by a Method of Differences. 
159 
sine of p-\-qz~b sine of p-\-zqz — &c. and likewise of, cos. of 
pz — cos. of p-\-q • £-f- cos. of p-\-%q . % — &c. 
Here in both, a, b, c, d, & c.=i, 1, 1, 1, &c. 1 therefore, in 
a , a', b',c', & c.=i, o, o, o, &c. /the first se- 
ries we have, s' or s . 2 cos. of -§-<72; == sine of ^>2; and .-. 5 = 
sine ^ 1 and for the second series we have, 5', 
z eos. of %qz 2 cos. or ±qz J ’ 
or, 5.2 cos. of -qz = cos. of p—^q.z, and therefore, 5 = 
cos. of p — \q.x 
2 cos. of \qx 
Cor . i. If p~q the first series will be, sine of pz— sine of 
<zpz+ sine of 3pz &c.== = £ tangent of \pz, and 
the second, cos. of pz — cos. of %pz -f» cos. of $pz &c. = 
COS. of — \px I 
2 COS. of Jpx ’ * 2 * 
Scholium. Though we have given two theorems, the one 
for a series whose terms are all positive, and the other for n 
series whose terms are alternately positive and negative ; they 
are both true whatever the signs of the terms be, provided 
that proper signs be used in the operation ; that is, if any 
term should have a contrary sign, to the sign of that term 
contained in the enunciation of the theorem used, then a con- 
trary sign must likewise be prefixed to it in the operation ; 
thus, for instance, if for a series whose terms are all positive 
we should use Theorem II. or for a series whose terms are 
alternately positive and negative we should use Theorem I., for 
a, 6, c , d, &c. we must write a , — b, c , — d , &c. and therefore 
a, a.', b', Sec.— a , — a + b, c+b, — d + c, Sec. — (suppose) a, —a 4 , b t , — c,, Sec. 
Qy cl > b y &c.— Sec. „ • ■ • • ciy &c. 
dy a , b , Sec. — a, — a^-pa, b, / -\-a ll >—.c ll ^~b uy Sec.^, - » 
Sec. Sec. Sec. Sec. Sec. Sec. Sc c. &c. Sec. 
Sec. 
k c. 
