408 PHILOSOPHICAL TRANSACTIONS. [ANNO 17 QS. 



xc , 75 .5 .25 



(3j? 1.05 1.3 1.55 



Excess of attraction of ndrg above Bbrg 23/4 .1614 .0813 



mdrp above nbrp 2374 .l6l4 .08 13 



— mesp above nasp 3705 .25 16 .1271 



Sum of these 8453 .5744 .2897 



Excess of attraction of Bbnfi above vdmS 5007 .3271 .1606 



Aanj3 above Eem$ 4677 -3079 -1525 



Whole attraction of the inside surface of the half box . . .1231 .0606 .0234 



It appears therefore, that the attraction of the box on x increases faster than in 

 proportion to the distance occ. 



The specific gravity of the wood used in this case is .6l, and its thickness is £ 

 of an inch ; therefore, if the attraction of the outside surface of the box was the 

 same as that of the inside, the whole attraction of the box on the ball, when 

 ex = .75, would be equal to 2 X .1231 X .61 X 4- cubic inches, or, .201 

 spheric inches of water, placed at the distance of 1 inch from the centre of the 

 ball. In reality, it can never be so great as this, as the attraction of the outside 

 surface is rather less than that of the inside ; and besides, the distance of x from c 

 can never be quite so great as .75 of an inch, as the greatest motion of the arm is 

 only 14- inch. 



XXII. An Improved Solution of a Problem in Physical Astronomy ; by which, 

 swiftly converging Series are obtained, which are useful in computing the Pertur- 

 bations of the Motions of the Earth, Mars, and Venus, by their mutual At- 

 traction. To which is added an Appendix, containing an easy Method of 

 obtaining the Sums of many slowly converging Series which arise in taking the 

 Fluents of binomial Surds, &c. By the Rev. John Hellins, F. R. S. p. 527. 



It was with much diffidence that I entered on a speculation which had engaged 

 the attention of such learned men as Simpson, Euler, and La Grange. Consider- 

 ing the great abilities of these men, and the length of time which Euler, in par- 

 ticular, appears to have employed on the subject, all that I at first expected to 

 effect was, to facilitate the summation of the slowly converging series by means 

 of which they had computed the perturbations of the motions of the planets in 

 their orbits, which arise from their actions on one another, by the force of gravity; 

 and that this might be done by a method which I had some time before discovered, 

 was evident, on inspecting their series. Here probably I should have stopped, 

 had not Dr. Maskelyne put into my hands a sheet of paper, written by the late 

 Mr. Simpson, which, though very ingenious, was by mistakes, which seem to 

 have entered in transcribing it, rendered unintelligible to some eminent mathema- 



