184 Mr. Hellins's new Method of computing 
this letter. As this method is general, for all slowly conver- 
ging series of the form abovementioned, (which is generally 
allowed to be the most difficult,) I am induced to present it to 
you, requesting that, if it meets with your approbation, you 
will communicate it to the Royal Society. 
I am, 
Rev. Sir, &c. 
'JOHN HELLINS. 
P. S. I need not observe to you, that it is not requisite to 
the summation of the series mentioned in this letter, that b 
should be less than a, c less than b, d less than c, & c. but only 
that the first, second, third, &c. differences of these coefficients 
should be a series of decreasing quantities : for, you well know, 
there are series of that form, which arise in physical astronomy, 
of which the coefficients are actually a diverging series, and yet 
the sum of the whole is a finite quantity. And the same thing 
is evident, from the bare inspection of the theorem which I 
shall presently use. 
1. The computing of the value of the series ax -j- bx*-\- cx % 
+ dx*-\- &c. ad infinitum , in which all the terms are affirma- 
tive, and the differences of the coefficients a , b, c, & c. are but 
small, though decreasing, quantities, and x is but little less 
than 1, is (as has been before observed) a laborious operation, 
and has engaged the attention of some eminent mathemati- 
cians, both at home and abroad, whose ingenious devices on 
the occasion entitle them to esteem. Of the several methods of 
