of the conic Sections . 
459 
vanishes together withy, and therefore needs no correction. And 
for this reason chiefly I have introduced it, as it affords us ano- 
ther mean of obtaining the value of the constant quantity d, by 
which the descending series are to be corrected. 
11. But the fluxionary expression y obtained in 
Art. 1, is evidently =y f [ee + = also toy s/[ee — ~~ j ; 
and this expression converted into series, by the Binomial 
theorem, becomes <y 
3-5 a *y 
2.4.6. 8e 7 (1+j'j’) 4 ’ 
aay a*y 3 a 6 y 
Tefi+yy) ~~ 24c 3 (i-f yy) z "" 2.4.6^ ( 1 +33O * ■ 
&c. Here again, denoting the fluents of, 
y 
1 +yy 
y 
y 
-- &c. by A, B, C, &c. we shall have 
(j+joO O+w ) 3 j > > > 
A = circ. arch, rad. i , and tang, y, 
y 
B = 
C = 
D = 
&c. 
2(1 +yy) 
y 
4 (i+yy) 2, 
y 
&C. 
+ ■ A, 
+ 
+ 
2 
iL 
4 
j£ 
6 
5 v 
And, by proceeding as before directed, we get (Theorem IX,) 
% 
ey 
aa 
2 e 
A 
2.4e 3 
■B — 
3 a 
2.4.6e 5 
3-5« 8 
2.4.6. 8e 7 
D, &c. 
And this series, it is obvious, will converge the swifter the 
greater y is, so that it will begin to converge swiftly when the 
preceding series begins to converge slowly. It is evident too, 
that this series vanishes together with y, and therefore wants no 
correction. Moreover, it has an advantage over the preceding 
series, In that the coefficients of it decrease by the powers of 
that is, by -~j. And it supplies us with a different exr 
pression of the value of d, as will appear in the next section, to 
which I now proceed. 
