120 Mr. 0. Heaviside. [June 15, 



Success of Alternative Method of Representation by Harmonic Analysis. 



41. We may then adopt another method. Thus, (44) and (45) 

 arise from 



u = i_ v - + v- 2 -V- 3 + .... , (70) 



V 4 + ..... (71) 



With unit operand, the u series is immediately integrable without 

 any obscurity, giving e~*. The v series leads to an unintelligible re- 

 sult. But let the unit operand be replaced by its simple harmonic 

 equivalent. Then 



. . . . ) V = o - cos mx dm 



= (l-V)*r^ = e-*, [(72) 



when x is positive, which is the required result. We are only con- 

 cerned with positive x, but it is worth noting that when x is negative, 

 this method makes v zero. This is also in accordance with the 

 analytical method, or (70) directly integrated, for we suppose the 

 operand to start when x = 0, and to be zero for negative x, which 

 makes u also zero then. 



42. As regards the derived formulae (39) to (42), although I have 

 not examined them thoroughly to ascertain limits within which the 

 suspected numerical equivalence may obtain, I find there is a rough 

 agreement between (41) and (42) when n = ^ and m = 3, even with 

 x = 1, and the convergency confined to the first three terms of v, the 

 results being 



16 \ 54 V 128 



which, when x = 3, give 



U =: 1'47, V = 1'41. 



Again, with the much larger value x = 9, we have 

 u = 3'88, v = 3'87, 



which is a very close agreement. 



This is promising as regards further numerical agreement when 



