VOL. LXXXVIII.] PHILOSOPHICAL TRANSACTIONS. 415 



series, which arose in art. 7, 11, and 16, were obtained ; the investigations of which, 

 because they would have detained him too long from the immediate subject of this 

 paper, if they had been inserted in it, are given in the following appendix. 



An Appendix to the foregoing Paper, in which the method of obtaining the 

 sums of the very slowly converging numerical series which are there used, and of 

 many others of that kind which arise in the fluents of binomial surds, is explained 

 and illustrated ; and some observations, tending to facilitate and abridge the com- 

 putations of the co-efficients a and b, are added. 



1. As the sums of the very slowly converging numerical series, which arose in 

 several articles of the preceding paper, are not exhibited in any book that has come 

 to my hands, and as series of that kind frequently occur, I conceive that the fol- 

 lowing method of obtaining their sums will be acceptable to the lovers of mathe- 

 matics in general, and particularly to those who have frequent occasion to use the 

 sums of such series. And having observed, while considering the literal expressions 

 in the preceding paper for the values of a and b, that others, no less accurate, 

 might be derived from them, by which the arithmetical operations would be facili- 

 tated and abridged, I thought these observations might likewise be acceptable to 

 those who are engaged in the theory of astronomy, and have inserted them also in 

 this paper; which therefore consists of 2 principal parts, the summation of the 

 slowly converging series, and the observations now mentioned. 



1 . The Summation of the slowly converging Series. — 2. But, before beginning 

 the investigation, it will be proper to premise a few particulars, an attention to 

 which will shorten and facilitate the operations now to be performed. 



lst That i-^ 1 " ^ being - iz^HjlM v 1 + ^( 1 -»). U _- / | vt. 



1st. mat j + v(1 _^ } being - l+v(l _ yy) X j +v(1 _^> » - 4+y(i^)) ; 



from which it follows, that h. l. of "" ) ~~ , is = 2 h. l. 2 r. 



i+V(!-j») . i + vti-jy} 



2dly. That the fluxion of h. l. ■ , is = — — £ — ~ — ^. For it is = 



' i+VO-yy) yV(i-yy) y 



the fluxion of - h. l. (1+4/(1 - yy)) = -£-^ x 1+v ^„y and if both 

 numerator and denominator of this expression be multiplied by 1 —"/{\ — yy), it 



will become -7/—, X l^&zM , wh ich is = — J r - I 



v(l — W yy * yV{i-yy) y 



Sdly. That the h. l. — — i . j 8 therefore = f-rrf r - '..fi = /^ + 



* 1 +V(i~yy) J yV(i-yy) J y 2.2 *~ 



w + 3 - 5 * 6 . 3 - 5 -7y* &c 



2.4-4^2.4.6.6^2.4.6.8.8' > 



4thly. That a being put =^/(i — yy), the fluxion of 

 i will be = l <££ + 5-i). For it will be \ - -2§- = ^ - -^1 = 



-y "i( 1 —yy) __ - n j> . ' (» - *) j _ j t_zJL _i_ n jrzl\ 



y"— IQ y" + IQ ~ yn + IQ ' yn— Iq q ^yn + 1 f yn - l'' 



5thly. That, when any quantities, as | » (~ -J- -■ -J- A) I, are circumscribed by 

 a parallelogram, it denotes that a substitution for these quantities has been made 



