On Beep Water Waves. 385 



It appears that if we take A = B so that half the electrons are 

 in crystals and half distributed at random, the calculated values 

 of Ifl/Igo agree fairly well with those found by Orowther. 



The excess of scattered radiation in the emergence direction 

 seems therefore to be easily explicable on the theory that 

 Rontgen rays are very short electromagnetic waves or pulses. 

 The only property of Rontgen rays which is not easily 

 explicable on this theory seems to be that of causing the 

 emission of high-velocity electrons. The most reasonable 

 way of explaining the emission of these electrons seems to 

 be that proposed by Planck *, according to which matter 

 absorbs radiant energy continuously but only emits it in 

 definite amounts inversely proportional to the wave-length 

 of the radiation. This view enables the ordinary electro- 

 magnetic wave theory of light and Rontgen rays to be retained. 



XL I. On Beep Water Waves. By J. R. Wilton, M.A., 

 B.Sc., Assistant Lecturer in Mathematics in the University 



of Sheffield f. 



SOME time ago, when studying Stokes's papers on Oscil- 

 latory Waves (Collected Works, vol. i. and vol. v.), 

 I noticed that the work of determining the coefficients in 

 the expansions of the coordinates might be very considerably 

 reduced, and that the order of magnitude of any coefficient 

 was determinate. These expansions may, without great 

 labour, be carried to a high order of approximation, and the 

 form of the wave may be drawn for a value of the amplitude 

 not far short of that for the highest wave. 



By a change in the notation we are enabled to treat the 

 progressive wave directly, and at the same time to retain all 

 the advantages of the customary method of reducing the 

 problem first to one of steady motion. The advantage of 

 the change will be more apparent to anyone who attempts to 

 discuss the stability of the wave. 



We shall use throughout the abbreviations : 



s = x + iy, z' = x-uy, 



v: = (f) -f-t^r, w' — (j) — i\jr, 



X v /7 X 



\lT IV , ., 2tt w' 



* Sitzungsberichte der Koniglich Prcussischcn Akademie der W 

 schaften, April 3 ; 1913. 



t Communicated by the Author. 



Phil. Maq. S. G. Vol. 27. No. 158. Feb. 1914. 2 C 



