i86 Mr. Gompertz on Series which may be summed 
nz— sine of n-\-q.z-\- See. will be 
cos. of ?i-\-q.z-\- &c.= 
cos. of n — \q.z 
as in Example i, 
2 cos. of \qz 
Theorem II. except when qz= some odd multiple of 180°, 
something else being in that case to be taken into considera- 
tion ; and thus are we to reason, in the failing cases of other 
expressions : but by the common rules for finding the value 
of an expression when the denominator and numerator vanish, 
we may find the value even in the failing cases ; thus by 
dividing the fluxion of the numerator by the fluxion of the 
some multiple of 360°, we shall get simply, o for the sum of 
therfirst terms of the series sine of nz-\- sine of n-\-q.z-\-SLc., 
and r for the sum of the r first terms of the series cos. of 
nz- f- cos. of n-{-q.z-\- Sec. when z— o or some multiple of 360°, 
that is, o for the sum of the r first terms of the series, o-j-o-j-o 
&c. and r for the sum of the r first terms of the series, 1+1 + 1 
&c. which is self-evident. And thus may we proceed in other 
expressions when the sum of r terms can be obtained by a 
general value. 
That these things should happen as above described, is 
likewise evident, from the investigations of the theorems ; for 
in Theorem I. for instance, we have s v = + 2 sine of iqz^ -1 
or + s . 2 sine of %qz\”, v being a positive whole number ; there- 
fore if the sine of \qz be = o, which will happen when qz~o 
or some multiple of 360°, it is plain that we should have 
denominator in the expressions 
— cos, of n + r — fy .z-f- cos. of n — fq.z 
z sine of \ qz 
and 
and then making qz= o, or 
2 sine of \qz. 
