442 Sir James Cockle on a Differential Criiicoid. 



viii. p. 26). The theorem is, if [l] r = l . 2 . 3 . .r, 



\n—\ n-\ 



,0 n - 1+, =H. 



[1]- 



a t i. n(n—l)..(n—r + l) ,, , ^ A1 



4. Let n r =— - — -, so that z,n =n r+l . Also 



r rii r 



let If and 77 each represent -y- j but let f denote -7- operating on 



a function of x, and let tj denote it operating on a function of y. 

 Let H. = <p(x), and let the successive differential coefficients <£'(#), 

 <j> n (x) 3 &c. be denoted by X p X 2 , &c. respectively. Also let 

 S = X 7 'XiX V 2 . . . , and let <r denote a finite integration, with re- 

 spect to n, which when applied to/(w)H operates onf(n) only, w 

 in H being treated as a constant. Also let 0=za%, no arbitrary 

 constant being added after the finite cr integration. Let a have 

 V for its inverse operation. 



5. It is known (see Camb. Math. Journ. vol. i. last page) that 



or, S denoting an aggregate of terms, 



\ fl<r/ \dxj ny 



And Mr. Walton has shown (Quart. Journ. of Math. vol. ix. p. 

 356) that 



/ d \ n P n ~ 1 / d\ r 



\dx) ~ G „_ r W/ 

 — entering into every term of each aggregate. 



6. Now 





the term #"X n being suppressed in the development of the dexter, 

 because -7- enters into every term. The development, thus re- 



(XX 



stricted, is 



(^-f^- 1 +..4-6> 7l - 1 7 ? )X>, 

 giving 



-\ — S6» r X w 4— ^ 

 dx n 



w- 



