Mr. Hellins on the Rectification 
452 
A = H. L. of y + +yy), 
^ yV(i + yy) — A 
' 2 * 
r __ y^V^+y y) — 3 B 
4 * 
p-— ^ 5 V(i+y.y) — 5 C 
E = 
See. 
y 7 V(' + yy) — 7 p 
8 9 
See . 
And, lastly, by multiplying these quantities by their proper 
coefficients, we obtain (Theorem I,) 
% 
A -J- 
ee 
2 
B — 
C + 
3 e 
D — 
3 - 5 ^ 
E, &c. where 
2.4 * 2.4.6 2. 4. 6. 8 
it is manifest that, unless the quantities B, C, D, Sec. decrease 
in the ratio of ee to 1, the series will at last cease to converge; 
or, in other words, if yy be greater than — — , the terms of the 
ee 
series, at a great distance from the first, will diverge. And, of 
the nine theorems now produced, this is the only one that I 
have found in any other book. 
3. But, by converting -/(1 -\-yy), the denominator of the 
fraction in the fluxionary equation in Art. I. into series, making 
1 the leading term, we have z =y s/ (1 + eeyy) x : x- 
yy 
+ 
3 r 
- -Ai^L 4 - Sec. and, by taking the fluents of 
2.4 2.4.6 ! 24.6.8 ’ J 0 
y v/(i 4- ee yy)> yyyV{ x 4- ee yy )> C 1 + eeyy), & c - anc * 
calling them A, B, C, Sec. we shall have 
-A- t/(x 4- eeyy) +~yxH.L.g + eeyy), 
A: 
B: 
C = 
D = 
E = 
&c. 
y ( i -f- eeyy) » — A 
\ee s 
y 3 (i+ eeyy)i — 3 B 
6ee 
y 5 ( 1 + eeyy)'* — 5 C 
8ee 
/(x-}- eeyy\ i — 7 D 
loee 
Sec. 
