600 PHILOSOPHICAL TRANSACTIONS. [ANNO 1 800. 



the values of the letters a and b. used in the former paper, make up the contents 

 of this paper. 



The h.l. £-2 , which was denoted by a, both in the solution of the 



2 cc 3 c* % *)C^ 



problem and in the appendix, is = h.l. -, — — — ^-^ — ' & c . and if, for 

 the sake of distinction, the Roman letter a be put for h.l. - , we shall have x = a 



CC %C^ 



— — — — , &c. (of which series, the first 3 terms are sufficient for the present 

 purpose) ; and this value of a being written for it in the expression « x 



3.5 

 8.8 



1 4- -re. 4- 



8.8 ' " v 4 4#8 



3 3 5 



(1 + - cc -f- — '— c 4 ), which occurs in the first theorem in art. 12, of the first ap- 



3 3 5 cc 3 



pendix, we have (l -f -cc -f Y~T c4 ) X (a — — — TrT *) '* tnat iS > by actual mul- 



tiplication, 



- zee 4- — — - ac 



8 1 8.8 



1 3 



-cc — 



r 16 



Now the terms — ±cc and — -^c 4 may very easily be added to the terms fee and 

 gc A , i.e. to 0*1 036802 cc and 0*o687064c 4 , which will then become — 0'1463J98cc, 

 and — 0*11 87936V; and, by denoting the co-efficients of these new terms by the 

 Roman letters — f and — g respectively, the first theorem in the art. before- 

 mentioned, or the value of a, is 



~~ *(a + b)% X J 



\- e — ice — gc* 



+ a -f -ace + -g^ac 4 . 



o 1 'i 3 5 21 



The expression a. (- + -rr7.ee + ■ ' ■ c 4 ), which occurs in the value of a', in 

 art. 12, of the first appendix, is = 



r 3 , 3.5 , 3.5.21 4 -4 r 3 . 3.5 . 3.5.21 4 



\Z + m cc + EH--S< I S i a + 47Ti acc + iTiTsI*' 



1 1 3 . f — S 3 19. 



3 19 



Here again the terms — -~cc and — — ->c 4 may very easily be added to the 



terms ice and kc*, i. e. to 00551 502 cc and 0*0408309 c 4 , and we have the two new 

 terms — 01323498cc and — 0.1076091c 4 . Let the coefficients of these two 

 new terms be denoted by the Roman letters — i and — k respectively, and the 2d 

 theorem in art. 1 2 of the first appendix becomes 



L ' = i x \ : 



+ i- + h - ice - kc 4 



■ CC ' 



3 , 3.5 . 3.5.21 



-a 4 ace 4- ac . 



4 ' 4. 12 ~ 4. 12.32 



The product of a. (2 4- \cc + -fVc 4 ), which is found in the 3d theorem of the 

 art. before referred to, is = 



