660 BELL SYSTEM TECHNICAL JOURNAL 



is traveling through Pq, along the line normal to ^o, with a speed 

 to be designated by u. At the instant to + dt it occupies a locality 

 where the value of W is given by the formula : 



f dW\ 

 W^ + dW = W^ +\VW\ds + [^)dt 



= PFo + u\VW\dt - Edt, (14) 



for in the time-interval dt it travels over a distance ds = udt along 

 the normal to the surface 5o, and along this normal the slope of the 

 function T^is equal to \^W\ , and meanwhile at each point of space W 

 is varying directly with time by virtue of the term — Et occurring in 

 the equation (9) which defines it. Now if the imaginary vehicle 

 happens to be moving with just the speed defined by the equation 



u = ElVW = EHlmiE - V), (15) 



the coefificient of dt in equation (14) vanishes; that is, the vehicle as 

 it moves outward keeps up with the prescribed value of W; but this 

 is the same thing as saying that the value given for u in (15) is the 

 speed of the wave-front. 



At this (if not an earlier) stage of the argument, one begins to 

 wonder what W "really is"; one turns back to seek the original defi- 

 nition of this artfully constructed function, so suddenly advanced from 

 an auxiliary to the central r61e of the theory; one tries to grasp it, 

 to form an image of it. I can do little to satisfy this very human 

 craving. I can point out that W is that quantity "action" with 

 which the Principle of Least Action has to do; this feature scarcely 

 makes it more conceivable, but at least enhances its prestige. I can 

 point out that since no one has ever seen what moves or is inside an 

 atom, the conception of waves in an intangible medium curling and 

 flowing around a centre is no more far-fetched than the conception 

 of intangible particles sailing in ellipses around a nucleus. (To this 

 one can reply that the planets in their courses supply a visible analogue 

 for the notion of revolving electrons, but no one has seen in the sky 

 the wave-fronts of the function W.) I can point out that for some 

 important purposes, notably the prediction of the Stationary States, 

 it makes no difference what the function W "really is" — no more 

 difference than it makes to the solver of a quadratic equation whether 

 the variable be called x or /, whether in the mind of the propounder 

 of the equation it stood for distance or for time. One might in fact 

 begin with the forthcoming equation (20) as foundation, laying it 

 down without introduction or apology; yet there must be deep-lying 



