150 Mr. Hellins on the Rectification 
of an hyperbola are 7 and 1, respectively; it is required to 
find the lengths of three arches of which the three ordinates 
are 10, 20, and 30. 
Setting aside the circuitous method of rectifying the hyper- 
bola by means of two ellipses, if one was to think of comput- 
ing these arches by means of a similar hyperbola, and by the 
theorem (£) given in Art. 9, (which is a very useful Formula 
within the limits specified in Art. 13,) he would quickly per- 
ceive, that the ascending series, by which the value of G is 
obtained, would converge very slowly ; and therefore would 
make choice of some other method. Theorem IV. in my 
former tract (see Philos. Trans, for 1802, p. 434,) is a very 
convenient form to be used on this occasion, and is as follows: 
Retaining the notation used in the beginning of Art. 2 g, 
[eVyy -f 1 
z = 
-A 
2e 
js. 4? 3 
B + 
:.4.6e* 
C - 
3-5 
2, 4.6.8c 7 
D, &c. 
d; 
where A 
B 
C 
|-f L 0— 1 
y : 
-Vlxy+O a_ 
2 yy 2 ’ 
-Vpy+O 
4 y* 
n — i/W -0 
4 ’ 
5_C 
6 7 
&C. &C. 
The first part of the work may be to find the value of d } 
which may easily be done by the pair of series given in Art. 
32 ; but, since a computation of it was made by two series, in 
p. 46 6, 467, and 468 of the volume here referred to, the 
ascending series being = 0*6360768, the descending series 
