CONTEMPORARY ADVANCES IN PHYSICS 659 



or, seeing that the quantity ^{dWIdx)- + {dWIdyf + {dWIdzf is the 

 magnitude \^W\ of the gradient of the function W, the gradient of 

 a function being a vector well known in vector-analysis and denoted 

 by prefixing the sign V or the abbreviation grad to the symbol of the 



function : 



|V1^|2 = 2m{E - V). (13) 



This equation governs the space-derivatives of the function W\ 

 it is complemented by the equation derived from (9) which governs 

 the time-derivative of W: 



dW/dt = - E. (13a) 



At this point the procedure of classical mechanics and the procedure 

 of wave-mechanics diverge from one another. 



Were we to follow the classical procedure, we should perform certain 

 integrations and other processes, and arrive in the end at equations 

 describing trajectories or orbits— in the particular case of an inverse- 

 square central force-field, at equations describing elliptical orbits. 

 The particular elliptical orbit to which the reasoning would conduct 

 us would be determined by the value which had originally been assigned 

 to the energy E, and the values which we attributed to the various 

 constants of integration supervening in the course of the working-out. 

 The function W, having served its purpose, would have vanished from 

 the scene, leaving with us the electron swinging in its orbit within 

 the atom or the planet in its orbit across the heavens. 



The procedure of wave-mechanics, however, is based upon the 

 observation that the equations (13) and (13a) together are the descrip- 

 tion of a family of ivave-fronts, traveling with the speed Ej -^Im {E — V) 

 through space. 



To display this aspect of the equation, let it be supposed at some 

 prescribed time-instant to the function W has a certain prescribed 

 constant value Wo at every point of a surface ^o; for instance, that 

 at time /o = 1 it is equal to unity all over the sphere of unit radius 

 centered at the origin. It is to be shown that at a slightly later 

 instant to + dt there is again a surface everywhere over which the value 

 of W is Wo, this not however being the same surface So, but another — 

 a surface Si so placed that from any point Po on ^o the shortest line to 

 ^i is perpendicular to ^o and its length is (£/V2w(£ — V)dt). 



This is easily shown. Imagine a vehicle ^ which at the instant to 



* I use this word instead of "particle" lest this entity be confused with the moving 

 electron to which the foregoing equations relate. The electron does travel along 

 a curve normal to the surfaces of constant W, not however with the speed u about 

 to be defined, but with a different speed related to u in a curious and significant way 

 (cf. the allusion on p. 695J. 



