96 PHILOSOPHICAL SOCIETY OF WASHINGTON. 
These are the values at which Bessel arrives by the analytical 
method. The ares f, g, A are always less than 180°, and the only 
difficulty is in counting the angles F', G, H. In the purely ana- 
lytical process we merely substitute so as to satisfy the equations, 
and the result is right if we pay attention to the algebraic signs; 
but in the preceding quasi-geometrical method we must be careful 
to count the angles F', G, Hin the direction of increasing right 
ascensions from 0° to 360°. The formule for computing the six 
auxiliary quantities can be found from the spherical triangles 
TOX, TOY, TOZ. In these triangles the angles at O are 
TOX = 180°— (a— NV) 
TOY= 90 —(a—WN) 
TOZ = 90+ (—WN) 
Hence, we have 
cos f = — sin J cos (a — NV) 
sin fsin F= —sin (a— N) 
sin fcos F= cosJ cos (a— WN) 
cos g = cos é cos J + sin ésin Jsin (2 — NV) 
sin gsin G = —sin dcos(a—WN) (6) 
sin g cos G = cos 6 sin J — sin 6 cos J sin (2 — NV) 
cos h = sin 6 cos J — cos d sin J sin (a2— NV) 
sin Asin H = cos 6 cos (2 — NV) 
sin h cos H = sin 6 sin J + cos 6 cos J sin (ec — NV) 
The computation of these formule may be changed by introduc- 
ing other auxiliary quantities, as is commonly done, but nothing is 
gained by such a change if the computer is accustomed to the use 
of addition and subtraction logarithms. 
By means of the spherical triangles we can find a number of 
elegant relations among the quantities f, g,h, F, G, H. But we 
have first 
cos f? + cos g? + cos A? = 1, 
or these are the direction cosines of the line drawn from the planet 
to the pole of the orbit of the satellite. 
The triangle XTY gives 
cos XY = cos XT cos YT + sin XT sin YTveos XTY, 
and we have AY = 90°, XTY = F — G, 
