by a Method of Differences , 
iyi 
a sine of p'z -f b sine of f q . z— a ' sine of p' -f s q .z — U sine 
of p' - j- %q . % -f &c. = s' 
a sine of p"z-\-b sine of p"+q. z—a" sine of p"-\- Tq. z— b" sine 
of /"+ %q . z -f- &c. = s" 
&c. &c. &c. &c. 
and in the second case 
a cos. of p'z-\- b cos. of p'-\- q. z—a ' cos. of p'-f- 2 q. z—b' cos. 
of p'-\- o,q . % -f & c - — s> 
a. cos. of p' r z -f- 6 cos. of p"-\-q.z — a" cos. of p" -j- 2 q.z — b" cos. 
of p"-\- 3 q . z + &c. = s " 
&c. &c. &c. &c. 
where the terms a, 6, a', b ', c', &c. <2, b, a " , 6", c", &c. &c. are 
formed by taking the alternate differences, as in the last 
theorem ; p', p ", p'" , &c. likewise as in that theorem, s'= s . 
2 cos. of qz, s"= 2 cos. of qz \ 2 , s"'—2, cos. of <jr%| 3 &c. 
This is plain by multiplying the series continually by 2 cos. 
of qz by help of lemma No. II. for case 1, and lemma No. III. 
for case 2. 
Example. Required the sum of the series, sine of z -j- sine 
of 2 z — • sine of 3 z — sine of 4 z &c. 
Here / == q — 1 a,b,c, d , &c. == 1, 1, 1, 1, l, &c. 1 .\ s' = s . 
a, b , a', b', &c. = 1 , 1 , o, o, o, &c. / 2 cos. of z 
= sme of 0% 4 - sine of z r. s = r -. 
1 2 cos. or z 
Scholium 1. As the two first theorems depend on the 
differences of the coefficients of the immediate terms or 
omitting none, the two last on the differences of the coeffi- 
cients of the alternate terms or omitting one term ; so we 
may give theorems for the differences of the coefficients of 
Z 2 
