378 A. A. Michelson and E. W. Mbrley — Influence of 



the beginning of this time the quantity of ether in the volume 

 BC, (if S=surface of the base of the prism,) is Sddt. At the 

 end of the time the quantity will be Sddt(l-\-J). Hence in this 

 time a quantity of ether has been introduced into this volume 

 equal to SdcltJ. 



It is required to find what must be the velocity of the ether 

 contained in the prism to give the same result. Let this veloc- 

 ity be xd. The quantity of ether (density =1 + J) introduced 

 will then be Saddt(l + J) and this is to be the same as Sddid, 



whence x= -. But the ratio of the velocity of light in the 



external ether to that within the prism is n, the index of refrac- 

 tion, and is equal to the inverse ratio of the square root of the 



,v»_ i 



densities, or n=</l-\-d whence x = — — which is Fresnel's 



n 



formula.* 



* The following reasoning leads to nearly the same result; and though incom- 

 plete, may not be without interest, as it also gives a very simple explanation of 

 the constancy of the specific refraction. 



Let I be the mean distance light travels between two successive encounters 

 with a molecule ; then I is also the " mean free path" of the molecule. The time 



occupied in traversing this path is t= h— , where a is the diameter of a 



v / v 

 molecule, and b=l— a, and v, is the velocity of light within the molecule, and v, 



the velocity .in the free ether ; or if ,«=— then t=- . In the ether the time 



would be t.= , hence n=— = — — — . (1) 



v t, a + b 



If now the ether remains fixed while the molecules are in motion, the mean 

 distance traversed between encounters will no longer be a + 5, but a+a+b+B; 

 where a is the distance the first molecule moves while light is passing through it, 

 and 3 is the distance the second one moves while light is moving between the 



6 d 



two. If 6 is the common velocity of the molecules then d= — a, and B= ~~~3) ^' 



The time occupied is therefore — + K or — + — - a . The distance traversed in 



v, v — v v v — a 



this time isa + 5+( — + a ) d ; therefore the resulting velocity v= r — I- 0. 



\ v v—uj pa o 



V V — d 



T, ft 



Substituting the value of n=- and neglecting the higher powers of — , this 



a + b v 



(2) 



becomes v= — + ( 1 :, ; ) ft - 



n \ n 2 a + bj 



But — is the velocity of light in the stationary medium ; the coefficient of 6 



• , ,. , n m 2 — 1 1 a ,„x 



is therefore the factor x= — 5 — h — s r- w 



ra a n l a + o 



It seems probable that this expression is more exact than Fresnel's ; for when 

 the particles of the moving medium are in actual contact, then the light must be 



n 2 — 1 

 accelerated by the full value of 8: that is the factor must be 1, whereas — - -j— can 



