PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. gQj^ 



Theorem. — If (p{6 + a) be an integral positive or negative function of 6 + a; 

 then vi^ill 



We shall be able to obtain an equation in other cases by means of the last equa- 

 lity but one in our process. 

 To prove this theorem : 



Let /; de(p(e + a)(^z- ey be denoted by P. 



Assume d=zy where ;2; is constant ; 



d6=zdy 

 and P=/Jzd'y(f)(yz + a)(z-6y. 



Let (p(ys + a) — '2A('yz + a)'^ 



F=lAz/o dy(yz + a)"'(z~ey 



= lAzP^^/,'dy(yz + a)"" {l-—Y 



= 2AzP-'^/,'dy(yz + a)"'(l-yy 



= 2 A zP+'^/o dy i y^z"' + m 7"^^ . z""-^ a. 



Now f;dyy^(ii-yy=JI^J^^ 



/m+p + 2 



by Euler's and Legendre's theorems. 



, ( Im + l . z"" 



P = 2A«P+i/jo + l { - 



i /m+p + 2 



^ m fm z""-^ . a m (m-l) /m-l. z"^^ 0^ \ 



/m+p + 1 1 . 2 /^Tip^ ■■■ I 



and since m/m = /m + 1 



(m~l)m/m — l = fm + l 



we get 



P = 2AzP^^/p + l/m + l\ ,——- :- + 



{ m+p+2 



lm+p + 2 /m + p + 1 



+ . h . -f. 



Im + p 1-2 /?w4-jo — 1 1.2.3 



