Mr. Hellins's improved Solution of 
Our assumed fluent, therefore, is 
o ! -~ 35 ... - 95 - - 75 . 105 4- U ( 9 A 5 f 105 ) 
X.\8,izjy 8 iz.i6y 6 4.8.8J 4 8.8.8 yyj ' * 32 yy • 8.8.8 j 
+ 8^7+7^ * ~T^7y * > which may be cor- 
rected in the manner shewn in the two preceding Articles, or 
more, expeditiously, as follows. 
It is pretty evident, from the correction of the fluent in the 
preceding Article, that the constant quantities which lie con- 
cealed in this fluent, will appear in those terms only, (when 
the radical quantity ^/(i — y y), and the logarithm u, is ex- 
pressed in series,) in which the index of y is o. Thus, the con- 
stant quantities will appear as below. 
The 5 th term of 
-3 5^0— 
yy) 
is 
35 
5/ 
*75 
8.12 y 8 
lO 
8.12 
A 
8.1631 s 
" 4.12.16.16 ’ 
The 4 th term of 
— 95v/(i- 
■yy) 
l O 
95 
X 
y 5 
_ 95 
j 2.16 y 6 
lo 
12.16 
16/ 
12.16.16 9 
The 3 d term of 
I 
>-< 
1 
■yy) 
is 
75 
75 . 
4.8. 8j> 4 
lo 
4.8.8 
X 
8j 4 
8.16.16 9 
The 2 d term of ' 
1 
** 
% 
0 
k*“> 
1 
-yy) 
1C 
105 
yy 
__ io 5 „ 
8. 8, Syy 
lo 
8.8.8 
X 
2 yy 
4.16.16 9 
and the terms in 'which the index of y is o, in the logarithmic 
part, viz. yy + /, &c. 
2.l6 
y~* + 
2.16 
y 
— 2 
j_i£L 
1 8 . 8 . 8 s 
are these two. 
27 > 4 __ 2 7 
4.16. i6y* 4.16.16* 
and 
5 yy 
8.16J3; 
8.16 
The sum of these six fractions is The equation of fluents 
therefore is 
Q 
+ 
1 “35 
95 
75 
105 \ 
4, u ( 9 . 5 , I0 S \ 
00 
fv 
CO 
12.16 y 6 
i6.i6_y 4 
8.8.83'j j 
+ 1 3 2 j 4 i 3 Z yy s 8.8.8-J 
35 
j 5 
37 
, i C2 . 3 . — - the series 
8,12 j s 
1 16 y 6 
4 ° 9 6 
