of the Hyperbola. 
119 
A = area of f the mid. zone of a circ. of which the rad. is 1, 
and sine u. 
B = 
A — k(i — uu 
3B — u 3 ( 1 — uu)i 
3 
5C — 7/ 5 ( I — UU )* 
-p 7 D — n 7 (i — uu ) I 
10 5 
&c. &c. And, multiplying these quantities by their 
proper factors, and placing them in due order, we have 
G— A — C4- Hi! d 4 - E, &c. 
1 •?- a ? . * * 24.6 8 24.6,8 ’ 
2 4 
is. Now, by comparing together the fluents denoted by A' 
and A, B' and B, C and C, &c. it is obvious that the arithme- 
tical calculation of the one will not, in any respect, be more 
difficult than that of the other ; and that A is always less than 
A', B than B', C than C', &c. And it is evident that each of 
the series denoted by E and G converges by the same geo- 
metrical progression, viz. s’, e 4 , s‘, See. So that the arithmetical 
value of any number of terms of the latter series will always 
be nearer to the value of the whole, than the arithmetical 
value of the same number of terms of the former series will 
be to its whole. And as to transformations of the expression 
u\/ (- — ~ into others, in order to obtain the fluent in series 
of swifter convergency, when the case requires it ; it is ob- 
vious that similar operations may be performed on the expres- 
sion Us/[ 
V \ 1— SE UU I 
In the application of E and G to the rectification of the 
hyperbola, the one is multiplied by e, the other by e 3 (see the 
