1 53 
of the Hyperbola. 
It is now obvious that, to obtain the length of any greater 
arch of this hyperbola, the values of the algebraic quantity 
e s/yy -j- 1, and A, the first term of the series, are all that 
need be computed ; for the value of d, once found, serves for 
all. And if seven more arches of this hyperbola, correspond- 
ing to the ordinates 40, 50, 60 , &c. to 100, were to be com- 
puted, this theorem would afford a striking instance of the 
great utility of descending series. 
38. A second example might be, to find the lengths of ten 
arches of an equilateral hyperbola, of which the semi-axis is 
1, when the ordinates are 1, 2, 3, 4, 5, 6 , 7, 8, 9, and 10. 
These arches may be computed by the new theorem given 
in Art. 9, in which the value of G is always given in ascend- 
ing series ; but that series, when y is much greater than 1, 
will converge very little faster than the powers of \ : where- 
as, by using the theorem given in the Philos. Trans, for 1802, 
p. 458, viz. 
u — 
3-5 
3 - 5-7 
% — 
2.yC 
2.4.7 “ 7 24.6,1 12*“ 2.4.6.8.i5it ,5} 
&C. 
d } 
(where z denotes the arch of an equilateral hyperbola of which 
the semi-axis is 1, zms = V 1 -|» zyy, and d is = 0-59907012 ;) 
the geometrical progressions which will have place in the 
series, for the respective ordinates, will be the powers of these 
fractions, viz. -~ 
and — 1 — . 
40401 
1 
17 
361 ’ 1089’ 2601* 5329’ 9801’ 16641 J 26569 
39. In these examples the use and advantage of descending 
series appear : more examples of their utility might be given ; 
and it might easily be shown, that there are cases in which 
