646 ■ THE GAMMA, OR FACTORIAL. FUNCTION. 



= (a — I, /3) + (o, /5) for all values of a, ft . (i) 

 and, integrating by parts, 



T i'^ I _ 



Avz"-^{i+zy+' = - [2"(i+^)/3+i]_'-^Jl_ Arz'ii+zy. 

 Now, if /5 > — I, and o is arbitrary, the limit terms vanish, and 



(o-I,/3+l) = '^.(«,y3). 



Hence, by (i), (o -!,/>) = ' — ~ (o, /3), whatever « may be, if 



/?> — I. 



Now, if o > - I, (a, ft) = (/3)-„-„ 



(--^^n)--:^^-- ri03+«+i)n(-a-i) -('')-" 



Hence it follows (if /3> — i) that {a — 2, /3) = (/3)-a + i, and so 

 universally. 



The follow^ing Propositions follow from those proved in this 

 paper, and are of some interest : — 



A. \ (>/')«_,(");=('" + ")« for all values oi a, if ;« + ;/+ i>o. 



PC 



B. {i-\-ii)" = \ (")« + ,""■*"', when 11 is any unit vector, if ;;>o. 



Defining fractional differenlialion by D"x"' = —, — ^;;^^ 



C. £)"(?«') ="y^(")e • D"-'u . Dh', but D';f{a+x) is «o/ /"(a+x). 



