C. X. 
19 
[658 
a (each mean- 
The relations 
f course to be 
ulæ — 4&! = Scq, 
658] ON SOME FORMULÆ IN ELLIPTIC INTEGRALS. 145 
And consequently 
(a 1} b 1} c 1) d u eùiflh., 1) 4 = (1, - B, -2G, -D, 0){x 1 -6, l) 4 . 
Hence also 
Ii = ^1 ~ ^di + 3Ci 2 = — 4BD + 12G 2 = a?I ; 
Jj = a 1 c 1 e l — ciydf — b{\ + 2b 1 c 1 d 1 — c^ = — P 2 +8 C 3 — 4>BCD 
= -B 2 + 8C s + G,{a?I - 12 G 2 ) 
= a?CI - 4(7 3 - D 2 
= a 3 J ; 
where, as regards this last equation o?GI — 4<C 3 - D 2 = a 3 J, observe that G and D are the 
leading coefficients of the Hessian H and the cubicovariant <X> of the quartic function 
U, and hence that the identity — <E> 2 = JU 3 — /P 2 H + 4H 3 , attending only to the term 
in x 6 , becomes — D 2 = cdJ — a?GI 4- 4G 3 , which is the equation in question. 
We thus have I X =I, J x — J\ viz. the functions (a, b, c, d, e)(x, l) 4 , (a u b lt c 1} d 1 ,e 1 )(x 1 , l) 4 , 
are linearly transformable the one into the other, and that by a unimodular substitution 
x 1 — px + cr, y 1 = px 4- a, where pa — pa — 1. It may be remarked that we have 
(«, b, c, d, e)(x, 1) 4 =(1, 0, G, D, E)(x + b, l) 4 ; and hence the theorem may be stated 
in the form : the quartic functions (1, 0, G, D, E)(x, l) 4 , and (1, —B, — 2G, —D, 0) (x ly l) 4 , 
are transformable the one into the other by a unimodular substitution : or again, sub 
stituting for E its value a 2 /—3(7 S , = — 4PP + 9G' 2 , the quartic functions 
(1, 0, G, D, — 4PP + 9G 2 ) (x, l) 4 , and (1, — B, —2C, — D, 0)^, l) 4 
are linearly transformable the one into the other by a unimodular substitution. In 
this last form B, G, D are arbitrary quantities ; it is at once verified that the invariants 
I, J have the same values for the two functions respectively ; and the theorem is thus 
self-evident. 
Reverting to the expressions for a 1} j3 1} y 1} Sj we obtain 
s № a A, . „ a — S./S — y 
= ’ Pi~yi= ( v ~ =—I ¥ » 
Hence also 
2\ 
v\ 
ßi Si 2fi ; 7i «1 - (A 2 v% - 
7,-8, = ^; = 7i _ 8 , 
2v 
OLi - 8 1 
ß — 8. y — 
- a. 
ßi-8 1 
y — 8. a. — 
ß 
a. — 8. ß — y , ß — S . y — a , y—S.a—/S 
= a i-S 1 .ß 1 -y ly ßi — S 1 .y 1 — a 1 , 7i-^i- a i~ßi, 
which agrees with the foregoing equations 1\ = I and Jj = J, since /, J are functions 
of the first set of quantities and I 1 , the like functions of the second set; in fact, 
1 = ^4 (P 2 + Q 2 + Br), and J = (Q — B) (B — P) (P — Q), if for a moment the quantities 
are called P, Q, B.