£3$ Mr. Hellins's improved Solution oj 
> 
swiftly as the literal powers do in that case. And, when the 
ratio of the two radii is greater than that of 2 to 1, his series 
will converge more swiftly. 
With great pleasure therefore I see, that, by one or other 
of these methods, some of the longest and most difficult 
calculations which formerly arose in the theory of astrono- 
my, may now be exchanged for others which are short and 
easy. ' 
It is with satisfaction also, that I perceive the facility of com- 
puting by the series I now present to you, is not at all less- 
ened by the more general notation you have given to the 
denominator of the fraction from which it is derived, at the 
same time that a more accurate result is obtained than M. De 
la Grange proposed. For, in the computations of which I 
have been speaking, he neglected both the excentricities of the 
orbits of the planets, and their inclinations to the ecliptic, as 
inconsiderable : you, finding the effect of these omissions to be 
greater than he imagined, have taken them in. Your other 
ingenious labours on this subject will be best described by your- 
self, and cannot fail of being gratefully received by all learned 
astronomers. 
With respect to the method by which the sums of the very 
slowly converging numerical series, which occur in the sub- 
sequent pages, are obtained, I need not say to you, that it is 
of extensive utility, and may be successfully applied in many 
cases. 
I have only to request, that, if the paper here inclosed meets 
with your approbation, you will communicate it to the Royal 
Society* For, although I think I cannot be mistaken re- 
