lm.-1 k (_)s—1 a Q)ps log P, (z)=m,m,logz—p = = log(a+Q)+ 2 = & We n=0 n =0 nm,=0 n=0 s=] Ss. m,—1 mM-1 x x k —)\s—1 +| Sue ace. | sid Ge ah ep ee 3 m=0 2=0 m=0 %=0_] s=1 S(a+ Ps Mr BARNES, ON THE CLASSIFICATION OF INTEGRAL FUNCTIONS 329 in the limit when & is infinite. When for n,=0, 1,...... ,>m-—1 =O! Ty cosane > M —1, la+Q). ae ae. ; = | is>1, or for the remaining combinations of n, and m, I—5 is >1, the series | | 7 |\@eaF arise as summable divergent expansions of log (1+), the validity of which was established in J. F. § 26. On substituting the asymptotic expansions of m,—1 Mz—1 m,—1 a 1 = 2 log(a+), > (a + 22), 2, =0 NQ=0 N,=0 Ns = . given in the previous paragraphs, we obtain, in the limit when & is infinite, the asymptotic expansion 7) 2 Sn" (a) d®™ (a 32° — - —p.—— ©,(S, a)s= >2 = log P,(z)= mym, log z — p : ae & (S, @)s=y + 2a 138 Fe (5 log Z Ne a k (—)s f ; x on (a) d”™ ¢ gpst2 S =o SOE " : ee sz 1b eee 2D Ste n! Pr, ae (@ = 1) (ps + Bis - © .{ 2 28.%@) [= at zap ss 1h) ese : Ee s “| n=0 n ! . da” (ps ae 1) (ps == sl) 5 xe On rearrangement we get, since mm,=.S,” (a) PF, A 2 2 ie Pi @)=p— Ge = og P,(z) =p aS £2 (8, @)s—o + yet C2 (— ps, a) . ae eee a Y (2) = [z , < +8, (a) F.] 5 log # + p 4 AoE © 2 log ge 7s 8. (a) F. 1 5 ae + 2S, (a) |= A Hl Lg Migs SS mae = 2Snz2” (a) d" / & (—) a ) > ar! ss a ce =o (n+ 2)! P, Fae Sp ples z=mw § 8. In this asymptotic expansion the coefficients of the terms .S,(a) for n=0, 1, 2...%, are summable divergent series in the limit when k=~, and must be ‘summed’ in the same way as the corresponding series which arose in J. F. § 52. In that paragraph it was seen that, if —a = ee Je pats {s+ (2/p)} ie ) au 4 2sin (277; /p) (3) Ties Vou. XIX. Parr III. 43 330 Mr BARNES, ON THE CLASSIFICATION OF INTEGRAL FUNCTIONS. Hence , opE Fe a ge eae oe 2 ieee ay plge+ 2 S(ps +1) (ps + 2) 2 Sethp 25 tS Sp? ow ( ie ap E NO ee ee aes s+(/p) s+(Qip)j 2 | ee ea ek Sipe OE pees p | ee x 798 z 2sin(27/p) 2 +58 az m2le ~ sin(a/p) 2sin (277/p)’ Differentiating with regard to 2 we see that k (-)> gst Tr gllp —peloga+pa+alogz+ > ~*——_=__—___.. pee ee 8 s=-k S(pS+1)2 sin (77/p) We have then on substituting the values of the summable divergent series rs) x _)\s-—1 log Py (2)= px b(8, Bow & Eps, a) . deal i 2/p +S, (a) F, E sin (77/p) 2 sin (277/p) ° z=mMea (a) walle se Es E (7/p) Snes (@) SS +2 ote es (n + 2)! F, E a= log )| - r log | Z=Mw 7z2lP =P E(8, Ano & OE (= ps, a) +280), Iain (2a/p) 2S, (a) azile 252”! (a) ““T! sin@a/p)? 2 8% for PF, []r-mo=0; F.[A]r-ma=—A, if A be constant with respect to z. § 9. In the proof of this asymptotic expansion we have had to limit ourselves to the inequality — 7 < arg (#°/z)<@ in which #= mo, or m.@, oY m,o,+m,. Now if, in the Argand diagram with O as origin, ,° be the point A, and ,? the point Ag, all points of the type m,o,+m,, lie within the angle A,OA,. Hence if A,’ be —@,, and A,’ be —o,?, all the zeros of P,(z|@,, @.) lie within the angle A,’/OA,’. The inequality —7™ instead of > = , and similarly >> instead of > > , as a con- 4 Oe m=0 nm=0 0 m=0 2 =0 venient abbreviation. ] For, if &(f) denote the sum of double extended Riemann ¢ functions obtained from (xv) in the double Maclaurin sum formula, we have 29(+9)=69)+ SO nye n c=™Mw eer y= | a fp (@) da. Hence, employing the same process as_ before, m—1 log F, (2 | @,, @2)= m,m, log z— YE log (a+ OD) 0 aS: SS 3s (>=) Oia O) +(23 “ SS) Gir " 0 s= = sz 0 s=1 sp* (a + Q) ms = xe =.S8," (a) log zF . 2 x=mMmw ms [de ik log $ («) ae 3 — {6 (g) + = y ee sto ae if w(ede] wo (-)§ ge (2 oSn”) Sn (a) P| d” (i = | > S Bs 1 s 2 =0 n! da” ae, 2 a) Lx=mw ; Thus we have the asymptotic expansion - {& (log 6) + & & log F (2) =~ & (log g) + & OF (g) - SO tog s sz aS a Pilar is log e—[° de | log $ (a)da+ 2 ) ff de fo == ts aa! n=0 n! z=Mw when the limit for /=o of the summable divergent series is taken. 43—2 332 Mr BARNES, ON THE CLASSIFICATION OF INTEGRAL FUNCTIONS. =.0; therefore in the limit when k= =~, k —\— 3 (> log z—log d(x) dv+ > i 2 ey 0. s=—k sz Hence, on integration, sz a) z log z— ih daz i log @ (w) da + ee il de | eee) dx = Ax+B, where A and B are constant with respect to 2. Unfortunately we cannot determine A or B unless we first expand log¢(#) and : ‘ c ¢*(x) in powers of z If $(x) =a? ji+ +S =| where p>2, and 4, &, ... are positive quantities arranged in ascending order of magnitude, we can obtain, as in the corresponding theory of J. F. § 58, approximations to A and B to any desired order. The work is laborious, and so obvious an extension of the previous investigation that it may be supplied by the reader. Similarly it does not seem possible to express in a convenient integral form the functions ¢,(¢°). They must be evaluated from the expansions of $*(#) as occasion arises. We have finally, when |z| is large and not within a finite distance of the zeros of F(z), the asymptotic expansion log F=— €, (og ¢)+ > s— == ae (9) ee roger 3 = Sn SO nS (4 +B] n=0 ns =— f.(log ¢) + > 5 On £. (6) — 8 tog 2 - B.S," (a) — 4S" (0). It is quite possible that infinite terms arise in the expansions of A, B, &,(log ¢) or £($%). In such cases the methods of the calculus of limits must be employed. An example is given subsequently when the asymptotic expansion of R,(z|@,, @) is deduced from that of Q, (Z| @,, @»). We now see that if the zero of an integral function of double linear sequence be given by a=(a+0) [ae aes A, eI (a+ Q)2 (a+ O)® when p is greater than 2 and the e’s are positive and in increasing order of magnitude, the term of highest degree in z in the asymptotic expansion of the logarithm of the function will be 20"! (a) 72"? 727? 2 sin (27r/p) ~ Q@,@. Sin (27r/p) Thus the power to which z is raised is the order of the function. When e¢, is >1, the next term is of order 2", Mr BARNES, ON THE CLASSIFICATION OF INTEGRAL FUNCTIONS. 333 Application to Functions of Zero Order. § 11. As was seen in J. F. § 64, we cannot in general, without the introduction of functions hitherto unknown to analysis, obtain the asymptotic expansions of the sequence of functions of zero order. We can however give an elegant form to the asymptotic expansion for the dominant function of the sequence, viz. : Be Zz F(z) =U E + Ga n=() For we have at once, on carrying out the process previously adopted, 8 (4+) oO Les = s—1 58 e iss BCE a rea = = ses (G40) * m-1 fe} log F(z) = nama log 2 — 2% (a +O) +22 > (-)> n=0 s=1 n=0 n= mee rx Sz . é 1 « Now | de | edz=——-——-, Ss _ and therefore, as we see by expanding e* and substituting in the double Maclaurin sum formula, SF etn) _ 3 80. (—7, Oe S: Sn (a) d” | an 0 r! aon: da” \s Ss? 8] \acanes Eee aS rr (2) br 25,” (a) ca 25") (a) i Sane @) p d” (es p20 (EL)! s s ay ol Hs dx” 18?{ lem NG ZS 8 Sir (@) _ ea _ 2819 (a) _ 2S”) (a) =o (r+)! G-e)d—e) gee Le ee = 2Sn” p d” (es therefore ae is = = } n=0 Pal — eas) al =e) nN: >| da” | s? x2=mw And (Ss - 33) eset) = 5 Sn" @) p | a = 0 0 n=0 n! | dar I! s v=meo ee 2 S,° (a d”® «& Also Ba 0) — Ca, iM) SS Se Sn ® p an = = n=0 n! = da 3 c=mw Therefore, in the limit when & is infinite, = ae ARG fs = 2Sn” (a) ip | a 28 log F@) 5 log ; e Re b: ( 1, «) rear n! Hs da” 3! v=mw —)s—1 pas Gyre k (-)s f S28) (a) d” / es Sy = > = be a F, =a = £2) > oe 2 en ee eS 6 tna cake ie ee) seme) 1 8 =e") — a4 48 334. Mr BARNES, ON THE CLASSIFICATION OF INTEGRAL FUNCTIONS e—1 pas log F(z) =— (— il: a+ 3 a eas — 2S," (a) F; [a log Z)r-me — Thus Se ele ehees 2 (a) d* ed rd k ez | > : - l > ; nae n! + ES 3! i 2, °8 : a -k =e \ zZ=Mw Now we know that*, if —-7r<@<7 = Get DCm ae ze) few 8 7 ED! Qa 2 Also (Gamma Function +, § 16) ioe ie! Ave 8; (2)=7-_t+y eb, . Ss (w@ +4) =(a@+ 4) — 8 (w+ 4 + 3B, (a +4) ca 1 Sa since B.=5- Therefore, when the limit for k= of the summable divergent series is taken 2 23 k —)s— gst —ait 5 get >, GIe aaa aoe Se +5 loge +5 (w= log 2) s=-— a L x , (logz)s a 7 = 5 L082) = ara ot log z+ 5% Thus log F(z) =— €,(—1, a)+ aaa 2S: '€ m 5 io (a) ie e is —e™")(1—-e#)e" ot d” (x sp care (logzP om) Acti emule 2 | {5 oe) +31 Pees 31/8 af ame - _ Sp : (a) s (-/te ASH (a) 2 = 2 Estee s(1 —e"*) (1—e™) a1 log z + = ) {(log z)? — 7? log z} — 2S," (a) oe ch at > which is the final asymptotic expansion required The zeros of F(z) are given by z=— e(a+Q) If w, = pe, @, = = p.e* and a=|a/e', they are therefore given by z= e! 41+, p; COs $;+M2P2 COS $e et(atm, 7, sin 6,+ mg, r sin 6,+-7) * See a note by the author in the Messenger of Mathematics, No. 379 (Nov. 1902) + Messenger of Mathematics, Vol. 29, pp. 64—128 Mr BARNES, ON THE CLASSIFICATION OF INTEGRAL FUNCTIONS. 335 Hence, in general, the zeros cover the whole plane near z=. The exceptional cases arise when 7,sin @,+7,Sin@, are submultiples of 27, when the zeros are distributed along certain sets of lines at infinity. In such cases the asymptotic expansion which has been obtained is valid. § 12. If we substitute e? for z, we obtain the function ce) e? at E + aa | with the asymptotic expansion S’@_$ (Se ai (a), aS (a) , (2 m= - one = 3 — az) — 8 @) Fe pull 2 + > sd —e)(1 em) eT 1 Z+ 31 (2 — 7°z)—.8,” (a) {5 4 ae The function has zeros given by =@4+04+(2n+1)m, n=—o...... Dera OP nee oO. The zeros thus form a triply-infinite sequence, with two single sequences and one double sequence in the set. If a=aqt+be @®,=( +d) Wo = Co ar a the zeros are given by Z= 0+ M+ MC, +6 1b, + md, + md, + (2n + 1) 7}. If ¢ and c, are not of the same sign, the function has zeros at every point of the plane (though of course the points do not form a perfect aggregate) and is non- existent. If c, and ¢, are both positive, the zeros of the function mass themselves together at infinity in that part of the z-plane on the right-hand side of the axis of y: if both negative, to the left-hand side of the same axis. Corresponding to every value of mc,+mc, there is a line of singularity perpendicular to the real axis. In the former case the asymptotic expansion exists only in the region at infinity on the left- hand side of the axis of y; in the latter on the right-hand side of that axis. In connection with the ‘imvestigation of the present and preceding paragraphs it may be noted that the application of slightly more general analysis gives as the extension of the investigation of 7. F. § 61 the asymptotic expansion log I [1 pares | Hes (412) _ g(a) log z n=0 | e" +nhw »\2 o (_\r—-1 ar +S, (a) ee ge eS Gye g Dees ( \euay 2 nay Te" —err” 336 Mr BARNES, ON THE CLASSIFICATION OF INTEGRAL FUNCTIONS. Functions of Finite Non-integral Order greater than Unity. § 13. We can now readily obtain the asymptotic expansion of _ Z ee ee Paced alan Qe (Z| m1, @2) = int | (1+ @iny) ?} Cae where p is <2, 2/p is not integral, and where p is an integer such that p+1> 2/p>>p. We have m—1 log Q, (z) = mm, log z—p == log (a + Q) 0 S (a+ Oy Be ye pS Ee (Ga Oe m™ + SS 1 sz* 0 ML il M8 a) m-1 3) (Gamer +(35-23) S s=p+ 18 (a + OQ) = 8," (a) log zF, B r=Mo 7) x Sin! we) [ a 1x a 3a? p [=e £2 (8, @)sao + 2. n B, | da” \2 eee 4 Meg 25S) sealers : ae 20) |e get + eres =a 1e ( Ps, a) ok =s F, di” (e ae 1) (ps+ Fe } a pa ells ~ € (ps, a) CMe [5 Pip (ee Hehe 3, nm! “ Ge es Dn 0 Sh (eye So” (a) =p — Pe aN 2 Pp os o, (s, Q)s=0 es sok fa ( Ps, a) 2 ! log a Tp wSi(a) ete Osea On) Ita SENG): ave BS; (2) This expansion holds, as before, for all large values of |z| such that z 1s not near the zeros of Q,(z|@,, @»). § 14. It is now evident that if we have a function G,(Z.@:, @) of order 2/p given by the product on (h+ sccot tgerm*~ taper Or” Mr BARNES, ON THE CLASSIFICATION OF INTEGRAL FUNCTIONS. 3387 where p is an integer, such that p+1>2/p>p, and $(«x) is such a function that >> oe is convergent, and >> of log G,(z|@,, @:) is given by oe divergent, the asymptotic expansion = G (lo ) ate >) ee (G ($°) esa @log2 — B.S,” (a) = AS, (a), sug apes 2! sz where A and B are determined by the equality x rc k a xz (_\8—1 fs 5 loge-| dee | log 6 (a)dx+ > | de { OP EO) dy = An + B s=-k When, |a+ | being large, ¢ (a+) admits an expansion in the form (i Co (a+Qy [2 ‘Geo Geos | ; where p is <2, 2/p is not integral and p+1>2/p>p, and 4, &, ... are positive quantities in increasing order of magnitude, approximations to A and B can by sufficient labour be obtained to any desired degree. The term of the highest degree in z in the asymptotic expansion of log G,(z) will aS” (a) mz 2 sin (27r/p) ’ and when ¢ is greater than unity, the next term is of order z?. Functions of Integral Order not less than Unity. § 15. The standard function of integral order greater than or equal to unity is ise ial Zz 2 (—z)? R, (Z| a, @2) = _ kt + @+ Oy + Op exp (a+ Oy Kiss. "5 (a+ Oy | where 2/p is an integer p. If we take the function Q,(z|@,, @.) and make 2/p decrease down to p we arrive continuously at the function #&,(z). [Did we make 2/p increase up to p we should have to insert an additional exponential factor in each term of Q,(z) before we could arrive at the function of the type R,(z).] Consequently we may put Pre where ¢ is a small real positive quantity. Then in the limit when e=0 we shall find that in the asymptotic expansion for Q,(z) infinite terms cancel one with another, and we arrive at the asymptotic expansion for R, (z). Thus the asymptotic expansion for log R,(z) will be given by 2 a 2, (-j Qs 28, (a) ence Oe S A EAL, pa Een PUD Say sz a ea 2! Lee pte BN n\n Pe A + 28s Cy oe (HERE) sf (a) = pre |: sin ( 5) )™ There are two cases to be considered according as p is odd or even. Vou. XIX. Parr III. 44 338 Mr BARNES, ON THE CLASSIFICATION OF INTEGRAL FUNCTIONS. § 16. If p be odd, the only terms which become infinite when e=0 are (-2)° » (a) mt p n(2e. a) +, S,' DD Seman EBe) These terms may be written (S22 (2-* 2e (2) ™ (— 2)? = (alle +p a) +28 (1) 55 ze Now (Double Gamma Function, § 55), when s = 2, f.(s, a)= me = +." (a) — S, (@) — 2 (m + m’) 7e28," (a). Therefore the two terms are equal to Bare — 18,9 (a) + yo (a) —2 (m+ m') eS," et on iD 18 +25, (a) | 2 (1 + elogs +.. a = c= 5 - 2So” (a) (1 _ = — oS," (a) + Yr." (a) — 2 (m+ m’) 78,9 (@ + 2S,” (a) = + 9S,” (a) et 9 (2) Q (2) os e z)P \- 2S) : (a) at oe + 5,0 (a) 82) + ae {yo (a) — 2(m + m’) mre 8," (a)}. We thus have the asymptotic saa itt Meola 2s log RB, (2) @1, 2) = = £2 (8, @)s—0 + a aaa g on pe a) ge ‘s a4 {Yo (a) — 2 (m + m’) mre oS, (a)} — 2S.” (a) = sin (p7/2)’ where 2/p is an odd integer p. § 17. When 2/p is an even integer p, the terms which become infinite in the asymptotic expansion which, when e=0, becomes that of log R,(z), are ae 2p ies dae (2 ) > por pee alee ance pte ogPte wz 7 +18, (a) 5" — 28," (a) 2 sin 7 (p+e) sin PT & 9 Mr BARNES, ON THE CLASSIFICATION OF INTEGRAL FUNCTIONS. 339 Now (Double Gamma Function, § 55), when s=1, oS,” a €.(s, a) =— — yp, (a) + 2 (m+ m’) TS," (a). Therefore the two terms, which were not introduced when p was odd, yield 2 *) (a) Reet € Fe ayer Sr) (p +) —wWe (a) +2 (m+ m’) we S,” (a) — 2 (—2)?".8,° (a) {1 + 5 log 2 9 =i (— 2)PP {5 (a) — Wr’ (a) + 2(m+m’) mS,” (a) -F 25,” (a) log zt : Hence when 2/p=p is an even integer, the asymptotic expansion of log R,(z|@;, @») 1s 3 Sn Gea 5 eL 25% ie) == 825 aot > BSS cee bs 3 1 bo s=-ptl sz* (EO) Serve j + ee {aro (a) — 2(m + m’) we 8," (a)} So" (a) = aN! (a) 2 2p + (— 2)” \- +S." (a) “ge Peay a z)P2 1,8," (a) =2 5.0 (a) lo z Pp 21 = 1 s 2 oa (— 2)? {yp.’ (a) —2 (m+ m’) re oS," (a)}, the double accent in the summation denoting that the terms for which s=0 and s=—p/2 are both to be omitted. The expansion holds for that part of the region at infinity which is not within the acute angle formed by lines in the Argand diagram drawn from the origin to the points —,?, —@,. The logarithms are to have their principal values with respect to some line, say that to the point —(#,+,), within this angle. § 18. It is now evident that just as we have derived the asymptotic expansion of R,(z|@,, @,) from Q, (z|;, @:), so we may derive that of the function H(cjo, 0-00 [f+ germ]? eae) t tape sO] of finite integral order p not less than unity, from the asymptotic expansion of log G,(z|@,,.) by making the order 2/p of the latter decrease down to p. In all infinite terms which arise we must employ the calculus of limits. § 19. It is interesting to deduce from the foregoing asymptotic expansion that obtained for the double gamma function in my memoir [§§ 78—86]. For this purpose we put p=2 and then we find p=1, and that log TIII l( apt )e-aratac=a| 0 19} 44-—2 340 Mr BARNES, ON THE CLASSIFICATION OF INTEGRAL FUNCTIONS. admits the asymptotic expansion é ba a as £2(S8, @)s—o + +3 f(—s, a)+.8/ (a) logz +2 (8. (a) (log z— $) + vr." (a) — 2 (m+ m’) eS," (a)} z {S,° (a) (log z— 1) + yy’ (a) — 2(m + m’) 2S, (a)}. Hence in the limit when a=0 Z za Ze Ss us log a — log I’, (z) — 7 Ya 2Ym = @ = (—)*1.812(@) , ~ Os o2(8, @)e=0 i ape Ie aSie(0)ilog =F 5 £8, (0) (log z — 8) + yo” (a) — 2 (m+m’) 7S, (0)} z {8,2 (a) (log zZ- 1) + vr’ (a) —'9) (m o m) 7 8, (0)}. Now Ya = Lt \n- vp.” OF =0 = Lt [7 J (a) Ya = Lt | a vi ae a) a5 £2 (8, @)s=o = Lt {- log a — log p2(@1, @»)}, a=0 =0 as we see in “The Theory of the Double Gamma Function” (pp. 329, 330 and 333 respectively). Hence, on substitution, we find 1h (z) eer (m+m’) oS)’ (0) ] °8 P2 (@, @») = 2aru(m + m’) S,' (0) + = a ae + log z |- - oS, (0) — 2.8,% (0) —.S/ Oo) +2(m+m’) me 5 25,9) (0) + 248, (o)} ar 52 8,0 (0) + 22S,” (0) = — Sy (z) {log z— 2au(m+m’)}+ = 2 3 Oe 7 - i =F Sy (0) +228, (0). In this formula log z has its principal value with respect to the axis of — (w,+.). There is thus complete agreement with the former result. [Double Gamma Function, p. 380.] Mr BARNES, ON THE CLASSIFICATION OF INTEGRAL FUNCTIONS. 341 § 20. We may now conveniently conclude the investigation of functions of double linear sequence. An attempt to apply the same methods to functions of non-linear sequence leads to analysis of great complexity: the study of functions of multiple linear sequence may be conveniently postponed till after the publication of the theory of the multiple Riemann € function and the multiple Gamma and Bernoullian functions. Part II. The nature of the Taylor's series expansions of Integral Functions. § 21. In the previous parts of these investigations we have considered the asymptotic expansion near infinity of integral functions defined as products of primary factors. But an integral function can be equally expressed as a Taylor’s series absolutely convergent over the finite part of the plane. And therefore to complete a discussion of integral functions we ought to classify functions defined in such a manner, and therefore to obtain asymptotic expansions for such absolutely convergent series as represent functions of finite order. Now when we are given a Taylor's series the problem of finding the asymptotic expansion is in the highest degree difficult. For from the expansion itself we know _nothing of the nature of the zeros of the function*: we do not know whether they form one or more sequences, whether such sequences be repeated or non-repeated, whether they be finite or multiple; and so far no method has been suggested for solving any of these questions which are intimately connected with the asymptotic expansion of the function. Suppose however, conversely, that we know the asymptotic expansion of an integral function f(z), then as was first remarked by Hadamard+, we may evidently apply Cauchy’s theorem and determine the nth coefficient a, of its Taylor’s expansion from the integral Lao) Qare | gh , which is to be taken round a circle of very large radius on which the value of f(z) is asymptotically known. It might be thought that in this way for the functions investigated in J. F. Parts III. IV. and V. we could theoretically determine a, to any degree of accuracy. This is however not the case, for while we have obtained complete asymptotic expansions for the logarithms of the functions in question, we have done nothing of the kind for the functions themselves. § 22. Let us consider as an example the function which has (J. F. § 50) the asymptotic expansion e™"/2rzt when —m E C22) Nexp— ey cory | . (278 -=0 sin (7/2p) § 26. So far the results are comparatively simple, but when p=4 and =e", complexity begins. 2 9 We have l+o+o?+o'=——, 1+@—o?—o=— , ie 1-a@ ena eee leona =e @ (1—@)’ ow (1—@*)’ 1 = a 2 l—o+a? Bes eo eee Ome (ae a Oe laa)? Qe? 1-—o-—@—o*= 2 toate ; 1-o w* (1 —@)’ 344 Mr BARNES, ON THE CLASSIFICATION OF INTEGRAL FUNCTIONS. and the remaining set of eight summations can be obtained from these by merely changing the signs. Hence 7 f — 1 > 2rm\—} 7 err 2 = cs — rire /2; P,(z)= Qn) =o |e ) fexp einen) (ermz)? + exp — (Bz/2p) (e™™ zy | ’ § 27. When p=5 and w=e™*, there are 32 summations. The result is 9 1 . 21 7 2rire\1/2 Pul®)= Gary 0 [eon je sin (wp) “> — exp (ze2"™)¥2e + exp ) com : — 7 ULF sin (37r/2p) sin (577/2p When p=6, terms of a different type appear in the exponentials, and the result is still more complex. It seems hopeless to try to obtain any general formula. Then too when we have the functions P,(z) in which p is odd and greater than unity, the method which we have employed is no longer open to us. We might reduce such functions to products of gamma functions; but of the asymptotic expansion of the gamma function we know only the first term; and since this term is infinitely many valued we can but anticipate that in the complete asymptotic expansion there are an infinite number of terms similar to the dominant one. For since the gamma function is a uniform function of z, the complete asymptotic expansion must also be uniform. [It may however be urged that each term contains a divergent series of true asymptotic type in its index, and that this series with the conceptions of J. F. § 30—35 may itself have a countervailing many-valuedness.] § 28. The gamma function itself well illustrates the difficulty of applying the process of § 21. If G(z)= re we know that asymptotically 2) —)yr G(z) = 2-743 e Aone + a 1 Bray r= the many-valued functions having a cross-cut along the axis of —1. If then G(z) admit the Taylor’s series expansion At+aZt+...... + An2™ + ...05- ; we have =) [pom fog Dares Saree a am=5 | 2 exp 4z—4 log array ariel ray z the integral being taken round a circle of very large radius from argz=—7 to argz=7. o (_)\r Since all terms of the series = gr Brn are very small compared with z—4log 27 r=l1 when |z) is large, we have approximately 1 = —— —z—m—4 ‘ =_ m= 5 | exp {z—4 log 27} dz. Mr BARNES, ON THE CLASSIFICATION OF INTEGRAL FUNCTIONS. 345 We may deform the contour of the integral so that it is as in the figure, for the subject of integration is uniform in the dissected plane. But when the integral is taken round such a contour it is obviously infinite: an absurd result. And a similar absurdity ee a Pe | : aang ein Ss will arise for all functions of order greater than unity when the asymptotic approximation to the function takes its largest value as we approach the line of zeros of the function, near which that asymptotic approximation ceases to hold good. § 29. Suppose we take however functions of order less than unity for which the first term of the asymptotic expansion of the function takes its least value as we approach the line of zeros. If the function be P,(z) and we apply the procedure of § 21 we find, for the coefficient of 2” in the expansion of P, (2), 1 [om 2 (-) dn = Qar)-P2 z—m—43 exp 4———__ gp 4. S m om || ) p {sin ™/p pics 828 P(ps)h de together with integrals of terms of lower exponential order. The integral is to be taken round a circle of very large radius from arg z=—7 to arg z=, the axis of —1 being a cross-cut of the subject of integration. Put now z2/°=2 and write A for = a sin T/p Then we have Qn = la Osha [arms exp ae oe F (ps) dz, Qare (2p? s=1 Sah the integral being taken along that part of the circle of large radius for which —t/p1, we see by the result of 7. F. § 59 that the previous investigation where adn =7°+b,n°-"+4+.... is only altered, as regards the dominant factors, by substituting eh at as . Gig Gyraoo mee ashen exp {-2(0; YB )h for oe, in the absolute coefficient of the asymptotic approximation for dp. If e,<1, we must in the fundamental integral replace 7 bya 1 7 nr = ST . a > eos San > ee EE sin 7r/p Y sin m/p px sin /((1 —e,)/p] This modification will not affect the dominant factor in the approximation for dp. By the formula (II) Tr pm Am = ( - p ) 1 ecote,/m+... | sin 1/p/ (pm)pm+e/2+h Therefore : Ne me t™ = mP (¢ sim ale) etotd, log m/mM+...” Te This formula is valid for all simple integral functions of finite (non-zero) order less than unity for which the nth zero when n is large is given by mP {1 + z + lower terms} . § 32. When the nth zero involves terms like b mP f+ ae where 0 1, and where the order 2/p of the function is non-zero and less than unity. Then the dominant term in the asymptotic expansion of the logarithm of the function is (§ 10) arz2P 2, sin (277/p)’ and the next term is of order 21°. Mr BARNES, ON THE CLASSIFICATION OF INTEGRAL FUNCTIONS. 349 Now (eae is real and positive when z hes on the bisector of the angle between 2 the lines drawn from the origin to @, and ,?. Within the angle formed by these lines produced backwards the asymptotic expansion does not exist. But the assumption that it does exist in such a region will, when we carry out the process of § 21, only add to the true expression for a,, terms of lower exponential order than the dominant one. Therefore by analogy with the former investigation when the nth zero is of the : b : form n° {i+ 2+ z where 01, we shall have pb?Pw,@, sin (27r/p) 2re p/2 \ exp {9 (m)} One mp ( where g(m) tends to zero as m tends to infinity, and therefore even if ¢ be <1, the asymptotic approximation to a,’ 1s still of the same type. If the dominant term of the general zero is of the type b(a + QO) [log (a + O)] where log’ « = log {log {log {... 2}... }}, we shall still have A 1 = me? exp { f(m)}, where, when m is very large, f(m) is also very large but not of as high an order as log m. § 35. For simple functions of repeated sequence a similar investigation can be undertaken in accordance with the developments of J. F. Part IV. Subsequently it will be seen that analogous asymptotic expansions to those of Part I. of this paper hold for functions of r-ple linear sequence; and that the order of a function whose nth zero is (a+ 7m@,+...+%,@,)? IS Fg while the dominant term of the asymptotic expansion is of order p/r. We have then the result that for all functions which have been investigated, ie. for all the fundamental types of integral functions, whose zeros are such that the functions are of finite (non-zero) order R less than unity, the asymptotic expansion of the m+1th coefficient a, of the Taylor’s expansion is such that gy 1" = mi” exp { f(m)}. 350 Mr BARNES, ON THE CLASSIFICATION OF INTEGRAL FUNCTIONS. If the zeros are truly algebraic, f(m) is finite as m tends to infinity: if they involve logarithms in the dominant term, f(m) may be infinite but never to so high an order as log m. § 36. We also see, however, the difficulty that meets us when conversely we have given a Taylor’s series in which a,’ has an asymptotic expansion of definite type, and when we wish to classify the function defined by the Taylor's series. The problem will be insoluble until we have examined all possible types of integral functions and the nature of the asymptotic expansions for a,” which can be derived therefrom. When and if ever this is done we can form a sort of standard catalogue with which any given series may be compared. Then we may hope to be able to say whether the function given by the Taylor’s series represents an integral function deprived of a dominating exponential factor (as in J. F, § 11) or not: we may also hope to be able to define the nature of the zeros of the given series, to say whether they be repeated or not, whether they be of linear or multiple sequence, and so on. § 37. So far our researches into the nature of functions of non-zero order less than unity suggest considerations which may be made more clear as follows. Just as we saw in J. F. Part I. that imtegral functions may be classified by the manner in which their zeros are arranged, so they may be classified by the asymptotic i value of 1/Va,, when n is very large. In order that the series = a,2” may represent n=0 an integral function, it is necessary that Lt |Va,|=0. This theorem, due to Cauchy*, n=2 is a limited case of Cauchy’s+ more general result that Lt Van, is the reciprocal of n=a is) the radius of convergency of the series > a,2". If then this series be an integral n=0 function, a,—’" must increase indefinitely with n. Now the most simple mode of increase is that, to a first approximation, it should increase like n°. More complex will be the cases when a,~’” increases like exp n, exp {exp n},... on the one hand, or logn, log {log}, ... on the other. When p is greater than unity it is probable that the first and most simple mode of increase corresponds to integral functions of finite (non-zero) order 1/p, or to complex functions caused by multiplying together a number of such functions, or to such products multiplied by functions of infinite order. [It cannot correspond to the product of functions of infinite order by exponential factors of the type exp {2°} for the latter are not uniform functions of z.] We are led to enquire whether it is probable that a similar statement will be true when p has any non-zero value less than or equal to unity. When a,” increases like exp n, exp {exp n},...we have integral functions of more and more rapid convergency. It is probable that such functions are formed from products of which the dominant function is of one of the succession of zero order. * Analyse Algébrique (1821). Résumés Analytiques (1833). + Forsyth, Theory of Functions, Second Edition, p. 47. See also Hadamard, Liouville, 4 Ser. t. vu. Mr BARNES, ON THE CLASSIFICATION OF INTEGRAL FUNCTIONS. 351 Similarly when a," increases like log n, log (log n),... we have integral functions of less and less rapid convergency. Again we are led to enquire whether such functions are formed from products of which the dominant function is one of the succession of infinite order. Such probabilities and an accurate determination of the results which they suggest seem to indicate the lines on which the theory of integral functions must advance. And the researches of the future must endeavour to take integral functions of every type of zero, $(m, %,...,) say, and try to determine the corresponding asymptotic expansion of ad, ”™”, when m is large. In this way we may hope to form the standard catalogue, in which, as progress is made, we may hope with more and more accuracy to place any arbitrarily assigned series. § 38. The first step towards the further development of such an investigation is the examination of the asymptotic nature of a,” for functions of finite order greater than unity. As has been indicated in § 28, we must seek some other method than that employed for functions of order less than unity. Suppose that we have the function Q, (z), which (J. F. § 65) admits the asymptotic expansion 1 jae (yy! Ae = —_—_—_— €X = >! ——— — Q, (2) (Qari zi XP (sin ree szé r(2)h. By this we mean that when j2| is very large, @,(z) can be expressed as the sum of a number of terms of which the one written down is of dominant order. Suppose now that the dominant term be expanded by the exponential theorem in powers of z. We shall get a series of which the dominant term of the coefficient of z”P is - m! (sin 7p)™" The lower exponential terms of the complete asymptotic expansion for Q,(z) will give rise to coethcients which are of lower order when m is large. We shall therefore have an expansion in which all terms involving powers of z of non-integral order vanish, for @,(z) is a function of z, and in which the dominant nature of the coefficient of 2” is, when m is large, given by 7 \mlp 1 Ca = =) T {(m/p) + 1}? which is approximately equivalent to sin 1rp\1/P One => me (7) 5 Tpe If now we have a function of finite non-integral order p greater than unity of which the nth zero is given by b np E +—+ =| ; ns 352 Mr BARNES, ON THE CLASSIFICATION OF INTEGRAL FUNCTIONS. where ¢, is positive, we see by changing z in our former function into 2/b,, that ag = mie i sin ae ™ pe and that the terms neglected in this approximation are of the type exp {/f(m)} where F(m) tends to zero with 1/m. § 39. Similarly if we have a function of order 2/p of double linear sequence such as was considered in § 14, we have the asymptotic approximation Gm = mpl? pb,*?w,@, sin (277/p) = 2ae p/2 . ) exp { f(m)}, where f(m) tends to zero with 1/m. And the results obtained in § 33 for functions of non-zero order less than unity hold zn toto for functions of finite non-integral order greater than unity. § 40. The results of §§ 38 and 39 may also be obtained by a modification of another process due to Hadamard*. We have, in the limit when ~ is infinite, for the function Q, (2) _(22)-# ie dé Gare Bon es Ee —sio Ff (2)! en ee oe ganado ne ana Cas Jie oe eran’ \e pp rP and therefore, if we put snmp th we see that, when m is very large, we have the asymptotic equality 1 sin 1p —(m+4)/e Om = Bar) OP) | a (m + »| zine 3 hn aa & mp\s 1 /8\ aig = [Tp ‘p.- (er? — pid) way (m+ 4)? \ ap ) F () es. dé. Now the subject of integration is, if p be not integral, at most of order (m+4) py: and the result of integration involves such a term multiplied by terms of lower order. exp Therefore, asymptotically, , 1 (m+4)/p , 1 Om = [mente] exp [pat +terms of lower order than ak , sin 7p p } so that l 1/p Gh IU = mip SS exp {f(m)}, where f(m) vanishes with 1/m. We thus obtain for the fundamental function the same result as before, and therefore the derived results for functions of the same order also hold. * Liouville (4), Vol. 1x. pp. 171—215. Mr BARNES, ON THE CLASSIFICATION OF INTEGRAL FUNCTIONS. 353 § 41. The determination of the asymptotic value of a,’ for functions of finite integral order is difficult. Take, for example, the standard function R, (2). Its asymptotic expansion is (J. F. § 70) at Pp Ean —) Ry (2) = Geyer | aPloge+ (“p24 8 OF RO, Sap EP and the integral which results on adopting Hadamard’s process of § 40 appears intractable. Difficulties arise too when we try to obtain the asymptotic value of a ,—”” as the limit of our former results. For R,(z) is the limit on the one hand of Q,,..(z), when ¢ is very small but positive: and on the other hand of Q, .(z) exp {C2 ce). Asymptotically it p-e therefore demands that we should consider not one dominant term as in the case of Q,(z) but the two terms —2z)/P ie ) See: Ce) i Taiits Qa And to these two terms we cannot apply the procedure of § 38. We may however say that asymptotically R,(z) lies between exp {(—z)t*} and GE GE I (m/p + €) ane I’ (m/p —)° exp {(— z)?-*}, and that therefore a,, will lie between We shall thus have to a first approximation a,,-’"=m'?. The results of §§ 38, 39 will therefore, as regards the first approximation, hold also when the order is integral. § 42. Let us now consider the asymptotic nature of a,’ for the functions of zero order which we have considered: this is evidently the next step necessary to establish the sequence of probable theorems enunciated in § 37. For the function : a z\ F(2)= tt (1+ 5), considered in J. F. § 60, we know that we have the asymptotic expansion f — 2 ™ S Gyr Cs log F (z) =4 (log z¥ —4 log z+ res lenge: where C,=-—e&)-. Hence, if I @)) 022 Gh SS oon PEG AES con 5 il Tr i) (-)*2 ¢, = + log z—m—% LY = Nae ws) dm = 5 2 expt + paar dz, the integral being taken round a circle of large radius |z| from one side to the other of the negative half of the real axis, which is a cross-cut to render the subject of integration uniform. Vou. XIX. Parr III. 46 354 Mr BARNES ON THE CLASSIFICATION OF INTEGRAL FUNCTIONS. Therefore E ae prlogr—m-* exp i2. hog —(m+$)+ io} m3 (0 = 2* = sre | a0. ake now logr=m++4; then, when m is very large, we have asymptotically 1 2/6) m 3 i Gy pee va [T exp |-$+ >> Basha; “ An = therefore in the limit when m=, m+4)* 1 72/6 e & Ame +4)? = on i [i exp \- + dé. We thus have asymptotically —Vm — p(m2)+3+4+F (m) an Ie. rT 3 where f(m) tends to zero as m tends to infinity. § 43. Let us next consider the function Fee =titt]1 + 30], ( ) ee earn whose asymptotic expansion was obtained in § 11, where it was proved that, when || is large, ee ey = ASE aot ee tog 2 ate {(log 2) — 7? log 2} AC) (loge): zy Pa m7) (-)' ew ora Gil +35 s(—e*s) (1 — em) 28” this expansion holding at all points but those within a finite distance of the zeros of the function. e therefore have, approximately, dz atayAl (log z)* — mw logz _ (log z uN = |—— zhS.°(@}/2! gees ae a = Ne ,@ = . os 60,05 s5(@) i Gg at _8@), 3 Ore 2 o4s(1—e"*) (1—e™) 2’ the integral being taken round a circle of large radius || Thus = Lt romeesi81/ exp Ee _ {(log r)’ + 3.0 (log r)? — 3@ log r r=2 — 168 — 7 log r — 76} + “sca = {(log r+ 216 log r — + 34 ee 2S,’ (a) /S2” ate < Giants go +0 ( 5 a lal dd. Mr BARNES, ON THE CLASSIFICATION OF INTEGRAL FUNCTIONS. 395 Put now (a) 5 .\2 — a2 Y (0) eg Ge LSe a log if Si i (a) ar a 6,0. 2! Then we have asymptotically when m is very large Vy (2) 2 , (log 7)’ — 7? log r 281 ae | log rp? e| _ oS: o 6,0 dn = pam+{S) (a)}/2! exp eo 6a@,@, 60,0, 2, Now the final integral is at most of order exp {klogr}, where & is finite when r is infinite. Hence, when m is large, we have to a first approximation i i exp | ee 4 6@,@. Now to a first approximation ras 3 (log 7)? aa and therefore ap, = 10% 7/6ee.—m — yams, Uhl) therefore Am” = ri = exp 2V2ma,o.. § 44. Thus we see that for each of the functions of zero order which we have considered a,” is asymptotically represented by an exponential of some power of m. As has been remarked, functions of simple sequence with zeros of the type ne” cannot be investigated until we introduce new functions into analysis: and consequently we cannot at present investigate the asymptotic expansion of a,’ for such functions. And similar remarks apply to higher functions of zero order. The investigations thus suggested will form the next step necessary to weld the probabilities of § 37 into a formal theory. Prolonged investigation will be necessary before such a result can be accomplished. My object however will be fulfilled if the very incompleteness of my work serves to inspire further research. An absolutely complete theory of integral functions will however be always an impossibility. Although all the natural functions of analysis are of comparatively simple type, yet it is always possible to form functions of almost infinite complexity. The catalogue which I have suggested would at most only include functions lying within logarithmic and exponential limits, as we might say. Just as there exists a domain of convergency on whose borders the logarithmic criteria of convergency fail*, so it seems doubtful whether we can ever hope to enumerate the possibilities which integral functions may present beyond the logarithmic or exponential limitst. * Du Bois Reymond, Crelle, T. 76. + Hadamard (J. c. p. 191) constructs a function whose asymptotic form lies beyond the exponential limits. 46—2 XV. On the rise of a Spinning Top. By E. G. Gator, M.A. [Received 29 January 1903.] WHEN a top in rapid rotation is placed upon the ground the point of contact proceeds to trace out a curve of greater or less extent, whilst at the same time the inclination of the axis to the vertical oscillates about a mean value which slowly decreases till finally the top may attain a steady motion with its axis sensibly vertical and the point of contact at rest on the ground. This motion is quite different from that of a top supposed to be moving under gravity with a point on its axis fixed. In the latter case the inclination of the axis to the vertical oscillates between two fixed limits which remain always the same. In order to explain the actual phenomenon it is necessary to take account of the fact that the lower end of the axis is a surface of small extent and not a mere point. A general explanation is given in Routh’s Rigid Dynamics, Vol. I, p. 165, 5th edition, and the question is also considered in Jellett’s Theory of Friction, Chap. vi. Jellett supposes the lower end of the axis to be rounded off into the shape of a small sphere and obtains a formula which he uses to account for the rise of the axis. He only proves that the formula is approximately true, but it is shown by Routh (loc. cit. p. 167) that it is an accurate first integral of the equations of motion. In this paper attention is mainly confined to the form of top considered by Jellett; the resistance of the air is neglected, and the assumptions made with regard to the friction between the top and ground are that it may be represented by a single force at the point of contact, and that when slipping takes place the direction of the force of friction is opposite to the direction of sliding or at any rate acts so that energy is dissipated. It is shown that dissipation of energy is in general an essential part of the pheno- menon; for the energy in the steady motion with axis vertical cannot in general be equal to the initial energy, so that this state of motion could not be attained on perfectly rough or perfectly smooth ground except under special circumstances of projection. What is proved is that, provided the initial spin about the axis of figure exceeds a certain limit, it is possible to assign a limiting value to the inclination of the axis to the vertical which can never at any time be exceeded. This limit depends on the energy, Mr GALLOP, ON THE RISE OF A SPINNING TOP. 357 and as the energy decreases on account of sliding friction the limiting inclination decreases till, when the energy is reduced to a certain value, the limiting inclination is reduced to zero. It appears therefore that, under the conditions assumed, a top started with sufficient spin about the axis of figure will never fall down to the ground. Thus the ultimate fall of the top, which takes place under actual conditions, is not to be ascribed to what is usually called sliding friction but to the resistance of the air and friction couple which, though they require time to produce much effect, eventually diminish the spin until it is insufficient to counterbalance the usual effect of gravity. We may briefly describe the circumstances of the motion under the assumed con- ditions by saying that the minimum energy consistent with Jellett’s first integral of the equations of motion is attained when the axis of symmetry becomes permanently vertical, and that in consequence of the dissipation of energy by sliding friction the energy must decrease, so that the axis is driven to take up positions continually approxi- mating to the position in which the energy is a minimum. The method adopted admits of immediate application to the motion of a heterogeneous sphere spinning on a horizontal plane, and is applied in §§ 5—7 to explain the way in which such a sphere tends to set itself when in rapid rotation. § 1. JELLETT’s Equation. Consider the case of a top rounded off at its lower end into a portion of a sphere whose centre is 0 and radius r. Then, if @ is the centre of mass of the body, we shall call OG the axis of the top and shall further suppose that the body is kinetically symmetrical about OG. Let the mass of the top be taken as unity, let O@=h and let the moments of inertia be C about OG and A about any axis through G@ perpen- dicular to OG. In fig. 1, which represents a vertical section of the top through its axis, let P be the point of contact with c the ground. Draw PM perpendicular to OG and GL per- pendicular to the ground. Let @ be the inclination of the axis to the vertical at time ¢ and let a be the initial value of 0. Let @,, @., ;, be the left-handed angular velocities of the top at time ¢, , about GC, w, about GA perpen- dicular to GC in the plane of the figure and @, about GB perpendicular to GA and GC. The simplest way of obtaining Jellett’s first integral = of the equations of motion is described by Routh, Rigid ener Dynamics, Vol. 1, p. 167, Ex. 5, 5th edition, 1892. 358 Mr GALLOP, ON THE RISE OF A SPINNING TOP. By taking moments first about GC, then about GV the vertical through G, and eliminating the component of friction which enters into the two resulting equations, we obtain on integration = Ala SINIOMs Cas (ki COSIG) tN et ocsc.cconscnecceas-cteceese tere (1), where k=h/r and N is a constant depending on the initial conditions. Assuming that the top ultimately attains a state of steady motion, spinning with angular velocity n, about its axis which is vertical, we may find the value of mn by putting @=0 in equation (1). We thus obtain C(k+1)n=N, and therefore if #, is the energy in this state of motion E, = }Cn? + gh(1— cos a) NV? = 3+ IFC $l (lL — C08) saciaciscsscismesces sanpcccdeees (2), provided the potential energy is reckoned zero in the initial position. Now suppose the top is started with initial kinetic energy 2. If the top is pro- jected in such a way that the axis becomes permanently vertical without any slipping between the top and ground or if the ground is smooth, H,=,. But if this state of motion is only attained when the ground ‘is imperfectly rough and slipping takes place, energy is dissipated and £,>£,. In either case E,¢ F,. If the initial motion consists in a spin 2 about the axis of figure and the axis ultimately becomes permanently vertical, an inferior limit to the value of nm may be obtained from this inequality. For in this case N=Cn(k+cosa), £,=4Cn* k+cos a Therefore mh=n Sie: Cn? (ke + cos a)? and 4Cn?>4 —— + gh (1 —cos a); - (k+1) and therefore 4 On {(k +1 —(k + cos a} > gh(1— cosa) (k +1); or dividing out by 1— cosa i ee ee ee (3). 7? Ok +1 + cos a) Unless the initial spin satisfies this condition it is impossible for the axis to become permanently vertical. It is obvious that if 7 is supposed to be small, so that & is large, the limiting value of n is large and tends to infinity as r diminishes to zero. This result explains why it is impossible to account for the rise of the axis to a position permanently vertical on the supposition that the lower end of the axis is a mere point. Mr GALLOP, ON THE RISE OF A SPINNING TOP. 359 If the ground were smooth or if no slipping took place there would be no dis- sipation of energy and the sign of inequality would be replaced by the sign of equality. Now it is a matter of common observation that the axis will ultimately become vertical whatever the initial spin may be provided it is sufficiently great; no nice adjustment is required, and it follows therefore that as a general rule the steady motion with axis vertical can only be obtained when slipping takes place on imperfectly rough ground, so that there is dissipation of energy. § 2. Minimum VALUE OF THE ENERGY SUBJECT TO JELLETT’S EQUATION. We next proceed to prove conversely that when the amount of energy dissipated becomes equal to #,—#,, the axis must have become permanently vertical. It will also be shown how to obtain a superior limit to the inclination of the axis to the vertical when the energy has been reduced to a given value between ZF, and £,. Let v be the velocity of G at any time; then 7 the kinetic energy is given by 2T =v? + Ao? + Ao? + Co. If the potential energy is reckoned zero in the initial position the total energy F at any time is given by E =T + gh (cos 6 — cos @). We proceed to find the minimum value of # for a given value of @ when », @,, @s, @; are unrestricted except by Jellett’s equation (1). It is at once obvious that v and a, must be taken to be zero, and that as regards w, and @, we require Ao,do, + Co,da, = 0 for variations dw,, dw; subject to —Asin 6dw,+ C(k+ cos 6) da, =0. Or vite Or 2. N —sin@ k+cos@ A sin? 6+ C(k+ cos 0)?" Hence These values of , and , satisfy the relation @) = @s —rsn@ h+rcosé@’ or, from the figure of § 1, On @s —PM GM’ Hence for variations of v, @,, @., w, subject to equation (1) the minimum value of E is obtained when the motion consists of a pure rotation about PG the line joining the point of contact with the ground to the centre of mass. The minimum of £ is then equal to gh (cos é — Cos a) qr S { k (4) BY = 26 C(k éy Se ry . It follows that, in the actual motion, at any time when the inclination of the axis to the vertical is equal to @, the energy is not less than the expression just found. It is on this property that the whole of the subsequent work is based. 360 Mr GALLOP, ON THE RISE OF A SPINNING TOP. § 3. Form oF THE GuIpDING CuRVE y= F(z). Writing cos @=<2, we have E>F(z), TI N? where ee) =e Now consider the curve y=F(«) for values of x between —1 and +1. Write for the moment A(1—2*)+C(k+a2P=f(a), ave o that F (x) =gh(x—cosa)+5—-—. so tha (2) = gh ( esse Therefore, differentiating with respect to a, 0 sei) F" (x) =gh -4N?* 7; cop aoa, alco ald de add F" (¢)= ye if (x)? —3f (2) f(@) | ah ee): Now, for the values of x considered, f(x) is positive; therefore F”(#) has the same sign as {f'@P-4f" @f@), which is equal to 4 {(C — A) «+ Ch}? —(C — A) {(C— A) a? + 2Chka + A + Ch} =3(C—A)2+6C(C — A) kx+ Ck (80+ A)—- A(C— A). In the last expression the coefficient of a? is positive; therefore the expression will be positive for all values of « if 3 (C — A) {Cl# (304.4) — A (C —.A)} > 902(C— APF, that is if Che (80 + A)—A (C— A) — 30%? > 0, or OF = (C= AN 0) coats scesenc corer eee eee (5). The last condition is satisfied provided C A, provided A ke>1— Ce Now in the case of an ordinary top & is a number considerably greater than unity and therefore this condition is satisfied; therefore #”(x) is positive when lies between —1 and +1. Hence F’(«#) increases as « increases, and will therefore be negative for all values of « between —1 and +1 if negative when e=1; that is, if gh—gues LO) <0. {Ff )/ This requires (1) that f’(1) be positive; that is CTD) SoA ic Gate be aes eee ae eae eRee (6) Mr GALLOP, ON THE RISE OF A SPINNING TOP. 361 and (2) that FQ) ghC?(k + 1)! > Ce aj A OO OEEEE EEE EEEHEHEEEEEHEEHEHEE HEE (7). If the initial motion consists of a spin n about the axis of figure N = Cn(k+ cos a) and the last condition reduces to n> GSD IRN SON 00. eee (8). It will be noticed that the conditions (5) and (6) refer to the construction of the top only, whilst (7) refers to the initial motion also. It will be proved below that the limit of » required by (8) is greater than that required by (3). This property may also be easily verified independently. When r is small and therefore & large there is no great difference between the limits of nm required by (3) and (8); in each case the limiting value of n is given approximately by the equation pie RON ae OL Ci It is now evident that the portion of the curve y=F'(#) which lies between «=—1 and #=+1 will slope down towards the axis of # as # increases, and will have the general shape of the curve CAB in fig. 2, provided the conditions (5), (6), (7) are satisfied. y Fie. 2. If a point P is marked down in the figure having its coordinates equal to (a, F) it cannot lie below the curve CAB since H>F (a). If through P a line is drawn parallel to Ox and cutting the curve in Q, the abscissa & of Q will be found from the cubic equation F (x)=. From the figure it is evident that this equation can only have one root between —1 and +1. Hence it follows that at any time when the energy is equal to £ the inclination of the axis of the top to the vertical must lie between cos € and cos1; that is, it must be less than cos™ &. Vou. XIX. Part III. 47 362 Mr GALLOP, ON THE RISE OF A SPINNING TOP. Now suppose that £, is the initial value of the energy, and let P, be the point whose ordinate is £, and abscissa cosa. Draw the line Q,P,R, parallel to the axis of « cutting AB in Q,; let & be the abscissa of @, found from the equation F (x)=). Then, if the ground is perfectly smooth or perfectly rough, # will be constant through- out the subsequent motion and equal to £,; so that the representative point P whose coordinates are (cos @, #) will oscillate in a portion of the line QR, between Q, and R,. Therefore the axis of the top will never make with the vertical an angle exceeding cos’. If however the ground is imperfectly rough and slipping occurs, there will be a loss of energy, and therefore # will decrease. The representative point must however always remain below the line Q,R, and above the curve AB, and therefore within the area Q,R,B; and hence in this case also the inclination of the axis to the vertical can never at any time exceed cos&,. If after any time the energy has fallen to 2’, then at any subsequent time the representative point P must lie within the area Q’R’B, where the abscissa &' of Q is found from #(«)=#’; and therefore at any subsequent time the inclination of the axis of the top to the vertical cannot exceed cos &. We thus see that as energy is dissipated by sliding friction the area available for the repre- sentative point becomes smaller and smaller, and therefore the limiting inclination of the axis of the top to the vertical also becomes smaller and smaller. If the energy falls to that represented by the ordinate of the pomt B, the representative pomt must be at B, and the axis is permanently vertical. Now the ordinate of B is 1 F(1)=gh (1 — cos a) ag obey’ which is the value given by equation (2) for #, the energy when the top is in steady motion with axis vertical. It has been proved already in § 1 that the axis of the top cannot become per- manently vertical until the initial kinetic energy #, has been reduced to #,. It has now been established conversely that as soon as sufficient energy has been dissipated by sliding friction to reduce the amount to #, the axis will become and remain vertical. It is obvious now that subject to the inequality (7) £, is the absolute minimum of the energy consistent with equation (1). For when we vary 2, @,, @, @; only, and assign a given value to 6, the minimum value of # is F(«x), where e=cos@; and it has been proved that F(x) diminishes as «# increases to 1, in which case @=0 and E=H,. We may therefore describe the motion of the top by saying that owing to dissipation of energy it moves so that the energy tends to a unique minimum. It should be noticed that when the initial motion consists of a spin about the line through the point of contact with the ground and the centre of mass, the inclina- tion of the axis to the vertical in the subsequent motion will never exceed its initial value, provided the initial angular velocity is so great that the condition (7) is satisfied. For in this case the initial energy is equal to the minimum consistent with Jellett’s equation, ie. L,=F (cosa); so that the point P, is on the curve y=F'(«) and &=cosa, Hence, if the initial motion nearly coincides with that just described, the inclination of Mr GALLOP, ON THE RISE OF A SPINNING TOP. 363 the axis will never exceed a limit not much greater than its initial value; for example, when 7 is small a spin about the axis of figure does not differ much from the proper motion for a minimum. An illustration of this remark is afforded in the next section. We can now also give a simple proof of the statement made above that the con- dition (3) is covered by (8). For when the latter condition is satisfied E, > F(cosa)>F(1)>&,. § 4. NumericaL EXAMPLE OF A SPINNING Top. To illustrate the preceding method we will consider the details of a special case. Let us take h=4cem., r=02cm., so that k=20; also, taking the mass of the top to be unity, let us suppose that A4=(C@=3. These numbers correspond to a small hollow top, not differing much from a sphere of radius slightly greater than 2 em., the centre of the spherical portion being rather more than 4 cm. above the ground when the axis is vertical. The value assumed for g is 981. Suppose that initially the axis of the top is inclined at 30° to the vertical and that the initial motion consists of a spin n about the axis. The least value of n required by (3) is about 166, or 26 turns per second, so that it is impossible that the axis should rise so as to become permanently vertical unless the initial spin exceeds this value. The least value of n required by (8) is about 171, which corresponds to 27 turns per second. We will take for our example n= 200, or about 32 turns per second. The initial value of the energy #,, estimated on the assumption that the potential energy is reckoned zero in the initial position, is given by E, = $Cn? = 60000. When the axis becomes permanently vertical the angular velocity is reduced from 200 to 1987, so that the diminution in angular velocity is insignificant, being only 0°65 per cent. The energy at the same time falls to E, =59763, and the loss of energy is therefore 237, so that the ratio of the loss of energy to the original kinetic energy is about 0°004 or 515. As soon as this small fraction of the energy has been dissipated by friction, the axis will have become permanently vertical. 47—2 364 Mr GALLOP, ON THE RISE OF A SPINNING TOP. The following table shows the variations in the limiting values of 6 as the energy decreases from EF, to £,. Loss of Energy | Value of ¢ | Limiting value of @ — —|— - aie | 0 08444 32° 23! | to (Z, — #)) 0-8593 30° 46’ | to (Z, — £,) 0-8743 29° 9 | ro (Z) — 41) 0-8895 o7° 19! st; (Ey — B,) 0-9048 25° 12’ | =5 (i, — Ex) 0:9203 23° 2 | 3%; (% —H) 0:9359 20° 38’ | to (EZ) — 4) | 09517 17° 53’ 3,(Z.—£,) | 0:9676 14° 37’ to (Z,)— #,) 09837 10° 21’ £, — £, / 1-0000 0° 0’ From the table we see that the changes in the value of & the limiting value for cos6, are roughly proportional to the loss of energy, so that the curve AB of the figure in § 3 is approximately a straight line. The limiting value of @ does not change so regularly because, regarded as a function of @, cos@ is a maximum when @=0°. When the top is first started it is obvious that @ will at first increase from its initial value of 30°, but the first line of the table shows that the value of @ can never increase beyond 32° 23’ and may never even attain this limit. Moreover the point of the top in contact with the ground is started so as to begin sliding at once; dis- sipation of energy will commence immediately, and therefore before the limit 32° 23’ could be reached the energy would be reduced and a smaller limit than 32° 23’ would be obtained from the equation F(x)=#. The small amount of the first oscillation in @ is remarkable. Again the third line of the table shows that when the loss of energy has reached 1(£,—£,), which is less than one thousandth of the original kinetic energy, the top will have completely recovered from its preliminary fall; the axis will be nearer the vertical than it was initially and its inclination to the vertical will never afterwards exceed 29° 2’, A few concluding remarks may be added here. It has been proved that the per- manent state of rotation with axis vertical cannot be attained without dissipation of Mr GALLOP, ON THE RISE OF A SPINNING TOP. 365 energy, and conversely that if a sufficient amount of energy is lost the axis must become permanently vertical. But it has not been proved that this amount of energy will necessarily be lost. Whether or not this will be the case will depend on the value of the coefficient of friction and other circumstances. If the ground is very rough, dissipa- tion of energy will be rapid at first, but perfect rolling will soon ensue and if the energy lost is less than #,—#, the axis will not be vertical but its inclination to the vertical will oscillate between tolerably close limits, so that the path of the axis will be scarcely distinguishable from a right circular cone. If the ground is nearly smooth, dissipation of energy will take place more slowly, but it will be a longer time before perfect rolling begins so that perhaps a tolerably smooth surface may be more favourable than a very rough one for bringing the axis permanently to the vertical. If however the coefficient of friction becomes infinitely small, the time required for sufficient dissipa- tion of energy becomes infinitely long, so that in practice the ground must not be too smooth; otherwise the air resistance and friction couple which have been neglected in our calculations will reduce the spin below the limit required by (7) before the frictional resistance has dissipated sufficient energy to bring the axis to the vertical. § 5. Motion or A HETEROGENEOUS SPHERE. The preceding method may also be applied to the case of a heterogeneous sphere spinning on a horizontal table, provided the sphere is kinetically symmetrical about a diameter and the conditions (5) and (6) are satisfied, namely that both the expressions Ch? -—(C—A) and C(k+1)—A be positive; for these conditions do not necessarily require that the centre of mass should be outside the sphere. When such a sphere is set in rapid rotation it is a known fact that the centre of mass rises until the diameter through it is vertical, and the sphere spins about this diameter with its centre of mass above the centre of figure. In discussing this problem we shall use the same notation as before, but it must be remembered that k is no longer a number of considerable magnitude. To fix the ideas, let us suppose that initially the line of symmetry is horizontal and that the initial motion consists of a left-handed spin n about the vertical diameter drawn upwards. Then V=An and £,=}3(A+h*)n* Also the energy #, when the motion has become steady with G vertically above O the centre of the spherical end, is given by I Zale 366 - Mr GALLOP, ON THE RISE OF A SPINNING TOP. The condition 2, > £, leads to L(A +h?) nr? >gh+4 ai > ( gn 2 C(k i 1s 2 or \4 +h C(k at n? > 2gh. The coefficient of n? is evidently positive since the condition C(k+1)>A has been already assumed. Hence the condition H,> #, requires that ’ 2Qqh en Ss deuce gwaddctee maee de Miaemecctent (9). The steady motion with G vertically above O cannot be attained unless this con- dition is satisfied. The condition (7) that the guiding curve CAB in fig. 2 may slope towards the axis of 2, as x increases, becomes sgh Eas 7 CEA Saonnosogtoncocassosaccdsesseasscosc (10). nr This condition includes the condition (9) that #, may exceed £,. For if (10) is satisfied LE, > F(0)>F (1), that is, E, > E,. It is now clear that, if the sphere is constructed so that Ck*—(C—A) and C(k+1)—A are both positive quantities and if n is great enough to satisfy the inequality (10), the method that was applicable to the top may also be applied to the sphere. Thus if at any time the energy of the sphere is reduced from #, to E ‘the line of symmetry OG must then and at all subsequent times make with the vertical an angle less than cos &, provided the equation fee ems : A (1—a*)+ C(k+ 2) EL=ghx+3 has a root & which lies between —1 and +1. When the energy has been reduced to E, the axis of symmetry must be permanently vertical with the centre of mass in its highest position. § 6. SPECIAL CASE OF A LOADED SPHERE. Let us consider as an example the case in which a uniform sphere is loaded by hollowing out a spherical portion and filling it up with a denser material. Let O be the centre of the original sphere, R its radius and M the mass of the complete sphere before being hollowed out. Let o be the centre of the loaded portion, r its radius ; and let m be the excess of the mass of the load above the mass removed. Let Uo=c. Mr GALLOP, ON THE RISE OF A SPINNING TOP. 367 me Me Then Cea Meem! 8 o° < Mem Mme? Y— 2 Ra 2amy2 29 oe Dae, C=2 MR? + 2m", A=2MR+2mr Lay ere Hence C<4A and the first condition, that Ch?—(C— A) be positive, is satisfied. To satisfy the second we require Mme . Y], — 1M! Pera Ck>A COM ea: m Cc Mne that is, ¢ MimR ce M+ m’ Cae or R~ 2MR?+2mr’ G Pi mr? or ip ts iu ee =) : Both conditions as to the construction of the sphere will therefore be satisfied by taking c<2R. Take now as a numerical example the case when R=10, r=2, c=2, the centi- metre being the unit of length. Let us also suppose the density of the load to be 21 times that of the rest of the sphere, as would be the case if the load were lead and the rest of the sphere were made of wood having a density 0:54. Then Vat and, if we take the total mass of the sphere and load together to be unity, we shall have h=:2759, k =:02759, C= 347, Aso 18, The state of permanent rotation with G in its highest position cannot be attained unless the spin n is greater than 1914, or rather more than 3 turns per second. If this state is attained and the final angular velocity is n,, then ™ _ 0-9866, n so that the loss in angular velocity is very small, about 13 per cent. The guiding curve of fig. 2 will slope in the proper direction provided n> 24:8, or rather less than 4 turns per second. Let us take n = 30, or rather less than 5 turns per second. Then £,= 15865, Z,=15471; so that when the amount of energy dissipated amounts to 394, which is about 23 per cent. of the original energy, the state of permanent rotation with G in its highest position will have been attained. 368 Mr GALLOP, ON THE RISE OF A SPINNING TOP. To obtain a superior limit to the angle @ which OG makes with the vertical drawn upwards, we solve the equation Ey a F (x), which reduces to he - 2gh A oe ae Clas Che AOR or, after multiplying by 100, 08225 — 51382? + 131-7x + 10:26 = 0. The root of this equation is — 00756 = cos 94° 20°. Initially OG is horizontal and 6=90°. After starting @ will at first increase but can never exceed 94° 20’ at any time during the subsequent motion. It would be easy to construct a table, as in § 4, showing the way in which the limit of 6 decreases to zero as the energy falls to £,. If the initial spin is equal to 30 as before but is about the vertical through @ instead of O, a still smaller limit is found for @. In this case H,=}An? and the equation to be solved is 0°82a — 51:27a7 + 131-32 + 2641 = 0. The root of this equation is — 0:02=cos 91° 9’, so that in this case when OG is below the horizontal through O, it can never be inclined at an angle to the horizontal greater than 1°9’, a very small limit, considering that the spin, less than 5 turns per second, is by no means great. If the initial axis of rotation passes through G and the point of contact with the ground, the axis of symmetry will never fall below the horizontal plane through the centre of figure. § 7. § MOTION OF A HETEROGENEOUS SPHERE IN WHICH THE CENTRE OF Mass COINCIDES WITH THE CENTRE OF FIGURE. The case in which the centre of mass of the heterogeneous sphere coincides with the centre of figure is easier to treat than the preceding, as the potential energy is constant. In dealing with it we shall suppose the principal moments of inertia at G to be all unequal, as there is but little gain in simplicity by supposing the body to be symmetrical about a diameter. Let these moments of inertia be A, B, C. Let @ be the inclination of the C-axis to the vertical at any time, and let @ be the inclination of the plane through the A and C axes to the vertical plane through the C-axis. Let @,, @, , be the angular Mr GALLOP, ON THE RISE OF A SPINNING TOP. 369 velocities of the body about GA, GB, GC. Then, expressing the fact that the angular momentum about the vertical through G is constant, we have (— Aa, cos $+ Ba, sin d) sin 6 + Ca; cos 0 = No... ceceeeeeceee cess (11). This equation takes the place of equation (1) of § 1. The energy # is entirely kinetic, and B=} + 4Ao? +4Bo? + 1Co, where v is the velocity of G. To find the minimum value of H# subject to equation (11) for given values of @ and @ we require v=0 and Ao,do, + Bo,do, + Ca,da, = 0, for values of da,, dw,, dw, subject to the equation —Acos ¢sin 6dw, +B sin ¢ sin do, + C cos Oda; = 0. Hence @, Ws @, N ~ cospsin@~ sin ¢ sin 0 ~ cos 0 (A cos? @ + B sin? d) sin? 6 + C' cos? 8” and the motion for minimum / consists of a pure rotation about the vertical through the centre of figure. The corresponding value of F# is 1 i" (A cos? d + B sin? #) sin? 6 + Ccos? 6 When ¢ also is allowed to vary, the minimum value of H occurs when ¢=0 or 7, provided A is the greater of the two quantities A and B, so that the A-axis must be in the same vertical plane as the C-axis. Hence in general N? La sin? 6 + C' cos? 0’ or, writing cos? @=«, E>F (a), y? where F (x)= 3 A+(C—A)a’ The curve y=F(«) is a hyperbola and, if C> A, slopes down towards the axis of x in the same way as the guiding curve AB in fig. 2 for values of w between «z=0 and «=1. Hence, as in the previous cases, dissipation of energy will, if sufficient in amount, ultimately bring the C-axis into the vertical. In order that this may take place it is necessary that the energy be reduced to N?2 20" Wor, bx.” Parr IUl. 48 370 Mr GALLOP, ON THE RISE OF A SPINNING TOP. Thus the effect of sliding friction is to cause the axis of greatest moment to approach the vertical. When the energy is equal to a given quantity Z, ae (C — A) cos O> oR -A4, so that the inclination of the C-axis to the vertical must be less than pact N?—2AE)t s 2H (C —A))’ which reduces to zero when NV?=2CE. In particular, if in the initial position of the sphere the axis of greatest moment C and the principal axis of next greatest moment A are in the same vertical plane, and if the initial motion consists of a spin about the vertical axis through the centre, then the C-axis will begin by approaching the vertical and in the subsequent motion will never make with the vertical an angle greater than its initial inclination. § 8. Morion oF 4 Sotip oF REVOLUTION ON A PERFECTLY RoUGH PLANE. The motion of a solid of revolution on perfectly rough ground may sometimes be reduced to quadratures by the followmg method. Using the figure of § 1, let the direction of the axis of figure be defined by the angle @ and the angle yw which the vertical plane through the axis makes with a fixed vertical plane. Let GM=£& MP=yn, PL=q, LG=p, so that p, g, & 7 are known functions of 6 depending on the shape of the body. Let @,, @:, @; be the left-handed angular velocities about GA, GB, GC, where GA is a line in the vertical plane through the axis of figure GC and is perpendicular to GC, and GB is perpendicular to the plane CGA. Then o,=—vWsin@ and @, = 6. Let wu, v, w be the component velocities of G, viz. wu parallel to PZ, v perpendicular to PL in the horizontal plane through G, and w vertical. Then u= pb, v = — £o, — 7s, w= 6. For the motion of the body we have first the equation of energy $(A+p*+ q@) & +4402 + $Co,? + 4 (Ew, + nos)? + gp = const. ; and by taking moments (1) about the vertical through G and (2) about GC we have f (Aca sin 0+ Cons CoS 0) = = Mg je cect ec etenptenies seeder (12), Mr GALLOP, ON THE RISE OF A SPINNING TOP. 371 Also, resolving perpendicular to plane CG@A for the motion of G, dv - ane up = F, dv oT or arth ae BODO UOOCOCOOOCOOONOOOCCUOOOOOOOOOCOUCOGIES (14). From (12) and (13) we have d = q d bs ai (— Aa, sin @ + Ca, cos A) + wat (Cos) = 0, and therefore ie Aa, sin 6 + Ca; cos @) + cr ‘s SQ, AO eee (15). Also from (13) and (14) da (dv Sc Tea (gp + POF)» do. d d or C aa n 7a — pnw, cosec 06 = — 7 Tr (Ew, + nws;) — pn, cosec @............ (16). The equations (15) and (16) are two linear differential equations for , and ao, with @ for independent variable. If @, be eliminated, there results for w, a linear differential equation of the second order. If this equation is solved , can be found by differentiation and the equation of energy then gives 6? as a function of 0; the connexion between @ and ¢ is thus reduced to quadratures. It appears from the form of the equations that the motion will be symmetrical about a vertical plane through the axis when inclined at a maximum or minimum angle to the vertical. For when @ has equal values on opposite sides of such a plane the values of w, and w, will be the same for the two values of @ and the corresponding values of @ will be equal and opposite. In the case of a circular dise the differential equation for , reduces to Legendre’s equation, so that , is of the form AP,+BQ,, where n is fractional. In the case of the top one first integral is known, and therefore the differential equation of second order reduces to one of the first order. The integration can be effected and the connexion between @ and ¢ reduced to quadratures, but the results seem to be very complicated. The results in these special cases have been already pointed out. § 9. SpectaL Property or a Top In Morton on a PerrectLty RouGH PLANE. In the case of the motion of the top when there is no slipping a somewhat curious property may be noted here. Writing for brevity V=gh (cos @—cosa), the equation of energy may be written (A + £24 9) + Aw? + Cw? + (Eo, + no;)?=2(E, — V), where #, is the initial kinetic energy. 48—2 372 . Mr GALLOP, ON THE RISE OF A SPINNING TOP. Also, multiplying equation (1) of § 1 by r, we find — Ano, + CEw, = Nr. Now square both sides of the last equation, multiply the equation of energy by Cé2 + An’, and subtract. We obtain (CE? + An?) (A + £2 + 7°) @ + (AC + CE + An’) (Eo, + 90,)? = 2 (CE? + An?) (EB, — V) — N°, or (AC + CE + An’) [(E* + 7°) 6 + (Ea. + 703)"] = 2 (CE + An?) (Ey — V) — Nr? + A (C— A) Now, if U is the velocity of G, U2 = (E? +m) + (Ea, + 90s), and therefore foe ee AC + CE? + An? Ne), A(C-A) CE + An?) * AC+ OB + An (z,- v= Sap vee eae (17). Hence, when the axis of the top is inclined to the vertical at a maximum or minimum angle and therefore @=0, T= 2(1 — gos cp ag) (ope ae) : : ~ AC+{CE ee Ada =| [,— F(a), where z=cos@ and F(z) has the meaning given to it in § 3, viz, NN? £@) So) G08) Faery en A Now it has been proved in § 3 that, provided 2 ghC?(k+ 1) N > Cuevas F(x) decreases as « increases, that is, as @ decreases and the centre of mass rises. Hence E,— F(a) increases as the centre of mass rises. Also C(k+.2)?+ A (1—2*%) increases as # increases, provided C'(k+1)> A, as is the case in all ordinary tops. Therefore a AC AC+r{C(k+aP+A(1 — 2°)} increases, as # increases and the top rises. 1 Hence both factors in the expression for U* are greater when the centre of mass is in its highest position than when it is in its lowest; and we obtain the curious result that when the top moves without slipping the velocity of the centre of mass in its highest position is greater by a finite amount than when it is at its lowest position. As a consequence of this result it follows that, when the spin about the axis of figure is so great that MW? exceeds the limit mentioned above, it is impossible to pro- ject the top so that on perfectly rough ground it shall attain the state of steady =_— —. Tae Mr GALLOP, ON THE RISE OF A SPINNING TOP. 373 motion with axis vertical; for if it were possible to do this the velocity of the centre of mass in the highest position for which @ vanishes would be zero and therefore could not be greater than the velocity im the lowest position. In the case when (>A equation (17) shows that the velocity of the centre of mass in every position is greater than in its lowest position; and in the case when (=4, so that all the moments of inertia at the centre of mass are equal, the equation shows that the velocity of the centre of mass continually increases as the top rises. The last statement may be perhaps true in all cases, but the equation (17) does not prove that it 1s so. On the other hand. when J is sufficiently small, it is possible to project the top so that it shall even on perfectly rough ground attain a state of permanent rotation with axis vertical. For consider a top rotating with axis vertical with angular velocity n, less than m, the least value of nx which will ensure stability in this position. The slightest disturbance will cause the top to fall down till @ attains a maximum value, @, say. By reversing this motion we see that it must be possible to project the top from any position in which the axis makes an angle with the vertical less than @,, so that it shall ultimately spin with angular velocity n about the vertical. This steady motion would however be unstable. Now in the motion of fallmg away from the vertical, when the disturbance from unstable steady motion is caused by slight deviations from the conditions necessary for such motion, we have N=Cn(k+1); so that there will be no positions from which the top can be started so as to attain the permanently vertical position on perfectly rough ground if N*>C?(k+1)n?, where m is the least angular velocity necessary for stability. The value of m? may be deduced from a result given in Ex. 1, § 252, Routh’s Rigid Dynamics, Vol. 11., 5th edition, and is given by Fe {A + (h+7r)*} gh Ny? = 4 —_—_ C+r(h+r) We may obtain some information as to the amount of spin necessary to maintain a top with its axis vertical from the figure of § 3. For, if the guiding curve slopes down as in that figure, it is obvious that the top cannot leave the position in which x=1 and therefore @=0. So that, substituting Cn(’+1) for N, we find that a top, having its axis vertical and rotating with spin n, will certainly be stable, whether the ground be perfectly or imperfectly rough or perfectly smooth, if gh(k +1) non a weenie eens ccctatereststerenscrssececeseaes It does not follow that the top will be in an unstable position if n? is less than the limit. The limit found in this way is not the same as that found from Routh’s result last quoted for the case of perfectly rough ground. As there is a difference, Routh’s result must give the smaller limit. For instance in the case of the top con- sidered in § 4 the limiting value of n found from Routh’s result is about 148 or roughly 233 turns per second; whilst that found for any condition of ground by (18) is about 170 or roughly 27 turns per second. § 1. AOA OOOO am iid =O ol GO Goto so @ XVI. On the Theory of the Multiple Gamma Function. By E. W. Barnes, M.A., Fellow of Trinity College, Cambridge. [Received 7 October 1903.] INDEX. Introduction . 5 - 3 3 é A 5 f O C : 5 oi - PART I. MouutreLE BERNOULLIAN FUNCTIONS. The fundamental series . : : : : ; : : : : c 2 . The values of the initial coefficients 5 é : Fi ; 3 5 : F : dj The fundamental difference equations for ,S,'(a) . The fundamental difference equations for 8, (a) . The reduction formulae Expansions for ,8,(a@) and ,S, (a+2) The integral formulae 5 The value of ,.S, (,+...+@,—@) The transformation formulae . The multiplication formulae . . 4 A The general linear difference equation with cheat coefficients . Differentiation with regard to a parameter Conclusion , : : z : : 5 - c c 5 . : c : : PART II. THe MourtieLe RieEMANN ¢ FUNCTION. The fundamental relation between the parameters Definition of the contour of integration. Specification of the many- Fvalced) fanetens Definition of ¢,(s,a) when a is positive with respect to the o’s. The fundamental asymptotic expansion . 5 Extension of the definition of ¢, (s, a) for all values of a The function ,»S_, (x) and an alternative expression for ¢, (s, a) This expression gives ¢,(s, a) as an absolutely convergent series Derived asymptotic expansions The value of ¢,.(0, a) PAGE 376 377 378 378 379 380 380 382 382 383 384 385 386 386 387 387 389 391 391 393 395 396 Mr BARNES, ON THE THEORY OF THE MULTIPLE GAMMA FUNCTION. 375 PART III. THE MottreLeE GAMMA FuNcTION AND AssocitateD MopuLar Forms, PAGE Re DEE eeicatoe = eed aoe. na sfol . A ee ee pr (@) § 24. Definitions of y,( (a). The associated asymptotic series. ; : : ; : 5 398 § 25. Expression of T-!(a) as a Weierstrassian infinite product . : : : ; 0 5 399 § 26. The difference relations te = = (2). The difference relation between successive r-1 multiple Riemann ¢ functions . : : : : ‘ : : ‘ : : 399 § 27. Other asymptotic expansions derived from 5 AL 6 . ; : . , . : 3 402 § 28. The nature of the infinity of ¢,(s,@) when s=1, 2,..., (r—1) . F : : i 3 402 § 29. The absolute term in the expansion of ¢,(s, a) for the previous values of s_ . t ; 403 § 30. The values of ¢,(—g, @) . : : ¢ : 4 ; : : ; : : é . 404 § 31. Summary of results : : : q : : : : 0 . : : : c 405 PART IV. Contour AND LINE INTEGRALS FOR THE MuxtreLe Gamma FuNcTION, 17s DERIVATES AND THE Associatep Mopunar Forms. § 32. The contour integral for log Pa c : : . ; : : § : . : 5 405 Pr § 33. An expression for loga . : . g : j : C : ‘ : . é : 406 § 34. The contour integral for logp,(o) . c : : : : : : ; : : : 408 § 35. The contour integrals for y,%(a) . : é é é : : : : : c 408 § 36. The contour integrals for the gamma modular ferns Yrq(o) F ° : < : : 409 § 37. Expression of logT,.(a) as a line integral 5 F : : . 6 : 5 j 5 409 § 38. Similar expressions for y,(% (a) F 5 : : 6 : : : 5 0 : : 411 § 39. Similar expressions for logp,(#) . : : . : . - : : : é 5 411 § 40. Similar expressions for y,q(@). : : : : ¢ : 5 : . : P : 412 PART V. THE PROPERTIES OF MuntipLe GamMMA FUNCTIONS AND THEIR ASSOCIATED Forms. The Multiplication Theory. § 41. An expression for T,.(ma) : A : : 5 7 : : ¢ c E ; ; 412 § 42. Determination of the constant ; : : : 413 § 43. The expression of p,(@,, ..., @,) in terms we gamma. functions of a half quasi- area : 414 ~ § 44. The expression of y,,(@,, ..., @,) in terms of the functions wy, ees, : 2 414 The Transformation Theory. § 45. An expression for I, (« et ee = 3 : ‘ ; : ; : . c 3 : 415 Pi Pr § 46. Determination of the constant - : . . : : : : ; : : : 416 § 47. Expressions for p, () and yrq ( ‘) ¢ . 5 ; : ¢ : ° : 5 : 416 § 48. The transformation formula for ¢,(s, @) : 5 . é : : ; 5 : 3 416 376 Mr BARNES, ON THE THEORY OF THE MULTIPLE GAMMA FUNCTION. The Integral Formulae. PAGE é § 49. The formula me WANG a1, 00) ON=W (A+@, | 01, @, @200., @;) : ; : : 417 r § 50. The formula ay, (a)+ 2 We (A+, | 1, ..., @s, ©, 00) Or) =Xy (A) - : . : = 418 = § 51. Determination of y,(a) . : ; - . : : : ; - ; : : . 419 § 52. Expression of [ ‘log, (¢-+a) dz in terms of (r+1)-ple gamma functions of specialised /0 parameters . Bs 5 : : F ci 5 - : : : c : - 419 § 53. The general analogue of Raabe’s formula : : : : : ¥ ; c : : 420 § 54. Derived formulae. - : ; : : 5 ; : : ‘ : : = = 420 The Asymptotic Expansion of the Multiple Gamma Function. § 55. Deduction from the contour integrals of the values of ¢,(s,a) when s is a positive or negative integer . : : . : : : : > : : ¢ : 3 421 —— 2 ae a) : : § 56. The value of = |s| tends to infinity, when R(s)>—-k . s : : c 422 ‘ Tr — : P T,. (+a) § 57. The asymptotic expansion of log Teta for large values of |2| : 5 C - - 423 PrA®, § 58. An alternative method of procedure. . . : : ; : : : ‘ : 425 The Transcendentally Transcendental Nature of Y, (2). § 59. Statement of the theorem : : : : ; ‘ ; : : A : ‘ ' 425 Addendum . : : : : 3 : : : : 5 : ; ; : 5 : 425 INTRODUCTION. $1. The present memoir concludes the series of papers in which I have tried to develop a general theory of gamma and associated functions. In the ‘Theory of the Gamma Function’* (1899) the simple gamma function of a single parameter (substantially Euler's gamma fune- tion) was defined, and its properties deduced by analysis intended to indicate the nature and existence of corresponding properties of similar functions of many parameters. On the one hand the properties were obtained by pure algebra, and on the other it was shewn that they could all be deduced from an extension of Riemann’s € function and an asym- ptotic expansion associated therewith. In the ‘Theory of the G@ Function’+ (1899) the theory of the double gamma function with two equal parameters was developed, and its connection with certain constants and functions previously introduced into analysis was indicated. ‘The Genesis of the Double Gamma Functions’} (1900) considered these functions qua functions of the ratio of the two parameters, and was followed by ‘The Theory of the Double Gamma Function’§ (1901) in which the theory of this and * Messenger of Mathematics, Vol. 29, pp. 64—128. + Quarterly Journal of Mathematics, Vol. 31, pp. 264—314. } Proceedings of the London Mathematical Society, Vol. 31, pp. 358—381. § Philosophical Transactions of the Royal Society (A), Vol. 196, pp. 265—387. Mr BARNES, ON THE THEORY OF THE MULTIPLE GAMMA FUNCTION. 377 associated functions was developed with considerable elaboration, The theory was first obtained by pure algebra, and then an alternative development was given by means of integrals connected with the extended Riemann ¢€ function of two parameters. It was shewn to be necessary to introduce Bernoullian functions and numbers of two parameters, and complete analogues were obtained of all the known formulae connected with the simple gamma function. More recently, in a paper entitled ‘On the Coefficients of Capacity of Two Spheres’* (1903), the properties of a function formed by combining two double gamma functions so as to be periodic in one of the parameters were briefly considered. The present paper extends the theory to any number of parameters. Its publication has been delayed by my desire to give to the formulae involved a form which it is hoped may be final. And consequently I have not merely extended the double gamma function by taking 7 parameters instead of two. Multiple Bernoullian functions are defined by and their properties deduced from the expansion of a certain infinite series instead of conversely from the difference equations which the functions satisfy. The fundamental asymptotic expansion which defines the 7-ple extended Riemann ¢ function is given in a much more elegant form than that given for the case r=2. And I have made a change in the definition both of the multiple gamma function itself and the associated gamma modular functions 7. (@, .-., @,) so that the numbers m and m’ which caused considerable complexity in the more simple case no longer enter. The simplification is obtained by giving a different signification to the many-valued functions which enter into the theory. On the other hand it did not seem necessary to build up again the gamma functions considered by a purely algebraical theory: the development from the asymptotic expansions of the contour integral theory is alone considered. I am not aware that any writers have considered the general theory other than Crani+, who has considered the existence of multiple gamma functions, and Pincherle ft who in a short note has indicated a few of their properties. Throughout the main part of the present investigation it has been necessary to use a symbolic notation to minimise the labour of dealing with the cumbrous formulae involved, I trust that this will present no difficulties to the reader. PARA MULTIPLE BERNOULLIAN FUNCTIONS. § 2. We define the r-ple nth Bernoullian function of a with parameters @, ..., @, as the algebraic polynomial ,S,(a@|@,, ..., @,) or more shortly ,S, (a), which vanishes when a=0 and whose first derivate ,S," (a) appears as the coefficient of 2” in the expansion —) ze—a2 — aye - r-1 4 bes con (ane nr "(a a ) é aK — ) = tO — A,(a) + > ( ) i v (2) on ai = r=1 * II {1 — e-} k=1 * Quarterly Journal of Mathematics, Vol. 35, pp. 155—175. + Crani, Giornale di Battaglini, Vol. 29, pp. 68—86. t+ Pincherle, Comptes Rendus, t. 106, p. 266. Wow, XOD<, IBA 100 49 378 Mr BARNES, ON THE THEORY OF THE MULTIPLE GAMMA FUNCTION. The expansion is obviously valid within a circle whose centre is the point z=0 and whose radius is the least of the quantities For within this circle Zee TI (1 — e~) k=1 has no poles, and may therefore by Taylor’s theorem be expanded in a series of ascending powers of 2. § 3. We may, in the first place, shew that A,(a) =,S,* (a), s=1, 2, ..., 7” For this purpose differentiate the fundamental expansion with regard to a: we obtain ~ es (-)? SPs (a)e" Sa ee it ON Sime Gea (-)'ze-@ is (—)' A’, (@) " aac A’ ,-a(@) - in aa (a) . r go gr 5s Zz {1 — e+} k=1 since A,(a) is evidently independent of a. Equate now coefficients of like powers of z in the two expansions and we obtain A; 91:(@) = A’,_,(@), Ofna Srey Pa and A, (a) = »S,"(@). Hence Ai,(a) = 8," (a). The fundamental expansion may therefore be written (—)'ze-@ = ¥ GES SO < (=) Sn! (2 I {1—e-e} =} Cae ee n! k=1 ; Note also that it has been shewn that SP ni(@) v (1) —————_—_———_ =, — 1 2 wae nie n aT Sn (a), nm 5) c] > § 4. We may next shew that ,S,,” (a) satisfies the system of difference equations Sn” (@ + ox) — Sa” (2) = 718," (@| a), «2, * 5 «2 @,), R= 1, 2, ..., 7 where the star « denotes that the parameter @; is to be omitted. We shall often write this system symbolically in the form Sn @ (a =F @) = Wn” (a) = rq") (a), in which » may be any one of the parameters @, ..., wy, and this parameter is absent from the function ,_,S,” (a). —— Mr BARNES, ON THE THEORY OF THE MULTIPLE GAMMA FUNCTION. 379 In the fundamental expansion change a into a+: then we have within the same circle of convergence, QUE. 2 OSM te) 2s (8,0 (a4 ©) on s—1 T{l—e-} s=1 Z n=1 m! Ul J k=1 On subtracting this expansion from the previous one we have (-)> Ze—t A & (-} 70, 8H) (a ot @) =a. pS, SD (a) i x ( at Sn!) (a + @) — Si, (a) =-2 ) c=) (= a. 1 nM: nm z \ = Ii* {1 —e-z) sal a n= 4 j the star denoting that the term (1 —e-) is to be omitted from the product. But by definition —)'—! zp—az ah =e y, (841) a = J. ( y ze s ( FS, oH (a) alts S (Se ‘raSn” (a) on Torey Se care Ton Ly ) k=1 Therefore on equating coefficients of like powers of 2 in the two expansions we have the required system of difference equations Sn" (a + w) — Sn" (a) = 8, (a). § 5. To obtain the system of difference equations integrate the fundamental expansion with regard Remembering that by definition ,S, (0) =0, we obtain satisfied by 8, (a), let us to a between the limits 0 and a. EFT _ Sy @)— 80) 5 yy, SO), = os—1 — 7 os Il {1 — eer} s=1 < nT n! k=1 : j oe mG) aie Hence ,S, (a+ w)—S,(a) is the coetticient of + in the expansion of n! (—)'7 {e7 (A+o)z _ GA WO weal) eee r r 2 T*{1—e} TT* (1 —eee} k=1 k=1 and this coefficient by the previous expansions is equal to _ 8S), (0) 7-1 NM, ron (a) + 0, pS@ 2 (2) = (2+ 1),Sr™ (a). We may readily extend this result and shew that Nate : n} < +Sn™ (a) 7) S,© (0) == (n—k)! rNn-k (a), if n>k>0. For from the fundamental result we have, if n> 0, Se? (a) =(n+k) (n+ k-1)\(n+k- 2)... (2 +1) Sn (@); and therefore, writing n—k for n, we have, if n>k>0, S (k+1) n! vi1) rPan (a) = (n—b)! rs n-k (a) miatatateteineiotetorelarereiviereiatesterstate taraietats (1). Integrating and remembering that ,S,_;(a@)=0, we have Si") (a) — »Sa® (0) = a ae Sy: (@), which is the required result. If in the identity (1) we make a=0, we obtain non (er) OG Sa Tee So S™,,-x (0) n! = (n=k=1)! rBn=ks if n>k>0. § 7. From the result of § 3 it is obvious that Sy")(a)=0. So that ,S,(a) is an algebraic polynomial of degree r+1. We can now see that ,.S,, (a) is a polynomial of degree (n+7). Mr BARNES, ON THE THEORY OF THE MULTIPLE GAMMA FUNCTION. 381 For putting L=n—1 in § 6 we have Sn (a) — 8," (0) =n! 5, (a). Therefore ,S,"~ (a) is a polynomial of degree (r+ 1), and hence Sn (@) is a polynomial of degree (n+7). By Maclaurin’s theorem we have, since ,S, (0) =0, , a Sn” 0) a Stn 0) ae +Sn(a) =,Sn (0) Tl + e+ eee a = Sn (0) 2 ‘iG a n(n—1) =,Sn O)5t+— 2! a+ ‘ Si 4 S’,-2(0) a +... + S1/(0) a” ry?) (0) qn >S, (3) (0) ot ie 9,7) (0) qty (n+1) (n+ 1)(n+2) aia: (n+ 1)(n+2)...(n4+r) a eep a+ MOE srs +, Ba ie Mee 7S?) (O)a"> oS (3) (0) are ¥s 8 So) (0) qtr n+1 (n+1)(1+2)° " (n41)(n42)...n4r)° A form of the expansion of ,S,, (a+) which is sometimes useful is ; nr S (7) (a) qi gtr n! Sara s SG) : gb (0): Sn (a+ 2) Gea da” an @ prea m 1Sn™t7-™ (a) =n (n— 1)... (+1) Sn (a), ' (n+r—m) ' so that Snot (a) gntr—m — Pryor att CRUST Sia!" (a) arte—m (n—r—my)! m! (n+r—m)! aa Sn” (a) qd gmtr m! da™(n+r)(n+r—1)... (W+1)° If nt+r>m>n Sm” (a) =m (m—1)... (+1) Sn" (a), zon Ret prs - s,m ( a) gnrr—m 5: Sin” (a) n! apes (vn +r—m)! m! (n+r— im)! n! ; Also Mme Eat aT {Sn (@)—-SO nee (0)}, and therefore we have the given expansion. 382 Mr BARNES, ON THE THEORY OF THE MULTIPLE GAMMA FUNCTION. The expansion may also be written rS’ngr (0) _ a Sn” (a) d” } aot n+1 Pits m! da™ (n +r) (n ra ea (n + 1) 4 Sn (@+2)+ From the expansion of § 3 we see that C 1 Sy? a)= aS ir+1) a= ———— =< rh ( ) rl ( @1@,~-. @; rSo?— (a) = +S" (a) =— ids ape LO saw, 5-9 (a) = 8," (a) oc (Yo,)a | YoP2+32o,0, Do” (a) =n," 7 (a) = — = aaa SS 3 te , JOT Oe eee nea eel Dy iPimeais. @ and the calculation of lower differentials can be readily performed if necessity arises. § 8. We next proceed to shew that nea (x) =r) Joa n+l penaete [--Suta) da = Both sides vanish when «=0; all we need therefore establish is that Seen 7Sn (2) =- ata (@) _ rBan, n+1 which at once follows from the result of § 6 on putting »+1 for » and k=1. If now @ stand symbolically for any one of the parameters @,,... @,, we have [Se (a) da= en =a Bes r—1 By .(@,, Rienttcieee @,) earn SP (o,, oe o,) on using the difference equations of § 5. We shall usually assume in such a formula that @ is the parameter omitted when we are dealing with functions of (*—1) parameters, and therefore the last formula will be written i Pe (a ) da =— Sons (0) Be ras ay (0) = ra Bus J0 nel (n+ 1)(n +2) — n+1 —o By... § 9. We may next prove the relation Sn (a) = (—)?*" Sp (@, +... +o, — a) — [1 —(-)""] Baas. For by § 5 Sn (@, +... +0,— a) - = (—)? 12” is the coefficient of pe qs Mr BARNES, ON THE THEORY OF THE MULTIPLE GAMMA FUNCTION. in the expansion m: {e(—a1- eos Wr) Z+0Z __ 1} 6s It (QL Gaee) k=1 le. im the expansion of : (-—)' e7&(-2) TI (1—e-?) (1—e- k=1 k=1 and is therefore equal to aS, (0) an eS ng (@) antes = oe +(- -) n Tee eee =(—)rn 1Sn (a) + peri ank — Bris; whence the required result. § 10. We proceed now to prove the transformation formula m,-1 m,—1 k o k o } q at 1 r Yr) SS Sour aon (« +—— + ...4—)@,..., or) k,=0 ky =0 m , = @ @,. / U 1 7 . ~ = ji, (« Penne on My v0 My» Day (Oy, «0s Op) — p Brin le 383 the numbers m being positive integers, in terms of Bernoullian functions of parameters @), 2...) Wy. By the fundamental expansion of § 3 @ @ Sa a — a my, Mm, (- ae is the coefficient of onthe in the expansion of Ms i Ws (- \" gw, { Ss S: Sts A b (—)y? evs ne ay ee r Ws ey er Sa Res ll {1-e rad II {1 — e~e?} s=1 0 Mise ty-le _(, kao Key wy ay -yo 2 {e (a+ ma + Mr ja ee kp=0 7 f -—wsZ) I {l—e s=1 Therefore @ QO; my,-1 My —1 / k @ ko. Sy (a — Br) x «> & Sn (a+ ——+ ... $=] @, ... @, my, Mey) k,=0 ky=0 \ my My | 384 Mr BARNES, ON THE THEORY OF THE MULTIPLE GAMMA FUNCTION. RTs + Now by § 8 rSn' (a) _ Sn (a) + Bue Therefore changing x into n+1 we have | @; @, @; @, Sei (aa) eee @) + Bus ee o-) dads Wal emt ue Nine es ky, m, m-1 My-1 “J AD. yey Sn (a+ ky=0 kr=0 ky,@, Saas ie) SMM eee Mon Dag @isies <5 @r)s ; which is the formula required. If now we make a=0, we obtain -1 m,-1 ~ ~ @, Qo, He " ON k,o, B sol or) = > D5 Sh (Ss SE oo Se eis U0 le (OD @ oY oye ae Pe aos PAT J8o Be Mp a 1(@), --- Y oe Mm, 5) My R=0 erie n Mm, mM, y) 1 Tren ( ? , r)> which expresses the Bernoullian number of @, @, Tie aie in terms of Bernoullian functions and numbers of parameters @;, ..., @,. § 11. From the preceding result we may readily deduce the multiplication formula. For, as in § 7, we see that @, «6. @ ySn (A | @), «+, @,) is a homogeneous function of degree (n+7) in G, @, «+, Wy. a\@ @ Therefore m” ,S. (= — ee or) = SA(alian ave 7 n m\m ? ) m mn ( 1> ? r ), @) @,; and me -Bra(=, 3005 | = Bay (a, «-- Oy). From the first relation we have one form of the multiplication formula, viz. : lo @ Sn (ma | @, ..., @-) =m" Sr (a — ee *r) 5 Putting now m,=...=m,=m in the result of § 10, we obtain Mm Sn (ma |e, .--) O-) +m" Bris (@1, «-., Or) m-1 ™m,—1 fain ese + k,o, 5 ON Fa ) +m rBay (@,, «+, @,), k=0 kyp=0 and therefore ; m,—-1 My—1 ko, te de k,o, Sn (me | a, <--5 O,) =m" 2. 27a (a+ Heit + er) ey) m + (m'*" — m™) ~ Buz (@1, «--, @y). Making a=0 we have my fe s e (ONS pp5 +hk,o, tee rPn = me . — -t.1)) Be ) = {—m" +m") Bai (@1, «++, @r)- k= tr=0 ~~ /' OO — — Mr BARNES, ON THE THEORY OF THE MULTIPLE GAMMA FUNCTION. 385 § 12. We may next prove that, to an additive polynomial of degree (7 — 1), ,S, (a) is the only algebraic polynomial which is a solution of the linear difference equation T F(at+o,+...+0,)- = f(ato,+...+% +... +0,) =1 r as +2 & f(ato,t+...¢% +... *% +...+0,)—... *=1 +=1 + (-)"> 2 F(a + @):) + (-Y f(a) = q”, c=1 In the first summation the star denotes that one of the o’s is to be omitted: in the second summation every two different pairs of w’s must be successively omitted, and so on. Symbolically+ we write the above relation in the form F,[f(a+2)Je-0 +(—) f(a) =a. If $:(a)= f (a+ w,)— f (a), gz (4) = $, (4 + 2) — f, (a), ; (4) = $2 (4 + @;) — py (a), the given difference equation may be written $3 (a+ @,)— $4 (a) =a”. Now [G@. F. § 11] the only algebraic solution of the last difference equation is Snrla | o,)+C,, where C;, is an arbitrary constant. Hence the difference equation satisfied by ¢,.(@) must be $2 (4 + @,4) — $2 (a) =, 8, (a | o,) + C,. Therefore we must have $2 (a) =.8S, (a | 4, @-) +C,a+C,, where the C’s are constants; for the difference between any two solutions of such a difference equation is a simply-periodic function of period @,,, which cannot be a polynomial. Proceeding in this way and remembering the difference equations of § 5, we finally see that we must have (OQ) =]AAG@ || hs Goon OAS Ob! SS ChE a0 SEL, whence the theorem. + The symbol is suggested by Cayley’s notation of matrices. Cf. D. G. F.§ 49, and C.I. F.§ 4. Note that when wy =u)=...=0,=1, F, [24], sat Leer (r- 1k” —— 2)k— + (=). Te = AO, cso | in the notation of text-books on the Caleulus of Finite Differences. Wor, D6 IBA IONE 50 386 Mr BARNES, ON THE THEORY OF THE MULTIPLE GAMMA FUNCTION. § 13. We may shew that é . ao ry) (@) = = pp“) (@+-@ ©, 0@,...,@,) [s=0,1,2,...,¢41)], Omce: i and ay ton (a) =— 2S n1(4+ @| @, @, ..., @,) (aay SH Bae Ca) where w denotes any one of the parameters @),..., @,. Take the fundamental expansion of § 3, (yee gOS @ 86a SO s=1 zs te n!} I(1—e-) ea Then on differentiating with regard to w we get rH ge (a-+w)z (—)F 5 SiS? (a) ce ayn o oS” ee (—) +1 Ze a < s=1 =e 7 el — e-*) tl el = ei) k=1 But the expression on the left-hand side admits the expansion tee (=) raulSneee! (a + @ | @,@),...@ ») ESS; S (= as 2” Sn. (a+ @ |@, @1, --- w,) s=1 aes n=1 ie Therefore, equating coefficients of like powers of 2, 0 . a0 S48) (a) = — 48,87?) (@+ @ | @, @, ..., @;), ome " ap” Sn’ (@) = — NryS’na (A+ @ | @, @;, ..., @;), which are the formulae required. § 14. We see by these results that by differentiating 7-ple Bernoullian functions with respect to one of the parameters we obtain (7+ 1)-ple functions in which the additional parameter is the one with respect to which we differentiate. We thus see that r-ple Bernoullian functions in which two parameters are equal are substantially (7 —1)-ple Bernoullian functions. It is for this reason that in the G@ function (which is sub- stantially the double gamma function with equal parameters) we were able to express all coefficients in the fundamental expansions in terms of simple Bernoullian functions and numbers. The results can be easily extended to the case of higher derivates with regard to the parameters. The algebra of multiple Bernoullian functions with equal parameters is needed in the investigation of the asymptotic expansion of repeated integral functions of multiple linear sequence *. As however it is not needed in the developments of the present memoir, it will not be considered further in this place. * For the definition of such functions see the author’s ‘‘Memoir on Integral Functions,” §§ 18—20. EEE Mr BARNES, ON THE THEORY OF THE MULTIPLE GAMMA FUNCTION. 387 PART Sn THE MULTIPLE RIEMANN € FUNCTION. § 15. We now assume that in the Argand diagram the points representing @,,@:, ..., @; all lie on the same side of some straight line through the origin. If, and only if this is the case, will it be impossible for mo, + mw, +... m,o,, for all positive integral values of the m’s, to represent any point as nearly as we please in the finite part of the plane (we assume that 7>2, and that at least three of the w’s are linearly distinct, so that there does not exist a relation Mo, + Mw, + mw, = 0, for finite values of the numbers m: otherwise the conditions binding the o's will be substantially those of double gamma functions *). By an easy extension of Eisenstein’s theorem+, we see that the series y a) oo 1 Sooo SS See m,=0 m,=0 (@ + (QD) yin where Q = mo, +... m,@,, is absolutely convergent except for the points a=— over the finite part of the a@ plane. The function represented by the series will have poles at the points a=—Q. These, on account of the assumption which we have made, will be isolated points in the finite part of the plane. They will mass together as we approach infinity within the non-re-entrant angle formed by lines from the origin to — @,, ..., —@,. § 16. Suppose now that all the w’s he on the same side of some line P through the origin. Further let a lie on the same side of this line P; we shall say that in this case a is positive with respect to the o's. Let 1/Z be a line at might angles to P drawn from the origin into the region in which the o’s lie, and let Z be the line conjugate to this line with respect to the real axis. The four figures represent different cases of distribution of the lines according to the position of the o's. Take a contour, which we may without confusion also call Z, which encloses the origin and embraces the line ZL. Then, when |2| is large, along this contour I&[(a+)z] will be positive, if QO = mo, + ... + m,0,, the m’s being any positive integers, zeros included. * Vide D. G. F. § 19. + Jordan, Cours d’Analyse (2éme Edition) t. 1, §§ 317, 318. ; 90—2 388 Mr BARNES, ON THE THEORY OF THE MULTIPLE GAMMA FUNCTION. For if @ be the: angle PO measured positively in a counter-clockwise direction from Ox [the angle is represented in figure (2), in figure (1) it is negative] the / /Y. (4) : 7 : *,° 5 : argument of 1/L is 6+ 5 measured in the same positive direction. Hence the argu- ment of LI is ae —6. Now the argument of a+ lies between @ and 7+6. Hence the argument of (a+ 2)z lies between pa and > and therefore 3 [(@ + Q)z] is positive. It is readily seen that the proof holds for all cases which can arise. Suppose now that (—z)* = exp [(s— 1) log (- z)], where the logarithm is rendered one-valued by a cross-cut along the axis LZ, and log(—z) is such that it is real when z is real and negative. Then will i (1—s) il s—1 p—(a+0)z = 2ar I De ae mone (a+ QO)’ where 1/(a +2) =exp[—slog(a+)], the logarithm being rendered one-valued by a cross-cut along the axis of —1/Z (ie. the negative direction of the axis of 1/Z), and log(a +) being such that it is real when a+Q is a real positive quantity. Mr BARNES, ON THE THEORY OF THE MULTIPLE GAMMA FUNCTION. 389 For the contour has been so chosen as to make the integral always finite, and the equality is known* to hold in the case of figure (1). Also as 1/Z starting from Ox passes round in the positive counter-clockwise direction with increasing argument, the relation continues to hold uniformly until LZ passes backwards over the negative half of the real axis. Then by definition (—z)*“ on the left-hand side of the relation is multiplied suddenly by exp {27(s—1)}. On the right-hand side however, since —1/Z passes in the positive direction over the positive half of the real axis, (a+ 0) by definition is divided suddenly by exp {27s}. The equality therefore remains true as 1/Z rotates right round to its initial position. § 17. When «@ is positive with regard to the o's, we define the r-ple Riemann € function by the relation dz, (6G, GA Ors cosy D>) = Wd —s) [ e-% (— z)s4 the contour LZ, and the expression (—z)' being defined as in the previous paragraph. The parameter s is any complex or real quantity. We assume that the contour Z encloses no poles of the subject of integration except the origin. By the result of the previous paragraph ag Priel 1 (1 =a s) - is eee Prict arr See _= = -{ (—2z) ee |) ee ede: m,=0 ; Myr=0 (a oF O) <7 JL m,=0 M,p=9 or, aS we propose to write this result symbolically, one el _Wd=s) | (—2)> pn} Snape aes (a =F QO) Qa L me 1 — e PR?) ===) ee ae ap i 2a eS ) ee a | | —¢e-** | é 8 pr — Pj Nwy,Z =(2((G,0)\\Grp oon OD) | e-2 (— z)1 eco de k=1 7 L Il {1—e-“#} k=1 vss SS | Paty ge legen k=1 1=1 Qar a eee k=1 r al s) sans + (— 5 | era (= 29 ____ a. 2ar L 7 f — 072 | II {l-—e aa In the double summation & and / must not take simultaneously the same value, and similar restrictions hold for the higher summations. * Vide G. F. §§ 22, 23. 390 Mr BARNES, ON THE THEORY OF THE MULTIPLE GAMMA FUNCTION. Take now the ¢th set of integrals J, (say) on the right-hand side. We have r r Ai ihc EP2 (Pi Phy FTP rye! (n= BE AGRO coat, Beh i ee ve II (1 —e-*) k=1 and therefore by the fundamental expansion of § 3 a ae Ts) (Oe oy a Se af Sa es) (Yr han 3 2 Ge [ce (a) + 1. § PHBA) (Emenee m=1 Mm: Hence : (m+1) ks -)r l= s 4ac S S Si ae oO oe 2S k=1 ke=1 | m=1 ; > ; = (* 3 pure.) ie S ily e! — 8) Sn'(a) 1 mai L(2—s—m) m! a som (w = Pewee) u= ees ee rd -s) i gen m=0 (» S Pe. eg) T(r—m—s+1)C(m+1) ul eit u ~) a ined (a) — ECs) ] as s+m—r — —_ m=7 (» > Dik) (—s—m+r+ LC (m+ 1) te for from § 6 Sm (a)= m! 7) (a), and therefore Sa Oy ST Sy see@y if r>m; and S’m-rn (a) = Sm? (a) PEE it mor. Thus r r (r ~ = CieLes 5 [see ra-s) : k=1 ke=1 | m=0 m. T( bad $ & STs (-s-—mtr+1)(r > PrwK, | = u=1 aj Sees | 5 ee Cre k=1 ke=1 L m=0 m! da” G21) G=2)e=niLss Pi, wy eS hd for = pa a see da™ (s —1)(s—2)...(s—7) Be Ge) es -(r=s—m+t])), T(1-s) astm ge str—m = (- yh (s— 1)(s —2)...(s—r) T(r—s—m+l1) Mr BARNES, ON THE THEORY OF THE MULTIPLE GAMMA FUNCTION. 391 Suppose now that F [6 (@) ]e=mo = b (1, @; + My@24+ ... + M,@,) Yr Yr — = d(mo,t+...+* +... 4+m,0,)+ 2 TY P(mo,t * ... * +... M,o,) ee Ber y ey = (mpop) in the symbolical notation introduced in § 12. Then we have pet ye & Sn!” (a) Sg a 2 , i Ss) eee nao (a@ + OY BGs ats) m=o0 =m! qd aster Ide E GEING=2).. €= 5 | siegepeneeeseee (A). The series is asymptotic: it must therefore be regarded as subject to the methods of summation which have been developed elsewhere. § 18. The preceding asymptotic expansion has only been obtained and &,(s, a) has only been defined when a is positive with respect to the w’s. The multiform functions which arise when s is not an integer are rendered one-valued by a cross-cut along the axis of —1/L. When %&(s)>7, we have on proceeding to the limit when s=o , m=0 (a+ Q) & (s, (), and we may evidently use this equality to define ¢,(s, a) for all values of a. In a similar manner we may use the asymptotic equality: pn-1 1 al 1S, (r) (a) qm” a Ss; ae Sesh } > v m f J x i ; m=0 (a at OF S (s; ®) ear me! x = (1 ig s) (2 = 8) poo (YP silon where the many-valued functions involved have a cross-cut along the axis of —1/L, and their logarithms which intervene are real for real positive values of the argument, to define £,(s, a) for all values of a when 3 (s) $7. § 19. We may extend the method given by Mellin} for the simple Riemann ¢-function and express ¢,(s,a) as thus defined by an absolutely convergent series of functions. We thus prove that ¢,(s,a) as defined by the asymptotic equality is independent of n and the p’s, a fact which however can be proved at once from the asymptotic series. + See the Author’s ‘‘ Memoir on Integral Functions,” + Mellin, Acta Societatis Scientiarum Fennicae, t. 24, Phil. Trans. Roy. Soc. (A), Vol. 199, pp. 411—500, especially No. 10, §1. §§ 36 and 37. 392 Mr BARNES, ON THE THEORY OF THE MULTIPLE GAMMA FUNCTION. We form the function ,S_,,(«), which is defined by the relation k+r-1 Sm (0) qm gr S s)= > z int 7s, k (2) Fe Ae anal : er m! dam where the many-valued functions have a cross-cut along the axis of — 1/Z. We have at once the asymptotic equality ry, ~) (w) = Ss, k (x) : ae merry om! = da” | (r—s)... (1 —8) 4 2 7S na (0) a” | ak. ~ QE Te ae aed If now & is a positive integer such that 3(s)>—4h, and | «| is very large, the last wie : ‘ ‘ 1 series involves only terms which vanish with eae Hence - : Ae il | F,[,S_s,4(@+@)]2—pno+ terms which vanish with 7 _ af & Su) a" (wt ay mes |, mt de® (rs). = alist [ vF+a a aS & a gers + | Ses rSm'” (0) a da 2) da? a | =i || 3, Ses Lm=0 m! dx™ (r —s) goo el = s) 4 2=pnw for when z=pnom the expansion d a d? DAO) = Oe le te ele cto Ce Pe ae, coat a is valid when all the functions involved have their principal values with respect to the axis of —1/L, since wherever a may be, if x be sufficiently large, (a +) is positive with respect to the a's. Therefore F, [,S—s,c(@ +2)}x-pnw + terms which vanish with : =F | 5 Sen (0) ae dé as r=pnw izo lmao om! «= (Rk—m)!) dat —s) ... (1—8) Now S Sm” (0) Gas . JS ce (0) a™ mao . m'! (k—m)! » ‘9 (k—m)! m! & 8+ (0) a™ _ Se” (a) 20 k ! m! k ! i Mr BARNES, ON THE THEORY OF THE MULTIPLE GAMMA FUNCTION. 393 Therefore : : were F,[ -S_s,x(@+2)]z-ynw + terms which vanish with = ss =. 7S” (a) d* ars | See lee) ue a dak. =e). 6) | err Bist 1 ¢ SSS Sa SS (GF a) m=0 (a a= Qs 5 c for all values of s and a, by definition. But F, [+S_s,4 (a+ E))| ere n—1 ‘ {F, [S_s,¢(a@ +04 i) \ es = ((=)¥ Skip (a+ Q)} = (-)’ a SLlap: (a), 0 | where the summation is r-ple and Q=mo,+...+ mex. : ; 2 : : F sini th ; We therefore have if we ignore terms which vanish with = the equality Se Q +42) 8 NS bey S_.e(. f(s, a=" rales —3,k (@ + 2+ 2)|r-4 +(—) r =s,k (@ + ~ (a+ OY ri—ak § 20. We now proceed to shew that this equality is not merely asymptotic, but that in the limit when n is infinite we have an absolutely convergent series: so that &-(s,@) may be defined absolutely, when 3 (s)+k>0, by the relation = ; a 1 ) Ae f(s, a) = =e Fe [-Ss4(@ +02 + 2)]p24+(—)" S64 (a + Q) — @+Oy| + (—)" »S_s,4 (a). We have : 1 F, [Sis (a +0 +2)}ru+ (-)" say: (a + Q)- (a+ OF ap [PEt Sm 0) a (a+ 04a a = m! -da™(r—s) ... (1—s) |e<. ys Sm” (0) qm (a ze 0 te br) ed | = il ce EO m! da™(r—s)...(l—s)|z-. (a+) ktr-1 of (r) (0) na —s) a(at+ Oy IP/Al — s) = STEN r—s—m wu TEE 5 m! {P- |(a+9) : R@=s2me1), UL (r—s—m) a (a+ 0)-*"=T (1 =s) to T@-s—m—1) tee r= 4 ON a+ Oyen Pd - a ist T(r7-—s—m+l1) (a+ Q)° The expansion is valid when Q=n( pro, t+... + ppo,x) if n be sufficiently large, for then, wherever « itself may be, a+ is positive with respect to the o’s, and therefore (a+Q2+x)*™ may be expanded in the given series and all the many-valued functions Vorseebx. Part ITT. 51 394 Mr BARNES, ON THE THEORY OF THE MULTIPLE GAMMA FUNCTION. involved will have their principal values with respect to the axis of —1/L. The assumption that n is sufficiently large involves the neglect of a jinite number of terms in the multiple series with which we started. Now F, [6p + % + ... + Cpt") z-0 = (—)"e. Hence the expression last written is equal to kt+r-1 = ) — = gitr (a+ Q)R=as 1 2 ae E de Peso)... (a+ Q)° m=0 Put now qg=m-+l and the last series may be written a ic T(1—-s) 4 Sm” (0) qaomtr cao (@+. OD) to om! (y—mtr)i Tes—9t rnc +F,| = Pd=s) SST Sm? 0) gyomtr oon goktr @+ ON yao mE (Gomer) P(—s—gt Jeu @+0F Now $ Sin'” (0) aim pe 4 -Syr*) (0) gmtr m0 m! (q —m+ r)! m=0 gq! (m + r)! == | -8@)- 8/0) 7 - = 5," 0) ees q _ a7 (r 1)! . Hence the above series may be written i & 1 T(1—-s) S, (2) Cae | Beg 30 G@+O)r qi! T(—s—qel)”* gae (aE Oy} "| gaktr (€+ O)7*T (—s—g41) nao mi (q—mtr)! fe= But Fe [pSo(@)|=. =0 by § 12) af ig =O; and, if g=0, TS Gb Hence the series may be written = T(1-s) aS 7S! (0) ahterte = > @+ DEFT (sk —r-qtl) nao m! (k+2r+q—m)! r= Se a Sm” (0) gktert+q—m i a Bs eke! (0) gmtr+g m=0 m! (k + 2r+q- m)! m=1 (k+r+q)!(m+r+q)! a 1 _ Siete (0) 2°84 = (k+r+q)! | -Seseea (x) — rSkirtg (0)-. (r+q)! ; the series may also be written an] Ms rs) Fo feats) o(k+r+q)! (a+ Ose T (—s—k-r—q4]) 0 (r+s)! 5 =a Mr BARNES, ON THE THEORY OF THE MULTIPLE GAMMA FUNCTION. Take now the first form of the series. It is evident that, for all values of q, (rr IC) al (0) aFteta—m —s-k-r—- 1)f,| . —--oH_ ( ga) | m=o Mi (k+2r+q— ila F reel Sin” (0) apk+2r+q—m-1 "| mao mi (k+2r+q—m—1)! Jr-u is always finite; let the greatest value of its modulus be 2X. Then for all but a finite number of values of 0 | F, [-S_s% (a +0 +2) ]r-w ar Cy rS_s,k (a +Q)— bel (a+ Q)s T(1—s) 1 if 2 “ re =e CEO Gee + ...J5 for, wherever a may be in the finite part of the plane, we can for all but a finite number of values of choose 7 a real positive quantity less than unity, such that esse PaO or” The preceding modulus is therefore less than | T(1-s) J |\T(-s—k-r+1) Hence the r-ple series 1 ‘ 1 Renicesosma E 1 > SEN Sy ae ag \F. Sere (a+ Q+ £) Jr—w a> ( ) AvLgp: (a aL Q) Ee - art is convergent with Rees mete Peal (a+ OYE | ’ and, since 4%(s)+k> 0, the latter series is absolutely convergent. Therefore we have deduced, from the definition of €,(s,a) by means of the asymptotic equality of § 18, its expression in an absolutely convergent expansion == m=0 {Ff So4 (G24 2Nlret (YF Sne(a+ ©) — Gr ayh +) Soa § 21. Take now the asymptotic expansion pn-l 1 na San? (a) Ee eee a 5 ee F m=0 (a + 0) Sr(s, es m=0 a etr m! da™ (—s+r)(—st+r—1)...(-st =|. in which all the many-valued functions have their principal values with respect to the axis of —1/Z. 51—2 395 396 Mr BARNES, ON THE THEORY OF THE MULTIPLE GAMMA FUNCTION. Make s=e, where e is very small, and expand in powers of e. The logarithms involved will all have their principal values with respect. to the axis of —1/Z. And we obtain Pips Pent 67S log (a +0) + © = S [log (a + 2)} ae ; e [ 3? = ¢,(0,a)+e E (E(CR = + rT E GAs «| Also ii ime loots i ] ‘y \ 2 ” (a) qm 1 € 08 2 5 WOE —.. sla ry ror y See Een ace = | 5 m=0 ™m!} [ie rn! pce be fi k=1 [=1 kit z=pnw where in oe simultaneous values of & and J are included. k=1 1=1 a Equate now coefficients of corresponding powers of ¢, and we find (r) — Pi Po... prnr =F, (0, a) + 8,” (a) F, be a Ss Ar, F ts | aS epomagbODe (1), iy (r a 1)! t=pnw "S log(a+.0)=-(F (a) + 5 Op, m=0 ray reo cu d™ ar ae sae (log z- 3 sie oe (2), =i oe E Dog (a+ oF =[F b6s, | =0 8=0 rn 3 4 a (a) p m=o) ure! &e. | Ty Staal e Ss F, | ae = + |(log 8 — 2loga S Spt? = atl Peer (3), m7 i) 1) ‘A ca F,.| —— =((), Slee eee (r—s)! x=pnw F, = ©) es Hence (1) becomes Again F,[a|,.=7!@,@,...@,, for by the relation just written F,[2],... can only depend on the product of the o’s from the manner in which the p’s enter into both sides of the relation, and therefore it is equal to the term involving @,...@, in (@, + @)+...+@,)’ which is 7! @,... @,. Hence the relation is equivalent to Ose cs 1 and So” (a) = 5 i 360 Oe the latter identity being in agreement with § 7. The former may by § 6 be written f, (0, a) = (—)" Sy (a). - - —- = ee Mr BARNES, ON THE THEORY OF THE MULTIPLE GAMMA FUNCTION. 397 PART TU THE MULTIPLE GAMMA FUNCTION AND ASSOCIATED MopuLarR Forms. § 23. Let us now consider the relation (2) of § 21 which may be written, if we leave unspecified the prescription of the logarithm on the left-hand side, « te} 7 m ar r fee a+9)=|-266 a) | es eee (2) 5 ee (log « - 37) s 0 m: k z=pnw m=0 =0 m= da” Pe k=1 It thus gives the asymptotic expansion of the logarithm of the product pyn— pyr Tl... Il (a+mo,+...+m,o,), m,=0 m,=0 when 7 is very large. We shall take ut ||, ‘eo(G a]. + log a+ =—log p, (a,, ..., @,), a=0 ——_ and call p,(w;,...,@,) the (r+1)th r-ple gamma modular form*. And then the _pre- ceding expansion may be written when a= 0 m-1 x m a tom da” r! the accent denoting that the term iii, to m,=m,=...=m,=0 is to be omitted. Now Sm” (0) = Sera (0) = Byres m! (m—r+1)(m—r)...1 (m—r)!? so that the last expansion can be written so that the coefficients are r-ple Bernoullian numbers. We further put “0 jet; Oe |i ir(s@) |] = log and define 1,(a) by this relation as the r-ple gamma function of a of parameters a, ..., @,- We now have LO, § Se" @p [2 (eg $1 AAC) ee m! ti da” (108 2 PEt z aeons n-1 log ai (a+ 0) =-— log m=0 and we see that Lt | 2 G(s; a)| + log a! = Lt log [al’, (a)] — log p, (w), =0 0s s=0 ) a=0 * We may also call p,(w,,...,,) the r-ple Stirling modular form (cf. D. G. F. § 64) as it arises in the generalisation of Stirling’s asymptotic value of n!. 398 Mr BARNES, ON THE THEORY OF THE MULTIPLE GAMMA FUNCTION. so that Lt log [aI (a)] =0 a=0 and therefore Lt on = il. a=0 Combining the two results just given we see that pn-l Y aed : / = S Simi (a) = mi (0) ce =( = *) log a + log ait (1 se 4)= = log T.— (a) + = = 1 FEI log x a an § 24. We proceed now to define the r-ple y-functions by the equality ds dat SOC |Okey Som Oh Saleh sage. Let us differentiate the preceding asymptotic expansion of pyr-l prn-1 log IT ... Il (@+m,o,+...+m,@,), m,=0 mr=0 with regard to a. It is readily seen by the theory developed in the author’s “Memoir on Integral Functions” that the process is legitimate*. Wietobtemn) if (q/— ly 2a oor se 1 il 2” Sy +a) (a) qd x r :) {1 (q —1)! SS ltl > 2h = CG D2. ee ae =~ WO 2 i roe ee = (loge — = i) loan’ since ,S,,"*9 (a)=0 if m=0, 1, ..., (¢—1). When qg>r the asymptotic series on the right-hand side of this equality vanishes when vn is infinite, so that we have ay oO 1 (r-+1) (Neo S ae. ————— 3 (a) ( ) fy m,=0 se (a sl Oy) We thus see that ,'"*” (a) can be expressed as an infinite r-ple convergent series. It is possible, starting with this definition, to build up purely algebraically a theory of r-ple gamma functions. This process was carried out for the case r=2 in Part II. of The Theory of the Double Gamma Function. We now define + the first 7 r-ple gamma modular forms Yn (@,, Py) @,), POOH) Yrr (@,, IR) @;), by the relations il Lt Jor (q-1)! fa +, «| =— Yrq (@), Of Uy) 00 a=0 so that on making a=0 we have the set of asymptotic expansions n=1 % (r+q) mM mr , rg vPE demos 2 Se Op [2S (loge ED] a nS =pnw a m! da” pec * Loc. cit. $ 17. See especially § 31. theory, we have modified the definition of these forms so + Apart from the new specification of the logarithms that for the double gamma function -y.;(w,, @,) and which makes a simplification throughout the present Ya(@,, #2) must be interchanged. Mr BARNES, ON THE THEORY OF THE MULTIPLE GAMMA FUNCTION. 399 § 25. We may now shew that rh ) oO a are Oa —)rar Es (a)= exp. Jer + 2! rat -- Die aie svt | a Il [ 1+ a) e229 ae r ver), m=0L\ / thus expressing I,1(«) as an integral function given as a product of Weierstrass’ primary factors. For we have pn—l + a pn-1 f a a (-)" a’) r / ibe: we ASB tose Se ee Ne pad | \ log a + log Ir (1 + 5) + Ean +50 7 OF + ayn Yer = log I’, (a) + = Sm ee = HOF E= Zi og 2 — S Al 3 k=) '/ =piiw m=0 dam ri \ r = 3) ! wep m=q m: at s Sm? (0) dm ar r il Ee da™ ry (log — k= = 3) ne And since ,S,,"*% (0) =0, when m=0, 1, .... g—1, we may write this expansion log T,(a)+ 3 1,8 (a)— Sn ()— 3S Sq"*9 (0)} F, Es 5 (loge — 3 Al Bae a || q=19- dan! pak £= Sin . When m = : a E (G=p=l el Deo. pao! ce i= GS |, -o..t Now a) ke ke F, eae a5 [os cre - (ona) hia | ui a] k : X=pnw =F, [27] 2—pnw oF Je [27 2—pnw = Fy [(x ate pro)")z—pnws and recalling the definition of the symbol F',[2%],-p,,, we see that the last expression vanishes. We therefore have asymptotically when n is very large the equality & (s, a) = 6, (s, a+ @) = (i (s, a) = — (Gas (s, a+ pro). Now when » is very large a+pno is a point positive with respect to the o’s, and therefore [§ 17] (1 —s) fe e7 (a Fpne)z (— a 27 f(s, a+ pno) = dz. = ie WUGlSsreco) k=1 But this integral vanishes when n is infinite. Therefore on making n infinite we obtain the absolute equality f(s, a+ w) — ,(s, a) =— 6,_,(s, a). We have thus obtained for the r-ple Riemann ¢ function a set of r difference equations valid for all values of s and a. We therefore have 2 tta+ “|, - |Z 66 a) = loa «| and therefore, by the definition of the multiple gamma function, log i. ats @) 1% log i) —— log Ps (a) Pr Pra a T (a+@) af I, (a) eae (a) Pr—i ; The functions T'_,(a@) and p, are functions of (r—1) parameters @,, ..., *, ..., @,, the star denoting that the parameter w has been omitted from the sequence. WO, SDK, IRA IDOI 52 402 Mr BARNES, ON THE THEORY OF THE MULTIPLE GAMMA FUNCTION. § 27. Let us next in the fundamental expansion of § 21 write q+e for s, g being a positive or negative integer, and expand in powers of e. We obtain pn-l 1 e log (a+ Q) 3 3 = taauR- (a+) + be (qa) tes &lqa) +n m=0 Sm" (@) p a4 ae A St aia ail P| Gop ep [~eeee + $ og ay —...} S Tr Tr l 1 l+e > +e pay tat | : x| 5 2a q k=l t= kK—q l—q 2=pnw Equate now coefficients of corresponding powers of e and we get pn—l if 2 Sm” Sm” (a) a™ a4 SS S en mao (@+0)2— a as sane ale Es E (r—q)...- pel pmllog(a+Q) a 2 Sm (a) | a” gr4 ee | pao) eg ee F. ine aa a= (EO ae ee dx SSS Agee ee: and so on. § 28. It is evident that these relations as thus written become nugatory owing to the presence of an infinite term if g=1, 2,...,7r. When however q has one of these values we have pnzl 1 elog@a+Q), ) == (eROyEe ea m=0 7S. ‘m' Sm” (a) ad” a4 f =. = +e ee S = + a br(q+ 6,0) “anal = = (r—q)...(—€)...(1—q) iy Ee 3: (log 2) a r aI ORE ie ip if see > * —— +. 4] . k=1 k-—q k=1 l=1 k-q q ‘l-q z2=pne the star denoting that the term or terms corresponding to k=q or /=q are to be omitted in the summations*+. : 1 If now we equate coefficients of => We see that «o Sant? (a) a™ (—)2a"-2 = f l= > —— EE ae \ebr q zs : 9); Ce m! te da™ (r os q)! (qg = 1)! one In the expansion all the terms vanish except the one for which m=r—g. We therefore have rS"),_9 (a) (—)" (r—g)! (q—1)P i {eE,(q+e,a)} = + By equating coefficients of « in this and the previous expansions we see that we may obtain the asymptotic -1 expansion of *s ¢ (a+), whenever ¢(x)=<* (log x)‘, t being a positive integer, and s any complex or real quantity. m=0 Mr BARNES, ON THE THEORY OF THE MULTIPLE GAMMA FUNCTION. 403 pn=1 il (—)a+ ws”) re a) > = Pe q Mend Cea cg = Li [Eg + 6a) q@-DiG-@! e=0 2 Sin (r) (a) | qm (- )rrar-4 @ 1 ) S F, s* | gears m! da™ (r —q))! (= 1)! flogse— =1 eae We see then, in the first place, that when qg=1, 2,...,7, &(q,q@) is infinite. When e is very small we have the expansion Ef (—)2*7 S”,_(a) 1 &F(q+te a= AGE iz +A,+Aje+.... Now rS,9(@) = (7 — g)! Si2 (a) by §6. Therefore the expansion may be written (- _— 27) (a) &(qte6a)= SiMe +A,+Aje+.... pn-1 § 29. Let us next compare the asymptotic expansion for = (a oy with that ob- m=0 \% tained in § 24, viz.:— pn-1l 1 1 oS ae SS SS) ) S@roe- qa © mi 1 Se (a) CHO ae k *) —)i - & ie — — = = ce) (q—1)! ma¢ m! E r! (log « = | error = a Se) > i 1 Sm tt) ) (a) OHS bi _)q-1 S ane : 4k ) (qg pare 1)! m=q m! i E url log ole oo Sy na *2 (a) qm ar = ; ae a2 m! Li lao pe Gl log | Zepne _ 3 SO , fam og m= =0 G +m)! P eeaem Ge > L=pnw es (Ga) Cae =e q q-q-1 1 = S rqt+m Ls 2a a l a = el Sen moo (Q+m)! E (r—9)! | = fhle le onGa{e=Ce) i ow girtal a) ar—4 ; q q-q-1 — atm ( _)\r-1 a ~ m=o0(qtm)! ” Liem ae |! reer 2(r—q+1)(r—-q+2) q-q—1.q—-2 cae x Gel) aah 7S, "*4) (a) +3@-q4+D(r—-qt+ 2-943) i r} r} _ = Sm "@) p d™ a’—log x m=0 m! i dam (r—q)! =pnw = G1 Tred gigal Gag)! ae +(—y",. sieen(a) | el Nae NO Ea Hen) = 52—2 404 Mr BARNES, ON THE THEORY OF THE MULTIPLE GAMMA FUNCTION Therefore ~— il BE (—)9 —————— cet | (@) Reg Fa) ky “Br Gao Gap Pa aoH ee a) =i a Se (r) ( a) qm (-)2 z4 ] = eee or - a= m! a5 FE (¢— 1)! (r—q)! (log #) z=pnw (aa) q me Boe! ea (Gn Ol =qal ~ 29+) r—qt+2) r! On comparing the two expansions we see that Sy) aS a aa =e ae = eh A, + (—)s** Gam = a) aS S,() >= rag le Sad carey pone feet es q-q-1 (SEG)! ae @= Se Oh =a 2r—qtl)r—qt2) r! & (—)2 ers KS +1) are gl One — gan 2 (a)} + ce pi? (| 3 WORE ae es Ae q-q-1 pee ent | r—-gt+l 2(r—q+)(r—qt+2) ~~ q(ir—gqt)(r—- G42)... 7] But Sie fe -q-1 (—)?%q.(q-1)...1 pile wees . a r=g4l 20 —g4l)@=742) ” '¢@=gP De =¢e2).F 6 rol. eee as we may readily prove by induction. (-)or> 5, (9) (a) qe} il ENN ie Hence ‘l= vv, (a) + G=DA ais (q—1)! We thus have, for ae SI t, (q+6a)=(-)"* an ae i = wv, (a) + nar pS, (a) PS +4 A,e+... § 30. Suppose next that s is a negative integer. Then, replacing q by —q, by § 18 we have the asymptotic expansion, when g is a positive integer, pn-1 oo S (r) m +q S (a+ Oy=f,(-qa+S SO w]e = m=0 m=0 7 Se GET GET come This expansion is however finite, since all terms for which m>(r+q) vanish. It must therefore be an absolute and not merely an asymptotic equality. m! We may write it so as to bring out the coefficients of the various powers of n in the summation = (a+)? in the form Si ‘m r+ ang ys r+q (a) or ” (a) qtr—m | d me ] a ans +q)! = ee. Fr | gam (+1)... (gtr) Jz=po Mr BARNES, ON THE THEORY OF THE MULTIPLE GAMMA FUNCTION. 405 e 4 pn-1 : Now when p,n=...=p,n=1 the r-ple series (a@+)% becomes a%. The ex- m=0 pression just written becomes, by § 7, : aN -) — 4 as Gal q, a)+F, | S(a+e)+ q+1 _j|2=o and this by § 12 = ¢, (—g, a) + (—),S, (a) + a? + (-)27 ae, q We therefore have £,(—q, a) =(-)" | S,(a)-+2 an (Oh = (-)" {Sq (a) a5 aS Sy 1 = Sano by § 8 § 31. We have now obtained the values of £,(s,a) for all integral values (positive or negative or cr of s. If we tabulate our results we see that Gs) — Fe ) (a), when s>7, Si) (a) (—)2 | q=1 ;| = ((— are Ds NY JL () , _)\r-1 7S) > fal = cemie= 7) gem Or eae as s tends to q, which may be any of the numbers 1, 2,..., 7, =(-) = Sas ao) when s<0. —) Soa, (a Since G(s;0) = fe ) s=—qg<0, and by § 26 f,.(s,a+o)— €,(s,a)=— 6,4 (8, a) we see that pS'qui (@ + @) = S943 (@) = aS gar (2) which is once more the difference equation of § 4. PART IV. ConTouR AND LINE INTEGRALS FOR THE MULTIPLE GAMMA FUNCTION, ITS DERIVATES AND THE ASSOCIATED MopULAR Forms. § 32. We next proceed to obtain contour and line integrals to express log I, (qa) and its derivates when a is positive with respect to the o's. In this case we have (§ 17) FG, a) ee a eee = dz, * TI (l—e™*) k=1 2a the integral being taken along a contour embracing the axis Z which is defined in § 16. 406 Mr BARNES, ON THE THEORY OF THE MULTIPLE GAMMA FUNCTION. We have also [§ 23] OG elon Ag oT NY en © pb, (o)" Hence — ik —az (_. 7 \e—1 ioe a Lit 2{, | eS fet, (0, a) Pr (@ e=0 € <7 L i (1 — e-™*) a, ps0 meee Ve 622 (Hye elas (= 2) ste. ao ah. e=0 (€ Il (a — er) k=1 We therefore have ye Pe ee edz (—)’ Si (a) = §(0, a) = aa Zz I (1 —e-***) k=1 and log ae = yf, (0, a) — | erie On Aa) Ay Pr (@ 2a} 1 il att e717) Thus ie Oo f © Cae log (= fear Le Pr (@) a7) L Fil lac a2} k=1 the logarithm having its principal value with respect to the axis L. Again we have seen [§ 23] that ae log {aI (a)} =0. Hence — log p,(@) = iit log ro + log a} = Lt {= | emmnlve ls es]anh Le ie al. aan Sin ieee ps k=1 § 83. We may now prove that, if a@ be positive with respect to the o’s, a If : dz leas {log (—z) + y} rim log a. With the notation of §15 suppose that arg l= 9. Then changing z into ze‘* the integral just written becomes l a Sac cael owl — ge") ahs and the contour now embraces the positive half of the real axis. The expression log (— ze*) has still a cross-cut along the axis of Z, and its imaginary part ranges from u(p—7) to v(p+7). The expression is therefore equal to log (—z)+vep where the logarithm has a cross-cut along the positive half of the real axis and is real when z is real and negative. Mr BARNES, ON THE THEORY OF THE MULTIPLE GAMMA FUNCTION. 407 The integral is therefore equal to t nis , az a fe ae? ‘log (—z) tid + ¥ taken along a contour embracing the positive half of the real axis, this line being a cross-cut for the logarithm. Now the effect of multiplying any point on 1/L by e* is to swing the line 1/L round until it coincides with the positive half of the real axis. The ’s then will all be situated to the right of the imaginary axis and hence ae is situated in the same region of the plane. We may then by an argument previously used [G. F. § 33] deform the contour of the integral till it embraces the line from the origin to 1/{ae}, the logarithm having its principal value with respect to this line and the formal expression of the integral remaining the same. Write now z for zae* and the integral becomes [ dz | e~ {log (— 2) = loga +9} =, the contour of the integral embracing the real axis which is a cross-cut for the logarithm. But = jen {log (—z)+y} =e 0. Zi Hence the original integral is equal to Joga. An alternative method of proof is the following :— N Let € be a point on the axis of Z very near the origin. Reduce the contour till it becomes the axis of I from « to ¢, a small circle of radius | «| round the origin and the axis of L from ¢ to a. dz a = = [es = flog (—2) +4} p= [ew tf cnet {log e+y¥ + u(@ —r)} dé a € ; Zz Qqrr- 0 a ) = ics = +log ¢+y+terms which vanish with e. When 7 is real and positive we have seen [G. F. § 28] Ie (R) e-? = =—log»—y+terms which vanish with 7, 1 where R denotes the positive half of the real axis. 408 Mr BARNES, ON THE THEORY OF THE MULTIPLE GAMMA FUNCTION. Put now az for z and the integral becomes 5 f@eé. a the integral being now taken along the axis of But this axis is within an angle T °) from the axis LZ, since a is positive with respect to the a’s. a5 dz z See £ ere > 1 Hence considering je along a contour consisting of the axis of — from ~ to a a infinity, a circular are at infinity, the axis of Z from infinity to altae where ¢=arg L, 2 _ surrounding the origin, we see that | and a circular are of radius =e SY ls i G)ewo-] | 7 la (Ly ere 2 +0 ( — arg =) + terms which vanish with 7. ob Z y Put now — er —6 = meade 1 and we see that | (L)e-* — +loge+y=t (arg aa $) —log a +u¢. Hence S5= || ue {log (—z)+y} =— log a. 2Qar L Zz j § 34. We now have Giselle ee 1 Oh) ee =3-| 4 a { -1} toga) +4 %. | II qd — eK) k=1 The integral has only been established when @ is positive with respect to the o's, but it is obviously continuous as a tends to zero. And therefore we have = log ease ye 1 t log p, (w) = =P II (1 —e-™) k=1 an expression for the (r+1)th 7-ple gamma modular form which is valid whenever such a form exists. § 35. Contour integral expressions for the functions y,? (a), which are valid when a is positive with respect to the o's, can be immediately deduced from the result of § 32. For we obtain, on differentiating with regard to a, v2 (a) = x| eH (— 2 {log (—2) + eae = II {1 —e-*} a7 k=1 Mr BARNES, ON THE THEORY OF THE MULTIPLE GAMMA FUNCTION. 409 The residue at the origin of e7a (— Zz) TT (1 —e-} k=1 is (- yas, (q+) (a), when gr7. Hence e7* (— 2) log (— z) ,(2 (a) = x te - dz + (—)y.S,'* (a) when g r. BED Meneseee k=1 §36. We proceed next to obtain contour integral expressions for the gamma modular forms Yrq (@). qg=1, 2, ...7. = gate) 24 Fyne DY. at For by § 24 Differentiate the result of $33 with regard to a, a process which is obviously legitimate, and we find SE @G= i b on at oe = ard (—.z)* (log (— 2) + 4} dz. Hence 1 5 = 919 (0) = 5 Lt | Ae) I a nee cee II {1 —e-%} k=1 And therefore, as the integral is continuous up to the limit a=0, r 1 = Yrq(@) = =| 1 — ——_——__| (- z)™ [log (— z) + y} dz, L TL {] pa e7 =} giving the required expressions, § 37. We proceed next to express log T’.(a) as a line integral. We have are pats ) log Er = =. | oa {log ( 23 dz Pri@ eee aD {1 — e-*} k=1 and by § 34 log p, On (ee : yk a a 8 Pr 207 u A “| II {1—e-#} 2 k=1 PON ae e Hence log I’, (a)=— 5 =! fae : Ny eis log (== DEY te ee II (i —e-*) k=1 the integrals being taken along the contour embracing the axis L. Vou. XIX. Parr III. 53 410 Mr BARNES, ON THE THEORY OF THE MULTIPLE GAMMA FUNCTION. Reduce the contour as in $33 to that part of the axis Z from infinity to ¢, a small circle of radius || round the origin and the axis LZ from e to », e being a point upon Z very near the origin. Then we obtain log T’,.(a) = [oi —— +i Lene | pa — acco __ | { == Z +1| {loge+y+e(A—7)} dé. “7 J 0 | tLe ee k=1 The second integral is, by § 5, when we neglect terms which vanish with e, equal to ele x (= ¥{.S,"(a)—,S. uO} S (-),8,(a) eles +-1)>| floge + y+ (0 —7)} dO 0 on = eo ese Sal n! = (—)' (log e + y — ur) [Sy (a) — »Sy (0) + (—1)"] he rn : «fri (a) — 8," (0 a +(-y 5 | “eae EX fe Qasr Or. eh | =(-Y {log e+ y — er} {, Sy (a) — ,Sy (0) + (— 17} soy Sp SL@=S2 +(—)’are {Sy’ (a) — Sy (0) + (— 1). e1(s—1) Thus be, (2)= | a Feit | = Tl (l—e-* Ps? el =F yt log e + y} {,S, (a) +(- 1)"} + 26 yee —az __ {s) — Y (8) 0 =[ [+t B00" oa SO) eee el rostete ‘ zie wz) oe Now if arg L=¢, we have [G. F. § 28], neglecting terms which vanish with e, OD ates 2 = — (log e +7) + op. Hence } lg eto) =| =. cA Olea 45+ 3 —)jrrs* = oe Cle Mee = ‘ — ze" —(- jy aS) (a) +1)" Sere yr 8 (a)}- Make now e tend to zero, and we obtain | S, (s+) = Ss (s+1) log T, (a) = ie ols oe eee {a de oe) ged +1—{1+(-)"-S,(@)} ea : + op (1 + (—)' Sp (a)}. Mr BARNES, ON THE THEORY OF THE MULTIPLE GAMMA FUNCTION. 411 3R (L) be positive, we may write this in the form log I. (a) = in (is) | 3 ar el Ee) Tl (1 — e-*) s=1 a Op +1—{1+(-)'S,(a)} e* | = § 38. Let us differentiate g times with respect to a: we obtain — az a Sy r+s+q—1 =tSh haces WG) == ia (L) e > G) — (a) a (Sexe) = (= years oe ene) (_ 2)21dz + wh (—)" Sy (a), when Gi— oe This formula may also be written Az r = Na! ESB) 76 yo@=Cyrs | a@/F=_4 2 Camo |B TI (l—e-e) 04) 7 +(-)- _1 So — eee) 292 dz+(-y ih So? (a). It may be modified so that ¢ does not occur when 3 (L) is positive. When qg>v7, the formula becomes Wy, («)=-| (iy eae TEA S II (1 — e-*) k=1 §39. Let us now transform in the same way the contour integral which expresses (§ 32) log \ ot , namely Bent / e— {log (— 2) + y} dz 277 | L rn ( =z) : Zz Il il = ier k=1 We find n e-w 1 S64) zZ log 2 [ (L) = = +(- y= x & yy a ) (a) +(e yo, Si (a) ee" | dz pr (@) J0 Tl fl =e= wo) Zz eZ k=1 + (—)' ep Sy (a), or, if (L) is positive, 2 me Sr (es \ sale aad) =. (L) ee +(-) = ee ay Si a = II {1 —e-»} oe k=1 Hence d na PE (\ St S81) We log p, (@)=— i Ge == FEY 2 = ee tee aa ae | 0 IL (1 —e-") on ‘ This line integral for log Ae) is valid whenever the o’s are so distributed that the function exists. Daz 412 Mr BARNES, ON THE THEORY OF THE MULTIPLE GAMMA FUNCTION. § 40. We may similarly obtain line integrals for the gamma modular constants Yry (@) G=150 25 ney t For we have (§ 36) t if Yrq(@) = =| 1 — —_—_—__| {log (— z) + y} (— 2)?" dz. L ll {1 —e-} k=1 And this contour integral when transformed in the usual way becomes . Nt 7 =a) ,S, e+) 0 (-y't3 Ae (ZL) co +(—\O2+ > ( zB : = (C ) a? II (1 —e-*) ae 5 k=1 (= )r- _ So) re iO) ene? ai dz+(— = td -S SD (0). When sae is positive we may write these integrals ri (3) —z yan] (Bde {= 1 - Og eo OFS 8," @, i (1—e-™) a e, : pe (7-41) ( 2 ya=[ (E)(-2)de {~~ — 1-204 oy S890), 0 Tl (1—e-*) i : k=1 and finally (r+) t= (L)(—2 yds |e 1 aio- ‘ (ee “*) Notice that, in verifying this formula for the particular cases of the double gamma function, we have altered the definitions of the gamma modular constants so that the numbers M, m, m’ in the case 7=2 now vanish, and y,,(@) is written instead of ¥,, ,.(@). PAR VE THE PROPERTIES OF MULTIPLE GAMMA FUNCTIONS AND THEIR ASSOCIATED FORMS. The Multiplication Theory. § 41. We next proceed to consider the multiplication theory for multiple gamma functions. We will prove that (—)71 Si (ma) m-1 m-1 m vod @ -O,. T’.(ma) = — ilies P.(a+? Je ee + Pee) mv ay pe 1 pi=0 where m is any positive integer. For we have seen (§ 25) that I (a) = exp. lyn + Site + wet ter a nm (1 +5) e =F a y 2 n=) the product being 7-ple. Mr BARNES, ON THE THEORY OF THE MULTIPLE GAMMA FUNCTION. 413 Therefore I’. (ma) m—1 j \ I V.(a+? Bee, aE eae p=0 me m / is a function of @ with no poles or zeros, whose expression in the form exp. {g(a)} is such that g(a) is at most a function of a of degree 7. We may therefore assume T. (ma) m= lox == =¥,,(a) + loo I 2 pr(@) Xe(@) er m me je (w+ eae in) pr (@) 5 @ = Change now a into a+ = and subtract, denoting any one of the parameters a, ... @,. Then, since [§ 26] V7 (a + @) a Dy (@) (Oia we have Xr (a + | — Vr (a) = — yy (a). This formula constitutes a system of 7 linear difference equations. Now x1 (a) = (—) log m 1S,’ (ma) [G. F. § 4] x2 (a) = (—)*-! log m 2S,’ (ma) [D. G@. F. § 65). Hence by the system of difference equations x3 (a) = (—)> log m Sy’ (ma) + C, where C is a constant, and therefore generally Xr (a) =(—)'* log m Sy (ma) + C. § 42. We can conveniently prove that C= 0 by considering* the case when a is positive with respect to the a's. For in this case [§ 32] T,.(a) c [ e&“ \log(—z)+y} dz og = ral aoe ; = aa =a A Zz ay jl —e- oe} : : 1) >@,. ae ‘ : Hence, since the points a +e tit Pre are all positive with respect to the a’s, i] log II vm p=0 pr (@) i [ enw \log (—z)+ y} dz re (ES) SF 5 — —@ “p-9\ m m 27 at ee iui {1 — e-?} k=1 —az { ( l de m4 u [ e-® {log (—z)+ 4} dz Dar Bie : zIl{l1—e 4 k=1 ? MECN / em ‘log (—z) +7 + log m|} dz 2 r piney zII {1—e-*} k=1 for the change of z into mz does not substantially alter the contour L. * The introduction of the integral forms is instructive but not necessary. The same argument as that used in § 51 can be employed. 414 Mr BARNES, ON THE THEORY OF THE MULTIPLE GAMMA FUNCTION. The integral just written is equal to log [. (ma) + (—)’ »Sy’ (ma) log m. We thus have xr (2) =(—)' Sy (ma) log m, and therefore the form given for the multiplication formula is established. § 43. Suppose now that in the multiplication formula we make a=0. Then re- membering that [§ 23] i log [T,.(a) a] =0 m—1 ee , @ wee @,. Aer pl (@) = mi 841 TTT, (ee rhe , p=0 Put now m=2: we have 1 pr?) (@) = 2-¥ 7 rS'O+1 TT’ TL, (Pet FP) 3) oN m If now we put + ee. + O> IP (3) hy Fe OOO) ee (tte) yi so that y,...,ys" are the 7-ple gamma functions of half quasi-period, we may write (ltrs@41 ne Pr (@,, erty @,) =2 27-1 Ly eee rYor—1 2" -1 thus expressing the (7 +1)" r-ple gamma modular function in terms of gamma functions of a half quasi-period. This formula is the complete generalisation of the relation [(4)=,/7 in simple gamma functions. § 44. If now we differentiate logarithmically the result of § 41 we obtain mar, (ma) = (—)"7,S8,'2 (ma)mlog m m S @) Pi®@1 Pr >) fre eee (a+ eae 5 q=1, 2,...,7, m—1 and mi, 2 (may= Ly (« SEE st oe q>?. =U m m Make now a=0, and remember that [§ 24] = am(o) = Lt | yn(a) + OGD") ‘ g=1, 2, ...,7. az Then we obtain, when q=1, 2, ..., 7, (int — 1) Yrq (1, «++, Oy) = (-Y 8, 9+) (0)m2 log m — 5 yy, (Re: + = ae Beer) p=0 and we find, when g>7, "Ss 1, (ee ain = se Ps) =(-M(g-D)i(m-1) 3... So m=0 mr=0 a2 Mr BARNES, ON THE THEORY OF THE MULTIPLE GAMMA FUNCTION. 415 The Transformation Theory. § 45. We next proceed to consider the transformation theory for multiple gamma functions. By the same argument as that employed in § 41 we have, if the p’s are integers, | ul k Tp @ T,(a|%,..., 2) Weiss 2 3 Pe (at 4... + er) —— 4 ero TT) AP Pr p (@: 2 =0 kr=0 Py (@,, ++. @,) 2 7 yprees Pr Pr or, as we shall symbolically write it, r, (a *) poi lr (« 353 =) P Xr (Ct) TL : (*) k=0 = Pr(@)” P where y,(@) is an algebraic polynomial in a of order r. Change a into a+, and we obtain by § 26 r= (« | =) o p-1 Ds (« + ss) l ae a ry =} °) ] Ml no og = 2 Xr (4) — x (a A + log IT EE) P ; : 3 ko . where the star denotes that in the summation the term corresponding to i is ) omitted, ‘We therefore have the system of linear difference equations typified by Xr (a+°) — Xr (d) = — Xr (@). Now for the case of the simple gamma-function [G. F. §7] yx, (a)=0, since Hence x:(a@) is a doubly periodic function of a satisfying the difference equations [ @ 2 | 1 m1) 2 ‘ =0, x. & = X2 (a) Xs (a + 2) — x2 (a) = 0, and therefore, since it is a polynomial in-a, it must be a constant. Hence y,(a) is a mere constant, 416 Mr BARNES, ON THE THEORY OF THE MULTIPLE GAMMA FUNCTION. § 46. We may prove that this constant vanishes by means of the contour integral for log oe which is valid when a@ is positive with respect to the o's. - pr (@, For rt p= ae _ hrwr ey (a age e-™ {log (— Ope = Gz (> par ) p-l p t = log II SR CKL Gea - - dz k=0 Pr = L zl a x e-"} k=1 « [ e flog (— z)+y} i gee. Dade dz z il le rat since ZL is defined with reference to the parameters precisely as with reference to Pr the parameters @,. The transformation formula may therefore be written symbolically IE. («|2) poune («+= 2) ST P / (=) r=0 Pr (@) § 47. If now we make a=0 in this formula, we obtain x (2: or) ~ [Pr (w) |PiPr Pi 7 P =p in (Aa ees ae) ’ k=0 Pr Pr which is the transformation formula for the (7+1)th 7-ple gamma modular form. We have also, on differentiating the concluding formula of § 46 logarithmically with regard to a, De ko (2) (a *) = > 7,9 (« +2 ) ; y P/ k=0 ¥ 1g which is the transformation formula for the function , (a). If, when g=1, 2,...7, we make a tend to zero in this formula we obtain [§ 24] @, ,ot) = Bo} & @, k, o, ey rq \@y5 «085 ,) —2 mY 2h qb us ae a Na ey ae Pr the transformation formulae for the first 7 gamma modular forms. § 48. The multiplication formulae can of course be deduced from the transformation formulae, though the labour is almost as great as that of the investigation previously given. The transformation formulae are all particular cases of the transformation formula for the Riemann €-function to which we now proceed. Mr BARNES, ON THE THEORY OF THE MULTIPLE GAMMA FUNCTION. 417 We have [§ 18] for all values of a, ,(8,a|@,,...,@,) defined by the asymptotic equality pn-1 1 a SG a) qd” aa str Si emp Ot 2 P| dee OTD em’ where the many-valued functions have their principal values with respect to the axis of —1/L. Hence, writing p,q, for p,, ete. h qr/ @ Y (r) 2) Ea 1 S Sin (« q/ F qm gost ser Mm, @, XD =e m | "| da™(1—s)...(r—8) |r= at +... +— c= pnw Nn qr AES Se at eo 1 =0 r= = = lo 2 1, 1-=0 m,=0 mr=0 (a +3 243m) q q=l iA lw qd” ast Yr as BP (e+2 iGaet es 22 SeereeNlaee : moe m! F, E (1 —s)...(r— Slo a IS In the above algebra the symbolic notation has been changed into the literal whenever it seemed that clearness was thereby gained. When the parameters are not indicated they are invariably o,,...,@,. The transformation formula, written at length, is | @ @ eM US if lo, ¢.(s,a ee) Po enables Sa (hee eee eee \ 1h qr 1,=0 1,=0 \. n From it the results of §§45—47 may be deduced. L.@, qr see Pe The Integral Formulae. § 49. We proceed next to consider the integral formulae for the »-ple gamma functions. In the first place we may shew that = ee) (| @i, ---5 Or) = Wr" (€4+ @| @, @),---, @,.) @ where » is any one of the parameters @,, ...,o,. r+1) \ (are rt For [§ 24] Ar) (QO aa Or ee (a+ Oy" where Q=7,@,+... +7",@,. Won, IDX, IPN Nuk 54 418 Mr BARNES, ON THE THEORY OF THE MULTIPLE GAMMA FUNCTION. Hence 6 r+) (q| S < (a a) Sam == 1) (aliays oeaiOe) =a aenece geese NE EES Ns os BF Cw, te ee ) m,=0 m,=0 (@ + 4 @, +... + M,@,)"? Now the series involved are absolutely convergent, and may therefore be summed in any order. Put a=a+m,o,+...+m,@, and consider the sum = mm, m,=0 (a an 1, @,)"** It is evidently equal to a = =- iz =e A te] ? m’=0 m/=o (A+ @, +m @, +m @,)"*? AS for the latter series is absolutely convergent, and there are in the double series m, terms for which a+a,+m'o,+m’o,=a+ mo. We therefore have eels kd kee eee (yeep Go, *” NPT n=O m’=0 m=0 meso (G+ @, + Mw, + M’@, + M.O2 +... + My@,)" (r-+2) HV (a + @) @), @), @o, eres @,). In the case when a is positive with respect to the ’s the formula is obvious on differentiating the contour integral for Ww,’ (a) with respect to o. g g Pp § 50. We may next shew that r ay,” (@ | @;,..-, @;) +z Os 11 (WT+ Og @), «.., Os, Ws, «++, Or) = Xr (a), s=1 where y,(a@) is an algebraic polynomial at most of degree r. FE = r+) (q | S (Sa or since i Oy ee Oe) — eo) Sa Vv, | 1» ’ ) ahs (a au oy is homogeneous of degree —(r+1) in a,a@,,...,@,, we have by Euler's Theorem A é . (ox +0 5 +...+ 0, <) vr") (a | @y, ..., Or) = — (r+ 1), (@| a, --.. @,), and therefore by the previous paragraph : (r+2) avy," (a| @,..-,@-)+ & Oy (G+ wy | @;, -.., Og, Ws, ---, Or) s=1 +(r+1),")) (a| @,,..., w,) = 0. Integrate this result (r+1) times with respect to a, and we obtain r ay,” (a| wy, 20, @r) + YS OsW pas (@+ Wy! Wy, ..., Wy, Ws, 5 @,) =r (a), s=1 which is the required result. Mr BARNES, ON THE THEORY OF THE MULTIPLE GAMMA FUNCTION. 419 § 51. We will next prove that Xr (a) = (—) ,.S,/(a). For x, (a@ + @) — x; (a) = (a+ @,) "9 (@ + @,) — ay,” (a) + = Os (We 4 (@ + @, + @s| @1;..-, Ws, Ds, -.+) Oy) = Wr" 11 (@+ @s| @;, ..., Ws, Ws, .-., Or)} =-— aw", (@) @,..., @) + a, (4 + @) — @," (a + @,) : — = as," (A+ @5| Os, ..., Os, Ws, ++, @r) by § 26 =2 s =— Xr (@| @, ..., @,). We thus have the system of difference equations typified by Xr (@+ @|@;, ..., Or) — Xr (A|@;, ..., Or) =— Xya(A| @, ... ¥ , ---) Oy), the star denoting that the parameter corresponding to w is to be omitted. Now it has been seen [D.G. F. § 70] that Alexeiewsky’s Theorem [Theory of the G Function, § 29] may be written ayy (a\w) + or,’ (4a + © @, ©) = 8, (a @). Hence Xi (4) = Sy (a @), and therefore by the above difference equations for the case r=2 X2 (a) = — Sy (4 @,, @.) + C, where C is a constant; and therefore in general Xr (a) = (—J7 SY (a), ..., @) + C. The constant C must vanish. For suppose C= (= AS) CA Greasy 2) Then, since y,(a) satisfies the difference equations Xr (a + @) — xr (4) =— Xr (A\@, * ..., Or), the difference y,(a)—(—)’,S,'(a) satisfies the equations f(a+o)—f(a)=—C,4(@, *, @;). But it is impossible that C,(@,,...,@,) should satisfy this system of 7 difference equations unless C,_, vanishes. Thus all the C’s must vanish, and we have x, (a) =(—)" 1,8: (a). § 52. We have now established the formula Zs ay,” (a\@,,---, Or) + D asp ,41 (A+ W5\@1,..., Ws, Os, +--+, O) =(—)" 1, S,'(a). s=1 Integrate now with respect to a and we obtain a log I’, (a) — I log 1, (a)da+ S w, log Dir (@ + Os) @1y -1+) Os) Oss v1) Or) =(S Gy. vy) s Zi Penn (GH ins aoan Oh Opeosenn 420 Mr BARNES, ON THE THEORY OF THE MULTIPLE GAMMA FUNCTION. Thus we may ewpress the integral of the logarithm of the r-ple gamma function in terms of (r+1)-ple gamma functions of specialised parameters. The formula may be slightly generalised as follows. We have (2 +a)v,2 (e+ a)4+ 2 Oh 24 (@ + 2+ Wy! @), 2-5 Wg, Wg, «++, O) = (—)7Sy (2 + @). Hence integrating with respect to 2, [, log T(z + a) dz =(z + a) log T(z + a) — a log’, (a) r Dea (E-* @|@i; ory Oey Osteen) La) 5S fz +1 | 1 $ J Ny = ea! w, log Le AN(Gllon ks, OnOneao@.) ieee), Fol i) Se We also have the formula ik log BAe oy =(z+a) log I,(z+a) —alogI,(a) — z log T, (z) 0 l, (2) Dy (a + 2 + os) V4 (5) Rada SS CEES) CEST se anime § 53. If we now make z=@ when @ is any one of the parameters @,, ...,@,, We obtain | tog T(z + a) dz= (a + @) logl.(a+ w) — a log I, (a) 0 Ta (@ + @ + @5|@), -.., @3, Os, «++» Or) s Y “ 1 Dn (4 + @s @,, .-+, Ws, Ws, ---, Or) a (Oy [Sy (a+ @) — rSy (a)] whence D1 (@| @s, ..-, @,) Pra (@, OSE) @,) |, test. +a) de =— a log + @, log p,(@,, ..., @) 0 s=2 : pm (a Do, «++, @,) Pr (@s, ses) Os, Wg, --e, @,) 1 GS hog Cp) OP) conn O)) aan (hy S555 @))) +(-)" [aS, (a Wo, wees @,)+ saplé (@., OOS} w,)], and therefore . |, 8 I’. (z)dz=a, log p,(@, ..., @,) + S w; log p, (Ws, «.+, Ws, Ws, --+, Wr) = w = ((@n4F @r)s the latter identity being the analogue of Raabe’s formula for simple gamma functions. § 54. Many other interesting formulae can be deduced from the previous results. Thus we have Av? (A)+ LT wsW psy (M+ Os Wy, ..., Wg, Wy, +00 Or) +(G—1) WF (a) =(—Y7 8, (a). s=1 Make now a tend to zero and remember that (§ 24) 1Gt Fa (a) aa G are (q = 1)! a=0 ai Ja Helos os 0 G= laziest a Mr BARNES, ON THE THEORY OF THE MULTIPLE GAMMA FUNCTION. 421 Then we have r S(G= 1) ¥x9 (@), a aey @,) 35 = Ws Yr (@; eeey ®,) — Y-42),q(@1, seey Dg, Wyy aeey w,)} = (—)',.S,'2 (0), s=1 I AS or | - a-q+1| Vrq (@,, see, w,) + (—)" 8,2 (0)= > @s Yir+i),q(@1) tery Ms, Ws, -eey @,). s=1 s=1 A large number of other results can be obtained by further applications of Euler’s Theorem for homogeneous functions. By making all the parameters equal we derive from the multiple gamma function a generalised G function for which all the associated Bernoullian numbers and functions can be expressed as derivates of the simple Bernoullian functions. Such special cases however as that of this theory may conveniently be left for future investigation. The Asymptotic Expansion of the Multiple Gamma Function. § 55. We have previously (§ 31) obtained the values of the r-ple Riemann &-function for integral values of the variable s, by means of its definition by the fundamental asymptotic expansion. As we proceed to use these values in the ensuing investigation we will verify the results for the case when a@ is positive with respect to the w’s by means of the formula _ wh =s) eat (ez) €,.(8, @| @,, ..., @) = ———— aa dz. 2a 4 TI (1 —e-") k=1 When s=0 or =—m, m being a positive integer, the subject of integration is one-valued, and the contour may therefore be reduced to a small circle surrounding the origin. Along this contour we may use the expansion of § 3. We obtain r—1 l Lr (dd ar mv) —m—2 f.(— m, a) =(—) os Ic 2) fe awe! 3° nv) Sina (a) Qere = (—yr S’m4 (4) Qar (m+1)! ~ m+1 ' r —)s i (s+1) (a) x“ _\n-1 Ky u ‘a gh 3 Ot es CFE Ss O27, =i! 2 n=1 n: =(-)" Again, when s is an integer and 0S,8*) (a) + c= (= coi {yr (a) + (—)'7 8,8?) = zs ate Finally, when s is an integer >7, _ © g(a) =o 7) (a). We thus have the set of formulae previously obtained. § 56. If Ws) is greater than 7, we have seen that 1 (8, a) = S ( m= ae + (a + 0)’ where QO =m,0, +... + M,o,. Zz Put now a = OQ 7 7mem and assume that a@ is positive with respect to the o’s; then if s=o +17, es z — GAS a). < 9 S r. a m\7 |, sin 7s m=0 If now w,|<7, it is evident that the series and therefore aa CAS; a) | will tend to zero like e-*'7!, as + tends to infinity, e being a real positive quantity. If s is large and very near or on the positive half of the real axis, it is evident that | 2°f,(s,a)| will tend to zero with =. if 7, is less than unity for all values of m. Again when r>%(s)>—4, where & is a positive integer, we have seen that [§ 20] f(s,a) = - = 4F {fel rS_s,% (@ +.O-2)[e=64-(—)) rS_s,z (a +0) — (a+ a} +(—)'-S_s,z(@), ktr-1 SS,” (0) q™ a’ Sass where rS sk (4) = fas el ie Fe E = oer =). For all finite values of the m’s the terms in the series for £,(s,a) are such that | f(s.) 2 | sin as in so far as those terms enter into this expression, tends to zero, as |7 tends to infinity, like e-¥mt—=I7!, Mr BARNES, ON THE THEORY OF THE MULTIPLE GAMMA FUNCTION. 423 Now there are at most only a number of terms depending on an algebraic power of 7 for which (1) either |a@+0|+7 when 7 is large, or (2) for which oe 1 F,, [S64 (@+ 0+ 2) 220+ icy Sask (@ + Q).— (a + OF does not admit of expansion in the form s rad = s) : fecce Sm” (0) gkhtr+q-m g=o (a+ Q)st+9 PD (=s—k—-r—qs+1) m=o Mi(kK+2r+q—mM)! |raw ” wherever a lie in the finite part of the plane. And therefore the sum of all such terms will tend to zero substantially like e-7/7 as |r| tends to infinity. We need therefore only consider the r-ple sum of the remaining terms for which this expansion is possible. Now the modulus of first term of this expansion multiplied by z* behaves when |r| is large like |r tr PETES, emt and the ratio of any term to the preceding when q is large behaves like The series is therefore convergent and behaves as much as we please like its first term , | : if we make —7 sufficiently small, |a+Q} » o But the 7r-ple series S [rl br te evar m=0 — ee ez : Bie Le I SIICE Tg =| and o +k> 0, certainly convergent and when multiplied by ae tends n7s to zero as |7| tends to infinity like e-*!7!, e being a real positive quantity. Therefore, if %(s)>—k and a@ is positive with respect to the o's, the expression | 2 6,(s, a) sin 77s tends to zero as |s| tends to infinity at all points except the zeros of sin zs, provided Zz , Eagan evm and z is such that m<1 and |Wn| 2 LT. (a) ; , 17 - See LOB din, oe (a) (m+ mn’) Aare —S\' (a) Mare of the previous memoir on the 2 1s 2. double gamma function ; (2) Yo (@y, @2) replaces rY22(@1, @2) + Sy (0) 2are (M+ m +m’); (3) Y22(@, @2) replaces Ya (@1,@2) + So”) (O) 2are (M+ m + m); (4) log p.(@,,@,) replaces log p.(@,, @2) + 2Mru oS,’ (0). Vor “eewPanr ITT. 55 XVII. On the Asymptotic Expansion of Integral Functions of Multiple Innear Sequence. By E. W. Barnes, M.A., Fellow of Trinity College, Cambridge. [Received 7 October 1903.] § 1. In the present paper it is proposed to apply the general theory of multiple gamma functions and Bernoullian numbers to the asymptotic expansion of integral functions of multiple linear sequence. The paper is therefore a development of the theory previously given in the authors Memoir on Integral Functions* and Classification of Integral Functions‘. § 2. When the typical zero of an integral function is itself a function of 7 numbers necessary to define its position in the series to which it belongs, the function is said to be one of r-ple sequence. When the general zero, a say, is a linear function of the positive numbers m, ..., m, so that a=f(a+mo,+...+m,o,), a, @;, ..., @, being arbitrary parameters, the integral function is said to be of r-ple linear sequence. We shall assume in the first place that the zeros are all formed according to the same law and that no zero is repeated a number of times dependent on the m’s which enter into its expression. Functions for which the latter phenomenon occurs it has been proposed to call repeated functions [v. J. F. Part 1v.]: when the zeros of such functions are repeated a number of times dependent on an algebraic aggregate of sums of positive powers of the m’s their theory can be deduced from that of non-repeated functions by coalescence of sets of the parameters. Such functions we shall term linearly repeated. When r>2 the points @,,..., @, must all lie on the same side of some line through the origin, otherwise [JZ G. F. § 15]} the function will not exist. We have previously seen that when @,/@, is real and negative a similar case of exception occurs for functions of double linear sequence. * Philosophical Transactions of the Royal Society, (A), + In this way throughout will references be made to Vol. 199, pp. 411—400. ‘The Theory of the Multiple Gamma Function,’ Cambridge - + Cambridge Philosophical Transactions, Vol. 19, pp. Philosophical Transactions, Vol. 19, pp. 374—425. 322—355. Mr BARNES, ON THE ASYMPTOTIC EXPANSION, Ere. 427 § 3. We write Q=mo,+...+m,, and then the general function to be considered will be x ( (-—z)? TH | 41 e fla: oye Tor @roy? |, Ee genie ae oe where the product is r-ple, and p is an integer independent of the m’s and such that 2 i 1 is divergent and S 2 sins Li sees n=o [f(a +O)? m=0[ f(a +O)" We therefore do not consider functions of infinite order, for which p must depend on the convergent. ms in order that the Weierstrassian product just written may be convergent. When the m’s are large and f(a +) admits an expansion in descending powers of (a +) of which the dominant term is (a + 9), the function is said to be of algebraic sequence. The order of such a function is defined to be r/p. When p>vr, the exponential terms vanish from the Weierstrassian product: in this case the order is less than unity. The general extension of the Maclaurin sum formula. § 4. We have seen [M/. G. F. § 18] that for all values of s, we have the asymptotic expansion BA il ~ = +s Sp ” (a) d™ a str m=0 (a+) br (8, a) pea m! r| Fa = 5) one one the summation on the left-hand side being r-ple, and the many-valued functions having their principal value with respect to the axis of —1/Z, which is defined in § 16 of the memoir just cited. ao Suppose now that ¢(z)= = a,e2* is an integral function of order less than unity, s=0 m-1 then the asymptotic expansion of ¥ ¢(a+), where Q=m,o,+...+m,o, and the n=0 summation is r-ple, may be written = Sn (r) (a) d” SS a A Z,+ 2 ra F, da® wv (2) Ten n=0 L. where wW (a) = if dz Ie dz ea ¢(x)dxz (the integration being repeated r times) and 0 “0 0 [M. G. F.§ 30] x= s I M8 me us S S ~)AgS ‘sii(@) er ( 8,a)= = et - By applying the fundamental expansion to each term of the series > a (a+ OY, it is s=0 evident that the expansion is formally true. Now [M. G. F. § 3] a ae is the coefficient of (—)"?2"+7~ in the expansion of (aye a . —“~__.. Ifa be the greatest of the quantities |@,|,..., |@,|, the expansion has a radius (l—e-™*) Qa of convergence equal to a 55—2 428 “Mr BARNES, ON THE ASYMPTOTIC EXPANSION OF Hence by a theorem due to Cauchy | il | = aaa = @ es ret) 2 | Qar- Thus the series for Z, is convergent if, when s is large, |a;|=1/(s!}*" where 7 > 0, that is to say if @(#) is an integral function of order less than unity. If ¢(#) is an integral function of order greater than or equal to unity, the series for Z, is in general divergent. But we may “sum” it by the usual rules and with this convention the formula is still valid. And if ¢(x#) be a possibly non-uniform function which admits outside a circle of finite radius its Taylor’s series as a divergent but summable expansion, the Maclaurin sum formula is still valid with the same convention*. Thus when ¢ (zx) =e we have for all values of |&), except those for which & les on the lnes drawn from the points + zl : @ away from the origin to infinity, Tox (¥ S cSt We me a (ey S Si") (a) sso (stl)! (1—e™*)... 1—err*) sz ks And by putting ¢(#)=log(+.) we may derive again the known formula LM. G. F. § 23] pn-1 Ihe(@) Se ASEe@) GHG. Gad || el = log(a+02)=—log ~~ + 27" F, loge — 3 5 | : m=0 ig ( ) 6 Pr (@) m=0 m! da™ r! i = kak Z=pno More generally if ¢(«) be given by a Laurent series convergent within a ring space whose centre is the origin, and if it admit the Laurent series as a summable divergent expansion within the simply connected area formed by drawing lines from the singularities of ¢ (2) outside the outer circle of the ring space in a direction away from the origin to infinity and from the singularities inside the inner circle of the ring space to the origin, and if the points @+m,o,+...+m,o, lie withim this area we have the Maclaurin sum formula m,—-1 Ny—1 & Sin” (a) qm zy ... 2 O(@+mo, +... +m,o,) =f, (b)+ 2 —_— F, | sv (2) ; m™m,=0 m,=0 m=0 m: da es where {,(@) is independent of the n’s and is in general given by a summable divergent ix x x series, and where yf (7) = da | Cities | $(x)daz, the integration being 7 times repeated. In the integration ¢ («) is to be expanded in its Laurent series and the lower limits are successively to be 0 or % according as the term to be integrated when integrated vanishes at this limit. Integral Functions of non-zero order less than Unity. § 5. The standard function of non-zero order less than unity we take to be rP, (2) = _ 4 + tay! where p>vr and the product is 7-ple. We proceed to find the asymptotic expansion of this function. The process is the same as that previously employed [/. F. § 50 and C. I. F.§ 7] and consequently only the skeleton of the analysis is given. * As has been shewn in a paper ‘On the Generalisa- pp. 175—188, where an error of J. F. § 41 and C. I. F. § 4 tion of the Maclaurin sum formula and the range of its is corrected. applicability,’ Quarterly Journal of Mathematics, Vol. 35, INTEGRAL FUNCTIONS OF MULTIPLE LINEAR SEQUENCE. 429 a If m, ..., m,, are values of M, ..., N, respectively such that | |my@;|, ... | My; @, | have a finite ratio when |z| and the m’s are all large, we obtain in vie limit when & is infinite m— m-1 k s—1 log ,P, (z) =m, ... m, log z — p S) es (a+ Q)+ 2 eis “s (a + Qs =0 s=1 x ie Me 3 ) s (— Wie yes n=0 n=0/ s=1 § (a te s(a + Q)% The summations with regard to n are r-ple: for values of the quantities n chosen from the 2 a+ numbers 0, 1, ..., m—1, for which ae | 1, or for values of n chosen from m, m+1,...0 ‘ Zz for which lee a> 1, the series are to be regarded as summable divergent expansions of the logarithms. m— aE 1 Substitute now the asymptotic expansions of > ‘(a+0 dye and = log(a+Q) given n=0 n= =0 in § 4 and we obtain ls (a) 2 Sn” (a) am: Zi log -P, (z) =m, ... m, log z — p |- log - - “ an oe Fe = ; (log 2 — 35) ; iE pr (@) ke (SaNe—3 qd” ps-+7" v ( ) S r Sn! Saw! (a) F eS Tig |: ro sz \s Gee) sia = one ra Ei ae™ (os ulin (es + 7)")s—me k (- ee ial Sn” (a) qd” aeapstr > > = £ a 12 n! ef, BE a = ps) «. (r a alee ; Now m, ... m, log z=,S,” (a) F, [Fe og ‘| [M. G. F. § 22.) OL 2hF [net eee GaN ee ne (-y7 = Ii (a2 aie Hence ; a0 ~ Se nS!" (a log ~P, (2) = p log eH) + 2 .(— ps, «) + (—) a Se (r) (4) p dq” px" ge Ti ke (- ) 1 pps +r H S go paste 5 # ; - n=0 «0! Feel r! aia ean =-k8(ps+1).. (pst rye ae z=mo Now, since 1 ele (2 (ps+1)...(ps+7r) pi (p—1)!(r—p)! (ps +p) n : : Gh : P the expression under the sign age ot this expansion may be written eR SG Be atte ae apie Z S S se r} SEA ee ae (p—1)!(r—p)! .=-x 8 (ps + pe! rt 087 qr toi r (—)P 2" ( ™p Zz pip p oe =— _ l — > = a > ; 7 | ) os lo += ylo Z, Pe Ee a eal p= p(p—1)!(r—p)! |pein Pp P Pp s 2 eI s 430 “Mr BARNES, ON THE ASYMPTOTIC EXPANSION OF on employing the formula (J. F. § 52), valid if —a 6. ps, a) (yO) bg & (— =e eae p=1 p! sin p/p (r—p)! ” We note that the concluding series may equally be written [I G. F. § 6) (—)?*"arzPPS,'?™ (a) 1 p! sin pr/p § 6. In obtaining this expansion it has been assumed that —7 o. (a +m)? | The latter function 1s one of the fundamental types of repeated integral functions of simple sequence: it is a slightly more general form of the standard simple repeated function of non-zero order less than unity given in § 77 of the Memoir on Integral Functions. By exactly the same analysis as that previously employed, we may shew that when |z| is large and z not near the line of zeros of F, ee INTEGRAL FUNCTIONS OF MULTIPLE LINEAR SEQUENCE. 431 Suppose now that in the function ,P,(z) we make all the r parameters equal to unity. Then the number of times the prime factor {1 + winx is repeated will be the number of ways of choosing m as the sum of r numbers, one being taken from each of the r sets 0, 1,...,m. It will therefore be the coefficient of #”" in the ex- 1 pansion of (l+a+a?+ =a: The number is therefore r(r+1)...7;+m—1)_ (m+1)(m+2)...(m+r—1) m! rs (r—1)! 5 ns § 8. We now proceed to shew that (@+1)@+2)---(¢+r—1) _ Can GH Yaak eee Bee (r—D! ee ee = Tene a when the parameters of the multiple Bernoullian functions are each equal to unity. We have seen (M.G.F. § 3) that ,S,°(a) is the coefficient of the absolute term in r (—)etr 7S, (P+) (0) _\r—-1 > p—az the expansion in ascending powers of z of ee And therefore ik e 7 er — he Oa) — a ee, ( ) Sh (a) are (dl = ¥y dz, the integral being taken round a small curve enclosing the origin. If now we put 1—e*=z2, the values z=0, z=0 coincide, and when z describes a small curve round the origin w does the same, and in the same direction. eee (78,0 (@) = zs je — x) dr ee (a—1)(a— 2)...(a—-r+ 1) TL a” (7-1)! . v+1)(@+2)...(e+r—-1 Thus Oris eye Ue a ee r ays} (p+1) Now (78.9 (@—a)= & Ceo oP, since ,S,°(a@—«) is a polynomial in (a—«) of degree r—1. Hence* (@ ate 1) (x afi 2) ie (@ pal 1) = (-)" 7S) (- x) = S (ye Say (0) “gp P—1, (r—1)! p=1 (Gil) The first few terms of the series are Ge (Beinn r(8r—1) a GDI a@—ait eer Gat Note that we also have Cab 42)... +r) we is - 7. — (r—1)! =(-)'7 8S,” {a eae: T (x+r) 3 A The coefficients are substantially factorial coefficients. See ———_____ in powers of 2x is ae = : T (x+1)V (r) Boole, Finite Differences, 1880, p. 113, where many refer- valuable in connection with many different investigations. ences to the earlier work on these coefficients are given. * The expansion of 432 Mr BARNES, ON THE ASYMPTOTIC EXPANSION OF § 9. .We see then that when each parameter is equal to unity ae) — ll {1 + m=0 z (=)-1,$,2)(-m) (a+ = the product being now simple. Hence by the result of the previous paragraph 7 1 Zz (=) Ptr S\PtY(a) eas C3) = Js st {1 a it @=DI (a+m)p-1 5 p=1 m=0 (a +m)P And therefore when |z/ is large we have the asymptotic expansion r (- 2a 9, (2) (a) S 2 Zz | > ee 4 o —— = San C=)! = (a +m)? log 41 + =e a r +r / Sp log be i — SC(—pn ace ee BO hog z as, o_ S242 (a), 0, (@)u.= =1 = i * pa p!sin a Now when the parameters @ are all equal it may be readily deduced from the asymptotic expansion which defines €, (s, a). that r orn (6, a)= 3 CPX sin (ae. +5—p. a), p= : so that on differentiating and making s=0, T@) ~% & = loo me Pr (@)o=1 p=1 (p— yr (a) "(1 =p, @) Therefore the expansion may be written Speed etree A) eh @=0! log F, (2) 5 (

us E Be exp (a + Q) (a a ml > where the product is 7r-ple and Q=n,o,+...+7,0,;. * Quarterly Journal of Mathematics, Vol. 31, p. 300. Vou. XIX. Parr III. 56 434 Mr BARNES, ON THE ASYMPTOTIC EXPANSION OF [Unfortunately the Maclaurin sum formula cannot be applied in the investigation pre- viously made for the cases r=1 and r=2 [/.F. § 60; C.Z.F. $11 and 12]. As the question of the behaviour of functions of zero order is very important and - as it might seem to the reader that the same methods ought to apply to functions of both zero and non-zero order, it is proposed to take up the analysis for the most simple ease and correct it. We consider then the standard ut function of zero order F,(2)= ae i} ata am ? the sole limitation on the variables being W(@) > 0. Z We have evidently, if eme| > z > | e(m-Deo| a —\s—1 m-1 m-1 atne ao (_\s—1 2 log F, (z)= > ( y > eletne)s_ > log é iL SS ( ) 2 S eriatme)s s=1 82 n=0 n=0 Zz s—1 s n=m m-1 Ss 2 nr Now S glatne)s — en TESS Sn’ (a) [ d is > ; n=0 WG Sy 7h ane ist ileaees is a result of the Maclaurin sum formula which can be interpreted by the theory of divergent series*. Hence we may put Peel ie 25 F412 Sa (a) fe me ee ae = et pe Se aes “ | eee] tm loge aes (esa The general process consists in rearranging the double series which here intervenes. When we are dealing with integral functions of finite non-zero order, the series which enter as the coefficients of the various powers of 1/z are truly asymptotic and satisfy Poincaré’s criterion that im any such series. the remainder after / terms have been taken, multiplied by the power of x which arises im the last term taken, tends to zero as |x|=|mq@) tends to infinity. Therefore it is possible to rearrange such a series, inverting the order of summation, and by so doing we obtain a new series in which the coefficient of S,’(a) is a summable divergent series in 2 which can be summed to a definite value: for the remainder after / terms have been taken of such a series can be made as small as we please by taking / sufficiently large. But the series < S, (a) E | n=0 nN. dx z=mw is not an asymptotic series proceeding by powers of 1/m. It may be written S Sn (2) on n=0 n! emo S And although such a series, qua function of s, is summable, this fact does not enable us to repeat the former procedure. We evidently must devise some other means for * See pp. 181, 182 of the paper quoted in § 4. At hilton INTEGRAL FUNCTIONS OF MULTIPLE LINEAR SEQUENCE. 435 attacking the problem. It is perhaps best solved by the ingenious process of contour integration devised by Mellin*. | We obtain, when |z| >| e%|, arg <7, R(o)>0, 9 Sy (a) log z + 8,’ (a) 5 4 (log z)) 2 = )\n=1 an bSe4 feotinayanrs eS Be 2 | nma1 nz 1 —erm 2 k Ir fv - (2) = eo es peace + log a | a ) e | a e This equality differs from the result formerly given by the addition of the last product term. The proof may be conveniently postponed to a paper which shall more fully develop the theory. It is important to notice that it is an absolute equality, and not an asymptotic expansion: it is therefore totally different from the expansions obtained for integral functions of non-zero order. § 13. In the case when a@=1,@=1, we may write the result in the form (| arg z|< 7) a A (peo aaa m™ 1 & cos (27s log z) log ow i er) o=1 52#(1 =e?) 2 2 cee 6 +12 s-1_-$ sinh 277s | We see therefore that, when |z| is large, the function log H i +31 behaves like n=1 ] PI : : : O62)" _— log 2 + terms of lower order. To this extent we do get asymptotic behaviour, so that there is no need to modify the analysis of the behaviour of the general co- efficient of the Taylor’s series for the standard simple function of zero order for which w is real, so far as such analysis was carried in O.J.F. § 42. Sufficient has now been said to shew that any modification of the fundamental procedure when applied to functions of zero order with zeros of the type — exp {a+ na, +... + 2,@;,} does not give asymptotic expansions but transformations of these functions. The theory of such transformations I reserve for the present. The fact that the fundamental pro- cedure leads to results of an entirely different character when applied to functions of zero order indicates yet again how necessary it is to discriminate these functions abso- lutely from functions of finite non-zero order.] Integral Functions of Finite Order greater than Unity. § 14. The same extensions of previous formulae which lead to the asymptotic expansion of log,P,(z) will now shew that we may obtain the asymptotic expansion of the standard integral function of r-ple linear sequence of non-integral order greater than unity _f Z Ree ee pele 602 (2)— l} an (ee ay 210 = Oyen wae aR * Acta Soc. Sci. Fenn. t. 29, No. 4. The result stated is slightly more general than that given by Mellin in formula (25) of the paper quoted. 56—2 436 Mr BARNES, ON THE ASYMPTOTIC EXPANSION OF where the product is r-ple, p2>p. The expansion is A(@) ae a en cs (=)™ 2m, Sym (a) log -Q, (2) = POE Gh oe 2 Se (— ps, a) + ( yet (@) logs ™m: ae + § 15. Similarly for the asymptotic expansion of the logarithm of the general function of finite non-integral order greater than unity Il ag 1+ sant ex | ar pee Geeks zie |; n= para) OP a@+a)t” *pe@+ OP) : F 1 ' 1 where p is an integer such that TEC Oye is convergent, and SICICES ODE di vergent, we may give a formal expression. It will only differ from that given for the On function of § 11 in that Ss £,(¢*) must be substituted for = 3G — (¢°). s=-p s=1 § 16. In the two preceding paragraphs we have assumed that r the order of the function, is not integral. But by making " decrease down to p, where p is an integer we can, by evaluating the differences of the infinite terms which arise, obtain the asymptotic expansion of the general function of finite integral order. When in § 15, ¢(a+) is an algebraic function of (a+) we can approximate to the terms which arise in the asymptotic expansion to any degree of accuracy by sufficient labour [ef. IF. § 58, 59 and 68]. We proceed to carry out the analysis in the case of the standard function Z aa Ee pie al ee LD ary ont eeny t+ per oyt| which is of integral order p= . the product being 7-ple. We have at once by § 14, log ,R, (z) = Lt FE e=0 L@, 2 &€ ye LG) ee one Se 4) ae (mtr OF) ma Oye mr m! sin aye (p +e) pte +(—)',S/(@) logz+ & m=1 There are two cases to be considered according as p is or is not a multiple of r. Suppose in the first place that p is not a multiple of r, Then log +R, (2) =" log => + (YS (loge © ae s=—(p-1) $2" mp mut (—)m+r mz” Sena (a) Ae) — 2 2 . mr net m! sin 22? = INTEGRAL FUNCTIONS OF MULTIPLE LINEAR SEQUENCE. 437 where f(z) denotes the sum of the two ultimately infinite terms ce z)? | , «| mzPrte AISA (r+1) (a) & - > a6 FAS see pte r! sin (ap+ qe)’ By the formula given ei (M.G.F. § 31) we have = yp 7S, 7+) ( ( +e) Gy Fs kee ai ait j| 2 BBE) 6 OF [ymrecrm sen co'S Ip 2? 8S," (a) S@= = rt (=)Pe {1 + elog 2} neglecting terms which vanish with e. ee Ke) (+a) logz 5 1 Hence fl) Fay Ye + aa [Ee & Th. We thus have, if p be not a multiple of r, T i (a — sr log -R, (2) =" log Sate ySi(a)loge+ 3" Os, (-<.2) mp r-1(_)mir,2r Se) (a) ptr 2 SL SS ee yee (@) =] . Np me m! sin 2@ 7 Ca? gnin(q) oee_ § 2 § 17. We consider in the second place the case when p is a multiple of 7 and equal to kr (say). Then in formula (1) of the preceding paragraph infinite terms arise for the values —s=k, 2k, ..., rk and m=1, 2, ..., 7 The sum of these terms is r —\km ~kin . Tam S (yee i kmr ' a) a IETS kr +e m (kr-+-e) a (—)"" are 7 7S, (+) (a) Be ee i= ; (kr + € m! sin mar | ——— FA > and is therefore [M. G. F. § 31} equal to te (- jem gkm (See ps ty (a) (kr +e) a1. bm (m— 1)! (— me) (=) aly (m) (a)+(— yr, ASS (m+1) Oe "> = 7 | (m — a r _\mt+r+mk ». mk S (m+1) 3 O ee AON as ae m! me ae r legate neglecting terms which vanish with «. And on reduction the infinite terms cancel and we obtain r (—) (ety) gkm Yr —) eH) m+r gkm v,™ (a) + = ( en WOE aes m! ‘ my Seinen (0) i168 z— 2 al 5 4 438 Mr BARNES, ON THE ASYMPTOTIC EXPANSION OF Therefore if p=kr, where k is an integer, we have the asymptotic expansion T.. (a) ——— pr (@) e (—)*H)m gkm r (—) =) m+r gkm 73 ee (m) (qa > mik Mee Vee mk sr Pa _\yr og sa (op (sr ) log ,R, (2) = lo + (—)" SY (a) log z Bo - c.( 5 a pS," (a) tt log z— = i} ; 1=1 the star indicating that the terms which correspond to s=0, —k, — 2k, ..., —rk, are to a m=1 be omitted in the summation. The expansion exists over that part of the plane at infinity which is not covered by zeros of ,R,(z), and the logarithms involved are to have their principal values with respect to the axis of — 1/Z. The reader will notice that the formulae agree with those for the special case r= 2 which has been previously investigated* [C. 7. F. § 16, 17]. § 18. We proceed to deduce from the foregoing formula the asymptotic expansion of logT',(a) which was obtained in a very different manner in the memoir on that function [M.G.F. § 57]. In the formula of § 17 we put p=7, so that #=1, and we have the asymptotic expansion ea Ze) ( Zz (— 2)" log at fp ap O\ (aa oer aso oy | pee Desc (ay leet a Sea ie we ee Gia ) Sy (a) log z+ ea g28 G, ( 8, ay aie (a) e < (-) gm Syn) (a) {log 2 = S i = 1 m=1 Mm! =1 1 If now a tend to zero the expression on the left-hand side of this equality becomes [M. G.F. § 25] log E> (2) = 2yn — ai Yr— a eae Lt \- log a Ss tet t} tf a=0 a (- ap rar We therefore have, in the limit when a=0, log T(z) =— log p,(w) + (—)” Sy’ (0) log z + 3 = f,(—s, 0) s=1 = ein m) (ee (m ae 1)! l < (x De m+ ( aa Ay 1 see m! {ye Cee a” | sere m! ac (0) eRe ee Now Lt {ym (a) + —— + aml =O, [M. G.F. § 24] a=0 apes v0 0) : and f,(—s, 0)=(-) ie [M.G. F. § 31] < 1 * In the results to which reference is made 7 has been by an oversight replaced by 2° y INTEGRAL FUNCTIONS OF MULTIPLE LINEAR SEQUENCE. 439 We therefore finally obtain Le) ae ra & (=) Sh48 (0) leg ae) a ) Si, (OSes Zs s(s+1) S (See zm (m-+1) ( as ae 1 = _— 1 aes m! ee) yes ei ae ann 7 (he? which agrees with the previous result. § 19. At this point we may conveniently conclude for the present this series of investigations. Sufficient results have been obtained to prove the statement made previously [C. I. F. § 35] and complete the initial stages of the general theory there outlined. We have, in fact, shewn that for non-repeated functions, with algebraic zeros, of 7-ple linear sequence and finite non-zero non-integral order R there exists an asymptotic expansion of which the dominant term is of order A, that is to say, the function behaves like exp [cz®+ terms of lower order when | z | is large in z, logz, &c.], where ¢ is independent of z. When the order of the function is integral but non-zero the dominant term behaves like exp [cz® log z+similar terms of lower order]. By coalescence of the parameters we may obtain similar results for the whole range of linearly algebraically repeated functions of multiple linear sequence and finite non-zero order. It has been shewn that it is possible to obtain complete asymptotic expansions for the logarithms of all these functions, and that the coefficients which arise can all be expressed in terms of functions derived from the extended multiple ¢ functions. The beginning of a similar theory for integral functions of zero order has been out- lined: the development of such a theory is still to be obtained by extended algebraic analysis. The functions which have been investigated are the most natural in analysis and their range is of no small extent. Much, however, remains to be done. We must in- vestigate functions of non-linear sequence, functions of linear sequence whose zeros are non-linearly repeated, functions for which the repetition is transcendental, functions of infinite order, and so on. The gradually increasing complexity of a typical zero and _ its repetition furnishes an infinite series of problems. GENERAL INDEX. Acta Mathematica 233 Adler 138, 141 Ampére 1 Annales de Toulouse 271 Annali di Matematica 137, 141, 143 Appell 141, 190, 233 Archimedes 158 Armenante 136, 137, 138, 141, 148 Aronhold 136 Baker, H. F. 233 Barnes, E. W. 322, 374, 426 Baxter 62 Berry, A. 249 Bertini 71, 73, 77, 135, 136, 137, 138, 140 Berzolari 137, 140, 142, 143 Bologna Rend. 141, 142 Bonnet 71 Bordiga 142 Borel 297, 300 Brambilla 144%, 143 Bravais 85 Brill 139 Bromwich 274 Budden, E. 157 Burnside, W. 17 Cambridge and Dublin Mathematical Journal 69, 140 Cambridge Phil. Trans. 83, 151, 157, 250, 426 Castelnuovo 253 Cauchy 351 Cayley 69, 132, 133, 134, 140, 142, 236, 239, 251, 253, 257, 385 Charged particles in motion 173, 177 Chrystal 158 Classification of quintic surfaces 252 Clebsch 134, 254 Crani 376 Crelle’s Journal 1, 7, 16, 85, 134, 140, 142, 252 Cremona 133, 134, 135, 136, 138, 140 Darboux 69, 141 De Franchis 250, 255, 259 Del Re 142 de la Vallée Poussin 308 Differential equations 1 Diophantine equations 111 —————— inequalities 83, 111 Dirichlet 85 Divergent series 297 ——_—_—_———— differentiation of 304 ——_———\—— integration of 307 Dixon, A. C. 1, 158, 190 Du Bois Reymond 355 Eberhardt 142 Edgeworth 23 Electric convection 173 Enriques 253, 271 Euclid 157 Factorial functions 190, 233 Forsyth 143, 213, 218, 323, 350 Gallop, E. G. 356 Galton 49 Gauss 85 Glaisher 31 Gordan 137 Hadamard 341, 350, 352, 335 Hamburger 1, 7, 16 Hardy, G. H. 297 Heath, T. L. 158 Heaviside 173 Hilbert 111, 112 Hill, J. E. 252 Hill, M. J. M. 157 Humbert 271 Hypergeometric series 151 Integral functions, classification of 322 ———1-—— of finite order less than unity 328; of zero order 333 — of finite order greater than unity 336; Tay- lor’s series expansion of 341 — of multiple linear se- quence, asymptotic ex- pansion of 426 Integration matrices 208 Jahrbuch iiber die Fortschritte der Mathematik 132 Jabresbericht der Deutschen Mathe- matiker-Vereinigung 137 Jellett 356 Jordan 387 Journ. de Math. (Liouville) 140, 271, 283, 308, 341, 350, 352 Journal Ecole Polytechnique 85 Klein 212 Lie 69, 73, 77, 80, 82, 138, 141 Logarithmic Potential 212 Lomb. Ist. Rend. 71, 141, 142 London Mathematical Society, Pro- ceedings 1, 17, 142, 233, 257, 310, 312, 319, 376 Lorentz, H. A. 173, 179, 188 MacMahon 83, 87, 94, 111, 235 Mathematische Annal. 69, 71, 111, 137, 141, 142, 143, 151, 252, 254, 271, 283 Mathews, G. B. 83 Matrices of infinite order 190 Matricial functions on a Riemann surface 190, 220 Maxwell 174, 181 Mellin 391, 425 Merrifield, C. W. 50 Messenger of Mathematics 334, 376 Meyer 142, 143 Minimal curves 69 —— surfaces 69 Multiple gamma function 374, 397; integrals for, 405; properties of, 412; asymptotic expansion of 421 Multiple Bernouillian functions 377 Multiple Riemann ¢ functions 387 Napoli Rend. 143 Neumann 212 Noether 252, 271, 283 Normal correlation 23 Orr 151 Painleve 271, 283 Pappus 158 Partition analysis 83, 112 Pearson 23, 48 Perpetuants 239, 246 Philosophical Magazine 23, 31 Phil. Trans. R. 8. 23, 24, 50, 65, 111, 323, 376, 391, 426 Picard 250, 265, 271, 278, 281, 283, 284, 291 Pincherle 376. Pliicker 132 Pochhammer 151, 152 Poincaré 188, 220, 233, 250, 341 Quarterly Journal of Mathematics 143, 297, 319, 376, 377, 428 Quintic surfaces 249 Quintics with two linearly indepen- dent integrals of the first kind which are functions of one another 271 Quintics with two functionally indepen- dent integrals of the first kind 283 Rational space curves of the fourth order 132 Retali 142 Richmond 61, 132 Roberts, R. A. 138, 142 Rohn 139, 143 Routh 356, 373 Salmon 77, 132, 133, 134, 140 Schlémilch’s Zeitschrift 142 Schwarz 190, 212, 213, 252, 253, 254, 256, 269 Seminvariants of systems of binary quantics 234 Sheppard 23 Simart 265, 278, 281, 283 Singular solutions 1, 21 Sitzungsberichte, Vienna 134 Spinning top 356 Spottiswoode 142 Stahl 142, 143 Steiner 133, 135, 137, 139, 140 Stolz 158 Stroh 137, 235, 239, 242 Study 142 Sturm 140 St Jolles 142 Thomson, J. J. 185 Walker, G. T. 173 Weyr 134, 136, 137, 138, 140, 145 Wirtinger 142 Yule 49 CAMBRIDGE : PRINTED BY J. AND C. F. CLAY, AT THE UNIVERSITY PRESS. CANDING SECT. JUL 19 1968 Q Cambridge Philosophical 41 Society, Cambridge, Eng. C1g Transactions v.19 Physical & Applied Sci. Serials PLEASE DO NOT REMOVE CARDS OR SLIPS FROM THIS POCKET UNIVERSITY OF TORONTO LIBRARY STORAGE we = ps oes Snead. sae A ipoetrdeen oe + ete ed be " ee “ irre ee) ea SARE Ray cape Cap pelea ten < han = SS wre aS Ae. | wae ERE Vedra LY