1 7^ Mr. Gompertz on Series which may he summed 
i 2 COS. of pz — COS. °{ p-\-q COS. of p-\-zq .z — 
*+3*| m cos. of^-j^^.s & c - 
Using Theorem II. to find the sum ad mjinitum, and expand- 
ing the coefficients, we have, 
a, b, c, d , &LC.—l*,t'-\-2tv-\- r v l ,t'-\-4 i tv-\-4p % ,t*-\-6tv-l r c)v',- 
t* -\-%tv i6v\ &c. 
a, a', b\ c', &c .=/*, 2tv-\-v*, ztv- f-3^ e , 2^+5^*, 
2^4-72/, &c. 
a, a", b" , c", &c.=/*, — t*-{-< 2 ,tv-\-v% 
2Z>“, 
2Z>% ^ 
2 z;% 
&C. 
a, a'", b"', d", &c. —t ', — 2I 4 -f* 
d <2.tV -\-V* , 
O, 
O, 
&C.- 
Therefore s the sum ad mjinitum = 
: [t a COS. Of p — 
-f? .* + 
2z* — 2 tv — z>*. cos. of p — \q . z-\-t — z>I* cos. of p-\- ~ 
2 cos. of -Iq^l 3 , but the sum of r first terms of the series is 
evidently equal to the sum ad infinitum + the sum of the series 
t-\-rv\ cos. of p-\-rq.z — t-^r-\-iar? cos. oip-\-r-\-i .q .z-\- &c. 
adinfinitum, which is found from the lastby writing t-\-rv for t, and 
p-\-rq for p, to be ££-j-rz>| a cos, ofp-\-r — 4 q-z-\-2,f-\-2.r — 1.2 tv 
4- 2 r 2 — 2r — 1 . z; 2 cos. of p -{- r — \q . % -f- ^ + r — iTzi] 2 . cos. of 
~b 2 cos. °f which added to, or subtracted 
from, the value above, according as r is odd or even, gives 
the sum of r first terms of the original series. 
Cor. If z= o, the cosine of any multiple of % will be equal 
to 1, therefore the sum of r first terms of f — £-j-z>j 2 -f-£4-2z>f 
