410 PHILOSOPHICAL TRANSACTIONS. [ANNO 1798. 



If indeed the perturbation which arises from the action of Jupiter on the earth 

 was to be computed, la Grange's series would be the best that has hitherto been 

 published for the purpose, as the literal powers of it would, in that case, be nearly 

 equal to the powers of -,V, while the literal powers in the new series would differ 

 but little from those of 44. So that, for computing the perturbation of each of 

 these 3 planets, we now have series converging so very swiftly, that the first 4 

 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 2 of the 3 are most easily cal- 

 culated 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 on improving astronomi- 

 cal 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 motions 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 -j^, which is a swift rate of conver- 

 gency. And, for the perturbation of the Georgium sidus by Saturn, and vice 

 versa, the series will converge nearly by the powers of -f, 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 2 planets differ from each other in the ratio of 2 to 1, 

 M. la Grange's series may be used with advantage, since the convergency of the 

 first 5 terms of it will then be nearly by the powers of -^\ the numeral co-efficients 

 of those terms converging as swiftly as the literal powers do in that case. And 

 when the ratio of the 2 radii is greater than that of 2 to 1, his series will converge 

 more swiftly. 



An improved Solution of a Problem in Physical Astronomy, 6?c. 



1. The perturbation of the motions of the planets in their orbits, by their 

 actions on each other, is a curious phenomenon, which, while it affords to the 

 philosopher a clear proof of the general attraction of matter, produces a problem 

 of no small difficulty to the astronomer; viz. to compute the quantity by which a 

 planet, so acted on, deviates from an ellipsis in its course round the sun : a pro- 

 is the greater, and c, z the cosine of the angle of commutation. Then will M. la Grange's series con- 



TT 



verge by the powers of the quantity — j and, since rr + rr = a, and 2Rr = b, in our notation, and 



R R 



bb 4rV* 

 the converging quantity in M. Euler's series is (nn) = — , it will be se r : and cc, by the 



° ° n ' aa (rr -f rr) 1 



CL "™~ h R.R — — 2R7* -4" TT ( R ~" t*f 



powers of which the new series converges, is = , = = - r-.. See the 



^ * a + b rr + 2Rr + rr (r + r) 1 



Memoirs of the Royal Academy of Sciences and Belles-Lettres at Berlin, for 1781, p. 257 5 M. Euler'« 

 Institutiones Calculi Integralis, vol. 1, p. 186} and art. 4, in what follows. — Orig. 



