of an Elliptic Function. 43 



But sin am {2 n (f> — 2 oi) = + sin am (2 w), 



sin «7» (2 « CO — 6 w) = + sin am (6 »), &c. 

 and 

 sm CO am {'21100 — 2(a) = sin«m(K + 2w — 2«c«) 



= + sin aw(K + 2«;) = + sin am ( — K — 2co) 

 " '*^.4:sina?»(K — 2«>— 2K)= + sina»2(K — 2co) 



= +s\ncoam{2ui)i smcoam{27iw —Q(a)=. + sin co aw (6 w), 



&c. 

 These constants therefore may be all expressed by the two 

 series 



sin^ a/w (2 co), sin^«m(4!co) sin^aw(«— 1) co, 



sin^co am (2 w), sin^ coam{4<co) sin^ co am (n — 1 ) w. 



Consequently the 4 co in this theory may be everywhere re- 

 placed by 2a>; and this reduction Jacobi has himself partially 

 made. 

 But 



. o ,^ , cos^ am (2 co) 1 — sin^ am (2 co) 



sm* CO am (2 co) = — r-o 1^ — r = , ,o . c, — ,^ r » 



'^ '' A2 a»? (2 co) 1 — A-^ sm^ am (2 co) 



• o ,. . 1 — sin^ aw (4 co) . 



sm^coaw (4co) = ^ . ^ rr^i &c. 



^ 1 — fc^ sm'^ am [4f 0)) 



And sin^awi (4 co), sin^ am (6 co), &c. may all be expressed by 



functions of s'm^ am (2 co). Therefore all the constants in this 



theory may be expressed by functions of sin^aTw (2 ca). 



Now sin^ am (2co) = sin^ am (2 co — 2 K) = sin^ am (2 co'), if 



co' = CO — K. Let 



2 r K + 2 r' K' \/^ . 



n 



and make n = 2^ — 1> ^ — i> = ^i ; we have 



,_ (2r-2p+l)K + 2r'KV^ _ (2ri + l)K + 2;-'K'\/^ 

 "" w n ' 



Hence the second form of co reduces to the first. 



Again, 

 sin^ am (2co) = sin^ aw (2co — 2K— 2K' V^) = sin^ am{2j)y 

 if ct/srco-K-K'^/^. Let 



2rK + (2r'+l)K''v/-l 



w = i ) 



n 



and make r/ = r' — ^ + 1, r^ and p remaining as before; we 



have in this case, 



, (2r-2p+l)K + (2r'-2» + 2)K'\/^ 



Ctt' = -5 ± i. '- 



n 

 ^ (2 r^ + 1) K+ 2 r^ K' a/^ 



