of the co?iic Sections. 
And hence it follows, that this very slowly converging series is 
i*57°79%2 x : 
+ 
3 - 3 -; 
3 - 3 - 5 - 5*7 
&c. by 
2,2.4 1 2.2,44.6 2.2.44.6.6.8 ’ 
which expression its value is easily attainable, and will be found 
to be = 0-59907012, 
I observe, in transitu , that the ratio of this slowly converging 
3 3,5 3 5-7 &c. to a series 
series, 1 — 
2-3 
24.7 
24.6. 1 1 
2.4.6.8.15 
of good convergency, is easily attainable ; by which mean we 
may likewise compute its value to any degree of exactness. 
18. A general expression of the value of d being found in 
Art. 15, by which it may be computed, whatever be the ratio 
of the two axes of the hyperbola, I might now proceed to show 
the use of the theorems by a few examples ; but, as the same 
series is attainable another way, and the same value of d is 
attainable also by different series, it will be no less curious than 
useful to show in what manner. 
19. The nth term of the series of quantities 
— r 
2 C 1 + O'JV ) ’ 4(1+330*® 
_ <y 7 ^ 
^ C ' enter the values of B, C, D, &c. in 
Theorem VIII, Art. 10, is evidently — , which, by the 
Binomial theorem, is = - --- -- x : y~~ zn — ny ~ zn ~ 2 > -- 1 
zn 
y 
- 2 n — 4 
„„ n + 1 n -\- 2 — 2 n — 6 Q 
n.—r~. y , &C. 
2 2 J ■* 
2 n 1 2 y 
+ n. 
n -\- 1 
ly" 
,&c. 
which, when y becomes immensely great, is barely = And 
the value of A, in this case, is y — the quadrantal arch of a circle , 
of which the radius is 1. Let this quadrantal arch be denoted 
by a ; then, by substituting for A, B, C, &c. their proper values 
as they thus arise, we have 
