350 Mr. G. G. Stokes on a difficulty in the Theory of Sound. 



laws which the actual motion tends indefinitely to obey when 

 the magnitude of the velocity is indefinitely diminished. And 

 if, moreover, we have every reason to believe that the first 

 order of small quantities contains all that are sensible in prac- 

 tical cases, we have arrived, not merely at abstract mathema- 

 tical results, but at important physical laws. Nevertheless it 

 will be admitted on all hands, that any apparent contradiction 

 arrived at by employing exact equations is deserving of serious 

 consideration. 



The difficulty already alluded to is to be found at page 496 

 of the preceding volume of this Magazine. In what follows 

 I shall use Professor Challis's notation. 



Without entering into the consideration of the mode in 

 which Poisson obtained the particular integral 



w=f{z-(a + w)t}, (1.) 



it may be easily shown, by actual differentiation and substitu- 

 tion, that the integral does satisfy our equations. The func- 

 tion/being arbitrary, we may assign to it any form we please, 

 as representing a particular possible motion, and may employ 

 the result, so long as no step tacitly assumed in the course of our 

 reasoning fails. The interpretation of the integral (1.) will 

 be rendered more easy by the consideration of a curve. In 

 fig. 1 let oz be the axis of z, and let the ordinatesof the curve 

 represent the values of no for t=0. The equation (1.) merely 



Fig. 1. 



Fig. 2. 



asserts that whatever value the velocity no may have at any 

 particular point when t = 0, the same value will it have at the 

 time t at a point in advance of the former by the space (a + w)t. 

 Take any point P in the curve of fig. 1, and from it draw, in 

 the positive direction, the right line PP' parallel to the axis 

 of z, and equal to (a + w)t. The locus of all the points P' will 

 be the velocity-curve for the time t. This curve is represented 

 in fig. 2, except that the displacement at common to all points of 

 the original curve is omitted, in order that the modification in 

 the form of the curve maybe more easily perceived. This comes 

 to the same thing as drawing PP' equal to wt instead of {a-\-<w)t. 

 Of course in this way P' will lie on the positive or negative 

 side of P, according as P lies above or below the axis of z. 

 It is evident that in the neighbourhood of the points a, c the 



