260 PEOFESSOR A. G. GREENHILL ON THE 



The function c is the one that should be tabulated numerically, as 



HM /ew H (K - u) lu , _ B (K - w) /M , . 



= HK/0K' HK /ed' OK 700 



Also 



or one of its transformations, depending on the region. 

 29. ,1=6, /= or f. 

 Derive from p. 3 in (3) 6, by putting c = a 3 (2a 1), so that 



_ 2 , y( + a) ,/^ - (2a - _l) . _+ 2_a* - } 



3 2* 



= 2 gin _ ] y (< - a) v/] 2e~ + (2a -^1+ 2^- a } 



3 ' 2i 3 



nt* + Za'_-ct dt 



~] a t ' x /{4/;' i -(r- + 2 3 -a 2 ) 2 } 



Next put 



6 + 1 , /r - 1 1> - I 



/1 = ^ + 3' ' == * 6T4TJJ ' ; -/>- + 3 



(6- l) 3 (^ + 3) (6 + !)(- > + 8) 4 



1G?> 



T /")\ C? 'i 



Then from (8), 23, in the region 3 > b > 1, 



so that there results the well-known relation 

 and in LKUKXDKE, 1, chap, vi., p. 27, 



/ 1 \7~ an J. "~~ ZitXj /n\ 



x = sin rf> = sn 4- K = --- - , K~ = (8). 



6+1 2.7T + x* 



The following table shows the six linear transformations of the division-values for 

 fji = G, obtained in accordance with the preceding theory by the substitutions 



6 + 3 



the accents are omitted from K by changing to the complementary modulus. 



The numerical value b = ^/3 gives a modular angle 75 ; while b = ^/5 corresponds 

 to the functions required for /A = 10. 



