[ 397 3 
» + t 
&c. J p x X g X y-i x &c. XI where n = 
7/ -f I /> + I + 2 T 
/>+ I 
p -j~ y. For this feries is the Tame with # — q x 
2 />+ I 
* &c. fetdown in Art. ift. as the value of the 
P + 2 
ratio of A Cy to HO; as will eafily be feen by put- 
ting in the former inftead of r its value i-x, and 
expanding the terms and ordering them according to 
the powers of x. Or, more readily, by comparing the 
fluxions of the two feries, and in the former inltead 
of f fubftituting - x *. 
p q p -f- r I 
* The fluxion of the firft feries is at r- x + q x i -P X r-f 
P + 'q - 1 
/> + i 
l q~ I P~b^q—z P+^-q—2. 
qx r x -f q x q— I X x r r + q X q- 1 X x r* . 
" — X 
p 4- 1 
H” 1 p+-: 
P'h 1 p+z 
p -f- 3 . 
+ g X g-i x q-q 3 X .v 7-^ “3 Sec. or, fubftituting - x for r , 
P+ 1 P+ 7 - />-H 
p q p+i p + i 
x r x — q x 1 x + qx 1 x — q X q — I X 
P + 1 /’ ft * 1 /’ ft * 1 
P+ 2 q — 2 P+ 2 i 7-2 
* ^ * + q x q~ 1 x .v * &c. which, as all the 
/> + 2 /> + ! 1 /> + 2 
terms after t he firft deftroy one another, is equal to x* r? x — 
X? X I — *1 q x = x? x X I — qx+qXq — I x z &C. — X? x — 
P+i p-bz 2 
qx x + q x q- 1 .v x See. — the fluxion of the latter feries 
p-bi _T ~ p + 2 
or of x — q x x See. The two feries therefore are 
p-b 1 p + 2 
the fame. 
Vol. LIII. 
3. In 
Fff 
