Theory of a Contractile ^Ether to Optical Problems, 525 



Let Ix -\-my + nz— Yt give the position of the wave-front 

 at any instant. Let X, p,, v, be the direction-cosines of the dis- 

 placement, and let © be the amount of the displacement. 

 Then „ ^ ~ ~ 



and hence 



X ^^ = (A ~ B) dX X d^ +fM dJ +V dT) + BX V 2 ®, (6) 



&c 



Now let 



©-©^(te+mjf+ns-VO. (7) 



then 



8 = et(/\ + m A t + nv) (8) 



Substituting in the equations we find 



\ Px V 2 = (A-B)l(l\ + mfjL + nv)+B\ 1 



m Y 2 = ( A - B) m(ZX + wi/a + hf) + Bp, > , . (9) 



v^V 2 = ( A — B) n (/X + m/A + nv) + By ) 



Y 2 {p x l\ + pym/jL + p z nv} = A{l\ + mfjL + nv). . (10) 

 Now put 



c?=B/p x ; & 2 =B//> y , c 2 =B//> g . 

 Then 



BX^-lW(A-B)/(/X + 7?2 A 6 + H 



B At ^ 2 -l) = (A-B)m(a + m/A + 72i/) |>-, . (11) 



Bi~-l]=(A-B)n(/X + ^ + 



?iv 



and 



ZX. 



BV 2 {5+^ + ^}=A(a + ^ + ,v). . (12) 



Multiply the first of equations (11) by I, divide by (V 2 — a 2 ), 

 and so on, and add the three. Then we find 



/ Z 2 ?7l 2 ft 2 \ 



= (A-B)(/\ + ^ + nv)( TJ -^ + ^-p + ? ^ ? J.. (13) 



Hence, and from (12), 



I 2 rn 2 n 2 _ A 1 



V 2 -^ V 2 — ft 2 + V*-c 2 ~~ A-B V 2 ' ' ^ } 



This is the general form of the equation of wave-slowness, 

 without any assumption as to the relative magnitudes of A 

 and B. If the sether be incompressible (Rankine, Stokes, 



