MR. IVORY ON THE THEORY OF THE ELLIPTIC TRANSCENDENTS. 367 
Wherefore we have, 
M 2 + (3 2 x 2 N 2 = (1 + x 2 ) p (1 - A- 2 z 2 ) R 2 ; 
and by converting (1 — k 2 z 2 )'. R 2 into a function of x 2 , we get 
M 2 + j3 2 x 2 N 2 = (1 + x 2 ) (1 + A' 2 x 2 ) (1 + c 2 x 2 ) 2 .... (1 + cVoX 2 ) 2 , 
k' 2 — 1 — k 2 , c^ 2 = 1 — k 2 sin 2 X 2 
By treating this equation as before, we get 
1 — tan 4/^—1 1 — x */ — 1 1— Jc' x V — 1/1— c q x \/ — 1 \ 2 
1 + tan 4> — 1 1 + tc V — 1 1 + fxy / -l \1 + c s x / 
and from this we deduce, 
p being an even number, 
4 1 = V + $ + <P2 + 94 • • • * + $p- 2? (11) 
tan <p' — k ' tan <p, tan <p 2 „ = c 2 „ tan <p. 
8. In what goes before, our attention has been confined to two related 
functions, which, for the sake of abridging, we have denoted by the prefixes 
H and K ; but as we shall have occasion, in what follows, to compare several 
functions differing from one another in their moduli and amplitudes, it will be 
proper to adopt the usual and more general notation, by means of the cha- 
racteristic F prefixed to the modulus and amplitude. According to this nota- 
tion, the equation (1) will be thus written, 
F(A,4)=pF(A,?) ; F(i,?) = jF(A,4). 
The modulus k being given, we can compute the amplitudes, X l5 k 2 , &c., at 
least by approximation ; and the amplitude <p being supposed known, the fore- 
going formulas will determine the modulus A, the multiplier (3, and the ampli- 
tude ; so that the function F (k, <p) will be reduced to the similar function 
F ( h , ip), of which the modulus h is less than the given modulus k. And in 
like manner as the three quantities A, (3, were determined from the two 
k, <p, we can deduce, from the two h, \p, three new quantities, A , (3 /} ^ /5 which 
will satisfy the equations, 
F (K (h, V; = 
3 B 
MDCCCXXXI. 
