A. Hall on the Secular Perturbations of the Planets. 371 
rise to the degree denoted by the number of planets considered, 
and in our solar system will be of the eighth degree. If we 
enote by 9,, 92,9; ----J, the roots of this equation the 
general integrals will be 
A=Ni sin (Git+81)-ENg sin (got+f,.)+ ----+Ns sin (gst-+6s), 
T=N, cos (g,t-+81)-+Ne cos (gat-+H2)-+-- ----+Ng cos (gots), 
W=N'sin (git-+61)-+ N’gsin (gat-+6)-+---.EN’esin (gets), 
U=N's cos (git-+-61)-+N'2 008 (gat-+00) + --.-+N'acos (gat +68), 
&e., &e. 
- 
1) 
The arbitrary quantities N,, N’,---- 8, 8, &c., are determined 
by the initial values of h and The solution for p and q is 
quite similar to that for A and 1. 
The conditions necessary for the stability of the system are, 
first, that the eight roots of the equation for g shall be real an 
unequal, in order that outside the circular functions there may 
be no terms containing the time as a factor or exponent and 
which would therefore increase indefinitely ; and secondly, it is 
necessary that the coefficients N may not be great in order 
that the excentricity may not increase so as to render pte ok 
oa 
the series which have been assumed in the solution to be rapidly 
certainty on this point an analytical solution is to esired, 
sidered. These coefficients depend on the assumed masses of 
the planets, and are generally determined by neglecting terms 
of the third order. The most complete investigation of this 
