866] 
NOTE ON KIEPERT’S L EQUATIONS. 
491 
and if as usual Jc — u 4 , X = v 4 , and M is the multiplier for the form 
dx _ Mdy 
Vl — x 2 .1 — Jc 2 x 2 Vl — y 2 .1 — \ 2 y 2 
then we have 
/= T u(l + 14m 8 + m 16 ), 
J = (! + O (! - 34m 8 + 2d 8 ), 
A = ^ m 8 (1 — u 8 ) 4 , Aj = (1 — 2; 8 ) 4 , 
and thence 
1 a/« 1+14u* + u M t (1 + m 8 ) (1 — 34m 8 + m 16 ) 
fs — & v 2 - ITS - > 73 — 4 q\9 5 
M 7 (l — 22 8 + 22 4 (1 — M 8 )" 
D _ V§ (1 - 2> 8 )^ 1 
22^(1 — U 8 )* M 
which last equation is the expression for L 2 in terms of the Jacobian symbols u, v, M. 
As an easy verification in a particular case, suppose n — 5. We have here 
v + m 5 
¿2 = ^ ( 1 “ v *t 1 m = v 0-~ u ^ 
(1 — 22 8 )^ M’ V—U 5 
u 6 — v 6 + 5u 2 v 2 (m 2 — v 2 ) + 4uv (1 — u 4 v 4 ) = 0, 
5m (1 + v?v) 
72 = i 
3/Q 1 + 14m 8 + 22 16 # 
u i(l 
and it should be possible, by eliminating u, v, M, to deduce hence the 7-equation 
X 12 + 107 6 — 12y 2 7 2 +5=0. (Kiepert, p. 428.) 
It does not seem in any wise easy to do this in the case of an arbitrary modulus; 
but writing the modular equation in the form 
(u 2 — v 2 ) (id + 6uH 2 + v 4 ) + 4<uv (1 — u 4 v 4 ) = 0, 
we satisfy this by 
uv — 1 = 0, u 4 + 6u 2 v 2 + v 4 = 0, 
or say by 
V = - , 22 8 + 6m 4 +1=0, 
u 
and the equation may be verified for this particular modulus. 
We have 
1 + 14m 8 + 2d 6 = 48m 8 , (1 - m 8 ) 2 = 32m 8 , 
and consequently 
72 = ^ v 7 2 • 
4822 s 
m^ (32m 8 )* ’ 
1_ 
2.3 
2> 
U 8 A 10 16 
m t .2'^ .u jr 
= l, (whence also 73 = 0). 
62—2