272 Prof. J. K. Young on some Properties derivable 



If we interchange x and a, the foregoing development will 

 become 



(^ + «) w = « ra + «(« + |8)»-^+ ? ( w "~0 ( a -f 2/3)»- 2 #(#-2/3) 



+ " (W ~^ W ~ 2) («+3/3)"-3^-3/3) 2 + &c. 



Having obtained these results for the expanded binomial, in 

 the case of n a whole number, it is natural to inquire whether 

 the theorem does not admit of a wider application ; whether 

 indeed it is not perfectly general for every function of (x + a). 



Let us then assume, from the law thus suggested, that 



<p(x + u) = A + Bx+Cx(x-2p)+Dx(x-3l3)2 

 + E l r(^-4/3) 3 + &c. 



.•.^^=B + 2C(^-/3) + D{(^-3/3r- + 2^-3/3)} 

 -f E{(#-4/3) 3 + 3.*(.r-4/3) 2 } +&c. 



^dl*^ = 2C + 2 -3D(*-2/3) + 2.3E 

 {(#-4/3) 2 + .*•(#- 4/3)} +&c. 



# ^y a) =2.3D + 2.3.4E(*-3/3) + &c. 



Nowthe right-hand members of these equations become each 

 reduced to its first term by putting a?=/3, # = 2/3, <r=3/3, and 

 so on in succession, Hence 



irfa'i \ *r \ ^ <W + «) ! rf 2 <p(2/3 + a):r(#-2/3) 



rf 3 p(3/3 + «) #(>- -3/3) 2 ^(4/3 + «) a?(#-4/3) 3 

 + {dspf 2.3 + (rf4/3) 4 ~ 2.3.4 + 5 



or, interchanging « and a, and remembering that 



d n <p( p + a?) _ a^z+jt;) 

 ~Jdp) n {dx) n ' 



we have 



«(«-'3j8)«< Py( g + Sj8) 

 + ~~2T3 ^ +&C,, 



