of the Hyperbola. 
*37 
denoted by the Roman letters a, b, c,* &c. respectively, we 
f * r 3 L 3- 5-3 3-5- 7-5-3 <> T „ 
| rl X • 2 „ , „„„ ’T" n . « j 7„+ A « A . CCC. 
shall have <7 = < 
2.4. zaa 1 2.4.6.4.2a 4 2.4.6.8.6.4.2a 6 ’ 
b 
L+^l+Mxn-^y-y.k 
The law of continuation ad infinitum is very evident in the 
first of this pair of series ; and although it is not so in the se- 
cond, still it is obvious that b is less than a, c than b, &c. ad 
infinitum. And since the numeral coefficients of each of these 
series may be expressed in decimals, and their logarithms 
written out ready for use, an arithmetical calculation by this 
pair of series, when a is considerably greater than 1, will be 
much easier than by either of the single series given in Art. 
25- 
But this pair of series may be transformed into another 
pair converging by the same powers of a , yet of a simpler 
form, and therefore more convenient for arithmetical calcu- 
tion. The operations to be performed on this occasion are 
as follows. 
30. The H. L. of ( s/ 1 -f- aa — a), which enters into the 
value of A, is = — H. L. [cl -f- v/ 1 -f -aa); and this logarithm, 
expressed in correct descending series, is — - H. L.2 a, — 
" 3 ZZjte ?*' &c * therefore, being = ~ H. L. ( V 1 + 
1 
2.2 aa 
2.4.4a"' 
aa 
a) , is = — ~ H. L. 2 a, — 
-f\ -f /) - — 3 4- 
a ' > / 2.2 a 3 1 
1 
+ 
3 
3-5 
2.2a 3 1 2,4.4a 5 
3-5 
7 , See. = 
2.4.4a 5 
2.4.6.6a 7 ’ 
2.4.6.61,7 ’ &c - x bein S P ut for 
- The values of these letters are | 4. 1 H. L. 2, -f H. L. 2, and - 1 4 ^ 7 
respectively. See Philos. Trans, for 1798, p. 538. 
+ See Philos. Trans, for 1798, p. 557 and 558; and for 1800, p. 87 and 88. 
MDCCCXI, T 
