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VIII .—Researches in Physical Astronomy. By J. W. Lubbock, Esq. V. P 
and Treas. R.S. 
Read February 9, 1832. 
In general, two methods present themselves of solving any mechanical pro¬ 
blem : the one furnished by the variation of parameters or constants, which 
complete the integral obtained by the first approximation; the other furnished 
by the integration of the differential equations by means of indeterminate 
coefficients, or some equivalent method. Each of these methods may be 
applied to the theory of the perturbations of the heavenly bodies; and they lead 
to expressions which are, of course, substantially identical, but which do not 
appear in the same shape except after certain transformations. 
My object in the following pages is to effect these transformations, by which 
d Ft 
their identity is established, making use. of the developments of R and r 
given in the Philosophical Transactions for 1831, p. 295. The identity of the 
results obtained by either method serves to confirm the exactness of those 
expressions. 
Integrating the equation 
_ Ji I 
2 d < 2 r 
£. 
a 
+ 2/d« +r (^)=° 
omitting the terms which are independent of the quantities h, and which 
result from the part of R which is equal to r - r ' . ? 0S J* —and the factor —. 
_n!_J ■ - wa - b,.+ JL cos i(nt-n,i) 
n}{i{n-n,) - n} t («—«,) «, 1,1 2 a, da J 
2(i+l)« 3 / n 2 ^ | a 3 
(i (n - «,) t «)(*('*- n /) + 2 «) i («-«/) 4 " a ‘ 
{i (w — n ,) + 
+ 
r 3 ^3 ,i 
^ b 3,i+l } ecos Qint-ny) +nt-^ 
