CONTEMPORARY ADVANCES IN PHYSICS 293 



are all ' the functions of the pair of variables x and /, in which these 

 variables appear coupled together into the linear combination {x — ct.) 

 Using/ as the general symbol for a function, we may write 



s=f{x-ct). (8) 



When such a relation prevails, any value of 5 which occurs at a given 

 place x at a. given moment / recurs at any other moment t' at another 

 place x', distant from x by the length (/' — t)lc. All of the values of 

 5 existing at t are found again in the same order at t', but they have all 

 glided along the x-direction through the same distance (/' — /)/c. 

 The form, the profile, the configuration of the string are moving along 

 with the speed c, although the substance of the string is oscillating only 

 a little, and not even parallel to the :x:-direction. Now this is the 

 property which to a certain degree of approximation ripples on water 

 display; this in fact supplies the elementary and restricted definition 

 of wave-motion, out of which by generalization and extension the 

 wave-theory has grown. 



Thus we see that there is reason for calling (7) a wave-equation, 

 and identifying the constant c with the speed of the waves. Yet there 

 are also solutions of (7) which are not of the form (8), and these do 

 not correspond to an unchanging profile of the string travelling along 

 with a constant speed, though by mathematical artifice they may be 

 expressed as a summation of such ; and nothing is easier than to find 

 solutions in two or three dimensions of the general equation (6) which 

 do not bear the least resemblance to a regular procession of converging, 

 flat, or diverging waves. The question then arises: is there a feature 

 common to all solutions of the "wave-equation" fitted to serve for a 

 general definition of wave-motion? 



I will now show — in the manner of Kirchhoff and Voigt — that there 

 is such a feature, and it is precisely the one already proposed as a 

 definition for wave-motion. If 5 be a function conforming to (6), and 

 U a function related to 5 according to (3), then the value of 5 at any 

 point at any moment is determined by the values of U and its partial 

 derivatives dU/dx, dU/dy, dU/dz, over any surface surrounding that 

 point.* The proof is long and intricate; bur for anyone who desires 

 appreciate the nature of wave-motion, it is not superfluous. 



To prove the theorem we have to manipulate the vector (call it W) 

 of which the components are : 



1 Exceptions being made for functions which do not have derivatives, and other 

 curiosities of the mathematicians' museum. 



* The necessary requirements for continuity in 5 exclude sources of light from 

 the region of integration. 



