162 Mr. T. H. Havelock on 



infinity; 7 and Y will vanish at the latter limit, and we see 

 from (24) that we may regard the mechanical force upon the 

 medium as a pressure upon the surface equal to the mean 

 value there of 



that is, equal to the mean density of the energy of the 

 vibrations in front of the surface. 



§ 5. This case may be worth considering directly. We 

 have in the free aether to the left of the origin 



Y=AcosA:(ci— .i')-f-Bcos{/c(c^ + .i')-f Sn . y. 



7 = A QO^ K{ct — X) — B cos \K\Ct-\-,v) + Sj- J ' 



representing the incident and reflected waves. 

 And to the right of the origin we have 



Y = C^-"*«^-cos {ic{rt — nx)-\-h'} 



y =?iCf-"^''^[cos {K{ct—mv) +6'} + a: sin {K{ct — mv) + 8^}] 



representing the transmitted wave. 



The surface conditions at x = are continuity of the tan- 

 gential components of ethereal displacement and magnetic 

 force ; these give 



A + Bcos8=Ccos8^ ^ 



Bsin8 = Csin8^ I 



A-Bcos8 = 7iC(cos8^ + A-sin50, h ' ' ^^^^ 



Bsin8=?2C (^ cos 8' -sin 80- J - 



And these lead to 



tan b = .,.-. ■ 79. ; tan 0' = 



mr) + 8^}]J ^^ 



©■= 



n^(l + F) + l-2n 



Also the true current v' is given by 



2)^= Real part of — ; — -s— 

 ^ 47r ^t 



==Real part of ^^'(^-/^)'- "^ i^,Q^ u{ct-.n,i-iic)\ 



= ^Ce-''''^'[2n'k cos {K{ct-na,') 4-^} 



-{n^(^l-k'-)--l}^[n{K{ct-7i.v)+S^\]. . (29) 



