256 
Proceedings of Royal Society of Edinburgh. [sess. 
From those we easily get the tangent of the true anomaly ASP, 
and thence the distance SP. Here the great part of the labour is 
in finding the logarithm of SH, the angle HSP from its log 
tangent, the log secant from the angle, and SP from its logarithm ; 
that is to say, in using the tables of the logarithms of numbers, and 
of circular functions to a considerable number of decimal places. 
This labour, repeated for each of the twenty thousand cases to be 
tabulated, rises to a formidable total. 
But if, on the perpendicular diameter AOA', we describe another 
ellipse having SS' for its minor axis, and consequently s and s' for 
its foci, and if from Q we draw the ordinate Q ph, we have, accord- 
ing to the properties of the ellipse, SP = 0 A =?ph, and conversely 
SP = OA ± PH. Thus the computation of the ordinates in the one 
of the two orbits gives us, with only the labour of writing the 
numbers in their places, the vectors of the other orbit, and we are 
now enabled to compute the true anomaly from its log sine. When 
following this course, it will be convenient to begin with the orbit 
e = 50°, and to take the others in couples, e = 49°, e = 51°, and so on. 
Our working formula then stand thus : — 
( 1 tcos p. cos e ) 
distances < # > whence, log distances , 
[ 1 =p sin^? . sin e J 
log sin anomaly = log ord. - log dist., whence anomaly . 
If it were proposed to make these computations with all the 
precision obtainable from our fifteen-place tables, it might be 
economical, even for this single piece of work, to interpolate the 
logarithmic sines for each hundred-thousandth part of the quadrant. 
log ordinates 
log sin jp + log sin e 
log cos + log cos e 
whence ordinates 
