496 ME. J. W. L. GLAISHEE ON THE THEOEY OF ELLIPTIC FUNCTIONS. 
the right-hand side of (24) must be of the form 
A 0 +A 2 cos — + A 4 cos — +&c. 
Now, <p being even, 
„ . ( . . 2rmx 7 
2A 2m =l <p(x)cos — dx 
2mm x 
But 
={ • • - + J- +f + I +-•} cos T dx - 
i <p(#)cos dx=( <p(|— p) cos d%, on taking x=%—(a, 
J-m ^ Jo P 
f <p(#)cos dx=\ <p(£+/^) cos : ^^-dj~, on taking x=%+(a; 
Jn P Jo P 
2A 2m =f {<p£ + <p(£-^)+<?5(£+^)+- • • }cos^~d^ 
Jo r 
C * , / x 2imx 7 . , p. 
= ) $(#) cos — — (M7= A 2m . g, 
Jo ‘ 
unless m=0, in which case 
2A 0 =Ao . [a, 
so that (24) is proved. Formula (25) may be either obtained independently by a similar 
and 
thus 
treatment of the integral 
2A m = f <p(x) cos ^ + l) ™ dx, 
or it may be deduced from (24) by writing therein 2[a for [a (remarking that by this 
substitution A 2m becomes A m ) and subtracting (24) from the double of the equation so 
formed. Similar processes apply to (26) and (27). 
The method by which the formulae (24) to (27) have been just obtained is the same 
as that by which Sir W. Thomson (Quarterly Journal of Mathematics, t. i. p. 316) 
deduced the theorem 
cos 7TX -\-e~^ 2 cos ---• +&C.1 . (28) 
■v/tt 
It was after reading Sir W. Thomson’s paper three or four years ago, that I made a 
list of all the suitable integrals of the form 
<p(#) cos nx dx 
that were given in Professor De Haan’s ‘Nouvelles Tables d’lntegrales definies : 
