414 PHILOSOPHICAL TRANSACTIONS. [ANNO 1798. 



(^TxTos^r = * ( 2A X COS * 2Z + B ( COS * * + COS> 3 ^ + C + C0S - 4 «) + » 



(cos. z + cos. 52), &c.) And the sum of the fluents on the right-hand side, 

 when z = «•, will become barely cz = ctt. Therefore, the fluent of the left-hand 

 side of the equation, when z = ?r, will be = c?r. The fluent of this fluxion, it 

 is evident, will consist of 3 parts, the 1st and 2d of which, n being = 4, are ob- 

 viously attainable from the values of a and b above found in art. 9 and 16; and 

 the 3d in series similar to those which have been given in the former part of this 

 paper. 



It is evident also that, if n be = 4-, all the 3 parts of this fluent are attainable 

 from the values of the 2 co-efficients already found, and c' would be = — 2a" + 

 \ (b'« - b). 



1 9. And in this manner may the other co-efficients, d, e, p, &c. be determined. 

 And since the cosines of 3z, 4 2, &c. are = 4x 3 — 3x, 8x 4 — 8,r 2 + 1, &c. respec- 

 tively ; and since x — a ~" . , it is evident that the numerator of the frac- 

 tion into which the fluxion in the preceding article is to be multiplied, will be 

 always of this form, viz. p -f- qvv + rv 4 -f- sv 6 , &c. ; from which it follows, that 

 if the values of a', a, a, &c. corresponding to n 3 n — 1, n — 2, &c. be computed, 

 the values of c, d, e, f', and all the rest, may be found in terms of a', a, a, &c. 



with the co-efficients a and b. But, since the easiest method, that has come to 

 my hands, of computing the values of c, d, e, &c. after a and b are found, is ex- 

 plained in M. Euler's Institutiones Calculi integralis, vol. 1, p. 181, I shall not 

 pursue this method any further ; but, having examined his process, and corrected 

 the errors of the press which occur in it, now give the equations expressing the 

 values of c, d, e, f, &c. which were obtained by that method. 



20. For the sake of brevity, let ^- = d ; then will the general values of c, d, e, 

 f, &c. be expressed by these equations : 



c = 



2«A — 2«?B 



D = 



(» -f l) b — 4dc # 



(n + 2) c — 6dp " __ (n + 3) d — 8cte . 

 E — - - ; F — - - , &c. 



n — 4 n — 5 



»— 2 ' »— 3 



where the law of continuation is very obvious. And the particular values of these 

 letters, when n = 4-, 4, 4, will be as expressed in the following columns : 



21. The solution of the problem being now finished, it may perhaps be satisfac- 

 tory to the reader to see how the sums of the very slowly converging numerical 



