ON THE ORBITS OF THE ASTEROIDS, 127 
the equations (1) reduce to 
Ai d 
7) 
S p= bh E, sin (gt + 8) + E, sin (mt Hp) +.... 
The integrals of these equations may be expressed in the form 
h= p sin tHe T sin (git +f) -+....-+ A sin (bt +B) 
(8) 
I = 2 eos (8148) pe eos (git + fi) + >. + Acos(bt4 B) 
i and B being arbitrary constants, depending on the values of h and lat a given epoch. 
The integral values of p and q may be obtained by a similar process. By putting 
1,=(0,11) M +(02)M --(03) M +....=3M l 
I, — (0,1) M, + (02) M", + (0,3) AE .... (9) 
&c. &c. 
we have from (2) and (5) 
dt 
a= bp Isa y E (ht 3-7) ken (ty) — A, 
The integrals of these equations are 
p= M sing peg sin (ht +7) + pigs (at +) +- + KE sin (bt 4- 0) | 
(11 
q = M eos y + yq cos (ht +7) keine t +72) db ses + Koos (—bt + 0) 
2 = bq + I, cos y + Å cos (kt + y) + Locos (lat + y) +.. d 
(10) 
K and C being arbitrary constants, depending on the values of p and q at a given epoch. 
To give a clear idea of a geometrical construction of these equations, suppose a 
sphere described around the sun as a centre, and let the radius of this sphere be taken 
for unity. Let the point in which the pole of the ecliptic (or other plane of reference) 
cuts the surface of the sphere be taken as an origin of co-ordinates. From this origin 
draw on the sphere a radius vector equal to M, and making an angle equal to y — 90° 
with the axis of w, from which we suppose the longitude to be reckoned. From the 
I, : 
end of the first radius vector, draw another equal to FEE and making an angle equal 
to yı, — 90° with the axis of x, and so continue through all the terms of the second 
member of (11) The end of the last radius vector, K, will be on the point in 
which the pole of the orbit of the asteroid intersects the sphere at the origin of 
