554 Mr. J. McCowan on the Solitary Wave. 



this, I have only to point out that I have nowhere said that it 

 was inadmissible. I am not therefore concerned with his 

 further statement regarding the expression of non-periodic 

 functions by series of periodic functions : Fourier's repre- 

 sentation of such functions in definite integral form is well 

 known. 



He goes on to say further, " I cannot agree with Mr. 



McCowan that the form which he proposes at 



p. 58 to substitute, that of a series involving exponentials 

 in which the coefficient of x in the index is real, is (at least 

 for my purpose) admissible." It is " inadmissible, on account 

 of the discontinuity of the expression." To this, again, I 

 must reply that I have nowhere proposed to use any such 

 series. To clear away whatever obscurity may remain, let 

 me quote the part of the paper " On the Theory of Oscillatory 

 Waves " which I have criticised in § 12 of my paper. In § 2 



he says ' f the general integral of (2) [-r-^ +-p! = 0, where 



<£ is the velocity potential,] is 



the sign 2 extending to all values of A, m, and n, real or 



imaginary, for which m 2 -\-n 2 = In the present 



case, the expression for <£ must not contain real exponentials 

 in Xj since a term containing such an exponential would become 

 infinite either for x=— go, or for ^=+co, as well as its 

 differential coefficients which would appear in the expressions 

 for u and v [the components of the velocity] ; so that m must 

 be wholly imaginary.''' 



The italics and the added explanations within square 

 brackets are mine. I understand this to mean that it is only 

 for the reason given that real exponentials are omitted. 

 Eemembering then the difference in the notation, that my 

 " imaginary m " is his real m, and that my z is his y, and 

 that my <£ + TJx is practically his <£, refer to § 12 of my paper. 

 I explain that he has " put aside imaginary values of the m 

 as inadmissible/' having " concluded that such a value would 

 imply infinite velocity, &c, either when a?=+co, or when 

 x=. -co ;" and then I go on to criticise this view, saying " but 



this is not necessarily so for in fact the value of </> 



given in (5) 



[ TT T-. sinh mx ~| 

 6 + TJx = l)a - r , 

 r cos mz + cosh mxj 7 



gives a well-known expansion 



[<j> + Ux = JJa + 2UaS(- Je- mix cos miz 



