Calculus of Operations to Algebraical Expansions. 355 



and if x be made equal to a, the second side vanishes except in 

 the term whose index is n— n : so that 

 <j)D(J^x— #) n .w)(when x=a in the operation) is 

 n(n— 1) . . . 1 , , , , 



And if u=<xx, it follows that 



</>D((#— a) n xx) (when #=# generally) =</>WD . ^, D being j-here. 



If </>D be simply D m , we have the well-known formula 

 ~D m (^(x— #) w W)(when #=fl)=m(m — 1) ...(m — n-\-l)^ m ~ n ^a. 



The expression {<£D.-^( c r, #)}w, considered as a function of a, 

 involves an idea of some novelty, which may possibly prove useful 

 as an extension of our means of expression. I shall not, how- 

 ever, pursue this branch of the theorem further than to observe, 

 that since 



{0D.^r(#, fi)}tt = ^r(V + 0)£D .U, 



we have 



• — ({£D.^ a) }ti)=^W(v + fl)*D.« 



= {<£D.^(^)}^ 



f «(*,«) being <*>fo«) . 



The other branch of the theorem, 



</>D{i|r#.w} (when-a7 = fl)=^r(v + «)(^-D) . w, 



possesses an immediate value by enabling us to arrive almost 

 instantaneously at the results of differential operations when the 

 variable receives a particular value, and to express such results 

 explicitly instead of merely indicating them. The formula is 

 one of great simplicity, particularly when a = 0. The following 

 examples will illustrate the nature of the process : — 



1. </>D(e w ^- a >%#) (when x — a) = e n v<£D . u=<j>{D +n) . %«. 



2. ^D{(A + A,(a?-fl)+A a («-fl) 9 + .. +A n (x-a) n ) X x} 



(when a?=a) = (A o <£D + A 1 0'D + . . + A n <f)(^J)) X a. 



3. Let 



(A +V+...+A„*r i =,-^+r^- + ..+ 7 ^-=2-5-, 



t — a. t — otj t — ct n t — a 



then 



^((Aq+Aj^— a) + .. + A n (x— fl) n ) -1 %#), when x—a, 



