270 Prof. J. R. Young on some Properties derivable 



: To determine the constant C, put x=0; and we have 

 w(n—l)^^, n(n— l)(n — 2) „ 1 



2 2.3 * +«c.j-, 



which, by (8.), is equal to zero, p being by hypothesis less 

 than m : hence if the formula hold for a value less than w, it 

 holds also for the next value, and so on for all values up to 

 m\ and as it obviously holds forp = by (6.), the formula is 

 general. 



If, however, we suppose j? to be so great as n — 1, then the 

 expression for C will be 



C^-K+^V- "^H*- 2 > 3» + &a}, (11.) 



in which, if we make /3=1, we have for C, as shown in (5.), 

 the value 



C=(-l)M.2.3.4...rc, 

 and therefore the general formula (10.) is also true. 



It thus appears that we may give to m in (9.) any positive 

 integral value up to n— 1 ; for this latter value the formula is 



0=x n - l -n{x + P) n - l + n V l ~ l > (a? + 2/3) ra " 1 - 



5^^(, + 3fl«- +& c, Y ' (12 ° 



and therefore, by transposing the (rj+l)th term, 



(-l) w - I (« + »j3)»- 1 = *«- 1 -w(a? + /3)»- 1 + ^^ 



2 > (13.) 

 (.r + 2£) n - 1 — ... to n terms. 



This expression will be useful in simplifying the investiga- 

 tion of the theorem of Abel already alluded to*. There is 

 only one step in Abel's process in which the need for any such 

 simplification can be felt ; it is that in which the constant in 

 the result of his integration is to be determined ; and where, 

 by the analytical artifice employed by Abel, there seems to be 

 something like an anticipation of the value sought. 



Abel's generalization of the binomial theorem is thus ex- 

 pressed : 



(x + u) n =x n + na(x + /3)"- J + H ^ n ~ ' u(a — c 2fi){x + 2/3)»" 2 



+ *-^<*- g) «(«-iW(«+g)~ + .... 



;va(«-(w — \)fiy-\x + {n— l)/3) + a(a— nB) n ~ l 

 * CEuvres Completes, vol. i. p. 31. 



(14.) 



