rOL. LI.] PHILOSOPHICAL TRANSACTIONS. * 447 



Now, it is obvious, that this value of h' must be equal to the value of h' in 

 the preceding article, when both series converge. 



Therefore %^ + ^^^- + '^-^^-^ &c. is then = x'a" - a'' + 



2 4 ~ 4 ' ^ 



Hence, by taking x equal to — 1, we find — 2s'' = Z?^ + ^ + 2s'' ; and, 

 consequently, s" = — — 4-. 



And, by taking x equal to -^ — , we find 



+ 7= X (iTF - 5^5-^+ ^^ -' ^^O = -^^' - 2^ + 2 ---^T+ ^ + 

 2s" = 2:^73^ — — ~ ; and, consequently, jr^i — p-5- + rry-, &c. = 4. - ^. 



17. Seeing that a" is = -, and s" = - — 4-3 it follows, from the last article, 



thatH ,s = ^V' + - ^ + - _^+ _-|.-__+^_ &c. 



Whence, by multiplying by -, and taking the correct fluent, we get 



11 .r q" _. 6i'ix bx'^ ^^ b f^ bx _\^^ ■^~' *~' ^~^ c 



H — —-+-4 T"*"4'^T""^^"4'~ lA3i "" FTS"^ "~ 'E^T'' ^^' 



And hence, by multiplying by x.v, and taking the correct fluents, we have 



H _ — - + — g g^ + T6 "•" ~8 " ■•"64"~3 ""9"'"'5-+1 l6 + 



_ I &C ■' "^i* 



3^.5^7' ~ 5'.7^9^' 



Now, this value of h''" being equal to the value of h"''' in art. 1 5, when both 



series converge, it follows, that - ^, ^^ ^^ + - ^^^^.fr + 5^.72.9. -? &c. is then 



x''q" ■, 6j+x 560:'^ j^ fcr^^ j^ 6j?'x j^ b ,c^ "^ _L 4 _1_ "^'^^ I 3a^ „/ 



— • — + — g- ~ "ej "T- Tg + ■~^~ "^64"~3~"9~'"'^"''~"*' TF""^^ • 



36* 



Hence, by taking ^ equal to — 1, we find -t is'" = — — |- -§- + 4s"'; and 



3a^ I 



consequently s'" = g— — ^. ,t 



Many other instances of the use of this method might be given ; but these 

 may suffice to enable the intelligent reader to pursue the speculation farther, at 

 his pleasure. 



The above subject is much further pursued by the ingenious author, in his 

 Mathematical Memoirs, p. 67, published 1780. 



LV. Conjectures concerning the Cause, and Observations on the Phenom.ena 

 ay Earthquakes ; particularly of that Great Earthquake of Nov, 1, 1755, 

 whichproved so fatal to the City of Lisbon, and whose Ejects were felt as far 



