148 



Proceedings of the Royal Irish Academy. 



The addition theorem is (Greenhill, p. 268) 



6 (u + v) 

 <j> {u + v) = 

 \p{io + v) = 



duipvxpp 



(pHt - (p''v 



(4) 



The denominators being all equal to ^w - ^jv. 



The addition of the half periods can be made by contour integration. 

 The following method is due to Mr. K. Eussell. It is assumed that e,, e^, e^ 

 are real, and Ci > e^ > e, so that I, m, n are all real. Since (ei e^e^co) = 

 (00636261) = (63006,62), if .(',£1,^2 be correspondents in the homographic 

 systems determined by the roots of Q; taken in these three orders, 



(««, 62 63 00) = (li 00 63 62 6|) = (I2 «3 00 61 62). (1) 



The AR of any four elements in any one of these systems is equal to that of 

 the corresponding four in either of the others. Thus [xei e-i co) = (^i 00 63 61) 

 = (ftifii ?i 00), 



?i - ^1 62 - e, 61-62 li - 63 x-Ci ?i - 62 X- e-i 



and 



6] - «3 62 - a; X- 61 Ci 

 (sB 61 62 00) = (5, 63 CO 62) = (63 ?2 62 co), 



Sa- e. _ 61-62 

 62-63 Ci — X 



?2 - 63 e^- X 



X- I 



fij X- e-i 



m 



or 



\/«8 - ?j = 



a/«i 



63 



and 

 also 



62 - 62 



?2 - g| 



62 - 63 



(?.!; 



62 - XT 

 Ci - g 



6a; - 62' 



or 



or 



.^Z -'^ — 62 



\/?2 - C3 = v/(e2 - 63) /- 



Va; 



(3) 



2v/a 



62. a; ■ 



a/«i - S2 = V^Cs 



f^Si 



fe - 63 . 



ll^v^gl - 61. Si - 62-^1-63 



f" <Z?2 



= - «. 



■■ 2\/e, - ?2 • «8 - ?2 • ?2 - C3 "" ^ 



Let - U represent the contour integral round any chosen path, starting 

 from CO, - till is the linear integral from tx> to 6, - wz is the integral whose 

 path is from co to c,, round a half circle in the positive direction round 6, 

 and on to 62. wi + &)2 + wj = 0. 



1°. If - Z7 be the integral round an infinite circle in the positive direction 

 of rotation, the integrand becomes affected by the factor i-^i-^, and the 

 integral changes sign. The functions 6, <p, ^ are each affected by the factor 

 e+i^, and also change sign. Hence 0, <j,, ^p are all odd functions. 



