Q A 
 
 
LIBR^RV 
 
 OK THE 
 
 University of California. 
 
 Class 
 
ON A CERTAIN CLASS OF FUNCTIONS ANALOGOUS 
 TO THE THETA FUNCTIONS. 
 
 f^^±-^%.. 
 
 [MlTBBSITT) 
 
 
 DISSERTATION 
 
 PRESENTED TO THE BOARD OF UNIVERSITY STUDIES OF THE 
 
 JOHNS HOPKINS UNIVERSITY FOR THE DEGREE 
 
 OF DOCTOR OF PHILOSOPHY 
 
 BY 
 
 j^BRi^HAM COHEN 
 
 BALTIMORE 
 
 1894 
 
 ^ B R A R 
 ^ OF THE 
 
 UNIVERSITY 
 
 OF . ^ 
 
 PRESS OF 
 
 THE FEIEDENWALD COMPANY 
 
 BALTIMORE 
 

. ,8B A R y 
 ^ OFTHt 
 
 UNIVERSITY 
 
 INTRODUCTION. 
 
 M. Appell, in a brief note in the " Annales de la Faculte des Sciences de 
 Marseilles," gives as an example of a function of three variables having a true 
 period and a quasi-period, analogous to the (9-functions, the function 
 
 where, in order that the series may be convergent, the real part of a is to be 
 negative. 
 
 This function evidently satisfies the conditions 
 
 <p{x-\-a, y-{-2x-^a, z-\-Sx -\-Sy j-a)=e"'+^''+^'J + ^'^(p{x, y, 2), 
 
 dx dyoz cy o^ 
 
 Moreover, as M. Appell shows, these conditions are sufficient to determine 
 <p{x, y , z) to within a constant factor. 
 
 The object of the present paper is to investigate the properties of this 
 function and of functions derived from it, as well as of others similar to it, 
 pointing out, as far as possible, their analogy to those of the ^-functions. As 
 is not surprising, some of the properties of the latter seem to have no analogues 
 in the case of the functions here considered. In such instances it has been 
 endeavored to assign the reason, as far as possible. 
 
 The great difficulty throughout the preparation of this thesis has been the 
 utter poverty of known theorems holding for functions of more than one complex 
 variable. As a consequence, this work has been rather of a tentative nature. 
 
 In this, the assistance rendered me by Professor Craig, at whose sugges- 
 tion this subject was selected, was invaluable, my appreciation of which it is 
 only proper that I express here. I desire also to acknowledge the debt of 
 gratitude I owe to both Professor Craig and Professor Franklin for their 
 interest manifested in my work throughout my entire connection with the 
 Johns Hopkins University. 
 
 .} >-^ n o - ^ ■; 
 
I. 
 
 hetf{xy y, 2) be a holoraorphic function satisfying the conditions 
 f{x-j-(o„ y, z)=f{x, y-^co^, z)=f{x, y, z -^ co^) =zf {x , y, z), (1) 
 
 = e <-.*,, 0,3 /(a;^ y^ z), \^) 
 
 (3) 
 
 a« 2riitOi dydz ' dy 27ti(02 dz^ 
 
 where w, , co^ , wg are any quantities, real or imaginary, 
 
 p any given integer, 
 
 a a constant whose real part is negative. 
 
 The most general entire function of a;, y, z satisfying conditions (1) is given 
 by the Fourier series, 
 
 f{x,y,z)=T 'I" "f a.z.,»e"''^<^*+4' + <-^'"^ (4) 
 
 i= — 00 /= — 00 »=_ 
 
 where Ck.i.m is independent of x, y, z. In order that /(a;, y, z) also satisfy 
 conditions (3) , we must have 
 
 k=zlm and Izizm? 
 or 
 
 kzzzm^ lz=znr 
 
 and we now have only the simply infinite series 
 
 • f{x,y,z)='^f C^e'-^-r'-^.'-r'-'^r'. (40 
 
 Finally, from (2) we have, on multiplying both sides of the equation by 
 
 am* + ap* + 2iri (-i p» + JL p» + ~p) 
 6 •>! "a "s 
 
 and properly collecting the terms 
 
 ^ a(m + p)* + 7ni[l (m + p)« + V. (m +p)> + ± (m + p)] am* + M(^m»+ JL m* + J. m) 
 
In order that this equation be satisfied, it is evidently necessary and sufficient 
 that, for all values of m, 
 
 Hence the most general function of a;, y, z satisfying the conditions (1), (2), 
 (3) will be given by 
 
 f{x, y, z)z=z X Ome 
 
 0), U] a>3 
 
 (6) 
 
 with 
 
 Oot Cm -|- p ' 
 
 Since by hypothesis the real part of a is negative, this function is holomorphic 
 for all values of a;, y, z. 
 If we write 
 
 -til {X , y , z) — ^ ^ "« "« "« - 
 
 J:i'i{x, y , z) — ^6 w, wj w, 
 
 Mp[Xf y, z) — 2^e »>» wu <"3 
 
 -tip\^> V} ^) Ze "I "a "a 
 
 . (6) 
 
 it is clear that /(a;, y, z) will be a linear homogeneous function of i?i , jRg, • • • • , 
 Rp, . . . . Rp. Moreover, the latter are linearly independent, as may be seen at 
 once from their development. They can, however, be replaced by simpler 
 functions. Write 
 
 ./^ msj. y 
 
 (p{x, y, z)=z z e ^ S ^u,, ^<03 '. 
 
 (7) 
 
 Then, for ^ , ft, v any integers, we have 
 
 tn (t -i- ^'^'^^ 11 -L t^^'^^ ^ _L ^"^3^ ^ am* + 2,ri (-?- ms 4- -^ w» + A m) + ^JL* (Xm» + M^' + vm) 
 
 Jr P P m 
 
 or 
 
Giving to )., ji, v each separately all values from to p — 1 we get p^ equa- 
 tions of the type (8) to be satisfied by the p quantities R . Of these p^ equations, 
 only p can be independent. Moreover there are p independent ones among 
 them, viz. as we shall see, those obtained by putting ).=. [x=zO and letting v 
 take all integer values from to p — 1 ; these are 
 
 VO) 2>rty Inipv 
 
 <p{x, y, s-|---«)z=6P B,-\- .... -\.e-T-Bp-\- .... -f i2p 
 
 P (9) 
 
 v = 0, 1, ....,p — \. 
 
 For the sake of brevity we shall introduce the following notation. Write 
 
 <p{x,y, z) = \0,0,Q)\ 
 <p{^+Y, y, 2) = [^, 0, 0] 
 
 Jr 
 jr 
 
 Also write 
 
 [0, 0, v] = C-(a5, 2/, 2)=Cv. 
 
 (10) 
 
 Our equations (9) may then be written 
 
 [0, 0, 0] = Co= i2x + i^2 4-----+ i2p + ....+i2p 
 
 [0, 0, i] = Ci = rii2i + r2^+..-- -hnRf>+-"'-\-rpRp 
 [0, 0, 2] = C2 = ;ii2i + ?1i22 + ....+r?Rp+-.--+?i^P 
 
 where 
 
 3iri 4fft 2pirt 
 
 ri = «* , r2=ep , — , Yp = e'j' , — , rp = i- 
 
 The independence of these equations follows at once from the fact that the 
 determinant of the system 
 
J= 
 
 1,1, 
 
 Ti } Tz , 
 
 11)72} 
 
 r 1 , / 2 ; 
 
 
 rP — 1 
 /p , 
 
 fp 
 
 rr^ 
 
 n 
 
 = n {xi—Ti) *4^i 
 
 is evidently different from zero, the y^B being the p^^ roots of unity. Of course 
 
 (p-l)(p-2) 
 
 the value of zP is (— 1) 2 ^p . 
 
 Denoting the minor of ^f in J by Ci_i, t, the determinant of the minors is 
 
 D = 
 
 ^0,0 , M, 1 ,••••» vp— 1, 
 
 M), 1 , ^1, 1 ,••••, ^p-1, 1 
 
 Solving equations (10) , we have 
 
 Ji2i= CcoCoH- Co, iCi + ^©.aCa^" • • • • 4" ^Vp-iCp-i 
 
 J/[2=Z C/i,oCo H" Ci, 1 Ci + Ci,2C2 ~l~ • • • • "h Ci,p-lCp-l 
 
 JMp=: Cp_i^ oCo "h ^-1, iCi ~r Cp-i, 2 C2 "i" ■ • • • "h ^-1, p-i Cp-i J 
 
 (11) 
 
 The remaining p'' — p functions \_X, //, v] for ^>, // = 1 , 2, , p — 1 and 
 
 v:=0, 1, 2, . . .. , p — 1 can now be expressed as linear homogeneous func- 
 tions of 
 
 C. . v = 0, 1, 2,....,p — l. 
 
 From (6) it is obvious that 
 
 }(0 A — 
 
 ^p(^+yS>, z) = e"''TR,{x, y, z)=f:R. ^ 
 
 Rp{^ + ^y y, z)z=e^' P Rp{x, y, z) = Rp 
 
 (12) 
 
Making these changes in (11) we get the following p systems of p — 1 equations 
 each : 
 
 n^Ri=Oo,o[^, 0, 0]+eo.:[A, 0, 1]+ .... +(7o,p_i[;, O, ;> — l] (130 
 
 r;jEp=c,_i,o[;.,o,o]+(7,_i,i[/(,o, i]+....+Cp_i,p_i[;,o,2)-i](i3,) 
 jR, = o^_,,,[?., 0, o]+c;,_,,,[;., o, i] + . . . -^o,_,,,_,ix, o, p-i](i3p) 
 
 X = l, 2, .... ,p-l. 
 
 The determinant of each of the systems of p equations obtained by taking the 
 P" equation of each of the above sets is D, which is different from zero since J 
 is. 
 
 Hence we can solve for the p^ — p quantities [?., 0,p']foT X=zl, . . .. , p — 1 
 and v=: 0, 1, . . . . , p — 1 in terms of R^, R^, . . . . , Rp, which, in turn, can 
 be expressed linearly in terms of 
 
 C. v = 0, 1, ....p — 1. 
 
 Thus, taking the system 
 
 n^R^ = Oo,o[.^, 0, o] + Co,,[;., o, i]+ .... 4-(7o.^_i[;., o, p-i] 
 
 f:jR,=a,,iA, 0, o]-hc,,,[;, 0, i]+....+ci,^_,[;, 0, p-i] 
 
 r:^ii,= q._a. o[^ 0, o]+G,_,. ,[/(, 0, 1]+ ... . +c;-i. ,>-i[^ o,p-i] 
 
 Ji^^=Q,_,,o[;,0,0], Q,_i,i[/,0, l]+....+C^_,,^_,[/,0,_p-l] ) 
 and remembering that the minor of Q^ ^. in D is ^-J^^^ Jf~^, we have 
 
 And finally, from (11) 
 
 P=l j=o 
 /l=:l, 2, ...., p-1 v = 0, 1, 2, ...., p— 1. 
 
 In exactly the same way we get 
 
 j[o, /., v]= V ' k r^rfi.-x.iZi (15) 
 
 /^=1, 2, , p — 1 v = 0, 1, 2, ,;> — 1 
 
9 
 
 or 
 
 j[o, ^, v] = 2rrr;q.-i.o[o, o, 0]+ 2 rrr;q,-i.i[o, 0, 1]+.... 
 + ....+2r^V;q,-i..[o, 0, ^]+ .... +2r?r^c;_,,^_,[o, 0, p-i]. 
 
 .-. .)[;, ;., v]=Vr^V;c;_,,o[/i, 0, o] + 2r:V;q._i.i[^, 0, i]+ .... ' 
 p=i 
 
 H- .... +2r^;r;q._i..[^, 0, ^] + .... + 2r^;^^c;_^^_,[/, 0,^.-1]. 
 
 Whence, from (14) we have 
 
 p=:l P' = l i = 
 
 + 2r^V^<^p-i.i22rA^'Vp'C'p'-i.,C, 
 + 2rj:V;C;.-i.222rfr^'^p'-i.,C,- 
 + 
 
 /, /i= 1, 2, .... , p— 1 v = 0, 1, 2, .... , p— 1. 
 If we write (14) in the form 
 
 ^ P=l j=0 ^ j=0 
 
 the determinant of any of the systems of p equations obtained by keeping / 
 fixed and allowing v to take all values from to p — 1 is seen at once on 
 writing it out, to be 
 
 or since Dzzz Jp~^ , 
 
 y(18 + 23+.... + P»)-P^ 
 
 /A jp 
 
 v[iP(P + i)]''=: J_| J(A) I 
 
 Similarly, writing (15) in the form 
 
 ^ p = l j=:0 J = 
 
 the determinant of any of the systems of p equations obtained by keeping /i 
 fixed is found to be 
 
 (l» + 2»+ ■ ■ . . +p^) — / (P-^) '^^-^' = 1 I ^(.>x) I 
 
10 
 
 Finally, (16) may be put in the form 
 
 1 k=p—i j=p— 1 
 
 k=0 j=0 
 
 (19) 
 
 The coefficients here are, to within the factor —^, the elements of the deter- 
 minant obtained by taking the product 
 
 We have thus expressed all of the p^ quantities 
 
 [/, /^, v] /, /^, v = 0, 1, 2, .... p — 1. 
 
 as linear homogeneous functions of the p linearly independent ones 
 [0, 0, v] = C. v = 0, 1, .... p — 1. 
 
 Hence we see that every holomorphic function of a:, y, z satisfying con- 
 ditions (1), (2), (3) can be expressed as a linear homogeneous function of these 
 p quantities. From which follows that there can be only p such functions 
 which shall be linearly independent. 
 
 We have obviously 
 
 :j{x, y, z) = <p{x, y, 2 + ^y') 
 0(^» y> 2 + W3) = ^^(a;, y, z) 
 
 CiCa;, y, z-^k'^-)=:._^_^{x, y, z) 
 
 If for brevity we write 
 
 ap* -\- 2-1 [^ p3 4- 1 / -I- i-^] =zE{p) 
 
 (20) 
 
 and if we denote the substitution 
 
 (x, y, z; X -\ r->, y -\- ^ « -\ Ap, z + -^^p H ?o' -\ ^-^p") 
 
 by Sp, we also have 
 
 S,:j{x, y, z) = e-^''"r^{x, y, z) 
 
 S^:j{x, y, z) = e-^^'^-"T:j{x, y, z) 
 
11 
 
 In general, if 
 
 Xx.^.J^, y', ^) = ^(^ + y, 2/ + -^' 2-f -^),then 
 
 Xx, ^, A^-hf'^i, y, z)=Xk. ^, X^, y+^9., z)=Zx. ^, >> y^ 2+^*^3)=/;,. ^. Z-^, y, 2) 
 
 /'<t^i 
 
 ;^<t>2 
 
 while the effect of the substitution Sp, where p^O (mod p), is to change 
 Xx, ^, vi^} y> 2) into some other function altogether, in general. 
 
 II. 
 
 Let us consider now, in connection with the function 
 
 Co = ^(a;, y, z)= X 
 
 am* 4. 2ni (jL m* + Ji m' + -L m) 
 
 (1) 
 
 711 = 00 
 
 the functions obtained by increasing zhy -^ , -~ and by '— respectively. 
 We may write these functions briefly 
 
 T 
 
 ^i^, y> ^) = <po{x, y, z)= 2 « 
 
 E(m) 
 
 m= — X 
 
 <p{^^ y, ^+ t) = ^1 (^' y, ^) = ^ C*)™ «^^™' 
 
 
 (2) 
 
 From what has preceded, it is plain that the functions obtained by adding to x 
 and to y, respectively, in (1) any multiples of the quarter periods corresponding 
 to them, will be linear homogeneous functions of the four functions (2). In 
 particular, it is obvious that 
 
 fi^ + '-^y y> ^) = <f{x, 2/+^S ^) = <P{x, y, z-{-"^)=z(p^{x, y, z) 
 and 
 
 f {^y y-\-"i, ^ + "^)=f («+?, y, 2+^')=^(a^+?, y-{-'^, z)=9^{x, y, z) 
 
 m 
 
 or THF. 
 
12 
 
 Further, we have manifestly 
 
 ^j(a5 + fl'i, y, z) = <f^{x, y-^fo^, z) = (pj{x, y, z -\- (Os)=:<pj{x, y, z) 
 
 S^<p,{x, y, z) = {—iye-^^^^(pf{x, y, z) 
 
 8^<p,{x, y, z) = {-iy^e^(p)<p,{x, y, z) 
 j=zO, 1, 2, 3. 
 If we apply the substitution 
 
 bi — {x, y, z, x-t^, y + ^^+ 4^' ' + ^+4^^47rt^ 
 
 (4) 
 
 we shall get entirely new functions, for 
 
 Si<fo{x, y, z) = e X e 
 
 m = — 00 
 
 >Ji<Pi\X} y} z) — e 2v W ^ 1 "» "s 
 
 ^1^2 (^) 2/) ^) — ^ -^ I — ■"■) ^ "> "^^ "' 
 
 This suggests the following functions, which may be written briefly 
 
 M5) 
 
 iP,{x, y, z)= S e«"" + *) 
 
 )H =: — CO 
 
 Writing 
 
 we find at once 
 
 ^i(a;, y, 2) = :S (*)"*«*""* + *' 
 ^2(3;, y, z) = V (_!)'» e&'('» + i) 
 
 RjTri 
 
 ei"* = />< 
 
 ^j(a; + wi, y , z ) = /i(/'j ^ ^ {x , y, z) 
 
 *Pi{^ ,y + <'>2,z ) = i</>j{x, y, z) 
 
 4>i{^ y y , z-\-f'>s) = — </'j{x, y, z) 
 
 ipi{x 
 
 y 
 
 , 2 + '4')=/^^j+i(^> y^ 2) 
 
 j=0, 1, 2, 3 and v''4 = ^''o 
 
 (6) 
 
 (7) 
 
 «>i 
 
 while ^j(x-|--^, y, z) and 9^^ (a; -j- — ' , y, 2) are entirely new functions not 
 expressible as linear combinations of any of the functions fj or (pj. 
 
13 
 
 It will be seen from these equations, that the periods of (^'j are not the same, 
 as those of (p^ ; for, in the case of the former, 
 
 8wi is the period corresponding to a^^ 
 
 4a^2 " " " « " y\ (8) 
 
 2^3 " " " " " z) 
 
 although each of the substitutions 
 
 {x, y, z; a; + 4wi, y, z), {x, y,z) x, y-\-2co^, z), {x, y,z] x, y, z-\-co^) (8') 
 
 operating on (p^ has only the effect of changing its sign. The effect of the sub- 
 stitution 8p on (pj{x, y, z) is the same as that on ^^ (a;, y,z), viz : 
 
 But while 
 we have 
 
 Si<pj{cc, y, z) = {—iye-^^'^(Pj{x, y, z) | 
 8piP,{x, y, z)=:{—i)^^e-^^P)<f>j{x, y, z) ] 
 
 Si</'j{x, y, z) = {—iye-^(i^<fj{x, y, z) 
 S2r + ,^j{^,.y, z)=:(-i)i('- + i)6-^(^ + i)^,(a;, y, z) 
 
 (9) 
 
 (10) 
 
 In general, the effect of Si is to change E{m) into jE'(m -|- 1), while Sp changes 
 E{m) into E(in -{-p). Hence the effect of 8p on our functions is the same as 
 that of 8^, to within an exponential factor which may be taken out from under 
 the sign of summation. Similarly 8^ changes E{m) into E{m -\- ^), and S^^^i 
 
 2 
 
 changes E{m) into E{m-^r -\-^). Hence we see here also that, to within an 
 exponential factor as above, the effect of 8-^ is the same as that of 81, and finally, 
 that of82rA.i the same as that of /8'|'"+^, or ot 8^8^, or of 8^ Si- 
 
 Changing x into — x and z into — z simultaneously has the effect of chang- 
 ing m into — m in (p , and m into — m — 1 in </> ; hence it 
 
 leaves (p^, (p^ and (^'q unaltered, 
 interchanges (p-^ and ^g, 
 
 changes (pi into — i(^'s , " 
 ^3 into i</>i , and 
 <p2 into — ^2 • 
 
 Consequently, as regards z and z simultaneously, we see that 
 
 f oj (fi} ^di <P\ fz, <P\ <pz are even, and 
 (pi is odd. 
 
14 
 
 Changing the sign of x or of z alone, or of y changes the values of all the func- 
 tions in such a way that no conclusions as to parity can be drawn in these cases. 
 In the case of each of these functions we may find those zeros which, like 
 the zeros of the ^-functions, cause the vanishing of the function by the cancella- 
 tion in pairs of the terms of the series defining it. This we can accomplish by 
 the examination of the function </>2 {x, y, z). We have, in fact, 
 
 ^2(0, y, 0) = 2 (—1)"* e«(« + *)^ + 2-» ^^ ('" + *)' , 
 
 m 
 
 Changing m into — m — 1 we get 
 
 ^2(0, y, 0)=-^{—l)-m-l ^im+i)* + 2.il.(m + i)^ 
 m 
 ;^ ^f l)»ga(m + i)«+2rriiL(m + i)» 
 
 Hence 
 
 ^2(0, y, 0) = 0. 
 
 Or, from (8') , we have more generally 
 
 ^^2(4^0; , y, Ic0s)=z0 
 
 where h and I are any integers and y is anything at all. 
 
 Finally, applying the substitution (Sj, (p^i^j 2/> z) is reproduced multiplied 
 by the finite factor ( — 1)' e-^(«' which is different from zero. Hence we have 
 
 The most general set of zeros of ^o is then, without loss of generality, from (8'), 
 
 X z=. 4Aa>, H i- a 
 
 y = y-\-2kco,-\-^^ V (11) 
 
 . — j.^ , 2^32/ , 2«;8a 3 
 
 From the second and fifth equations of (7) and the second equation of (10) 
 we can write the following table of zeros where, as above, 
 
 h, k, I, q, are any integers, 
 y anything whatever. 
 
15 
 
 UNIVERSt i . ^ 
 
 Zeros 
 of 
 
 2/ = 
 
 (4/i +.2)^.^1+ ^9 
 
 ^0 
 
 or 
 
 . , , 2awi 
 
 
 3, + 2A,„, + ?^=5« 
 
 '"'= + 2^^> + '^=«' 
 
 
 (4A + l)». + 2^"", 
 
 y + 2*<«, + ^=3' 
 
 
 ^1 
 
 or 
 
 4"^! H — z:r^ 9 
 
 Tl% 
 
 y + 2^.., + ?^^-^9^ 
 
 7 I n ^(>q I 2ac(jo s 
 
 
 y'^2 
 
 4Awi A H a 
 
 7rt 
 
 2/+2^c.,+-^-^;^9^ 
 
 TTt 
 
 ;,03 + 22/^^5 + ?^-«9^ 
 
 (4A + 3)..,+ 2^^9 
 
 S-"3 
 
 or 
 
 4Awi -|- ^^- 5 
 
 71% 
 
 3/ + 2^.., +^^9^ 
 
 + f^7 I OQliOo q 
 
 2^W2 + -^ 9~ 
 
 
 2a 
 
 ko,-\-2y^q-\-~^"q 
 
 Tti 
 
 2ac 
 
 {l + i)co, + 2y'"^q-\--^q' 
 
 m 
 
 <fo 
 
 J , 2acoj2rA-l 
 
 , , , Sa(Oof2r 4-1 
 2/ + ^'^'^2+-^(— f- 
 
 m 
 
 (,+,,.. + 2,5(2r±i) + 2^(?r.±i; 
 
 ^1 
 
 , , 2acoif2r 4-1 
 
 , , , daco^f 
 
 3aw2/2r+l 
 
 (, + .),,+2,5(?£+>)+?^3(2^; 
 
 f2 
 
 , , 2awW2r-\-l 
 
 , , , Saco2f2r-l-l 
 
 <Ps 
 
 , , 2acoJ2r-^l 
 
 OT 
 
 , , , 3aw2/'2r4-l 
 V + w^UaH- — T-^i — J — 
 
 ^''^^ + 22/^("-f~) + ^-^'(^~ 
 
 (/ + |).3 + 2,l^f2r±i) + 2a.3^2^ 
 
16 
 
 Putting h=zk=l=q = r:=0\\e get the following simple zeros 
 
 
 Zeros of 
 
 xz= 
 
 y= 
 
 2=0 
 
 </'<> 
 
 2fui 
 or 
 
 
 y 
 y 
 
 
 
 Wg 
 
 2 
 
 S^l 
 
 or 
 
 
 y 
 y 
 
 
 
 Wg 
 
 4 
 
 ^3 
 
 
 
 y 
 
 
 
 </'s 
 
 or 
 
 
 y 
 y 
 
 
 
 3ft>3 
 
 4 
 
 <fo 
 
 
 y + t7 
 
 2 "^ 47rt 
 
 ^1 
 
 
 Saw2 
 ^ ' 47:i 
 
 (t»3 I awg 
 4 ~'"4;:t 
 
 <fn 
 
 a Oil 
 
 Tti 
 
 , 3aci).2 
 
 4;ri 
 
 <Pz 
 
 Tii 
 
 , 3a Wo 
 
 y + 4./ 
 
 3^3 , aa>g 
 4 "^47:* 
 
 The zeros of ^p{x, y, z) might have been gotten directly, as follows : 
 
 am* + 2>rt (— m« + J^ »»»+ - m) 
 
 ^o(«, y, 2) = 2« 
 
 a (^ - TO)4 + 2.ri [^ (^_TO)B + iL(^_m)»+ i-(M-m)] 
 
 = 2e' 
 
 The corresponding terms of these two series will be equal but of opposite 
 sign for those values of a;, y , z which make the exponents of e in the two cases 
 
17 
 
 differ by an odd multiple of m for all values of m. Such values of «, y, z 
 will evidently cause ^o{^) 2/j ^) to vanish. We are to have, then 
 
 a (u — my 4- 2m [- (u — mf -\- ^ (u — mf -^ ^{u — m)^ 
 
 — \am' + 27ri (5 m^ + 1 m^ + - m)] = (2)fc + 1) :ri, 
 
 Oi\ (02 Ws 
 
 which on reduction becomes 
 
 (;. - 2m) { (!^ + a/,) (2m' - 2m;. + ;, + ;?) + (^;.' + 2^^ + M^ I 
 
 l «*! Wl tt;2 Wg J 
 
 = (2ifcH-l);:i. 
 
 This condition is satisfied if 
 
 // is an odd integer, 
 
 Ttix , hti 
 
 ^ + "'' = T- 
 
 TZIX 
 
 2my 
 
 0) 
 
 27iiz 
 
 ^''' + ^/'+^ = C/''^. + 2(/ + 2/' + l)]f 
 
 (12) 
 
 where X and /? are any integers. 
 
 To obtain the zeros of ^o(^) 2/> z) we need only put 
 
 y anything, as before. 
 
 2 wa ^ 27ri J '^ 
 
 (13) 
 
 This can be readily verified, for, putting these values in ^ (a;, y, 2), we have 
 
 ^ e '-V 2 TTt / ^ <uj ^ V 2 a)5^27rt/ J 
 
 TO = CO 
 
 a{m* - 2raV + w^lS) + 27ri 1- w (m — /oi) + 27rt (A m^ + ^ + 2p 4- 1 „ \ 
 ZIZ ^e "a ^2 2 / 
 
 __g-7^v>( ;^\mg''('"-f)'-|-«("»-^')'*''»+2Ti^(»i-M)w 
 
 because 
 
 gjri (Ama + A + 2p + 1) m / -lyn 
 
 since for X even, /m^ + / -|- 2,o + 1 is odd 
 
 X odd, /m^ + / -|- 2^0 -j- 1 is even or odd according as m is even or odd. 
 
18 
 
 When m is replaced hy /j. — m, the expression above is only altered by having 
 ( — 1)"* replaced by ( — 1 )''-'". Since /Jt is an odd integer, 
 
 i. e. for the above values of the variables, the function is equal to its negative, 
 and is hence equal to zero. 
 
 In the above it was stated that y may be taken arbitrary. As a matter of 
 fact, either y or z may be so chosen, since these two variables are, from (12), 
 subjected only to the one condition 
 
 27Ti . 27ti /: 1 o I i\ • I J 
 
 (14) 
 
 If z be taken arbitrary, our zeros will be 
 
 Ttl 
 
 y^/A+2/>+l 
 
 apr 
 
 2/^ 
 zz=z anything 
 
 2m fjLo)^ 
 
 (16) 
 
 For, on substituting these values, we get 
 
 am« + 2.i [(A ~ ?^)m»4- (^^t2p+ll + aM^__^\^,^J.„-| 
 ^e L\2 nlJ ^\ 2/1 ^ 27ri m«.^ <«. J 
 
 m 
 
 = e 1° ^e 2 2 /ncoj fi 
 
 The effect of changing m into // — m is to replace the factor e*"**"' in the above 
 by 
 
 g[A (M-m)» + (A+ 2p + 1) /Li-2 (A+ 2p+ 1) m] Tfi 
 
 which may be written, for brevity, e^"*. 
 If A is even, L is odd 
 
 ; is odd, e^""''' = (—1)" , and e^"' = (— 1)*" ~ " • 
 So that for all values of / 
 
 pLiti — gAm'rri 
 
 Which shows us, in the same way as before, that (15) is also a set of zeros. 
 
 The fact that z in (13) and in (11) contained the arbitrary quantity y might 
 have also assured us that we could so choose y as to give z any value we please, 
 and still have the resulting value of the function zero. 
 
 The zeros of ^j, ip^y (fz and those of (/>o, <pi, ^2> <ps can also be calculated 
 directly in the same way we have just found those of (po{x, y, z), or they can 
 
19 
 
 be derived from those of (p^ in a manner similar to that used in obtaining the 
 zeros of all the rest from those of (['2. 
 
 A comparison of our set of zeros for ^0 obtained by the two methods, which, 
 in fact however, are the same in principle, will manifestly show them to be 
 identical, if account be taken of (3) . 
 
 By the first method, the simplest zeros were first obtained, and from these 
 we determined the most general zeros, by observing what operations could be 
 performed upon the function without altering its value, except, perhaps, as to a 
 finite factor difierent from zero. By the second method we obtain the most 
 general set of zeros at once. The simplest zeros are then gotten by putting 
 
 I say that (13), for example, is the most general set of zeros possible of the 
 kind here considered, for all the operations which leave the value of the func- 
 tion unaltered, or unaltered except as to a finite factor other than zero, are there 
 provided for. 
 
 Thus / so enters, that a change in it by an even integer amount corre- 
 sponds to a change in x and z by some multiple of Wj and Wg respectively ; 
 while the change in k will be an odd integer when x and z are increased or 
 
 diminished by the same odd multiple of -^ and -^ respectively, which by (3) 
 
 does not alter the value of the function. 
 
 The presence of () permits a change in z alone by any multiple of Wg . 
 
 y so enters in the value of z that if changed by an integer multiple of 
 <t»2, z will be changed by an integer multiple of Wg, and if altered by an odd 
 
 multiple of ~, the effect on z will be to change it by an odd multiple of-^. 
 
 Finally, fx being an odd integer, it represents the result of the operation of 
 8p, where p is any integer, the effect of which is, as we saw, to leave the value 
 of the function unaltered except as to a finite factor other than zero. 
 
 In the zeros we have found, namely those which cause the cancellation, in 
 pairs, of the terms of the series, only y or z (but not both simultaneously) may 
 be taken arbitrary, while -x cannot be taken arbitrary at all, since it enters alone 
 in one of the equations of condition (12). No way of discovering other zeros 
 has suggested itself; and it remains a question whether other zeros exist or all 
 the zeros are confined within the above restrictions. 
 
 
20 
 
 The quotients 
 
 ^1 
 
 fa 
 
 III. 
 
 S^o 
 
 <P^ 
 
 92 fi ^2 ^8 ^5 
 
 are doubly periodic functions, the substitutions 
 
 f2 
 
 fa 
 
 ^2 and {x, y, z) x -\- aco^ , y -\- fico^ , z + y^o^) leaving ^ 
 
 fa 
 
 ^4 " (ir , y , 2 ; a; + aio^ , y + M > ^ + r^'^s) " — and ^ 
 
 fa fs 
 
 unaltered. 
 
 ^1 " {^, y^ z) «+8«Wi, 2/ +4/5^2, z 4-2^^3) " 
 
 fo 
 
 f2 
 
 f2 
 
 ^4 " {x, y, z) x-\-^aco„ y + ^i^co,, z-^r'^^rw,) " ^and% " 
 
 fa 9 
 
 «, /3, ^ being any integers. 
 
 If now we turn our attention to the derivatives of these quotients with respect 
 to X, y or 2 and inquire whether, analogously to the elliptic functions, these 
 derivatives are expressible in terms of any combination of the quotients them- 
 selves, it would seem that such is not the case. 
 
 We shall first consider the derivatives with respect to 2 . For convenience 
 of reference, the following tables may be of service. 
 
 We saw, page 13, that 
 
 Consequently 
 
 fo) faj i'di fi fs) ^1 ^^3 are even as to x and 2 jointly, and that 
 yAg is odd. 
 
 af 3f a d4'^ d (f 1 f s) d (^1 ^s) are ^aa ^nd 
 87' W Tz' ~W^' -^^- are oaa, ana 
 
 dz 
 
 IS even. 
 
 000 o 
 
 ^^<f^{x-\-io,,y,z)z=z^^iPi{x, y-\-co^, z) = ^^(pj{x, y, z -}- (0^)= ^<fj{x, y, z) 
 
 dz Loz ('H 
 
 fj 
 
 ' '^^ 82 ~ 
 
 -E{k) 
 
 OZ 
 
 Loz (Oi "^^ J ^ dz Loz ws J 
 
 i = o, 1, 2, 3. 
 
21 
 
 The effect of adding multiples of (o^, w^ and tog to x, y and z respectively in 
 
 ^pis the same as given in (7) in the case of (p^. 
 Cz 
 
 Let us now consider 
 
 dz \(fj (pi 
 
 If the numerator of the second member is expressible rationally in terms 
 of ifj and <f'k U ) k=zO, 1, 2, 3) it will be a linear combination of quadratic 
 functions of these, for it has no infinities for finite values of the variables, and 
 the effect of the substitution ;S'i is to reproduce it multiplied by e~'^^'^\ More- 
 over it is. even as to x and z jointly, and is changed in sign when z is changed 
 into z-f-Ws- Finally it is reproduced multiplied by the factor e~^^(*^ when 
 operated on by 8^ . 
 
 Of all the 36 combinations of <fj and (^'i. taken two at a time, only one, 
 (fo (po satisfies all these requirements. Besides 
 
 is zero whenever (fo and whenever </>o is. Hence it would seem that we could 
 write 
 
 (T ^'^'^ — d' ^^^ — A(D Lb 
 
 where, since this relation must hold when operated on by S^ and when x, y, z 
 are altered by multiples of (o^, co^, (o^ respectively, J. is a constant and equal to 
 
 ^,(0, 0, 0)^v'^2(0, 0, 0) 
 
 ^o(0, 0, 0) <f>o{0, 0, 0)' 
 So that we have finally 
 
 ' <p2 {x, y, zj 
 
 dz[_(p-i {x, y, z). 
 
 (f,{0, 0, 0) 1^,(0, 0, 0) 
 
 O^ : <Po{x,y,z) (po{x,y,z) ,.. 
 
 ^o(0, 0, 0) <Po{0, 0, 0) • (pi{x, y, z) ' ^'' 
 
 If now we apply the substitution 
 
 {x, y, z) x'-fwi, y, z) 
 
 we shall get 
 
 T^ri ( ^-^ ^2 (0> 0, 0)1-^2(0, 0, 0) , , ,^ 
 
 d_ [ <pi{x, y, zn _ ^ ^ > 'dz^'^ ' ' <fo{x, y, z) </>,{x 
 
 dzLf, {x, y, z)_\ ^o(0, 0, 0) ^o(0, 0, 0) ' ipl{x, y, z) 
 
22 
 
 which must hold for all values of «, y , z. Substituting the first set of zeros 
 for ^8 given on page 16, this equation is satisfied. But using the second set we 
 get 
 
 ^2(0, 0, 0)^J,{0, 0, 0) ^,(0, 0, 0)^^/.,(0, 0, 0) 
 ^o(0, 0, 0) ^o(0, 0, 0)-^8(0, 0, 0) ^Ao(0, 0, 0) 
 
 which is manifestly not true, for 
 
 ^i(0, 0, 0) = c^8(0, 0, 0) 
 while 
 
 ^o(0, 0, 0)d^ip,{0, 0, 0) 
 
 as may be seen from the definition of these functions. Hence we must conclude 
 that a relation of the form (1) does not exist. 
 Again, if we consider the numerator of 
 
 dz \(fj (ft 
 
 it will be found that of all the combinations of <pj and ^j only <po (p2 satisfies all 
 the conditions that it does. Moreover, the numerator vanishes for all our zeros 
 of <po and of (/'^ . But on writing 
 
 d_ r s^o(a;, y, z) ~\ _ ^ ip^{x, y, z) (f'^{x, y, z) 
 dzL(p2{x, y, z)J (fl{x, 2/> 2) 
 
 where, as before, B can only be a constant, we shall find, on substituting the 
 first set of zeros for ^'o , 
 
 ^JO, 0, 0)^J'2{0, 0, 0) 
 
 ^ = ^o(o, 0, 0) </'o{o, oro)' 
 
 Using the second set of zeros for ^o> we get 
 
 fo(0, 0, 0)^s^2(0, 0, 0)^ 
 ^ = ^2(0, 0, 0) s^'olO, 0, 0) 
 
 These two values are not the same, since 
 
 ^.1(0, 0, 0)ziz<pl{0, 0, 0). 
 
 Hence we conclude that no relation of the type (3) exists. In the same way, 
 the expressions for the derivatives of all the various quotients of a ^' by a f in 
 terms of the quotients themselves break down. 
 
23 
 
 The difficulty seems to lie in the fact that the fundamental periods of ^^ are 
 multiples of those of ^^. For we can pass from (p^ to v''j + i by either of the two 
 substitutions 
 
 {x, y, z; x-\r^o^, y, zj, [x, y, z; x, y, z + —-)• 
 
 At the same time (fj from which ^^ was derived as the result of the substitution 
 S^, is left unaltered by the first of these substitutions, and is changed into' 
 ^,+1 by the second. 
 
 All our relations, arrived at in a manner similar to that indicated above, 
 broke down when subjected to this test. 
 
 It should be noted that this test does not apply in the case of the 6- and 
 ^-functions. As no general theorem analogous to that made use of in this 
 connection in the case of the ^-functions could be established for the functions 
 here considered, the above method was employed, to show that no such relations 
 exist. 
 
 In a similar way, no quadratic relations between any of the <pj and ^^ 
 satisfying all the tests at our command, could be found. 
 
 The objection, above mentioned, as holding good against the relations 
 between the derivatives of the quotients and the quotients themselves, do not 
 seem to hold in the following cases, where only the quotients of the (fj or of the 
 <p)c are involved separately. Thus*, it was found that 
 
 8 
 
 dz 
 
 ' (pojx, y, z )' 
 
 ^zi^, y, z):^^(po{x, y, z) — (po{x, y, 2)^-^2(0;, y, z) 
 
 and 
 
 '<po{x, y, z)' 
 
 <pl{x, y, 2) 
 ^0(0,0,0)1^.(0,0,0) ^(,/^,,)_^.(,,^,,) 
 (*i{0, 0, 0) — }«|(0, 0, 0) • <fi(x,y,z) ^' 
 
 4'2{^, y, z)^J'o{x, y, z) — <Po{x, y, z)-^^(pz{x, y, z) 
 dz L^2 {x, y, z)J (pl{x, y, z) '■ 
 
 s^o(o, 0, 0) 1^3(0, 0, 0) 
 
 qz (pi[x, y, z) — ^^^3(3;, 3/, z) ,gs 
 
 -^1(0, 0, 0)-^U0, 0,-0) • ifUx, y,z) '^ ^ 
 
 which may be derived from (4) as the result of operating with 8^, together with 
 the other relations derived from them by all the operations at our command, 
 satisfy all the tests that were applied to them. 
 
 These results are given here, not as proved relations, but as such which 
 have not been disproved. 
 
24 
 
 The derivatives with respect to x and y are more complicated than those 
 with respect to z. Hence it will be at least no easier to establish relations 
 involving the former than to establish any involving the latter. Thus 
 
 
 ^^^0_2^'^_£(1) '"^ 
 
 92/ 
 
 C02 
 2zi 
 
 m 
 
 2 gE(jn+l) 
 
 m= — 00 
 
 = tlll!e-^(i) 2 (wi_i)2esw 
 
 _g-£(l) 
 
 . (On Wo 
 
 Wo J 
 
 Similarly 
 
 _ .-Eii) \'d<fo 2^8 d(fo , 2;rt 
 Ley fOo dz (t>2 
 
 ]• 
 
 9fo 
 
 3a; 
 
 ^ i^ __ g_£{i) rSfp 3 ^ 3^0 I 3w8 9^0 27ri n 
 
 '^" L9a; W] 3y wj 3z Wi °J 
 
 In some respects the functions 
 
 0O=Z<Po(p2 0,Z=Z(p,<ps ¥o=i(f'o<p, ?P*i=^i^8 
 
 are simpler than the (fj and (/>)c. Thus, from what was seen before, 
 
 00, 0^, ¥1 are even as to x and 2 simultaneously, and 
 ¥0 is odd. 
 
 S,0o = — e-^^(^^0, 
 
 </>j (a; -|- w, , 2/ + W2, z + ws 
 
 0j{x'{-oj^, y, z 
 
 0j{ X, y, z + ws 
 
 ¥){x-\-(o^, 2/4-^2, z + ws 
 
 ^^(a; + wi, y, z 
 
 ^j{ a;, y4-"^2, 2 
 
 r,( a;, y, z + wj 
 
 5-1^0 = — «-'^*" ^0 
 S, W, = e-2^(^' '/i 
 
 Si¥o = — e-^^^^^0o 
 Si¥,= e-2^'*)(l^i 
 
 = ^j{^, y, 2) I 
 
 — 0){x,y,z)) . 
 
 = —i¥j^i{x, y, z) >| 
 = i¥j^i{x, y, z) 
 
 . .^=: ^;(a;; y, z) [ 
 
 i = o, 1 
 
 i = 0, 1 
 
25- 
 
 while, as before 
 
 ^.(^+'f, y, ^)aiid r,(a.+ ^|, y,z) 
 
 are entirely new functions, not expressible in terms of 0^,0^, W^, W-^. 
 
 9^_^ 3^2 J- t. ^^0 
 
 fo 
 
 8^2 27ri^ 
 
 ~] g-2£:(i) 
 
 ^2 
 
 3fo 
 
 La^" 
 
 2;ri 
 
 '«] 
 
 er 
 
 "iE (1) 
 
 >«t + ^^t]+S'-^^>»^^ 
 
 a^ L as co^ J 
 
 Similarly 
 
 dz Laz (Os J 
 
 (72 L az wg J 
 
 ^1 
 
 ?I^= V2^(i)r^x_4!E! 
 
 and 
 
 Q a</>o_ 
 
 dz 
 
 '^^ ar— 
 
 \_dz W3 J 
 
 L a^ W3 J 
 
 g-2£(i) 
 
 L az W3 J 
 
 e-2^(i) 
 
 _ OZ Wg ^J 
 
 If by aid of these formulae, we attempt to express the derivative with respect to 
 z of the quotient of any two of our functions (Pq, </^i, Wq, W^ in terms of any or 
 all of the quotients, we will meet with the same difficulties as before. Thus if 
 we take, for example 
 
 dz\(Poi 'PI 
 
 we shall find that of all the combinations of our functions only 0i W^^ behaves 
 exactly like 0^ '^ — ^^o ^^ when put to the test of all the above operations, 
 
26 
 
 and besides, the numerator of our expression for the derivative vanishes for all 
 the known zeros of ^^ and of W^ . But on writing 
 
 3. r ^o(a^> y, 2) 1 _ f ^i{x, y, z) ¥,{x, y, z) ,«, 
 
 dzL0o{x,y,z)J-'' 0l{x,y,z) ^""f 
 
 where, as before, C must be a constant, we shall find that, according as we use 
 the first set of zeros or the second set of zeros of ?Fo (a; , y , 2) , which are those 
 o{</'o{x, y, 2) and ^'2 (x, y, 2), 
 
 0,{o, 0, o)|n(o, 0, 0) 
 
 or 
 
 0,{O, 0, 0) r,(0, 0, 0) 
 ^</>,(0, 0, 0) 1 2^0 (0,0, 0) 
 
 c= 
 
 <i5'o(0, 0, 0) r,(0, 0, 0) 
 
 But these two are not the same, since 
 
 (Pl[0, 0, Qi)z^0\{O, 0, 0). 
 
 Hence we conclude that no relations of the type (6) exist. 
 
 Quadratic relations between 0^, 0^, W^y W^ also seem not to exist for the 
 same reason, viz. because the periods of </^o and 0^ are smaller than those of Wq 
 and ¥^. 
 
 It may be mentioned in this connection, that the following symmetrical 
 quartic relation between the functions <i^o> ^m ^oy ^1 was discovered in the 
 course of the work, which, as far as could be tested, satisfied all the conditions 
 imposed; viz. 
 
 0l0i—¥l ¥l — Al0'o-\-0\-n—n'] 
 when ^ is a constant whose value can be obtained readily. 
 
 IV. 
 
 Consider the holomorphic function 
 
 f,{x, y, z)z= n"(l-|-e2«(2fc + i)* + 2-[?^ (2*4-1)''+ ^(2t+i)+*^J) 
 where the real part of a is negative. It is obvious that 
 
 ■1 -p C nil Wj W| 
 
27 
 Again, writing 
 
 k 
 
 we see at once that 
 
 • SJ,{x, y, z) = ll+e-'^-'^^^'£ + l + t'U{^> y> ')' 
 
 Finally, writing 
 
 F{x, y, z)=zfi{x, y, z).fz{x, y, z) 
 we have 
 
 s^F{x, y, ^)= z^^^,,.:' .:-.:: f{^, y, z) 
 
 or 
 
 8,F{x, y, .) = e-2«-2--(^+?, + l^' F{x, y, z). 
 
 We also have 
 
 F{xV^, y, z) = F{x, y-\-'^, z) = F{x, y, z + '^) = F{x, y, z). 
 
 If we put 
 
 2a=:A, |^=i?o '^=ii,, '^=Q, 
 
 our function becomes 
 
 X{x, y, z) = 
 
 n[(l+ e^ (2fc+l)3+2.f [|-(2fc+l)^+ ^(2^+1)+ ^-^]) (1+ ^Ai2k+l)^-2.ii± (2fc+ip-X(2fe+l)+ ^j)-| . 
 
 Now we have 
 
 X(x-j-i2i, y, z)=zX{x, y-\-ili, z) = X{x, y, z + £^s) = X{x, y, z) , 
 and, denoting by Tp the resulting form of Sp, viz. 
 
 ^p-{^, y, 2, ^-\—^^p, y-r-jj^P-^'-^P>^-t-jj^P-t-^p-t-^^P) 
 
 we have 
 
 T,Z(a., 2/,.) = 6-^-2'^'(^.+4 + 4 X{x, y, z). 
 
 The function X{x, y , z) which resembles the functions already considered, 
 in being periodic, and in being reproduced to within a factor on being subjected 
 to a linear substitution, seems to differ from them in not satisfying any simple 
 differential equation or equations. There seems also to be more freedom in 
 obtaining the zeros of this function. Thus, while in the case of the functions 
 
28 
 
 already considered, only y or z separately could be taken arbitrary, these two 
 variables having only to satisfy one condition, and x had to be chosen subject to 
 an independent condition, our present function X {x , y, z) vanishes whenever 
 either of the following conditions is satisfied : 
 
 A {2k + If + 2;ri [^(2^ + 1)^ -f |^ (2^ + 1) + ^J = (2/ + 1) ;ri 
 or 
 
 A {2k + If - 2m [-| {2k + 1)^ - 1 (2^ + 1) + j^ = {21 + 1) Tzi 
 
 where I is any integer, positive, zero or negative, and k is any positive integer, 
 including zero. The analogy, however, between our functions is made the 
 more striking by noticing that the zeros we found for <fo{x , y , z) are also such 
 for X{x, y , z). But all the zeros of the latter are not included in the former. 
 It may be interesting to note still further the similarities and the dissimi- 
 larities existing between our two classes of functions. For the sake of brevity 
 write 
 
 E, {k) = ^ (2^ + If + 27Ti [| (2^ + If + I (2^ + 1) + ^2 
 ^2(^) = ^(2^ + l)« + 2;ri[|(2A; + l)2 + ^(2^ + l)+^]. 
 Increasing zhy -^ we get a new function 
 
 And as before 
 
 X{x^^,y,z) = X{x,y-\-^,z) = X{x,y,z^^-^) 
 
 = X{x-^^^,y^^,z + ^-^)=zX,{x,y,z), 
 
 = X(a;, y, z) = Xo(a;, y, z). 
 The effect of the substitution Tp is to change 
 
 E^ [k) into E^ {k-^p), and E^ {k) into E^ {k—p) 
 
 — Q-{E,{Q)+E,(l) + ....+ E,(p-l)]X^{x, y, z). 
 
29 
 And, in the same way 
 
 The effect of the substitution 
 
 I^-[x,y,z,x-\- ^^.,y-t ^^ -t 27r*'' + ^+X^^ 
 
 is to change 
 
 El [k] into El {k + ^), and E^ {k) into E^ [k — 1) 
 
 thus giving rise, as before, to entirely new functions, when applied to Xo{x, y, z) 
 and Xi{x, y, z) : 
 
 Tr Xo{x, y, z) = S,{x, y, z) = ni{l + e^^^^ + i)) (1 +e^>(^-J))] 
 
 and 
 
 T^Xiix, y, z)= Si{x, y, 2) = 11 [(1 - e^. <*= + i)) (1 _e^.(«:-4))]. 
 
 It is noticeable that instead of the periods of S (x , y, z) being multiples of 
 those of X{x, y, 2) we have 
 
 ^(3^+4-. y, z)=S{x, 2/ + ^, z)=3{x, y, z + i2,)z=E{x, y, z). 
 Again, 
 
 ^o(a^-, y, z-\-~')=:Si{x,y,z), 
 
 ^li^, y, z-^-^) = Sa{x, y, z); 
 
 but there seems no way of passing from S^ipCy y, z) to Si{x, y, z), or vice 
 versa, by a change in a? or 3/ . 
 
 As in the case o? X{x, y, 2), we have 
 
 Tp3o(x, y, z)=e-^^^(i) + ^^i^) + ■^■■+E^ip-i)]3o{x, y, z) 
 Tp3i{x, y, 2) = (— l)^e-[^.(i) + ^.(i) + ----+-^.(p-i)]S^(a;, y^ 2). 
 Also 
 
 ^i ^oi^, y, 2) = n [(1 +e^'('= + i)) (1 -f e^»(«:-i)J] 
 
 = e-^^^'^X,{x, y, z)=TiXo{x, y, z) 
 T^ Si{^, y, 2)= — e-^'WXi(a;, y, z)=:TiXi{x, y, z). 
 
30 
 
 But we had, by definition 
 
 T,Xo{x, y, z)z=So{x, y, z), T^ X^{x, y, z)=E,(x, y, z). 
 
 Hence we see that 
 
 TlX{x,y,z)=T,X{x,y,z) 
 
 and similarly 
 
 TlS{x,y,z)=T,£(x,y,z) 
 
 i. e. the effect of two successive operations of T^ is identical with that of a single 
 application of Tj . 
 
 Changing the sign of x and z simultaneously interchanges E^ and E^. 
 From this follows that X^ and X^ are even as to x and z simultaneously. But 
 for S {x, y, z), we have the values changed, thus 
 
 ^^{—x, y, — 2) = j-^^^^^ziy,-o(a^, y, z) 
 
 77 1 \ 1 — C^'<-i>;r, , 
 
 ^x{—^y y^ — ^) = l_,g,(-i) -i(^> 2/, 2). 
 
 In conclusion, it may be mentioned that by taking the logarithmic deriva- 
 tive oi Xq{x, y, z) with respect to z, we shall obtain a new function analogous 
 to the Z-function of one variable. Thus, writing 
 
 X,{x, y, z)j=nU+«'^^'*+"*"^*'^4<'*+'^+2«^<'^+^>'+'^''4<'^+^'co8[27r;^ (2fc+l)2-|-^] 
 
 1 "Cj 
 
 we have 
 
 ^ ^=A(x, y, z) = 
 
 - fc=oo 
 *=3 *=0 
 
 ^A (2t +i)»+2,i^(2fc +1) sin [2;r ^ (2A; -f If + -^ ] 
 
 This series is uniformly convergent, since the real part of A is negative, 
 and therefore represents a function. We have, evidently, 
 
 A [x -\-Qu y,z) = A {xy y ■\-Qi, z) = A {x, y, z +.Qi) = A (x, y, z) 
 
31 
 
 and 
 
 a r.-A-'. 
 
 
 ,_^-2.i,^ + X+_l) 
 
 Another differentiation will give us a doubly periodic function, for 
 
 |j(ar, 2/, z)z=^^A{x-^i2„ y, z) = ^^A{x, y -j- iJ,, z) = ^J{x, y^z-i-I^,) 
 
 and 
 
 n^J{x,y,z) = ^J{x,y,z). 
 
 The successive derivatives also have the same property. 
 
 UNIVERSITY 
 
 i^LIFORH^> 
 
Biographical Sketch. 
 
 The author, Abraham Cohen, was born in Baltimore, Md., September 11, 
 1870. His elementary education was gotten at Scheib's Zion School, where he 
 was enrolled from 1877 till 1883. He then entered the Baltimore City College, 
 and upon graduation, in 1888, was admitted as a candidate for the degree of 
 Bachelor of Arts, in the Johns Hopkins University. This degree was conferred 
 upon him in June, 1891, and in the fall of that year he re-entered this Univer- 
 sity, as a candidate for the degree of Doctor of Philosophy, selecting Mathematics 
 as his principal, and Astronomy and Physics as his first and second subordinate 
 subjects respectively. Upon receiving his Bachelor's degree he was awarded a 
 University scholarship, which he resigned to accept an appointment as Assistant 
 in Mathematics. This position he held for two years. During the past year he 
 has held the Fellowship in Mathematics. 
 
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 on the date to which renewed. 
 
 Renewed books are subject to immediate recalL 
 
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