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A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. [578
4. I shall ultimately, instead of k, A introduce Jacobi’s u, v (u — \/lc, v=\/\); but
it is for the present convenient to retain k, and instead of A to introduce the
quantity ft connected with it by the equation A = kCl 2 ; or say the value of Cl is
= v 2 -r- u 2 . The modular equation in its standard form is an equation between u, v,
which, as will appear, gives rise to an equation of the same order between u 2 , v 2 \ and
writing herein v 2 = ftw 2 , the resulting equation contains only integer powers of u 4 , that
is, of k, and we have an ft&-form of the modular equation, or say an ft/t>modular
equation, of the same order in Cl as the standard form is in v; these ft&-forms for
n = 3, 5, 7, 11 will be given presently.
5. Suppose then, ft being a constant, that we have identically
this implies
(In fact, if
2i =
53 =
n
B <W-1>
— 23* •
21*.
then
2i = a 4- cx 2 + ... -f qx n ~ 3 4- sx n ~ l ,
53 =b + dx 2 4-... 4- rx n ~ 3 4- tx n ~ 1 ,
21* = s + qk 2 x 2 + ... + ck n ~ 3 x n ~ 3 4- ak n ~ 1 x n ~ 1 ,
53* = t + rk 2 x 2 + ... 4- dk n ~ 3 x n ~ 3 4- bk n ~ l x n ~ l ,
and the assumed equation gives
1 _ k? _ k n ~ 3 , _ k n -'
a ~ CIB(«- 1 ) C ~ fipM r> q -fltow d ’ S ~ CIB {n ~v bi
that is,
, Cl , Me 2 Clk n ~ 3
° - /¿Un-1) s > ( ' ~ ^j(n-i) 9» • • • » r - ¡¿hm-i) c >
Cl
_ Clk n ~ 2
* ~ a 5
and therefore 53 = T ,, — 2i*.)
[fit (W—1) 7
3323* \
From these equations = H 2 > that is, = ^, as it should be; so that ft signifying
as above, the required condition will be satisfied if only 21 = JiB^-C) ’ or substituting
for 21, 53 their values, if only
(P- 4- 2PQx 2 4- Q' 2 x 2 )* = ClB (n ~ 1( (P 2 4- 2PQ 4- Q 2 x 2 ),
where each side is a function of x? of the order ^ — 1), or the number of terms is
\ (n 4-1), the several coefficients being obviously homogeneous quadric functions of the
£(w4-1) coefficients of P, Q. We have thus |(w4 1) equations, each of the form
U = CIV, where U, V are given quadric functions of the coefficients of P, Q, say of
the ^(n4-l) coefficients a, /3, 7, 8, &c., and where ft is indeterminate.
