194 Mr. Gompertz on Series which may be summed 
+ r - t ~~ l • r =Ll &c., if we now multiply the first of 
p+ zq.q.r+z—p 
these by sine of ±qr—p.z, and the second by cos. of jqr—p . z, 
and take the difference we have 
sine of lqr—p. z X cos. of qr— pz — cos. of ~qr— p. z xsine of qr— bz , _ . 
qr _ p ‘ r L sine 
of %qr — p.z. cos. of pz -f- cos. of \qr — pz . sine of pz 2 -r- p — 
x [sine of ^qr — pz . cos. of r-f 1 . q — p. z — cos. of 
^qr — p.z. sine of r+ i . q —p.z^ — r — x [sine of \qr Sp. z 
. cos. of p -|— q . z -j- cos. of \qr — p . z x sine of p -f- q . z'J &c. 
qr-p 
which by trigonometry is reducible to — — ~ -f- /J ?rg 
sine of + i . g r sine of 7 . -lr + i . z & c __ ^ 
+ r 
r+ i_/> 
P + ? 
sine of 
T^~pT ; ov sine of te _ r sine of £ -iL±_Li£ = M sine of • « - 
p .qr-p 
p + q.q.r+ I -p 
qr- 2 p 
the same as Landen finds, page 83, Mem. 5. We may farther 
add, that when series are obtained from others having failing 
cases by substitution, as in Scholium in. to Theorem I. or as in 
Theorem V. and VI. regard should be had to those failing cases, 
according to the manner of substitution, in order to find the 
failing cases in the new series. We might proceed to many 
more examples, in finding the sums of new series from others 
multiplied by fluxions, or we might give examples of finding 
the sums of new series by throwing others into fluxions : but 
my chief object in these latter examples was to obviate any 
difficulty that might appear in choosing the cases for the cor- 
rection of the fluents. There are other inferences to be drawn, 
which I may perhaps consider at some future period. 
