VOL. LXXXVIII.] PHILOSOPHICAL TRANSACTIONS. 40Q 



ticians who had perused it; in which state it had remained 36 years. On perusing 

 this paper, the first thing that occurred to me was, a different method of finding 

 the fluent, from that which had been used by Mr. Simpson ; by which means, 

 series converging by the powers of £ were obtained, while the series brought out 

 the common way lost all convergency by a geometrical progression, and a compu- 

 tation by it was more difficult than the computation of the length of a quadrantal 



1 3 3 5 



arch of the circle by the series 1 + — - -f- ■ + 4 " g , &c. I afterwards 



discovered the method of transforming that series which had lost all convergency 

 by a geometrical progression, into another in which the literal powers decrease very 

 swiftly; which is the improvement now offered. 



In comparing the series here produced, for computing the values of a and b in 

 the equation (a — b X cos. z) -" = a -j- B • cos. z + c . cos. 2z + d . cos. 3z + 

 &c. with those which have been published for that purpose, by Messrs. Euler and 

 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 1 , the 

 literal powers which have place in Euler's series, would be very nearly equal to the 

 powers of -^-; the literal powers which have place in 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 -g^-* 



M. la Grange has indeed, by a very ingenious device, obtained a convergency in 

 the numeral co-efficients of the series that he uses, which, for the first 5 terms of 

 it, is nearly equal to the powers of -^; but this convergency becomes less and less 

 in every succeeding term, and the co-efficients approach pretty fast to a ratio of 

 equality; so that, to obtain the sum of the series to 6 places of decimals, he pro- 

 poses to compute the first 10 terms of it. The case in which those co-efficients 

 have that convergency, is when n (which answers to his s y ) is = — ->., a case 

 which does not often happen; however, from the values of a and b, when 

 n = — 4-, he derives their values when n = 4-, -§-, &c. by another very ingenious 

 device, worthy of that skill for which he is justly celebrated. But by the method 

 now proposed the chief part of the convergency is in the literal powers; and such 

 a difference in the numeral co-efficients, for a different value of n, does not 

 take place. 



For Mars's perturbation of the earth's motion, the literal powers by which the 

 3 different series converge, are nearly as follows: 



M. Euler's, S f -f; 



M. la Grange's, \ by the powers of ) 4-§-; 



The series now proposed, ) t~rs- * 



* For obtaining nearly the different rates of convergency of the literal powers in the 3 series, it will 

 be sufficient to consider the distance of the 2 planets of which the perturbations are to be computed, as 

 = V"( RR + rr — 2ftr x c, z,) where r and r denote their mean distances from the sun, of which r 

 VOL. xviii. 3 G 



