47 1 
of the conic Sections. 
it need not be repeated again. I shall therefore, in the remain- 
ing examples, show the convergency of these new series in 
most of the different cases that can occur. 
Example ii. 
24. Given a = 1, andy = 1, to find %•; 
This arch, it is obvious, may be computed by Theorem IVth, 
Vlth, Vllth, and some others ; but the Vllth is the proper one 
to be chosen on this occasion, as the series there given has the 
swifter convergency. 
Writing, then, 1 for a, and 1 for y, in Theorem VII, we 
have (by Article 1,) ee = aa+ 1=2, and, (by Art. 7,) uu 
== 1 -|- eeyy = 3 ; and then, (by Art. 9,) 
+ 
U = y / 3 = 17320,5081, 
2.4.7 
3-5 
2.4.6. XI 
3 - 5- 7 
2.4.6.8.15 
3 == 0'0320,7501, 
u~ 7 — 0-0011,4554, 
—11 
u 
u 
— I 5. 
3 - 5 - 7-9 u - } 9. 
2, 4. 6. 8 10.19 
3. 5. 7. 9.1 1 
2. 4.6. 8. 10. 12.23 
U 
- 2 3 
0-0000,6750, 
0-0000,0481, 
0*0000,0038, 
= 0-0000,0003; 
sum of the neg. terms — 0*0332,9327; 
sum of the series -j~ 1-6987,5754; 
correction of the fluent — 0-5990,7012 = — 
the difference of which is -f 1-0996,8742 = %. 
3 P 2 
d, (by Art. 17;) 
