MR. IVORY ON THE THEORY OF THE ELLIPTIC TRANSCENDENTS. 361 
the theorems of M. Jacobi ; the other three, (3), (6), and (8), have been added 
here. All the formulas will be true in the circle, if we make ft = 0, and derive 
the arcs &c. from the subdivision of the quadrant, in like manner as 
they have been obtained from the subdivision of the definite integral ft 
The coefficient (3 is the expression of the first ratio of the nascent arcs and <p ; 
and it is equal to p in the circle. 
All the formulas are, however, imperfect in one respect : they all suppose 
that the amplitudes 7q, ft 2 , &c., derived from the subdivision of the definite 
integral are known. By means of these amplitudes, the general solu- 
tion of the problem has been deduced from a particular case : but the formulas 
cannot be considered as complete till all the coefficients have been expressed 
in functions of the modulus ft ; and, with respect to this point, the researches 
of analysts have not yet been entirely successful. 
5. Having now investigated the relation between the arcs and p, we have 
next to demonstrate that the equation (1) is true when these amplitudes are 
substituted in it, and a proper value is assigned to the indeterminate modulus 
ft ; but this requires some preparation, in order to avoid complicated operations. 
First, p being an odd number, we have. 
sin \p= 
/3 .*P 
R 5 
COS 4 1 = 
*/ 1 — z 2 - . Q 
R ; 
R, P, Q, representing the rational binomial products in the denominators and 
numerators of the equations (2) and (5) : we therefore obtain, 
R2 = (3 2 2 2 P 2 -j- (1 — * 2) Q 2 . 
This equation has been found on the supposition that z is less than 1 ; but, as 
it contains no radical quantities, it will be true for all values of z. We may 
therefore substitute for z ; and, in the resulting equation, the symbol z will 
still represent a quantity unrestricted in its value. Now, the substitution of 
^ for z being made, we shall obtain, 
R 2 — (3 2 ft 2 z 2 P 2 = (1 — It 2 z 2 ) R' 2 , 
3 a 2 
