5G8 Lord Rayleigh on a Physical Interpretation of 



aerial vibrations. Let us consider the most general vibra- 

 tions in one dimension f which are periodic in time 2tt and 

 are also symmetrical with respect to the origins of f and t. 

 The condensation s, for example, may be expressed 



s = b - i rb l cos f cos t-\-b 2 cos 2f cos 2^ + . . . , . . (2) 



where the coefficients b , b u &c. are arbitrary. (For simplicity 

 it is supposed that the velocity of propagation is unity.) 

 When t = 0, (2) becomes a function of f only, and we write 



F(f) = 6 -f6icosf + 6,cos2f -f . . ., . . . (3) 



in which F(f) may be considered to be an arbitrary func- 

 tion of f from to it. Outside these limits F is determined 

 by the equations 



F(-0=F(f+&r)=F(fl (4) 



"We now superpose an infinite number of components, 

 analogous to (2) with the same origins of space and time, 

 and differing from one another only in the direction of f, 

 these directions being limited to the plane xy, and in this 

 plane distributed uniformly. The resultant is a function of 

 t and r only, where r= \/'(^ 2 +// 2 ), independent of the third 

 coordinate z, and therefore (as is known) takes the form 



s = a + a 1 J (r)cost + a 2 J {) (2r)cos2t-\-a s J {3r)co$3t + . . ., (5) 



reducing to (1) when £ = 0*. The expansion of a function 

 in the series (1) is thus definitely suggested as probable in 

 all cases and certainly possible in an immense variety. And 

 it will be observed that no value of f greater than ir con- 

 tributes anything to the resultant, so long as r < ir. 



The relation here implied between F and / is of course 

 identical with that used in the purely analytical investigation. 

 If cf> be the angle between f and any radius vector r to a 

 point where the value of / is required, g=rcos<fi, and the 

 mean of all the components F(£) is expressed by 



f(r)=-C F(rcosf)<ty (6) 



The solution of the problem of expressing F by means of 

 /is obtained analytically with the aid of Abel's theorem. 

 And here again a physical, or rather geometrical, interpreta- 

 tion throws light upon the process. 



* It will appear later that the «'s and 5's are equal. 



