[ 2 S 3 
ire. ss A-\- — 4 - ~ 4 - "5 &c. where A—i, B=z 
z z z z i 
=P Ay C= ire. as before. Which equa- 
tion or theorem anfwers in cafe of a defeending 
feries. 
COROL. 3. 
Hence, if each of the quantities n , J>, ire. be 
taken equal to unity, and their number be denoted 
by^j then will 14- — xi+ ^ xi+2! xi+£ ire. 
z z z z 
^=A-{-S. + — -2 -b — j &e. Which equation, multi- 
z z z 
plied by .s*, gives z+a x z- f/3 x z+y x 3+J ire. 
—Azr + Bz v - v + Cz*-* + c Dz*r~i &c. 
Whence it appears, that z — a x z — /2 x 2 ; — y x .3 — 
ire. is = Az* — Bz ^ -I + Cz Dz * -3 
Where ^=i, B=PJ, C= 2 ^Z^f, ®== 
<PC-gB+R 4 &c 
(as before) ; P being in this 
cafe = fum of all the quantities a, ( 2 , y, ire. 
.£= the fum of all their fquares \ R = the fum of 
their cubes, &c. &c. 
CoROL. 4* 
Since «, ( 2 , y , are the roots of the equa- 
tion, z v — Bz *- 1 + Cx*- Z — Dz v ire. =0; it 
D follows. 
