354 MR. IVORY ON THE THEORY OF THE ELLIPTIC TRANSCENDENTS. 
when the second becomes respectively equal to the several known amplitudes, 
bj ^15 
2. As we shall have occasion to refer to the formulas for the addition and 
subtraction of elliptic functions, it will be convenient to premise them. 
Let a and b represent any two amplitudes, and put 
K (a) + K (b) = K (s) 
K (a) - K (b) = K (<r) : 
then, according to the formulas of Euler *, 
• sin a cos b V 1 — 1c 2 sin 2 b + cos a sin b */ 1 — lc 2 sin 3 a 
S 1 — P sin 3 a sin 3 b 
sin <t = 
sin a cos b 1 — k 2 sin 3 b — cos a sin b \/ 1 — /c 2 sin 8 a 
l — /r sin 3 a sin 3 b 
From these we immediately deduce. 
sin s sin a 
sin 3 a — sin 2 b -j~ 
k 2 sin 2 a sin 2 b 
(A) 
It may be observed that if a — X m , b = \ n ; then s = X m a = X m _ „ : for 
it is obvious that 
K K (A n ) — (ni -f- n) K (Xj) — K 
K (X m ) — K (X n ) = (m — n ) K = K (X m _ n ) 
3. In order to avail ourselves of the analogy between the elliptic functions 
and the arcs of a circle, we must take that view of the matter first suggested 
by M. Abel. Let 
u 
dip 
*/ 1 — k~ sin 3 <p 
K(?>); 
then, as u is a variable quantity depending upon the amplitude <p, reciprocally 
this latter quantity will depend upon the first ; which dependance we shall 
express in this manner, 
<p = amplitude of u — A u , 
sin <p = sin A u. 
* TraiU; des Fonctions Elliptiques, tom. i. p. 22. 
t This equation is called by M. Abel “ la propridtd fondamentale.’ 
