464 Lord Rayleigh on the Character of the 



Now this is precisely the condition of things that would 

 result from the arbitrary distribution of a large number of 

 impulses, in each of which the medium is disturbed according 

 to a defined law. A simple case would be to suppose that 

 each impulse is confined to a narrow region of given width, 

 and within that region communicates a constant velocity*. 

 An arbitrary distribution of such impulses over the whole 

 length would produce a disturbance having, in many respects, 

 the character we wish. But it is easy to see that this par- 

 ticular kind of impulse will not answer all requirements. For 

 in the result of each impulse, and therefore in the aggregate 

 of all the impulses, those wave-lengths would be excluded, 

 which are submultiples of the length of the impulse. The 

 objection could be met b}' combining impulses of different 

 lengths ; but then the whole question would be again open, 

 turning upon the proportions in which the various impulses 

 were introduced. What I propose here to inquire is whether 

 any definite type can be suggested such that an arbitrary 

 aggregation of them will represent complete radiation. It 

 will be evident that in the definition of the type a constant 

 factor may be left arbitrary. In other words, the impulses 

 need only to be similar, and not necessarily to be equal. 



Probably the simplest type of impulse, (l>{x), that could at 

 all meet the requirements of the case is that with which we 

 are familiar in the theory of errors, viz. 



<\>{cc) = e-''-' (8) 



It is everywhere finite, vanishes at an infinite distance, and is 

 free from discontinuities. A single impulse of this type may 

 be supposed to be the resultant of a very large number of 

 localized infinitesimal simultaneous impulses, all aimed at a 

 single point [x = ()), but liable to deviate from it owing to 

 accidental causes. I do not at present attempt any physical 

 justification of this point of view, but merely note the mathe- 

 matical fact. The next step is to resolve the disturbance (8) 

 into its elements in accordance with Fourier^s theorem. We 

 have 



treated as infinitesimal), the resulting curve will have the required pro- 

 perty, and would exhibit a possible form for complete radiation. It 

 seems not unlikely that the law is here the same as that obtained below 

 on the basis of (8). 



* The reader may fix his ideas upon a stretched string vibrating 

 transversely. 



