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XL. On some Properties derivable from the development of 

 a Binomial ; with a simplified proof of a remarkable Theo- 

 rem of Abel. By J. R. Young, Professor of Mathematics 

 in Belfast College*. 



THE following properties of the binomial development, 

 although for the most part new to me, yet lie so near the 

 surface, that I can scarcely suppose that they have hitherto 

 been overlooked. It is probable, however, that there may at 

 least be something novel in the method of investigation I have 

 employed ; and this, added to the circumstance that the proof 

 of Abel's remarkable generalization of the binomial theorem, 

 and of the development of the function <p (# + a), is a little 

 facilitated by this method, may perhaps justify the publication 

 of the paper in these pages. 



If we apply successive differentiation to the formula 



i-i \„ , n(n — l) n(n — l)(n — 2) „ ; 

 (1— x) n =\ — nx + -*-— — '-x* J ~\ 'aP + bc, 



regarding n as a positive integer, and always multiply by x 

 previously to each differentiation after the first ; then, upon 

 making x equal to 1, we shall obtain these results, viz. 



0= -.+ ±=» 2- "0-')(*-2) 3 + &c . . . {1 .) 

 0=—+ "±=D g- »(—'>(—«> g» +&e . . (2.) 

 0= -n+ "Cj " 2°- "("-')("- 2 ' 33+&C. . (3.) 



0=— n + 



_v >_c>n-\ v r\ i3«- l -f-&c. (4.)f 



2.3 

 as also 



(-l)'l.«.5.4...»=-» + g^y- " (, '- 1) l"- 2 ) 3- + &c. (5.) 



In this last formula n may be any positive integer without 



* Communicated by the Author. 



t Certain expressions analogous to these are well-known, viz. 



0=l-n.2-+ ^2l3 m -&c, 



where, as above, m is less than n ; but whether the forms in the text have 

 also been given or not I am unable to say. By differentiating oftener than 

 once, before multiplying by x, different zero-expressions may be obtained, 

 as also by multiplying by powers of x, instead of by x simply. 



