152 Mr. Gompertz on Series which may be summed 
Example 2, Required the sum of the infinite series, cos. of nz 
-j-cos. of n-\-qz-\-cos. of n-\-iq . z &c. 
Here writing n in the room of p' we have 
a, a', b', c', & c.=i, 1, 1, 1, &c. | therefore s" or — s'. 2 sine of 
a, a'\ b" , c", &c.= i, o, o, o, &c. j \qz = sine of p"z = sine of 
n — -Irq . z s the sum == : p — . 
2 1 2 sine of qz 
Cor. 1. If n — ^q, we shall have cos. of nz - {- cos. of 3/12 -f 
r o sine of n — n.z 
COS. Of KUZ &C. = : -n = O. 
^ 2 sine of ±qz 
Cor. 11. If n — q, we shall have cos. of nz -{- cos. of 2 ?izp 
r 0 sine of \qz T 
COS. of <ZHZ &C.= — — ; fj — = \. 
o 2 sine of ±qz 2 
Example 3, Required the sum of the infinite series, sine of 
nz -\- 4 sine of npq.z-\-g sine of n-^-'iq ,z-\-i 6 sine of npgq.z 
&c. Her e p=n 
and a, b, c, d, &c.=i, 4, 9, 16, 25, &c.‘ 
a, a', b\ c', &c. = i, 3,5, 7, 9, &c. 
a,a",b",c",&c.= 1,2,2, 2, 2, &c. 
a, a!", b", c 1 ", &c.=i , 1,0, o, o, &c. 
therefore s'" or — s . 
2 sine of ±qz\ 3 = cos. 
of pl"z- }-cos. of p‘"-\-q 
.z — cos. of n — Iq.z 
-f cos of n — ±q . z, and therefore s the sum = — 
cos. of n—\q . z-fc os. of n — \q . z 
2 sine of ^qz'f 
Cor. 1. If n—\q , we have, sine of nz-\- 4 sine of §n%- f 9 sine 
COS. of ZHZ+I 
r 0 COS. of 2WZ+ I 
of KHZ & C. = — - ■ -i— V-- 
2 sine of wzp 
versed sine supplement of 2?zz 
2 sine of «z] } 
2 sine of «z) 3 
Cor. 11. If n=q, we shall have, sine of nz-\ -4, sine of snz 
2 . os. of inz 
. . „ . cos. of — -f cos. of \nz 
+ 9 sine of 3 nz &c.= 
because, cos. of — nz = coa. of + 
2 sine oi 
