VOL. LXXIV.] PHILOSOPHICAL TRANSACTIONS. 50,7 



p j p .p + l.p + i.p + 3..p + z p+z.p + z — 1 .p + z— 2.p + z — 3 .. z + r+l .- 



C0 r.r+l.r+2.r + 3..r + z = r . r+1 . r+2 . r+3 p—l ' P r 



be a whc'e affirmative number ; but this latter quantity has the formula above- 

 mentioned az m -\- hz'"- 1 -f- cz'"-i -\- &c. ; and consequently, if the sum of the 

 series a + bx s + cx ls + dx">' + &c. = p be known, by this method can be de- 

 duced the sum of the series a + p -bx> + p -^±±cx" + P-P + * -P + -?tfe3» + &c. 



1 r ■ r . r + 1 r . r + 1 . r + 2 



m m 



Exam. 1. Since (a + #)" = a"(l + - X - + - X — - — a~ 2 x' + - . _ — . 



v ' ' v n a ' n 2n ti 2n 



~ n a-ix 3 -{- &c.) ; multiply the successive terms of this series into the terms 



3« 



m _i.1L— 



of the series 1 , -, P — , &c. and a series is deduced a" + p —- a " ' x -\- 



' r r . r + V r . n ' 



m 



p ' p — - -- ' "' ~ " x " + &c. whose sum is known, if the sum of the series = 

 r . r + 1 . » . 2s ■ 



m 



(a + x) " is known. 



Exam. 2. If the series begins from the / + 1 th term of the above-mentioned 



m 



binomial theorem a* + - « + "~' x -j- &c. viz. the series be h X (1 + 

 1rJL+Jh . ■ + 'l^L£±An . ? + ^ii+_ 3 - • -! + &c) ; of which let the 



(l + 2) n a ' (I + 3) n a l ~ (I + 4) n a' ' ' 



respective terms be multiplied into ] , p -, p ^- ~-— , &c. there will result a series 



whose sum is known. 



Exam. 3. From the rule, first given by me, for finding the sum of the terms 



at h distances from each other, the sum of the series 1 + '~j~-^r~ X 

 m _ (/ + 2) n m — {l+h)n xk m- (I + h + 1)« m - (I + k + 2) n 



-jr+H)— ' ■ (/ + h + 1) n X a" "*" X (l + h + 2)n * {l+h + 3)n 



m - ( -f 2 i)ii x — ^ where p denotes the co-efficient of the preceding term, 

 (I + 2h + l) n a"' ~ r ° ' 



can be deduced ; and consequently the sum of the series deduced from multiply- 

 ing the successive terms of this series into the quantities 1, f- »5~f"+1 i' &c - re_ 

 spectively. The general principles of this case were first delivered by Mr. Ber^ 

 noulli, Mr. de Moivre, Mr. Euler, &c. 



12. Assume the series a + bx" + ex*" + &c. = p ; multiply it into x*~\v, 



d find the fluent ; then will X -x'p — \fx>v = - ax" + — j— *«•+• -f 

 L_ c ^=-+2« -j. & c . ; multiply this equation into x & ~ a - l x, and find the fluent of 



an 



the equation resulting, which will be X - X \ x s p — - • jJ xS P ~ ~ X JZIZ x3 ~ a 



/•» + ? • r-~ f*> = \ • s «* + -ir: • ft: fa ; + ' + its • ^u 



cx 3+z, _j_ g5 C# . divide by ^^ and there results i . J-p + ^ . —jrj x ~J x 'P + 



