372 MR. IVORY ON THE THEORY OF THE ELLIPTIC TRANSCENDENTS. 
and, by combining* the two equations, observing* that (3/3 ' = p, we get 
F (k, r) = p F (k, <p). 
If p be an odd number, the amplitudes are obtained by the formulas (2) and 
(17), viz. 
sin = (3 sin <p X 
1 - 
sin"' <p 
sin 3 A 0 
1 - 
sin 3 <p 
sin 3 X p — i 
1 — 1c 2 sin 3 sin 3 A 2 1 — /c 3 sin 3 <$>sin 3 A;, _ i 
sin 2 yp 
sin r = |8' sin ^ X tan „ 
1 , , sin* \f/ 
1 + 
tan 3 
1 + 
1 + 
sin 2 \{/ 
tan 3 fj. p - i 
sin 2 yp 
tan 3 j up - 2 
When a multiple is required, we pass directly, by means of the two equa- 
tions, from the given amplitude (p to r which is sought. In the case of an 
aliquot part, the amplitude r being given, the solution of the second equation, 
of which p is the dimensions, will determine sin ^ ; and the amplitude (p which 
is sought, will then be found by solving the first equation, which is also of p 
dimensions. From the nature of the second equation, it has only one real 
root, and p — 1 impossible roots, for every real value of sin r ; and therefore 
it follows from the first equation, that the amplitude <p of the function-^F ( k , r) 
admits in all of p 2 values, of which only p values are real quantities, and the 
rest impossible. 
If p be an even number, the expression of sin 4* will contain radical quan- 
tities, but instead of it we may take the value of tan in the formula (8) ; and 
the two equations for the amplitudes will be. 
tan 4* = 
/3 tan <J> 
1 - 
tan 3 <p 
tan 3 
1 - 
tan 9 <p 
1 — 
tan 2 <p 
tan*A, 
Sin r = 
/3' sin yj/ 
1 — 
1 + 
tan 3 <p 
tan 3 Ag 
sin 2 rp 
tan 3 1 «, 2 
1 — 
1 + 
tan* <p 
tan 3 A p - i 
sin 2 ^ 
tan 3 fa, - 2 . 
1 + 
sin 3 \J/ 
1 + 
sin 3 \J/ 
1 + 
sin 3 
tam/Xj * 1 tan 2 w. 3 ' 1 tan 2 p p - i 
from which the same general properties may be deduced, as when p is an odd 
number. 
