170 Mr. Gompertz on Scries which may be summed 
, % -f 2 r — n sine of p" + sq.z + r — n sine of p"-\- 3:/ . z .*. by 
restoration and division we have s the sum = \ji sine oi p — <zq 
.z~\-n-\-r sine of p — q . z zr — n sine of pz -f- r — n sine 
of p -f- q . sQ ~ 2 sine of qz\* : had we used Theorem I. we 
should have gotten a more simple valuation ; namely, s = 
- . i' ne , °J t~ ^ ,z which is reducible to the other by 
— 2 sine of 
multiplying the upper and under terms by 2 cos. of %qz( by 
help of lemma No. II. and III. Had the terms been alternate 
positive and negative we should have had 
a, b, c, d, e, See. =?i, — ?i-\-zr, — n-\-^r,Sec.' 
a,b, a!, b c r , &c . = n, — n-\-r,-\- 2 r, — 2r, -|- 2 r, &c. 
a, b, a" y b", c", 8 cc. = n i —n-\-r, 2 r — n , — r — n, o, &c. 
and therefore s — [n sine of p — zq.z — n -{- rsine of p — q.z- f* 
2r— - n sine of pz — r — n sine of p-\-q-z]~. 2 sine of qz[ z . 
If we had used Theorem II. we should have obtained s = 
~ '^ sine of ^ which is reducible to the other by 
2 cos. of \qz\ 
multiplying the upper and under terms by 2 sine of ±qz\ , by 
help of lemma No. I. and II. 
Theorem IV. 
If there be a series, a . sine of pz-\-b . sine of p-\-q . z — c . sine 
of p-\-vq . z-~d. sine of p-\-%q . % -f- & c - = s or a cos. of pz 
-j- b cos. of p-\-q. z — c cos. of p-\-zq .% — d . cos. of p~\~ 3 q-Z 
4. &c. = s the signs of the terms changing alternately two 
by two ; then in the first case 
