180 Mr. Gompertz on Series which may be summed 
sine of ry -}- &c. will likewise be expressed generally in terms 
of x and y, which is case the first. 
Again, if we have the sum of the series a cos. of pz -f b cos. 
of qz Sec., expressed generally by z; by writing x — y through- 
out for z we shall have the sum of the series tf.cos. of p .x — y 
-f- b cos. of q. x — y -j- c cos. of r. x — y Sec. expressed gene- 
rally by x and y and by writing x +>' for % throughout, we 
shall have the sum of of the series a cos. of p . x -| -y -j- b cos. 
of q . x + y + c cos - °f r. x +y & c. expressed generally in 
terms of x and y, and consequently the half sum of the two 
which by lemma No. III. is equal to a cos. of px . cos. of py -f 
b . cos. of qx . cos. of qy -f- c cos. of rx . cos. of ry Sec. will be 
expressed generally in terms of x and y ; and the half diffe- 
rence of the two which by lemma No. I. is equal to, a sine of 
px. sine of py -{- b sine of qx . sine of qy -{- c sine of rx. sine of 
ry Sec. will likewise be expressed generally in terms of x and 
y. O. E. D. 
Corollary. From the sum of the series, a sine of pz + b sine 
of qz -j- &c. having obtained the sum of the series, a cos. of 
of px. sine of py -f- b cos. of qx. sine of qy Se c. if a' be put for 
a cos. of px, b' for b cos. of qx, c r for c cos. of rx, See. this 
series will be reduced to a' sine of py -{- b’ sine of qy -j- c' sine 
of ry See. which is of the first form of this theorem, and con- 
sequently from it may be deduced the sum of the series a' cos. 
of pzu . sine of pv -j- V cos. of qw . sine of qv -{- c cos. of rw . 
sine of no Sec. and therefore its equal the sum of the series 
a cos. of pw . cos. of px . sine of pv -f- b cos. of qw . cos. of qx . 
sine of qv -f- c cos. of rw . cos. of rx . sine of rv -J- Sec. in terms 
of v, w, and x, but if a' had been put for a sine of py, b * for 
