a Problem in physical Astronomy . 531 
nearly equal to the powers of while the literal powers in 
the new series would differ but little from those of So 
that, for computing the perturbation of each of these three pla- 
nets, we now have series converging so very swiftly, that the 
first four terms are sufficient for the purpose. 
These indeed are the perturbations of motion, arising from 
the actions of the planets, which the inhabitants of this globe 
have most frequent occasion to compute. And, since two of the 
three are most easily calculated by the method explained in the 
following pages, I am not without hopes that I have rendered 
an acceptable piece of service to astronomers in general, and 
more especially to those who are most intent upon improving 
astronomical tables. 
But it may be proper to remark, that the use of the new series 
is not confined to the computations just mentioned, but may 
successfully be used in computing the perturbations of the mo- 
tions of other planets. For instance, in the computation of 
the perturbation of Saturn’s motion by Jupiter, (and vice versa,) 
the convergency of this series will be nearly by the powers of 
T j, which is a swift rate of convergency. And, for the pertur- 
bation of the Georgium sidus by Saturn, (and vice versa,) the 
series will converge nearly by the powers of -i, which is also 
swiftly. 
And it is further to be remarked, that in the last instance, 
and indeed whenever the radii of the orbits of the two planets 
differ from each other in the ratio of 2 to 1, M. De la Grange’s 
series may be used with advantage, since the convergency of 
the first five terms of it will then be nearly by the powers 
of yj '■> the numeral coefficients of those terms converging as 
3 y 2 
