of the conic Sections , 453 
And then, by multiplying these quantities by their respective 
coefficients, we obtain (Theorem II,,) 
2=A-}B + -^-C H-D 4 .- 5 S-E, &c. 
2 ' 2.4 2.4.6 1 24.6.8 * 
which series will converge till y becomes greater than 1 ; and 
consequently is a better series than that above found, which 
ceases to converge wheny becomes greater than But, when 
/ ' e 
y is much greater than 1, each of these series will diverge very 
swiftly; and, notwithstanding they are of that form which 
admits of a transformation to others which will converge, still, 
even by that means, their values will not be obtained without 
great labour. But here we shall have the pleasure of finding 
series which will quickly answer the purpose. For, 
4. By converting the denominator, y‘ (yy + 1), into series, 
making yy the leading term, we get z = y y (eeyy + 1 ) 
1 — _I_ + _J - _T5 , —3.5-7 
y 2 y 3 ‘ z.\y 5 2 .±.nv 7 » 
X 
*, &c. 
24.63/7 * 2.4. 6. 83 s 
And here, again, by denoting the fluents of + ^ 
yV(eeyy+ 1 ) y V(eeyy +i) 
T 
y > -, &c. by A, B, C, &c. respectively, we 
shall have A = \/ (eeyy + 1) -f- H. L. dS ee w + *) - ,\ 9 
ey 
eeA 
B 
{eeyy + i)r 
2 yy 
4 
— (eeyy -f x)l 
43 4 
D— - 
— ( eeyy + -)% 
6y 6 
E — - 
— (eeyy + i)! 
&c. 
83 s 
&c. 
and then, (Theorem III,) 
z== 
B+- 1 -C- 
1 24 
2 
eeB 
4 
3 eeC 
6 
$eeD 
8 ~“ 
3-5 
24.6 
D + 
3 - 5-7 
24.6.8 
E, &c. 
MDCCCII. 
