282 Mr. L. de la Bive on the Introduction of 



the electron is udt ; and on projecting it on the radius vector 

 EP , we have 



EX = —dr = udt cos (ur), 



reckoning r in a direction from E towards P . On inserting 

 for dr its value given by (7), we find 



dt = dt(l^ cos (ur)\ (8) 



I call t the time uf transmission ; and the relation (8) shows 

 that for a point P the ratio of the elementary variations of 



t and t is constant. The ratio -^ which may be termed 



JDoppler's Factor, is the ratio of the duration of a phenomenon 

 of emission to the duration of the same phenomenon when 

 transmitted, such duration being supposed sufficiently short 

 in comparison with the time required by light to travel from 

 E to P . Its value becomes unity when the velocity of 

 propagation is at right angles to the radius vector, since in 

 that case the source of emission neither recedes nor approaches ; 

 and it would become zero if u were equal to v and the angle 

 between the two directions were to vanish, since the source 

 itself would then, with all its consecutive emissions, arrive at 

 the same instant at P . 



Now that the ratio -77 , which we shall denote by K, is 



known and has been shown to be constant whatever the value 

 of r, and that equation (II.) as well as the other equations of 

 the system relate to the transmission space, we see that, 

 replacing, for the sake of greater clearness, the vectorial by 

 the cartesian expressions, 



1 d*4> p/-0 d-<f> d 2 <j>i 2 



r 2 dt ~ Ld,c 2 dy? d—J 

 or, putting 



dt = Kdt, d t r = I\dd: dy Q = ^Kdy 3 dz = Kdz, 



J. d*$ 

 V dt 2 ' 



aar ay* dz- J 



The first member of the equation again assumes the form 

 which lends itselE to the change of variable and the appli- 

 cation of Beltrami's theorem, but it becomes necessary to 

 examine whether this theorem is not modified by the change 

 £ variables. 



