of the conic Sections . 
4 6 5 
And, these values being written for A, B, C, &c. in the Theorem, 
it gives, in tins case, 
% 
ey 
aa 
ze 
■a 
2. t .\ e 3 
oc 
3 3 a 
2. 2. 4. 46c 5 
■a 
3-3 S S a 
2. 2. 4. 4. 6. 6. Be 7 
Oi 
, &C. 
And, since this Theorem also always gives the correct value of 
%> we shall, by comparing the expression now obtained with 
those which were found for z, in the like case, in Articles 12, 
1 3 , 15* 1 &> an d 19, see that we have now got another general 
expression of the value of d, viz. & x 
3.3 5,5a 8 
aa 
ze 
+ 
a - + 
+ 
3 3 “ 
+ 
2 . 2. 4^ 3 1 2,2.4.466 s 
, &c. in which series ee is = aa + 1, and there- 
.2,4 4 6.6. Be 7 
fore it must always converge. Yet it should not be hastily 
concluded, that this expression of the value of is always pre- 
ferable to that which was obtained in Articles 15 and 19; for, 
when a is a large number, the powers of • 
the series converges, will decrease very slowly. 
aa 
ee 
a a 
I cl CL 
-, by which 
21 . However, when it happens that a is a large number, the 
value of d may be obtained by means of two series, which, in 
that case, will converge pretty swiftly ; or indeed by means of 
three series, each of which will converge about twice as fast as 
either of the two series. But, for the sake of brevity, I shall at 
present describe the method of computing the value of d by two 
series only, and so conclude this section. 
The series proper to be used on this occasion, it is obvious, 
are those which are given in Theorems II and IV, Articles g 
and 5; and the value of v to be assumed, is with which va- 
V e 
lue each of the series will have nearly the same rate of con- 
vergency. As this will best appear by an example, I will give 
one, taking <2=7. Now, with this value of a, we have ee = aa 
+ 1 = 50, andy = • = 0-141421356 ; and,, by 
