490 
[866 
866. 
NOTE ON KIEPERT’S ¿-EQUATIONS, IN THE TRANSFORMATION 
OF ELLIPTIC FUNCTIONS. 
[From the Mathematische Annalen, t. xxx. (1887), pp. 75—77.] 
It appears, by comparison with Klein’s paper “ Ueber die Transformation u. s. w.,” 
Math. Annalen, t. xiv. (1878), see p. 144, that Kieperts L made use of in the 
Memoir “Ueber Theilung und Transformation der elliptischen Functionen,” Math. 
Annalen, t. xxvi. (1886), pp. 369—454, is, in fact, the square of the multiplier, 
“für das durch UA normirte Integral,” viz. considering the general quartic function 
(a, l) 4 = (a, h, c, d, e)(x, l) 4 , and the transformed function (a 1} ...)(y, l) 4 , then 
we have 
D VEdx _ VK x dy 
V(a, ...)(x, I) 4 V{ch, ...){y, 1 ) 4 ’ 
where if 
I = ae — 4<bd + 3c 2 , 
J = ace — ad 2 — b 2 e + 2 bed — c 3 , 
and similarly J 1} J 1} are the invariants of the two functions, then A, A x are the 
discriminants 
A — I 3 — 27 J 2 , A, = A 3 - 27 A 2 , 
and the y 3 of Kiepert’s equations are 
72 = I -j- \/ A, 7 3 = J -r- \/ A, 
whence 
72 3 - 277a 2 = 1. 
In particular, if the forms are 
1 — x 2 .1 — k 2 x 2 , and 1 — y 2 .1 — \ 2 y 2 ,