81 
“ A Proof of the Addition Theorem in Elliptic Integrals/’ 
by R F. Gwyther, M.A. 
The quadriquadratic equation, symmetrical and of e\en 
order 
A + B(# 2 + f) + 2Cxy + D ny* = 0 
may be used to perform the actual addition of two elliptic 
integrals of the normal form. 
ft y x r b 
Taking the integral / ■ „■ = / — j—r, 
& b J </{ 1 -^ 2 )(1 - Pa?) J A {x.k) 
o o 
we will use the above equation to obtain a substitution 
giving as result an integral of the same form, and having 
for its lower limit the value a. 
Solve the quadriquadratic for x and y respectively 
(B + D f)x + Cy=± yj - AB + (C 2 - B 2 - AD)^ 2 - B% 4 
(B + + Gx — J - AB + (C 2 - B 2 - AD)* 2 - BD* 4 . 
Now choose the constants in the equation of transformation 
so that the radicals become A (g.k) and A (x.lc) respectively. 
This requires 
- AB = 1 ; - BD ^ k\ &c. 
Also the equation connecting the variations of x and y is 
dx dy 
or, 
(B + D* 2 )y + Cx 
dx dy 
■ + ■ 
(B + Dy 2 )* + Gy 
= 0. 
= 0 , 
A {x.k) — A {y.k) 
In considering the limits of y corresponding to the given 
values of x, the lower is to be a , and we will call the upper 
c. This limit is to be expressed in terms of a and b. 
From the solved forms of the quadratics we get 
x = o ) x = b) 
y~ a ) V = g) 
Ca = + A {a.k) 
Ba=l 
(B + D6 2 )c + c& = A {b.k) 
with BD = - h 2 . 
^putting 
simultaneously^ 
