5gS PHILOSOPHICAL TRANSACTIONS. [ANNO 1784. 



1 L_ x - ? ' C&v = -.7«+ —r"" • T~i — ''' r '' + & c - : ar, d > n general - . 



(8 /3 - * ./ H a/3 ' « + » /3 + « o a 



- I &c. /J + -.— — .— — .r— /*"/> + 1--T-— -T-— -&c. a:-' 3 A-> + 



/8 " y ^~«" M _/3«— y ./ rn /3/3— «/3_ 7 ./ ' ~ 



L . — . — , he. x~ < / x p = - . - . - he. a -\ -- . — — . — — &c. bx" 



y y — a y — /3 ' J * /S V a + « Z 3 + « V + « 



_l - 1 — . . , &c. ex 2 " + &c. whence the law of continuation is 



* u — 2« /3 + 2» v + 2k 

 immediately manifest. 



Hence, if no two quantities a, (3, y, J, &c. be equal to each other ; and the 

 successive terms a, b, c, d, he. of any series a + bx" + ex-" + &c. = p be di- 

 vided by « . (3 . y . J . &c. ; (* + «) . (j3 + w) . (y + «) • (* + «) • &c. ; « + 2n . 

 (3 + 2ra . y + In . § + In . &c. &c; and in general by « + nz . (3+raz . y + «z . 

 J + wz . &c. &c. ; then can the sum of the series be found from the fluents of 

 the fluxions x"p, x B p, x>p, x*p, he. as has been observed in the Meditationes. If 

 two are equal, viz. a = (3, then also the fluent of the fluxion - I x p is required. 

 If three are equal, viz. a, = (3 = y ; then it is necessary to find the fluent of the 

 fluxion - j- f x°v ; and so on. 



1. Let p — — x— : ; ; and if the differences of the quantities a, (3, y, $, he. are 

 divisible by n, from the fluent of the fluxion x*p can be deduced the fluents of 

 all the other fluxions x s p, x y p, &c. ; and in general, if a — (3 is divisible by n, 

 then from the fluent of the fluxion x'p can be deduced the fluent of the 

 fluxion x n p. 



1. Suppose p = the terms of the binomial theorem expanded according to the 



dimensions of x, viz. (a + bx") - = a* -) a~ ' bx" + &c. beginning from the 



first or any other terms ; then, if «, (3, &e. divided by n give whole affirmative 

 numbers, will all the fluxions x*p, xPp, xVp, &c. be integrable ; and if the differ- 

 ences of the quantities a, (3, y, S, he. are divisible by n, from the fluent of the 

 fluxion x x p can be deduced the fluents of the fluxions xPf, xVp, he. If p denotes 

 the sum of the alternate or terms whose distance from each other are m, of the 

 binomial theorem, the same may be applied. 



3. Ifp = (a + bx" -+- cx z ")'; and a, (3, y, S, &c. divided by n give whole 

 affirmative numbers, then from I x x p can be deduced all the remainder fx$p, 



/xvp, he.: and in general from two can be deduced all the remainder. To find 

 when the sum of any series of this kind can be found, add together each of the 

 fluents, which can be found from each other, and not otherwise, and suppose 

 their sum = O; and so of any other similar fluent, and from the resulting equa- 

 tions can be discovered when the series can be integrated. 



13. If the general term of a series contains in it more variable quantities, z, v, 



