Mr. O. Heaviside. [Feb. 2, 



014: 



In a precisely similar manner we may show that 



0*0 



Further modifications are confirmatory. Thus, by making use of 



we can shift ^ and similar functions back and forth. Also using the 

 stock formulae 



_P__ = e at J*- = e -at, (23a) 



p a p + a 



we have the following transformations, 



(24) 



p \p-a p- 



In this we may nse the result (22), and so come round to (21) again. 

 From the above we see that 



"Ufor ^ 



and further, by shifting the exponentials to the right, to make them 

 the operands (instead of t\ 



\ i / . \ 



-; (26) 



p-a 



and now further again, by employing (23a) in place of the exponen- 

 tials, we obtain 



I o(a0 = (=f Y JL. = _* , (27) 



\p + aj p a (p z a z )* 



which is an entirely different kind of operator, since the square of p 

 occurs under the radical sign, instead of the first power. But (27) 

 may be readily tested and found to be not wanting. For expand by 

 the binomial theorem, thus, 



This may be immediately integrated, giving as result the series 



...., : - (29) 



