U 8 
Mr. Heluns on the Rectification 
I SS UU 
■ uu 
eux,') and e x the fluent of . 
’J ^/(i—auu) 
And 
(=euV( l iE: 
it is obvious to every competent judge that the arithmetical 
work of computing the value of G, with any given values of 
b and u, will be as short and easy as the computation of the 
elliptic arch denoted by E. Yet, for the more ready compa- 
rison of the series, with each other, which arise in taking these 
fluents, I will here set them down in the Newtonian form, 
which undoubtedly is the most convenient for arithmetical 
calculations that has yet been discovered. 
10. E is 
3 - 5 * 8 “' 
ll\/ { I -~EE Ull) U _ a UU e 4 M 4 
\/{l — UU) VC 1 — uu ) * 2 24 
2.4.6 
u Ull 
&c, ; and denoting the fluents of —— - — :, — 
^ C ‘ ky A', B', C', & c - respectively, we have 
A 7 = the circ. arch of which the rad. is 1, and sine u. 
A ' —u\J ( 1 — uu) 
B' = 
C' = 
D' = 
3^'— ( 1 — uu) 
4 5 
5C' — M S \/ (l—liu) 
6 3 
£/ _ 7P ’-u^^i-uu) ^ 
8 
&c. &c. And then, multiplying these quan- 
tities by their proper factors, and placing them in due order, 
we have 
E = A 7 
11. G is 
il B' - — c 7 
2 2.4 
uyj ( 1 — uu) 
V( I— EE uu 
Ji- D' - -HE: E 7 , &c. 
2.4 6 2.4.6. 8 ’ 
EE UU , 3 e 
~*T 
u v/( 1 — mm) x : 1 + “ : 4 " 
* 4 - 
3 - 5 £ “ 
+ 
3 - 5 - 7 £ " 
, &c. Now, denoting the fluents of « 
2.4.6 “ 2.4.6 8 
— uu ) , u uu y 7 ( 1 — ) , u mV ( 1 ~~* uu ) 5 & c * by A, B, C 
&c. respectively, we have 
