414 
ON THE DETERMINATION OF THE 
[476 
37. I remark that as to the elliptic and parabolic orbits, I have preferred using 
Lambert’s equations, and I should have done the same for the hyperbolic orbits, but 
for the absence of a table (see post, No. 39). As it is, for the few hyperbolic orbits 
which it was necessary to calculate, I have used the foregoing formula (*) : a table of 
hyp. log tan (17r + ^u), u — 0° to u = 90°, at intervals of 30’ to 12 places of decimals, with 
fifth differences is given, Table IV. Legendre, Traité des Fonctions Elliptiques, t. II. 
pp. 256—259. 
38. The other set of formulse may be written : 
Ellipse, r, r' the radius vectors, y the chord. 
2a. cos ^ = 2a — r — r' — y, 2a cos x = 2a — r — r' + y. 
3 s 
Time = 2tt a ' sm X + sin %')• 
Parabola, r, r, y, ut supra; 
Time = K r + r' + y'f — (r 4- r’ — y)^}. 
Hyperbola. 
2a cosh x = 2a + r + r’ + y, 2a cosh x = 2a + r + r — y. 
3 
Time = 2^ a "(~X + X+ sinh %“ sinh X)> 
where cosh, sinh, denote the hyperbolic cosine and sine of x> viz.: 
cosh % = i (e x + e~*), sinh % = \ (e* — e~*). 
3.9. The logarithms (ordinary) of the functions cosh sinh and of tanh ^ are 
tabulated by Gudermann, Crelle, tt. vm. and ix. from ^ = 2'000 to ^ = 8'00 at intervals 
of '001 and subsequently of '01 to eight places of decimals. I do not know why the 
tabulation was not commenced from % = 0, but the omission from them of the values 
0 to 2 rendered the tables unavailing for the present purpose, and I therefore, for the 
hyperbolic orbits, resorted to the first set of formulse. 
40. As regards the elliptic formulse it remains to be explained how the values 
of %, x are be selected from those which satisfy the required conditions 
2a cos x — 2a — r — r' — y, 2a cos x — 2a — r — r' + y. 
It is remarked in Gauss’ Theoria Motrts, p. 120, that % is a positive angle between 
0 and 360 ; x a positive or negative angle between +180°, — 180°, viz. x i s positive 
1 I rather regret that I did not use the foregoing formulae in all cases.