314 PHILOSOPHICAL TRANSACTIONS. [ANNO 1798. 



first 8 terms of the series, as they stand, and then compute the value of 



X X* X^ x^ 



g — -jq + — ~~ "12 * &c * ad * nmiitum > by the same theorem, the values of d', 

 d", tTj, &c. will be — , j^yy, gj^jl^ &c - wnich is a series decreasing, for a 

 great number of its terms, much more swiftly than the powers of — , and, in the 



10 



9 



first 7 terms, much more swiftly than the powers of ~. The value of the series 

 I ' To "" To * ("io^ + 17 ' (jo^ ~ I? * ("io^' &c ' is therefore = the series 



9 * IP + 9^ ' ^ + ***rt * kj# + 9^Ti~2 ' ^ + &c * the first ^ terms 

 of which converge above 14 times as swiftly as the other; or, in other words, the 

 first 7 terms of it will give a result much nearer the truth than 100 terms of the 

 other. And if, instead of the first 8 terms of the proposed series, the first 24 

 terms were computed, as they stand, and then the value of the series 



i5---S+|--!5> &c - b y its ^ uivalent '^-TTT+-i3l6--(TW + 

 jnw-(TTW + imm* -irhy + &c - the ra P id decrease of the co - 



11 2 



efficients — -, g , g , &c. compounded with the decrease of the powers 



of ' . > (in tn e present case = the powers of -^,) produces such a very swiftly 

 converging series, that 8 terms of it will give the result true to 1 1 places of de- 

 cimals. On these principles Mr. H. proceeds to compute the terms of these 

 series, and to collect them together ; in doing which he employs some ingenious 

 contrivances, to methodize and facilitate the operations. After which the ingeni- 

 ous author concludes the whole process as follows. 



The values of the several parts, into which the proposed series has been resolved, 

 being now so far obtained that we have only to multiply each by its proper factor, 

 viz. the numerical value of x s , a: 16 , x 3z , &c. and add the products together, to get 

 its sum ; this therefore is now to be performed. And, in this part of the calcula- 

 tion, several multiplications may be saved, and no larger factor than x 8 be used, by 

 attending to the method described by Sir Isaac Newton, in his Tract De Analysi 

 per iEquationes infinitas ; p. 10 of Mr. Jones's edition of Sir Isaac's Tracts; or 

 p. 270, vol. 1, of Bishop Horsley's edition of all his works. The manner in which 

 this is to be done will appear, by collecting the several parts from the preceding 

 articles, and exhibiting them in one view, thus : 



a + B.r 8 + c* 10 + (« + d) X tf 24 + (g + e) X x 32 + yx" + (J + p) X 

 a 48 = the sum of the proposed series. Now, 



1st. Calling I + p, z', and multiplying by x 8 3 we have z x 8 = 0*26584,70242 

 X 0-43046,721 = 0-11443,84267,9.— 2dly. Putting y -f z' x 8 = z", and multi- 

 plying by x 8 t we get z" x 8 = 0*15172,24360,1 X 043046,721 = 0'0653 J, 15337,2. 

 "— 3dly. Putting 6 -f e + z" x 8 = *'", and multiplying by .r*, we get z f,/ x 8 == 



