320 Mr. R. Har greaves on Radiation 



and the above takes the simpler form 



■Vn" 



/Vn" w\ „ 

 {Yn-w)d n^ -y-^) d n m 



U ""V J 



(v-u) 3 — 7? — u'V ' ' ' ' ^ 



the transit-velocity, or component of ray-velocity, depending 

 only on n". Introducing the fourth powers of p and p", 



p^-\Yn~ic){Y-V)dn=p"^-t(^ - ™^Q -^dn" . (12) 



With the wave-length as variable, since p"/£l=:27r/\", 



\ : \" = l: l"=m : m" = Y-wn :0,-wn" 



V TT" 

 = V-U:X2-U" = V-U: - — %, (13) 

 and ^ P 



X -4 ( V n ^) ( V_U)tfn=X"-^ 



We now give a transformation of a different kind, in which 

 the ray-cosines define direction, and a period-number is used 

 which has reference to a moving standpoint. For free sether 

 thisisw = p(l-U/V),andforthe dielectric «r" =p"(l-U'7Jl), 

 and by (9) w = w". With (11) the transfer from p to vj or vr" 

 is given by 



f(J n - w )(l- U/Y)dn/Y d = ^(Yn - w)dn/(V - U) 3 



-"•(^-^7(J-7)" < 15 > 



Ray-cosines (Z r w r ?? r ) are connected with the wave-normal 

 cosines by the scheme 



Y r l r =Yl~ u, Y r m r =Ym—v, Y r n r =Yn — w, 

 V„ 2 =V 2 -2VU + 2w 2 , Y, 2 + 2V r U r =V 2 -^ 2 , 1,(16) 



V r 2/ r Z=V-U, Ytl r l=Y r +TJ r = VV«-2M 2 + U r ^ 



Y r being a ray-velocity and U r standing for %l r u. In a 



Y^ ?/ 

 dielectric — , -,, ... take the places of Y, w, ... The 



.Pi* . 



radical is the wave-velocity in the direction of the ray. 

 Comparing \/V*-2w 2 + IJ7-U r and VV 2 -^ 2 + U r 2 + U r , 

 which are the numerical values of V r , with V — U and V + TJ 

 belonging to translation with or against the wave, the radical 



