on the Undulatory Hypothesis of Light. 475 



atoms contained in a slice of the medium of sensible thickness, 

 their mean effect would be inadequate to produce the observed 

 amount of retardation. 



Accordingly the calculation of the total retardation will have 

 to be made in the following manner. Let the velocity of the 

 aether at the given distance X from the origin at the time t be 



m sin — — ; and at a greater distance x, reckoned in the direction 



A, 



of propagation, let the velocity be msinf — 6). Then, if 



we take a slice of the medium such that its limiting planes are 



at the distances x— ~ and x-\-- from the origin, u being very 



small, from what is proved above the mean relative velocity of 

 the sether and the atoms will be 



, . (Zirht a , A 

 m cos 9 sin I — u + <p h 



and to this quantity the mean of the reflected velocities will be 

 proportional. We may therefore suppose that the retarding 

 effect of the reflexions, from all the atoms of the slice is, at the 

 given distance X, 



hum cos <f> sin ( — 6 + (f>\ 



the coefficient h being a function of a?— X. If the whole of 

 that portion of the medium from the atoms of which the re- 

 flexions are effective at the same distance X be divided into like 

 slices, the total retardation may be represented by the summation 



S . hum cos <£ sin ( — r- — # + <£), 



in which h and are different for the different slices. But the 

 terms of this series are such that their sum is known to admit of 

 the form ^nrbt , 



Km cos <£ sin( — c' + cp ). 



This may consequently be taken to be the amount of the retar- 

 dation which the aether suffers at the position where the velocity 



is m sin — — . But the condition of diaphaneity, before stated, 



requires that this retardation should have a constant ratio to the 

 actual velocity of the aether. That this may be the case we must 

 have — c'-\-cj) = 0, which equation may be considered to be the 

 theoretical expression of the condition of diaphaneity. Both <f> 

 and d are quantities depending on the number, magnitudes, and 

 arrangement of the atoms of the medium ; but for carrying on 



212 



