a Problem in physical Astronomy. 533 
specting the utility of the invention explained in it, yet such 
is my respect for that learned body, that I am unwilling to 
send them any paper of mine, on so difficult and important 
a subject, till it has been examined by an able judge of the 
subject. 
I am. 
Rev. Sir, &c- 
JOHN HELLINS* 
An improved Solution of a Problem in physical Astronomy, &c. 
1. The perturbation of the motions of the planets in their 
orbits, by the action of one upon another, is a curious phseno- 
menon, 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 quan- 
tity by which a planet, so acted upon, deviates from an ellipsis 
in its course round the sun : a problem which hath called forth 
the skill of several of the most learned philosophers and astro- 
nomers of the last and present age. 
A preparatory step to the solution of this problem is, to find 
a convenient expression for the reciprocal of the cube, or ra- 
ther of the 7 i th power, of the distance of any two planets. Such 
an expression was first given by M. Euler, in series proceed- 
ing by the cosines of the multiples, in arithmetic progression,, 
of the angle of commutation ; but the calculations of the first 
