CONTEMPORARY ADVANCES IN PHYSICS 301 



conforms to (45), it follows from (43) and (44) that 



^77 -1^2 



dx j \ dy / \ dz 



U- (^) 



The quantity in brackets, being the square of the magnitude of the 

 gradient of cp, shall be denoted by the usual symbol | V^|^- Then for 

 the square of the phase-speed we obtain wVlv^'K and for the phase- 

 speed itself: 



c==+n/\v^\, (47) 



The quantity n is 2r times the frequency. Also, it is the time- 

 derivative of (p, as follows directly from the definition in (41) ; conse- 

 quently: 



c = = (d<pldt)l\v<p\. (48) 



This is an equation with a very important meaning, which will now be 

 displayed. 



Select any point in the medium and any moment of time ^o, and de- 

 note by <po the value of (p prevailing then and there. Singular cases 

 excepted, there is an entire surface containing the point in question 

 everywhere over which (p has the same value (po. This surface is by 

 definition a wave-front. Call it ^o. At a slightly later moment 

 ^0 + dt, there will also be a surface everywhere over which the value 

 of <p is (po. It will however not be the same surface ^o, but another — 

 a wave-front Si so placed that from any point Po on So the nearest 

 point on 6"! is reached by measuring the length cdt along the line 

 normal to ^o, in the sense in which (p is decreasing (the sense opposed 

 to the gradient of (p) . 



To see this, imagine a particle which at the instant /o is travelling 

 through Po along the direction normal to ^o in the sense just stated, 

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

 a point where the value of <p then prevailing is given by the formula: 



(49) 



(po -{- d(p = (po — \^ip\ds -{- i -^ ] dt 

 = <po — u\'7(p\dt + I — j dt, 



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

 the normal to the surface So, and along this normal the slope of the 

 function (p is equal to JV'/'I, and meanwhile at each point of space (p 

 is varying directly with time at the rate {dcp/dt). Now if the imagi- 



