OF ELLIPTIC FUNCTIONS. 
427 
Observe that the general equation 
w(l— w 8 ) E,(w, + m) 
m 8 ) It(w, v) 
no longer contains the functions F'w, F 'u, which enter into Jacobi’s expression of M 2 . 
Theorem in connexion with the multiplication of Elliptic Functions. 
Article Nos. 37 to 40. 
37. The theory of multiplication gives an important theorem in regard to trans- 
formation. Starting with the wthic transformation, 
1 — y 1— x /a — fix + yx *— . . A 2 1— # /P — 2 
1 +y 1 +x \u + ^x+yx^ + . . .) ’ 1 + x \P + Qx) 5 
we may form a like transformation, 
\-z 1-y / u'-p'y+ 7 y-... y l-w p'-Qy y 
1 +z l+y\a! + P'x+vy + .. .) 5 l+y\B' + Q!y)’ 
such that the combination of the two gives a multiplication, viz. for the relation 
between y , z , deriving w from v as v from n, we have w=u ; and instead of M we have 
M', = H- -i-p ; that is, we have 
dx M dy 
VT — a? 2 . 1 — a 8 a? 2 vT— y 2 . 1 — v 8 y z 
d y M 'dz 
Vl — y 2 . 1 — v 8 y 2 Vl — z*. I — u 8 z z 
and thence 
dx ± ^ <fe 
Vl-x 2 .! -u 8 x*~ Vl-z 2 . 1-it 8 ? ’ 
or writing #=sn 6, we have z= + sn^; + is here ( — )“, viz. it is = — for w=3 or 7 
(mod. 8), and = + for w = l or 5 (mod. 8). 
Now in part effecting the substitution, we have 
l—z 1— x /P — QaA 2 /P' — Q 'y\ 2 
I+^ _ r+F\P + Q^J * \FTQ'2// ’ 
where y denotes its value in terms of x. 
And from the theory of elliptic functions, replacing sn ?i0, sn 0 by their values + 2 , x, 
we have an equation 
1-z 1-x /A-Bx+Cx^-^y 
1+z 1 +x \A + Bx + Cx 2 + . . .J ’ 
where A — Bx-{-Cx 2 — . . . , A-j-B#+C t £ 2 +. . . are given functions each of the order 
i(n 2 — 1) ; viz. the coefficients are given functions of 7c, or, what is the same thing, of u\ 
Comparing the two results, we see that in the wthic transformation the sought-for 
function, cc,-\-(3x-\-yx 2 + . . . of the order \{n— 1), is a factor of a given function 
A+Btf+Gr 2 -}-. . . of the order \{n 2 — 1). 
