ME. J. W. L. GLAISHEE ON THE THEOEY OF ELLIPTIC FUNCTIONS. 495 
Double this result and subtract (20) from it, and we have (18). In a similar way 
(23) follows from (21). 
The converse proposition is not true, viz. given the value of <px—<p(x — p ) — <f>(x+[t) + &c., 
we cannot deduce the value of <p#+<p(,r— jU/)+<P(#+i M ') + & c - 
§ 7. The results admit of being connected directly with Fourier’s theorem in the 
following manner : it is of course well known that every integral of the form 
or, let us write, 
gives rise to a series 
and that similarly from 
there follows 
J <p(x) cos-nx dx=A' n , 
x nwx , . 
1 <P(tf)cos— - dx=A n , 
Jo ^ 
<p#=- < A 0 + 2 A,cos — + 2 A 2 cos — — |-&c. > ; 
\ . ni xx , „ 
1 <p(x) sin — dx= B„ 
2 (_ . 7 r ^ . 2ttx , D ) 
^=- jB, sin— +B a sm — + &c. J- ; 
and it will now be shown that if <px is an even function of x, and if 
f <p(x)cos r ^-dx=A n , 
then 
<px -t- <p(x - ( a) + <p (x + (t ) + <p (x — 2(t) 4- <p(x+ 2p) + &c. 
and 
|A 0 +2A 2 cos^+2A 4 cos— +&c 
<p#— <p(#— |E a) — < p(#+^)+ <p(#— 2^) +<p(^ -f- 2f/>) — &c. jA, cos ^ + A 3 cos ~+&c. 
also, that if <px is an uneven function of x, and if 
then 
and 
. > . mrx 
<p(x) sin — dx-. 
■ B„, 
<$>x + <p(x - (b) + <p (x - 1 - yj) + &c. = £ | B 2 sin ~ + B 4 sin ~ + &c.| , 
<px— <p(x— p) — <p(^-j-i«')+&c.=-iBi sin— -j-B 3 sin^^+&c.l. 
fj, ^ [A. [A j 
(26) 
(27) 
It is sufficient to prove one of these formulae ; take (24). Since <px is an even function, 
<px-\-$(x— ^)+<p(#+^)+<&c. (which call -tyx) is a periodic function with period and 
