332 Mr. T. Muir on Sylvester's and other forms of 



and the left-hand member being also evidently 



Jo (*+i) w+i J (*+ir +i 



we therefore have 



Jo (*+l)" +l 



and hence, writing d>(n) for I x"~' ) - <fo, there results 



J, («+ir +i 



Now the denominator on the right is the same function of w-f- 1 

 that the left-hand member is of n; and thus by continued sub- 

 stitution we have 



<P(n+l) 2m + \ ^ ; , (n + 2)(n + 3) 



2m +. 



If w be now taken equal to 1, the left-hand member here becomes 



J (* + l)-+> lfeH -J/(*+l)-+» lte; 



1— a? 

 or, after integration and substitution of y for > 



J. "J - 5? 

 1 b 



2m 1 ^-^l+y)" 1 ^ — 1 

 and thus we have 



2m f>-(l +2 /)-V y = l + ^ 1_2 23 (I.) 



« ° 2m + — — 3 . 4 



2m + — — 



2m + .^ 



Putting in this general result m = J, we obtain, as desired, 

 2 i + x + ^ 3.4 



