534 ikfr. Hellins’s improved Solution of 
two coefficients in it were very laborious, requiring the sum- 
mation of series of the common form, which converged very 
slowly. Afterwards, other series were discovered by other 
authors, whereby the same coefficients might be computed with 
less labour ; the best of which, that I have seen, appear to be 
those that were pointed out to me by Dr. Maskelyne, in- 
vented by M. De la Grange, and published in the Memoirs 
of the Royal Academy of Sciences at Berlin , for the year 1781. 
Yet, the calculation of the two first coefficients, A and B, for 
the perturbations of Mars, Venus, and the Earth, by his me- 
thod, is not shorter, if it be so short as by my method, to the 
investigation of which I now proceed. 
PROBLEM. 
2. To determine the values of A, B, C, D, &c. in the equation 
A+B . cos. z -j- C . cos. 22 +D . cos. 
z being the arch of a circle of which the radius is 1, and b less 
than a. 
First, to find the coefficient A. 
3. The fluent of the right-hand side of this equation is A % 
+ B . sin. £-f -JC . sin. + | D . sin. 3 z -f . sin. 42?,* &c. 
which evidently vanishes when z = o ; and, when 2 = 3 * 1 4159, 
&c. the arch of 180°, it becomes barely = A 2, the sines of z, 
qz , 3 z, &c. being then each = o. If, therefore, the fluent of 
the first side of the equation be taken, the increase of it, while 
% increases from o to 3 • 14159 &c. = ?r, will be = ?r A; and s 
consequently, A will be determined. 
• See M. Euler’s Institutiones Calculi Integralis , Vol. I. p. 150. 
