NOTE SUR L’ADDITION DES FONCTIONS ELLIPTIQUES. 
548 
[94 
et les valeurs correspondantes de S (u — v), G (u — v), G (u — v) se trouveront en échangeant 
les signes de A', B', G', S', T, U'. 
[Reverting to the functions sin am, cos am, A am, or say sn, cn, dn, instead of 
S, C, D, and introducing Dr Glaisher’s very convenient notation s 1} c 1 , d x for the 
sn, cn, dn of u, and s 2 , c 2 , d 2 for those of v, the formulae just obtained may be 
written 
sn (u + v) = 
s-fi^d-2 + sXxd-L 
1 — k 2 s 2 s 2 
Sx 2 — s 2 2 
s x c 2 d 2 s 2 Cxdx 
SxCxd.2 + s 2 c 2 dx 
CiC 2 + SxdxSod.2 
Sxd^ + s 2 d 2 c x 
d x d 2 + /; 2 5 1 c 1 s 2 c 2 ’ 
cn (u + v) = 
Ci c 2 — SxdjS.d* 
1 — k 2 s x s 2 
SjC^, — s 2 c 2 dx _ 1 — s 2 — s 2 2 + k 2 Sx 2 s 2 
SxC 2 d 2 — s 2 Cxdx CxC-2 + SxdxS. 2 d 2 
CxdxC 2 d 2 — k'%s 2 
d x d 2 + k 2 SxCxS 2 c 2 ’ 
dn (u + v) = 
dxd-2 — k 2 s x CxS 2 c 2 
1 — №s 2 s£ 
s i d\ Co s 2 d 2 Cx 
SxC. 2 d 2 sXxdx 
CxdxC 2 d 2 + k' 2 SxS 2 _ 1 — k 2 s 2 — k% 2 + k% 2 s 2 2 
c x c 2 + SxdxS/d, dxd-2 + k 2 SxCxS 2 c 2 
viz. we have thus a fourfold representation of the addition-equation for each of the three 
functions.] 
De ces formules il peut être tiré un grand nombre d’équations identiques; par 
exemple celles-ci : 
(A 2 — A' 2 ) = KP, &c., S 2 -S' 2 =QR, &c., 
(B + B') (G — G') = K (S + S'), &c., (B - B') (G + G') = K(S-S'), &c., 
BC-B'G' = KS, &c. B'G — BG' = KS', &c., 
(S + S') (T + T') (U + ü , ) = (S-S')(T-r)(U- U') = PQR, 
S'T'U' + S'TU +STU + STU' = 0, 
ST U + ST U' + S'T U' + S'T' U = PQR, 
(.A - A') (S + S' )= {G-G') {U- U), &c., 
(A + A') (S - S')= (C + C')(U+ U'), &c., 
(.A — A') (T - T') = P (C — C'), &c., 
(A + A') (T + T') = P(C + C'), &c., 
(A - A') ( U + U) = P{B- B'), &c., 
(A+ A') (U- TJ') = P (B+ B'), &c.,