and Electromagnetic Theory. 321 



plays a part corresponding to that of V. For the transfor- 

 mation use the variables (>?,. <£ r ), (;i<£>), where tan<£ r = ?w r /Z r , 

 tan (j> = ni/l ; the Jacobian after some reduction gives 



*\dn d<j> ~ d<f> dn / { J ' 



Hence 



^\Yn-w)dnd(j)/{Y-V) s = ^Y r n r dnd^/(Y-Jjy 

 =W n r dn r ^ r /{V 2 -^M 2 + U r 2 } 2 , 



since by (16) V(V-U)=V r \/Y 2 -Zu 2 +U*. 



Apply to the third member of (15) with d(f> appended a 

 like transformation, and we get 



p\n - ic/Y)(l- U/V) dn d<f> = ^ n r dn r d<f> r /{ 1 -tu 2 /Y 2 + U r 2 /V 2 } 2 

 =#*V« n r " dn r " d<f> f "l{l-2u*/v?V* + TJ r " 2 /fi 2 Y 2 } 2 . (17) 



§ 5. Consider first the statical case, in which ot coalesces 

 with />. It is then indifferent whether p" or \" is used as 

 variable, and the form of equations (12) and (14) suggests 

 for the ratio of energy-contents in the two media 



x : %" = V/X 4 : V"/\" 4 =p 4 /V 3 : p" l /Y nz , . . (18) 



with V" for Y/fi : since p—p 11 or ^,X"=X, %" = yu, 3 ^. W© 

 may then write p/6=p"/d" or its equivalent \6 = tu\"6". 

 This is consistent with the use of "Wien/s formula in the shape 



V"-y'3 dp"f(p"/6") or V"X"- 5 d\"f(Y"/X"6"), 



for the result of integration through an infinite range of 

 p" or X" is proportional to 0" 4 /V" 3 , and the condition of 

 statical exchange Yn(d i /Y*)dn = Y"n"{d" i /Y" 5 )dn" makes 

 0=6>" follow from ndn/Y 2 = n"dn"/Y" 2 . 



When there is motion the use of (p, p f ) or (X, X') in reflexion 

 is indifferent. But there is this real difference between the-' 

 original and reflected waves on the one hand, and the re- 

 fracted wave on the other, that for the former the moving- 

 standpoint represents a momentary phase, while for the latter 

 it is permanent, and moreover is that with reference to which 

 we naturally seek to interpret matters. Now an argument 

 llx' — Yt becomes llx — (V — U)£, when by the use, of 

 x' = x + ut, . . a change is made to a moving standpoint; the 

 change alters the period not the wave-lengthy and this is 

 equally true if the variable velocity 12 is used instead of V. 



Phil. Mag. S. 6. Vol. 9. No. 51. March 1905. . \ Y. ) .. r ' r ] 



"■• . . 



