m 
To adapt the formulae to elliptic integrals, and ordinary H and 0 functions, the 
radical must be brought to the form Vx. I — x. 1 — k 2 x. Writing for this purpose 
a, b, c, d = — k 2 P, 0, 1, y,, (/ = oo ), 
substituting also ~y for u, and ikl.A, iB 1 as usual) for A, B respectively, 
we find \fa — x .b — x.c — x .d — x = I s/ x . \ — x .1 — k 2 x; and then 
dx 
2du = 
x. 1 — x . 1 — k 2 x 
A = Cl, B = Cl \/x, G=VL\/\.-x, D = l-k 2 x. 
fi is in this case = A, a ^-function: and in the equation for Ail, writing A in 
place of il, the equation becomes 
A d 2 A - (dAf = $ A“- {- 2x- + x(- k 2 P) + K ] , 
viz. replacing by a new constant, = X suppose and finally putting I — oo, this is 
A d'A - id A ) 2 = A-(X - fr-x) (duf. 
The differential equation is satisfied by x = sn‘ 2 w, giving 1 — x = cv?u, l —k 2 x — dn 2 w; 
and the equation for A then is 
d~ log A = (X — № sn 2 w) (du)' 2 , 
or say 
. -r hMfi - lc 2 du du sn 2 M 
A=Le J o J o , 
viz. by properly assuming the constants L, X, we shall have A= Jacobi’s function : 
B C kD 
and then snw = -j, cnu = -^, dnu= , which will give the ordinary expressions of 
sn, cn, dn in terms of If, ©.