1&8 Mr. Gompertz on Series which may he summed 
be then neglected unless in the case above mentioned of a, b, 
r, Sic. being converging, in which circumstance it will have 
no failing case : but had the coefficients a , b, c, d, &c. been 
alternately -{- and . — , the failing case would not happen when 
qz— o or a multiple of 360°, for then there being no reason 
for taking A of one sign rather than the other, it should 
therefore be taking equal to o ; but it will happen when sine or 
cos. of *sr -\-qr.z is alternately positive and negative, by conti- 
nually increasing r by 1 : for then the coefficients of the terms 
of the form, sine or cos. of Ttr-\-qr.z being alternately positive 
and negative ; and likewise the terms themselves alternately 
positive and negative, the whole values resulting from them 
will have the same determinate sign, and this will be when 
qz~iSo° or some multiple thereof. And if a , b, c, d, Sic. 
be positive and negative according to some other law, the 
failing cases may be found by the like reasoning ; which is 
likewise applicable to the other theorems. 
These remarks pave the way to the correction of fluents 
necessary in the application of the doctrine of fluxions to 
these series. 
1. In Example 2, Theorem I. if for nz we write and 
for, q we write 1, we shall have cos. of £4-2+ cos. of ^-J-2 z 
-f- cos. of Sic. = 
sine of k 2; 
2 sine of fs 
— sine of k. 
cos. of \z 
2 -sine of fs 
cos. of k. sine of 
2 sine of fs 
sine of k. cos. of \z 
2 sine of fs 
i£i£L. Multiply 
both sides by z, and find the consequent fluents, and we shall 
have sine of v-f e-f 
c 
sinc.ft+ ? « . .sineoft + j* 1&G _ fl „ ent of 
2 * n 
z r COS. of lz 
7 smeo{h A^m 
cos. of k 
jzQ, which because cos. of 
