352 Mr. G. G. Stokes on a difficult}/ in the Theory of Sound. 



gative value of -y- ; w here, as in the whole of this paragraph, 



denoting the velocity when / = 0. 



By the term continuous function, I here understand a func- 

 tion whose value does not alter per solium, and not (as the term 

 is sometimes used) a function which preserves the same alge- 

 braical expression. Indeed, it seems to me to be of the ut- 

 most importance, in considering the application of partial 

 differential equations to physical, and even to geometrical 

 problems, to contemplate functions apart from all idea of alge- 

 braical expression. 



In the example considered by Professor Challis, 



w?=?rcsin — {z — {a-\-iso)t}, 



where m may be supposed positive j and we get by differen- 

 tiating and putting t = 0, 



dw 27rm 2tt^ 



-t- — cos — , 



dz A A 



the greatest negative value of which is ; so that the 



A 



greatest value of t for which we are at liberty to use our results 



without limitation is - — , whereas the contradiction arrived at 



by Professor Challis is obtained by extending the result to a 



larger value of t, namely — . 



Of course, after the instant at which the expression (A.) 

 becomes infinite, some motion or other will go on, and we 

 might wish to know what the nature of that motion was. Per- 

 haps the most natural supposition to make for trial is, that a 

 surface of discontinuity is formed, in passing across which there 

 is an abrupt change of density and velocity. The existence 

 of such a surface will presently be shown to be possible, on 

 the two suppositions that the pressure is equal in all directions 

 about the same point, and that it varies as the density. I have 

 however convinced myself, by a train of reasoning which I do 

 not think it worth while to give, inasmuch as the result is 

 merely negative, that even on the supposition of the existence 

 of a surface of discontinuity, it is not possible to satisfy all the 

 conditions of the problem by means of a single function of the 

 form f{z— (a + w)t}. Apparently, something like reflexion 

 must take place. Be that as it may, it is evident that the 

 change which now takes place in the nature of the motion, 

 beginning with the particle (or rather plane of particles) for 



