of the Hyperbola. 
135 
series, each of them converging by the powers of But the 
advantage of computing by such a pair of series, instead of 
the single one above described, is less than might at first be 
imagined ; for, in order to have a result true to the same 
number of figures, about the same number of terms must be 
computed, whether of the single series or of the pair. Since, 
therefore, the advantage obtained by such a transformation 
lies not in the literal powers, it can have place only in the co- 
efficients ; and there it may be very considerable. 
I now proceed to the investigation of a pair of series for 
computing the value of the constant quantity d, each of which 
converges by the powers of 
29. If the transverse and conjugate semi-axes of an hyper- 
bola are denoted by a and 1, respectively, the ordinate byy, 
and the corresponding arch by % ; and if the eccentricity, 
V aa -j- 1, be denoted by e, and 1 -f- ee yy be put = uu ; then, 
as I have shewn in the Philos. Trans, for 1802, p. 457, will 
g — + aa + \ A + ~ B + c -f D, &c, 
' [_-d; 
where A = - H. L. 
a u 3 
— \/ (mm -J- chi') 
A 
2 aa uu 
2 aa? 
— v/ (mm-H aa ) 
3B 
4 aa m 4 
4 aa 3 
— yj (mm + mm) 
5C 
6 aa m 6 
6 aa 1 
&C. & C. 
and (i,'the constant quantity to be subtracted from the series, 
is (see p. 462 of the Philos. Trans, for 1802, and Art. 25;) 
