( 738 ) 



sions and represent hencefortli by 6 the whole electric force. Then, 



t 



if (§.<^ ziz bco«2,T— , we shall have 



T 



T^w^^-'i^i'-vy"' ■ ■ ■ <"" 



integrated over all directions on that side of F where 2' has positive 

 values. We therefore obtain the resultant Inminous vibration in an 

 arbitrary point P as the sum of small vibrations, belonging to a 

 great number of systems of plane waves of all possible directions. 

 These vibrations differ from each other in amplitude and in phase,. 

 The changes of phase are determined by those of the quantity 



TV' 

 Since TV means the wave-length in the crystal for the direction 

 considered and z' = r cos ?, the phase will vary very much by small 

 variations of S, i.e., of the direction of the wave system in question. 

 There is one direction for which 



Jv 



takes a maximum value. This is the direction of the wave-normal 

 OQ to Avhich OP corresponds as first ray. Indeed, z'/TV is 

 proportional to the time in which the vibrations of a certain wave- 

 system arrive at P and this time is really a maximum for the 

 system whose normal is OQ. We shall prove, that the resultant 

 vibration at P is the same as it would be, if we had only to do 

 with wave systems of this latter direction and of directions in the 

 immediate vicinity of it. To this effect we shall fix our attention 

 on an arbitrary normal ON, making an angle <J> with OQ, writing 

 i|j for the azimuth of the plane NOQ with respect to a fixed plane, 

 which passes tlirough OQ, and for which we might take the plane 

 POQ- We shall not however introduce t)> and $ as variables but tp and 



u = — cos s, 

 V 



if F"„ is the velocity of propagation of the plane wave, having OQ 



for its normal. Further we put 



2nt , 2jr/- , . ^^^ , , 



= h , = a , any = sin <J> — da d\b , 



T TV, '' du ^ 



Then 



2t 



/r n-ttbx'T ^ ö<J> 



I sin ( qu — k) sin — du d\b , . . . (11) 



