370 MR. IVORY ON THE THEORY OF THE ELLIPTIC TRANSCENDENTS. 
express succinctly the conditions necessary, in order that the transformations 
in the equations (1) and (12) take place. A little attention will show that the 
formulas (14) and (13) are entirely similar, the quantities (3', h', k' occupying 
the same places in the first, that (3, k, h do in the other. From this we learn 
that the equations (1) and (12) will still be true if we change (3, k, h for 
(S', h 1 , k 1 , respectively. Thus we have, 
F(tf,4) = /3'F(A',*>), (15) 
the letters 4 and <p, it need hardly be noticed, although used on a former occa- 
sion, here express simply the variable amplitudes of the related functions. If 
therefore we divide H' = F ( y) intop equal parts, and put p 1} p 2 , ^ 3 , &c., 
for the respective amplitudes of — IT, 
— II', — H', &c. ; we shall have by the 
p p J 
formulas (4) and (9), 
sin sin //. 4 sin p 2p _ 2 
^ — sin ! x 1 sin sin p 2p _ ’ 
p / 
k ' = h! . ksin sin jO -3 . . . . sin (M 2p _ 
(16) 
The multipliers [3 and f 3 ' being similar functions, the first of the amplitudes 
?q, A 2 , X 3 , &c., and the other of the amplitudes (Jj v p 2 , (Jj 3 , &c., the equation 
(3(3' — p, expresses a curious property of those functions. 
And, in like manner, if we change (3, k, h, respectively for (3 ', h', k' in the 
equation (12) ; or, which is the same thing, if we derive an equation from (15) 
in the same manner that (12) was obtained from (1), we shall get 
F (k, t) = (3' F (h, ff ). 
l + 
l + 
• ni „• w tan 3 a, tan~a 4 
sin r = (3’ sin <7 X 
Sin"' (7 cin- /t 
1 + 
sin- o- 
1 tan 9 1 u.p — l 
07) 
sin 2 cr 
tan 3 tan 3 ^ tan 3 [x, p — 2 J 
Although, in the investigations of this §, we have supposed that p is an odd 
number, yet it is obvious that they will succeed equally when p is an even 
number, the formula (8) being used instead of (7). 
The analysis by which the equation (12), of which those that follow are con- 
sequences, has been deduced from the equation (1), is precisely that by which 
