764 PEOFESSOE BOOLE OX THE COMPAEISOX OF TEAN’SC-EXDEyTS, WITH 
factor does not exist in the term ( 1 — — Again, there cannot be a term of 
A 
the form — in the development of the function in descending powers of a\ Hence the 
result of the operation 0 is 0, and we have, finally, 
^ C dx . ^ . 
= 
The above constitutes in reality what is usually termed the fundamental theorem for 
the comparison of elliptic functions of the first order. The equation ( 3 .), arranged 
according to the powers of x, gives 
[iv^ — c‘)x^ + 2vtvx^ + (-y^ + 2 w + c* + 1 + 2 1 ’ = 0 . 
If we then represent the simultaneous values of x by x^, x^, X3, we have 
— 2vw 
Wi ”P ^2 “ 1 “ ^'3 ^2 ^2 
v' + 2 w + + 1 
— ^2 g 2 
X 1X2X3 ^2 . 
2v 
whence, ehminating v and w, we find 
( 1 — ^)( 1 — a;2)( 1 — ^’3) = (2 — — ^*2 — ^3 + C^XlX^2Xiy (0 . ) 
Now this is the form to which the known relation connecting the amplitudes 
the modulus c in the equation 
±F.( 9 )±F.( 4 .)±F.(^)= 0 , 
viz. the relation, cos c^cos (p cos — sin p sin -\/l — sin^ <r, is reduced, when we therein 
make sin<p=^i, sin-;|/=r2, sin <7=^3, and rationahze the resulting equation. The signs 
of the integrals will of course be determined by the signs of the fimction 
To obtain a formula for the comparison of elliptic functions of the second order, we 
must deduce a finite expression for 
subject to the relation ( 2 .). We have then 
sfv/ ""l" dx= sf dx 
J V 1—^2 Jl+VX + WX -‘ 
r_l-£^n 
J Ll + (1 
2{\ + vx + WX -) {xdv + x-Bic) 
+ vx ivx^) — ( 1 — ( 1 — c^xr) 
Here, as before, the effect of that part of the operation 0 which depends upon 
l-\-vx-\-wx^ is 0, so that we have simply 
( 1 — c^x^) X 2 (a’Su + x'^hv) 
(I + vx + wx'^ — (1 —a’®) (1 —c^x^y 
