a Problem in physical Astronomy. 529 
n quadrantal arch of the circle by the series 1 -j- ~ ~1 — - — h 
3 2 4 * * 5 
& c ' Afterwards, I discovered the method of trans- 
forming that series which had lost all convergency by a geo- 
metrical progression, into another in which the literal powers 
decrease very swiftly; which is the improvement I now offer 
to you. 
In comparing the series here produced, for computing the 
values of A and B in the equation (a — b x cos. z)~ n = A -f- B . 
cos. z -j- C . cos. 2Z + D • cos. gz -|~ &c. with those which have 
been published for that purpose, by Messrs. Euler and De la 
Grange, it will appear, that those cases which were the most 
difficult to be computed by their methods, are the most easy 
by mine. For instance, if Venus's perturbation of the motion 
of the Earth were to be computed, (and vice versa,) the literal 
powers which have place in M, Euler's series, would be very 
nearly equal to the powers of T 9 ^ ; the literal powers which have 
place in M. De la Grange's series, would be nearly equal to 
the powers of and, in the series now produced, the literal 
powers would decrease somewhat swifter than the powers 
of sV 
M. De la Grange has indeed, by a very ingenious device, 
obtained a convergency in the numeral coefficients of the series 
that he uses, which, for the first five terms of it, is nearly equal 
to the powers of ~ ; but this convergency becomes less and 
less in every succeeding term, and the coefficients approach 
pretty last to a ratio of equality ; so that, to obtain the sum of 
the series to six places of decimals, he proposes to compute 
the first ten terms of it. The case in which those coefficients 
MDCCXC VIII. g Y 
