352 Mr. C. J. Hargreavc's Applications of tfie 



This theorem is not derived from any principles peculiar to the 

 Calculus of Operations; nor does it in substance contain anything 

 not previously known. Its characteristic consists in the use of 

 the general functional form <f>(D) in lieu of particular algebraical 

 forms, and in expressing the connexion between the different 

 sets of operations employed by means of <j>{D) and its derived 

 functional forms </>'(D),<//'(D),&c. In passing from <j>to <£',</>",&c, 

 we cause the functional form $(D) to undergo the same changes 

 in form that the quantity <j>t does in the course of successive 

 differentiations with regard to t, an independent variable. In 

 other words, for this purpose, viz. that of obtaining convenient 

 modes of expression, w r e treat D as if it were a quantity, and we 

 make it an independent variable ; and it is by an operation of 

 differentiation with regard to this supposed variable that we 

 express the connexion between c£D, <f>'D, &c. If we denote this 

 operation by a symbol having the same relation to D that 1) has 

 to x, and apply the established theorems of the Calculus of Ope- 

 rations to the new symbol, we shall acquire enlarged and more 

 convenient forms for the expression of complex operations ; and, 

 if necessity or convenience should require it, there appears no 

 reason against extending this idea without limit. 



In establishing a notation, it seems desirable that any symbol 

 expressing a derived function should have some affinity with those 

 already adopted ; but the calculus of finite differences and the 

 calculus of variations have already appropriated most of the 

 available types. I propose to denote the operation of passing 

 from any function of D to its derived function by the symbol y : 

 and, should occasion require it, we may distinguish the successive 

 grades of this operation by Vu Va> • • • V n '> eacn s y mD °l deno- 

 ting a differentiation in which the next preceding symbol is the 

 independent variable. 



At present we confine ourselves to the first symbol ; so that 

 we have 



^D)o 1 .^ = v(^D) ; 



and 



(«o + «iV + • • • a n y n )<l>T> = a <l>I) + a l <l>'D+ . . . +0«</> w D, 

 e ±w V(0D) = tf>(D±w), 

 y- n (</>D) = <£ n D, or nth integral of </>D, 

 ( v + c ) - 1 (<£D) = e-^y^^D^D, 



and so on for other forms of 1/ry. 



It is scarcely necessary to remark that the symbol v obeys 



