896] 
which is 
A TRANSFORMATION IN ELLIPTIC FUNCTIONS. 
V v (A) 
and similarly the whole coefficient of dx 2 is 
1 
(- x 2 ) [D s/(A) + B VC#)}; 
x 3 v(4) 
Hence the terms in dx x , dx 2 are together 
OO № ^{A) + b^(E)}. 
1 (x x dx 2 - x 2 dx x ) —) + B V W . 
X 3 
V(H) '~ m/ x 3 
and since the terms in dy X) dy 2 are of the like form, we have 
dz = j—^ (Xidxa - x 2 dxj) + (y x dy 2 - y 2 dy x )\ x D + E 
and combining herewith the foregoing value 
V(4^ 3 — Iz - J) = 
we have the required formula 
D^(A) + E^(B) 
A 3 
dz 
x x dx 2 - x 2 dx x y x dy 2 - y 2 dy x \ 
VCB) } 
\J(4>z 3 — Iz — J) ( *J(A) 
As a very simple verification, suppose a = l, b=c = d = e = 0; then 
(A, B, C, D, E)=(xA x x %, x I 2 y{\ x x y x 3 , y x % 
and if A — x x y 2 — x 2 y x as before, then 
rr> %/jj 2 
z _ Xl V 1 = 0 2 
(x x y 2 - x 2 y x f 
where 
a — ~ 1 _ _ V± 
^i2/2 - x 2Vi V 2/l 
Also / = 0, J=0, and consequently 
dz _ dz _d6 _ x x dx 2 — ic 2 f^i V\dy 2 — y 2 dy x 
V(42 s — Iz — J) 2 V(^ 3 ) # 2 ^i 2 yi 2 
which, in virtue of the foregoing values of A, E, is 
x x dx 2 - x 2 dx x y x dy 2 — y 2 dy x 
V(A) 
V(^) 
31 
I remark that an even more simple transformation from the general quartic 
radical to the Weierstrassian cubic radical was obtained by Hermite; this is alluded