$ 4.8 
Mr. Hellins’s improved Solution of 
I. The Summation of the slowly converging Series. 
s. But, before I begin the investigation, it will be proper to 
premise a few particulars, an attention to which will shorten 
and facilitate the operations now to be performed. 
I _ v/(l _ yy) u . I — A/ (I 
yy) 
x 
is 
That + "( 7 _ bein g r + vr—,T) - . + 
; from which it follows, that H. L. of 
i -f ,/(i — yy) 
i — A /(i — yy) 
j + ^/(i — yy) 
IS 
2 H. L. 
i + ^/(i — y yY 
2dly. That the fluxion ofH.L.~ 
is 
y 
y 
y 
yy 
+ — yy) y /( i — yy) 
For it is == the fluxion of — H. L. J l + v/(i— y y)) 
; and, if both numerator and de- 
~ r “ v{—yy) 1 + V(i —yy) 
nominator of this expression be multiplied by l — v/(i — yy), 
it will become 
yy 
y/(i—yy) 
i — >/(i — yy) 
yy 
, which is 
y 
yV^—yy) 
S dly. That the H. L. is therefore = 
-f± 
J y 
yy , 3/ i 3-5/ , 3-5-7/ M 
2.4.4 ■ z.4.6.6 * 24.6.8.8 : 
4,thly. That, O being put = 4(1 — yy), the. fluxion of 
■ff will be = + jEr.)- For it will be |r - * Jf $' 
— yy - »i Q!. — y ny (i — yy) — n ‘y 
/CL 
y 
y-FCL /-‘CL 
y n 4 - 1 cl / + 1 cl 
+ 
y nJ t l 
(n —1 )y 
‘CL 
— n , n — 1 ( 
Q^\y 4. i "l yn _ 1 |* 
4 + j- + h 
\y*. ‘ yy 
5thly. That, when any quantities, as 
are circumscribed by a parallelogram, it denotes that a substi- 
