154 



To reduce (89) substitute 



t = (a ^ (] 4R^{a))'(pu — (1 24)R''(a)) (40) 



Where a is a root of R(t). In this case take a = 1 d. Then equation (40) 

 may be written, 



t-- {(1 d) + ((— d 2)/(pu — pv)) (41) 



where pv = d ^/ * 



Since (pV) ^ = 4p3v — g^pv — gg — i (2) ', =) 2)d5 and (— p'v) pu — pv) 

 = {G'/(7){xi. -f V) — (a\'a){\x — V) — '2,'y a) (v) (Note 1) we have, remem- 

 bering tlie relation (dt du) — (R(t))' ^ + v= (— 1/((2)' -) (((2)' ^^ j^ 



a' 



— (U + v)(fT^ /fT)(u — V) — 2(rT' rr)v)du -f iV + V = ( — (d— (2) ' =('T7rr)(v')u) 



a 

 — ((2)', •'=)/2 log ((T(u — v),ifT(u + V) + iV 



(The above is equation 42.) 



Note: Schwarz, Foruieln der elliptscheu Fuuctiuneu, p. 13. 



