74 Proceedings of the Poyal Society of Edinburgh. [Sess. 
Substituting these values for the integrals, and rearranging, the integral 
becomes 
I r{ - t{\ + COsh2 rj) + 1{Z;2 + cosh^ rj)} 1 
L + x{ - cosh2 y{) + k^ - k^ cosh2 ^'2 cosh2 rj + k^ k'^)]\ 
+ J r {( - ^ + I + W){ - COsh 2 yj) + F (1 + A '2 COsh 2 ^)} 
+ sn^ x[{ - t + |A’2)(A:‘^ + A;2 cosh2 yj + k^ k'^ cosli^ yj -k'^ k'^) 
+ k\ -k‘^-k‘^ cosh2 ry) } _ 
The coefficient of k^l cosh^ t] is 
+ J + - \k'^) sn^ X . 
But since t is a root of the quadratic 
3^2- 2^(1 +F) + A:2 = 0, 
we have 
t-\- \k'^ _ 
t’ 
and therefore the terms which involve I cosh- rj may be put in the form 
(A;2 sn^x-t) x a quantity which does not involve x . 
Consider next the term^s which involve I but do not involve rj. They 
are (omitting the factor I) 
-t + JA^2 4- 8n^ x{tk‘^ - + \k^^ k'‘^) . 
But again from the quadratic 
32^2 - 2^(1 + ^.2,^0 
we see that 
^A;2-1A:4 + 1A:2A^ 2 
-t + \Jc^ t ’ 
and therefore the terms which involve I but do not involve rj can be put in 
the form 
(A;2 sn^ X -t) x a quantity which does not involve x . 
A similar method shows that the terms which involve J can likewise be 
put in the form 
(^2 sn^x-t) X a quantity which does not involve x ; 
and therefore, finally, the entire integral can be put in this form, which is 
the required result. 
The above analysis shows that to verify the theorem by direct integra- 
tion, even in the simple case of ?i = 2, is a somewhat difficult task, requiring 
the use of the recurrence-formulae between the integrals of elliptic functions, 
and also a considerable amount of algebraical work connected with the 
quantity t. 
