from the development of a Binomial. 271 



where #, ct, and /3 are any quantities whatever, and n any po- 

 sitive integer. 



That this is true when w=0, is obvious; and it may be 

 shown, as follows, that if it be true for n=m, it must be true 

 for n = m + 1 ; and thence that it is true generally*. 



Let 



(x + u) m =x m +mu(x + fi) m - l + 7 ^^u{u-2P){x + 2fi) m - 2 



+ m(m-l)(m^) u{ ^ mx+ ^ )m _ 3 + 



A . 6 

 w2a(a-(w-l)j3)'»- 2 (a? + (?w-l)|3)+a(«-^/3) m - 1 ; 

 then, multiplying by {m+l)dx, and integrating, we have 



(x + a) m+1 = z m+1 + (m+l)u(x + P) m +± — — a(a— 2/3) 



(x + ZP)™- 1 (ot+1)«(«— j»j8) w - 1 (a? + 0i!3) + C. 



To determine C, put x= — a; then, dividing by —a, 



0= (_ a )m_ (w + 1)( _ a + ^m + (^+ 1 ) ?W (_ g4 ,2^- 



r; 



(m + l)( — a + w/3) w - 



M (, g )«^( m+ l)(-. g + < 8)» + ^ 'Y (-a + gffl 



a 



(m+ \)m 

 a. 



to m + 1 terms. 



But, by (13.), the right-hand member of this is the develop- 

 ment of 



(-i) m (-a + (»i+l)/3) m , or (a-(m+l)/3) w ; 



.-. C=a(a— (m + \)fi) m .. 



Hence the theorem has place for n=m+ 1, and thence gene- 

 rally. 



The theorem (12.) is included in this of Abel, as that illus- 

 trious analyst has noticed ; but the other analogous theorems 

 in this paper, where the exponents are different from n— 1, 

 are not implied in the theorem of Abel, nor have I ever met 

 with them. 



It is probable, however, that some works on finite differ- 

 ences may contain these expressions ; but this is a point about 

 which 1 have not thought much inquiry necessary, as my chief 

 object is to show, in detail, how a remarkable theorem in the 

 higher analysis may be arrived at by only the simplest prin- 

 ciples of the calculus, combined with common algebra. 



* The process, up to the determination of C, is Abel's. 



