Mr. 0. Heaviside. [Feb. 2, 



Interpretation of Vanishing Differential Coefficients. 

 18. It should be noted that when we say that 



V ^ = l, (46) 



for all values of n, the right member is really aj/|0, and means on 

 the left, and 1 on the right side of the origin of x. That is, it is the 

 limiting form of the function x n l\n, when n is infinitely small positive. 

 It is convenient in the treatment of equations of the form F = Y/ to 

 have the function / zero up to a certain point, with consequently F 

 also zero, and then begin to act. Similarly, the expression v/jo, or 

 Vl or ~V| 1 although it has the value zero for all positive values 

 of x, is infinite at the origin. But its total amount is finite, viz., 1. 

 Imagine the unit amount of a quantity spread along an infinitely long 

 line to become all massed at the origin. Its linear density will, in 

 the limit, be represented by, as (12) is derived from (11), 



1 f 

 = - 



Tjfl 



cos mx dx. (47) 



It is zero except at x = 0. But its integral is still finite, being 

 Vl or 1. If we draw the curve y = x n /\n, with n infinitely small, 

 consisting of two straight lines, with a rounded corner, the curve 

 ^derived from it by one differentiation will nearly represent the func- 

 tion vl, being nearly all heaped up close to the origin, and of integral 

 amount 1. Similarly v 2 ! means a double infinite point, v 3 ! a triple 

 infinite point, and so on. But it is the function \7l that is most 

 useful in connexion with differentiating operations, whilst the others 

 are less prominent. 



But when n is taken to be infinitely small negative in. y x n \\n^ 

 then y drops from oo to 1 near the origin, or the corner is turned the 

 other way. That is, the function x n is unstable when n is zero. It 

 is the difference of the curves y = x n with n infinitely small positive, 

 and the same with n infinitely small negative, that makes the 

 logarithmic function when infinitely magnified. But we should try 

 to keep away from the logarithm in the algebraical treatment of 

 operators. 



Connexion between the Factorial and Gamma Functions. 

 18A. It will be seen by (42) that our factorial function is the gamma 

 function of Euler somewhat modified and extended. Thus, when n is 

 greater than 1 we have 



(48) 



