130 Mr. Hellins on the Rectification 
v^2o),the fraction — — -- 1 will be less than — 
and less than 0-381966 ; and consequently that the hyperbolic 
arch AD may be more easily obtained from the series denoted 
by G, in Art. 11, than by either of those denoted by <p in Art. 
18. And hence we may derive another expresssion of the 
value of d , in the following manner. 
24. Since uu is universally = ^^—7, (see equation £3^ in 
Art. 5, where e denotes the eccentricity ;) by putting xx — 1 
and writing 1 -j- nn instead of ee , we shall have 
- ““ - 1 Let the 
UU 
, uu 1 
— 7-7 7— — s, and ee uu = — = — - — 
» + v( l ~h nn ) ee 1 +«K-fwV(i +km) 
value of G, corresponding to these values of u and e, be de- 
noted by r; then, by the equation (£) in Art. 9, the hyper- 
bolic arch AD = H, is = <?V — • er. But, since V was put = ux , 
(see Art. 5 and 6,) it will in this case be = 1. Writing there- 
fore e — sF for AD, and 1 for m, in the equation at the end 
of Art. 17, we have d—n + 1/(1 + nn) — 2£ + 2 er = 2i=r 
+ n — v/ ( i -}- nn). 
It now appears, that d, the difference between the length 
of an infinite arch of an hyperbola and its tangent, or asymp- 
tote, may be computed by means of one series converging 
swifter than the powers of even in the most disadvantage- 
ous case ; so that a dozen terms of it will be sufficient for all 
common uses : but, that a series of such convergency was 
attainable in this case, appears not to have been observed by 
either of the writers before mentioned. 
25. If the transverse and conjugate semi-axes of an hyper- 
bola are denoted by a and 1, respectively, the ordinate byy, 
and the arch by z; and if v /( aa -|- 1), the eccentricity, be 
