MR. IVORY ON THE THEORY OF THE ELLIPTIC TRANSCENDENTS. 365 
have the same dimensions and the same absolute term, they are identical. 
The equation (1) is therefore demonstrated when p is an even number. 
7- The transformation expressed by the equation, 
/*’#• d-\> ^ d tp 
J 0 V 1 — k 2 sin® \|/ J Q a/ 1 — k' 2 sin® <p’ 
has now been demonstrated for any number whether odd or even, the constant 
jS being determined by the formula (4), and the modulus h by the special for- 
mulas in § 5, or, generally without distinguishing whether p is odd or even, by 
this formula, 
h — k p . (sin sin X 3 sin X 5 . . . sin X 2p _ 1 ) 2 , — - (9) 
the sines multiplied together being those of all the odd amplitudes less than 
1 80°. The relation between the variable amplitudes 4 1 and <p is expressed by 
the several equations in § 4. 
In order to render the solution of the problem more complete, it may be 
proper to add a useful method of computing the amplitude 
In § 5 we have obtained this equation, 
R2 = £2 Z 2 P 2 + (! _ *2) Q2. 
And, if we represent by N and M the products of the binomials in the nume- 
rator and denominator of the equation (7), we shall have 
, 3 tan <p N 
tan 4 = — yf — . 
3C 
Let x — tan <p, then z 2 — sin 2 <p = n — - — ; and, observing that 
X “j - CC 
1 — 
tan 2 K 
sin 2 x n 
it will readily appear that 
3 x N 
(3zP = ~? -, 
, 2 \ ’ 
(1 + x~) 
Jl - z* . Q = 
M 
p 
( 1 + tf 3 ) 2 (1 + as 9 ) 5 
These values being substituted in the foregoing equation, we get 
M 2 + (3 2 ,z 2 N 2 = (1 + x 2 ) p . R 2 : 
