by a Method of Differences. 
Here/>'= n, therefore, 
a, a 6 \ c\ &c.= 
155 
a y a!\ b\ c\ &c.= 4 ,=f>, f 
5-6 
5.6.7 
$.6.7.8 
5.67.8.9 
34 ’ 
34 -S* 
3.45.6’ 3.45.6.7 
±5 
2.5.6 
2,5.67 
2.5.67.8 
34 ’ 
34 - 5 * 
34.5.6 ’ 3.4 5.6.7 
-2.5 
2.5 
2.5 6 
2.5.67 
34 ’ 
34 - 5 * 
34.5.6’ 3.4.5.67 
2 ’ 
1, 
O, 
O, 
&c. 
&c. 
&c. 
&c. 
therefore j iv or 5'. 2 sine of sine of p iv z — f sine of 
p iy -\-q .2+ sine of y> iv .-|- 2^ . %, and therefore s' the sum sought 
^sine of p"z — sine of p ' v sine of p iv -\-2q.z 
z sine ot 
T^) J 
=|j| sine of n — \q . z 
■ — j- sine of n — ^.2+sine of n-\-j : q.z '2 ; 2 sine of \qz ]*. 
Note. The series might have been written thus, cos. oi 
nz 
w.g 
-f- — cos. of n-\-q ,z-\- ~ cos. of n-j-2qz & c. 
^ T „ . . 4'sine of — znz— 4- sine of o-f sine of 2 hz 
tor. If 2n=q } s becomes — 
6.7 
§- sine of 2nz 
2 sine 
of nd 
2.sme of nz\ s 
sine of nz .. cos. of nz j cos. of nz 
2 sine ot nz\ 
6 sine of nz\ 
5.6.7 
, for the 
sum of the series, f cos. of nz- j- cos. of %nz -j- cos. of 
gnz &c. or its equal, cos. of nz- j- ■ - cos. of 3 nz-\- ~ cos. 
of gnz &c. N =4.5 cos. of nz-{-g. 6 . cos. of 3 nz &c. 
sine of nz\ ^ 
Scholium ir. It is not always necessary for the differences 
of the coefficients to become equal to o to obtain the sum of 
the series, as will appear by 
Example 6 , Required the sum of the infinite series sine of 
pz- j-g sine ot p-\-q.z-\-g* sine of p-\-2q.z-\-g* sine of p-f^q.z 
& c. 
