DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 351 
Substituting this value of cost in (143.), and writing F, Q', R' for the coefficients of 
powers of sinX, the resulting equation will become 
i2=(d?VP'+Q' sin’^+R' sin‘X- . 
J Jk[a^ cos^A + ¥ sin^A ) 2 cos t 
(148.) 
As the first of these integrals is precisely similar in form to the integral in (123.), 
we may in the same manner reduce the expression into factors. Accordingly let 
F+Q' sin^?i+R' sin^X=(a+j3 sin^>i)(7— j8 sin^A) (149.) 
Writing a, |8, y instead of A, B, C, and following step by step the investigation in 
Art. XXXV., we shall have, as in (126.) and (128.), i//, m, and i, being the amplitude, 
parameter and modulus. 
As 
^ ,, a + /3_^ ^ .2 ^ fu + y\ 
tan -4/= tan X, m= — —q, ^ =- ( — —n )• 
(ty—(f]^^ (3=a^ — and y — a=a^—b ^ — 
(150.) 
(151.) 
we shall have the following relations between the constants a, (3, y, m, and A, B, C, w, i, 
in (150.) and (128.), 
/3=B, a=C— B, 7=A+B, a+7=A + C, 
y— ^=A, a+^ = C, 7 — a— /3+C— A-B= 0, ^ 
•2 ■■■ <^(‘‘ + 7 ) B(A + C) .2 ._. 
— (A + B)C~*’ « + 
Hence the moduli are the same in the two forms of integration, and the parameters 
m and n will be found to be connected by the equation m-\-n—mn—i ^ ; . . (153.) 
m and n are, therefore, conjugate parameters, as they fulfil the condition assumed in (1 .). 
The amplitudes <p and are equal. 
A + B 
In (126.) we assumed, tan^(p=— tan^0; and in (150.) tan^^/ 
« + /3 
tan% but 
tanA=| tan0, as in (146.), whence tan^4'=^ (A tan^ip. 
In (152.) we have found and A+B=7, 
An 
whence tan^^/^^— tan^^. But AC=Z»^P, and ay—o?]i^, 
as shown in (125.) and (151.), whence 
4/=(p (154.) 
We shall now proceed to find the value of the second integral in (148.). 
From (147.) we may derive tanV= 
{¥— sin^A cos^A 
cos^A + ¥ sin^A) 
(165.) 
Differentiating this expression, reducing, dividing by cost, and integrating, we 
finally obtain 
, r dA {¥■ cos‘*A — ¥ sin'^A) 
JcosT {¥ cos^A + ¥ sin^A)t ’ 
(166.) 
MDCCCLII. 
2 Z 
