412 PHILOSOPHICAL TRANSACTIONS. [ANNO 1798. 



f. 



2'vV 



— 2 



"% x ] V(vv-cc) 



— V°-r f 7 V i 3w , 3.5e< , 3.5.7v 6 - . 



(. "*" V(^ - cc) ^ "+" *T "T" 476 "•" TT6TS ' &c *' 

 And the fluents of these several terms being taken, and collected together, 

 then the whole multiplied by the common factor (a + b) * the fluent sought 



will be 



'2 j^(vv — cc) 



(a + b) T X < 



+ * + i* + 5 Av + i4*',te. 



4 ' 4.6 ' ^4.6.8 



6. We must now inquire what value this series has when z = O ; in which case, 

 X being = 1, vv is = — — ft = cc. And it will appear that, with this value of vv, 

 every term of the series vanishes, so that the fluent needs no correction. If, there- 

 fore, we compute the value of this series when z = tt, i. e. when x = — J , and 

 w = ~^- b = 1* we sna N have the value of A?r, and consequently, a will be deter- 

 mined. But, with this value of v> the terms C, y } $ t &c. lose all convergency by 

 the geometrical progression, v f v 3 , t> 5 , &c. and the computation of the value of the 

 series, by the common method, would be more laborious than the computation of 

 the quadrantal arch of the circle, by the series 1 -J '-I 1 '— — , &c. 



n J ' 2.3 T 2. 4.5 ^2. 4. 6. 7' 



Here then we are stopped. But, by contemplating this series, expressed in terms 

 of a and c, and by making various ingenious transformations, in this and the 7th, 

 8th, and 9th articles, Mr. H. at length obtains 



8 - 2cc 





* (a + 6)t 



8 — 5cc 

 [+•(!_ CC) . g + X + y.CC + VC *) 



10. The value of a when n = 4 being now found, let us next investigate the 

 value of it when n = -f- ; which, for the sake of distinction, in a use to be made 

 of it in a subsequent article, Mr. H. denotes by a'. Here, by proceeding in a 

 way similar to the foregoing, in this article, and the 11th, 12th, 13th, the au- 

 thor at length obtains for the value of the arc in this case, the following form, viz. 



96 — 23cc 

 128 — 84cc 



,+ V(l - cc) . (i±^- C + / + /cc + „V). 



2dly, tojind the Coefficient B. 



14. Multiply the equation in art. 2. by 2 cos. z = 1x, and it gives _ = z 



(a X 2 cos. z -f b X 2 cos. z X cos. z -|- c X 2 cos. z X cos. 2z -f- d X 2 cos. 

 z X cos. 3z, &c.) ; which, because 2 cos. z X cos. mz is = cos. {m — l) z -f- cos. 

 (m-f l) z, will be = z (2a . cos. % -\- b (1 + cos. 2z) + c (cos. z -f cos - 3z) -f D 



(cos. 2z + cos. 4z), &c.).* And, by taking the fluents, we have / J* ss 2a . 

 * See Simpson'a Miscellaneous Tracts, lemma i, p. 76. 





