1904.] Train of Waves emitted ~by a Hertzian Oscillator. 75 



X-X = n(p-p )-m( 7 -y Q ), a - a = m (Z - Z ) - n (Y - Y ), 

 Y-Y = /( 7 -y )-7i(a-a ), - j8 = w (X - X ) - Z (Z - Z ), 



Z-Z = m(a.-a )-l(f-p ) 1 7 - 7o = / (Y - Y ) - m (X - X ), 



W. 



These equations may be expressed in words in the statements that 

 the components of electric and magnetic force along the normal to 2 

 are continuous, and that the discontinuities of the tangential 

 components of the electric and magnetic forces are equal in magnitude 

 and are directed along lines at right angles to each other in such 

 a way that the discontinuity of electric force, the discontinuity of 

 magnetic force, and the normal to the surface, in this order, are 

 parallel to the axes of x, y, z in a right-handed system.* The case in 

 which there is no electric or magnetic force on the side of 2 towards 

 which it advances is included by putting (X , Y , Z ) and (a , /J , 70) 

 equal to zero, and the case in which the field on this side of the 

 surface is electrostatic is included by putting (a , j8 , 70) equal to zero. 



The conditions (4) have been established by a rather troublesome 

 process which may be replaced by the following simpler argument : 

 The ordinary equations (1) of the field fail at the surface of dis- 

 continuity 2 through the infinity of some of the differential coefficients 

 ex/Of", .... Consider the axis of x to be parallel to the normal to 2 at a 

 point P. Then as 2 passes over P the state of the medium at P 

 changes from that expressed by (X , ... a , ...) to that expressed by 

 (X, ..., a, ...). Suppose the change to take place in a very short 

 time 8t, and multiply both sides of the equations (1) by c8t. Then in 



r}X 

 the left-hand members we must write X - XQ for 8t, and similarly 



Of 



for the other quantities of the same kind. Again, we may put c8t = &c, 

 where 8x is the distance over which the small part of 2 near to P moves 



in the interval 8t ; and then the limit of ^- c8t or ^- 8x is the differ- 



ox ox 



ence of the values of {$ just before and just behind the surface 2, or it is 



/?o - P. The limits of such quantities as ^-0$, in which the differen- 



oy 



tiation is performed with respect to any co-ordinate other than x, are 

 zero. 



From the six equations (1) we deduce in this way the six 

 equations 



x-x = o, Y-Y O = -(70-7), z-z = A>-A 

 a - ao = o, -OS-A>)= -(Z -z), -(7-70)- YO-Y. 



* These results were given effectively in a paper by the author in ' Proc. 

 London Math. Soc.' (Ser. 2), vol. 1, p. 37 (1903). Equivalent conditions appear to 

 have been employed by O. Heaviside, ' Electrical Papers,' vol. 2, pp. 405 et seq. 



