164 Mr. Gompertz on Series which may he summed 
or not, it will follow that the series, sine of pz — r sine of 
s i ne of p-\-<2.q.z &c. will be equal to sl L e -^~] =^ 
and the series, cos. of pz — r cos. of p-\-q.z-\-r. r -~ cos. of 
p-^-zq.z &c.= whether r be a whole positive 
M 2 COS. Ot 1 
number or not. 
And in order to prove this requisite, we shall first premise 
that if we have the sum of the series p m — r .p-\-q™-\-r . r -~ , 
— r • “Y~ • &c. whatever r may be, (m, p 
and q being given quantities) expressed by a series, A- x 
A-J-Br-j-CA &c. of finite terms in which the functional values 
of p and r are distinct, A, B, C, &c. being given quantities 
independent of r, we may likewise find the sum of the series 
p m+l — r.p-\-q] m + 1 -\-r. r -~-p-\-^q\ m+l &c. for this series is 
equal to pp m — I'P-fqp-f q\”-f r - ~j~ -p-f^q . p-^-^qf Scc.=p x 
p m —rf>-\Yj\ m -\-r . A±A .'p+zqf Scc.—rq Yp-\Yj> a — r-f fp+2 q\ a -f- 
r+i r -yrP+ 3 <l | w -— r+i .L±f . ./>+4g|» &c. but - r . 
/>+ 9 |"+r. AAA ^4-29']“ &c. is equal to ~ x A -J- Br -j- Cr See. 
and if in this we write r+i for r and for we shall 
have the sum of the series p-\-q \ m > — r- |~i l±i 
& c -— jqrr x A^^.r+i+C^r-j-il 2 &c, A, B^ C / &c. 
standing for the values that A, B, C, &c. become by writing 
p-\-q for p ; and this may evidently be reduced to an expression 
of the finite terms of the form ~ x A'-^BV+CV* &c.and con- 
