55° Mr. Hellins’s improved Solution of 
cover the constant quantities which lie concealed in some of 
the terms on the first side, is to convert that side into series, by 
the binomial theorem ; which will then be as follows : 
— ■ v/(i— yy) 
2 y* 
\yy 
y ~ 1 + iy- + tV + -hr + ji ?f, 
— iy~ a + I + -h /+ -h y\ 
4-7r + 
2 y 
2 yy 
+ iy~* 4 iy~‘ 
-|H.L. 2 + T V/+ T f ¥ /,6? C . 
4-fH.L. — ; — ~ : 
1 4 i + yy) 
The sum is = * * + T V — I H - L- 2, + 1 5 < r /+ ¥ V<r/> &c. 
which evidently differs from the series on the second side by 
the constant quantity — |H.L.2. We therefore have, by 
subtracting this constant quantity from the first side, 
— - ( i ~yy ) 
2jy 4 
3 a/ (I —yy) 
4 yy 
I 
zyy 
7 
16 
+ “ + 
Which, when y becomes = 1, becomes 
# * -J- H. L. 2 
II 
r : 
3 -5 yy , 3 - 5 - 7 ^ . 3 - 5 - 7 - 9 y 6 & 
4 . 6 .a' l " 4 . 6 . 8 . 4 ' 1 'i, 6 -. 8 .io. 4 [ ’ - 
3-5 
2 
-I- X I_ l _9 
I 2 16 T k 
+ 
J 
3-5-7 
+ 
3 - 5 - 7-9 
4.6.2 1 4.6. 8.4 1 4-6.8.10.6 
which is the series denoted by x' in Art. 12. of the preceding 
paper. 
4, If the equation of fluents in the preceding Article be di- 
vided by y , and if y be then taken from both sides 
, we shall have 
4.6.2 16 
of it, and u be written for H.L 
"/{ l — y y ) 
2y s 
1 
s^i'—yy) , 3*0 
4 j 
+ + 
sy I 
16J 
>= 
1 4* -v/ ( 1 yy ) 
3 - 57 / , 3 - 5 7 - 9 / 
+ 
'4.6. 8. 4 1 4.6.8.10.6 1 4. 6. 8. 10. 12. 8 3 
z y3 I 2^,3 
And, if this equation be put into fluxions, and Q be written 
for y/ for the sake of brevity, there will be 
