CEETAIN APPLICATIONS TO THE THEOET OE DEFINITE INTEGEALS. 
7f;5 
Cj. denoting the coefficient of - in the development of the function in descending powers 
of X. Now developing the numerator and the denominator separately, we have 
2 {(fx^ — 1 ) y ) = 2 c^x*l w + 2 c v . . 
( 1 -j- + wa^y — (1--^)(1— c V) = {'uf— (f)x* + 2'ym‘* . . 
Dhiding the first of these by the second, we have a quotient 
2c^lw . / 2c% Ac^wvlw \ 1 I £, 
W‘- 
Integrating the coefficient of -, which is a complete difierential relatively to v and w, we 
have 
^f\/ ^\lS ^‘^=^^|^4-constant=c"^,a,’, 
X; 
( 6 .) 
The signs of the three integrals under the sign 2, are of course determined by the signs 
of the calculated values of ; h- 
1 + w + wx 
AY e might in the same way deduce the known formulae for the comparison of elliptic 
functions of the third order. Or we might at once investigate a formula for the com- 
parison of elliptic functions of every order. For the latter purpose we should have to 
evaluate the expression 
{a + bx^)dx 
J (1 + nx'^) V'" (1 — (1 —c^x^)’ 
under which the three canonical forms of elliptic functions are comprehended, in sub- 
jection to the condition (2.). By the general theorem of transformation this value is 
J®[( 
a + bx^ 
2[\-\-vx-\- WX") [xZv + x'^lw) 
( I + nx'^)[\ ■\-vx + wx'^) 
which may be reduced at once to the form 
! + bx^' 
I -^nx^ 
(1 -\- VX -\- ivx'^y — —x^){\—c^x'^)'’ 
2x^v + 2x^w 
T +vx + wx‘^)‘'‘- — (1 — (1 — cV) 
( 7 .) 
'fhe rest of the solution involves no difficulty. We must develope the entire function 
following 0 in ascending powers of x -\-^=^\/ — 1 and x — ^\/ — 1 successively, and 
take therein the coefficients of 
and 
From their sum we must 
X -= V' — 1 
V n 
subtract the coefficient of ^ in the development of the same function in descending 
powers of x, and integrate the result as a complete differential with respect to y and iv. 
The above results are entmely founded upon the assumed theorem of transformation, 
y/(l-x^)(l-c^x^) = 1 -yvx+UK^. 
But any other transformation which would connect x with two new variables, through 
