and Electromagnetic Theory. 325 



main fact indicated, and the second and third members of (17) 

 differ only geometrically. When the middle term is used 

 we are in effect interpreting from a moving standpoint 

 radiation which is proceeding in free aether, without re- 

 ference to reflexion or retraction. II: we form a collective 

 relation as before and take first the simpler case u = v = } we 

 have 



p 4 ) (n - w/\ ) (1 - icn V) dn = ^\ q n r Jn r ! { 1 - ic 2 (1 - n r 2 )/V 2 } 2 



or =^ 4 j 1 v^/(L-ir/V 2 ); 



the use of the kinematical cosine v= (w— w/V)/(l — wn/V) 

 being easier. The result is 



In the general case a double integration is needed, viz. : — 



*JR" ~ vX 1 - v) dnd f = ^ fJ^'^Vt 1 ~ v* + tCP 



but as each of the variables n r $> r has full range^ i. e. to 1, 



V 2 



and to 2tt, the integral on the right is ct 4 (l — ^r\ * na * 



on the left being unchanged. This gives a mean alteration of 

 period due to Doppler's effect and a diminished rate of 

 approach of energy. If we are dealiug with refraction B 2 is 

 used under the sign of integration, because the transformation 

 applies only to a part of the energy, and the integral is more 

 complicated. But the ct thus obtained is also the us' 1 of the 

 refracted ray ; there is no further change. 



Through the factor belonging to Doppler's effect a proper 

 contribution to mechanical work is levied on the section of 

 the incident energy which suffers refraction. Again, since 



P (i-u/v)= 5 y(i-u'/v) = w , 



the process of reflexion can be divided into two, the first a 

 change from p to vr, the second a change from ot to />', each 

 accompanied by a levy to mechanical work. Thus in effect 

 a levy is made on the whole incident energy, and a further 

 lew on the energy of the reflected wave, and each is associated 

 with a change of standpoint. 



The pressure, as pointed out above, can be found (in its main 

 term) by the use of a statical value of A 2 and B 2 ; the collective 

 yalues of 7/ and of -Brand V 7 " require a knowledge of the way in 



