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medium, arid out of one medium into another ; but he had 

 not attempted to account for his hypotheses, nor to connect 

 them together by any known principles of mechanics ; and 

 the only evidence in favour of their truth, was the truth 

 of the results to which they led. He had observed, how- 

 ever, that these hypotheses were not independent of each 

 other ; he had ascertained that the laws of reflexion at the 

 surface of a crystal were connected with the laws of propa- 

 gation in its interior ; and he had thence been led to con- 

 clude that all these laws and hypotheses " had a common 

 source in other and more intimate laws not yet discovered.'* 

 He became impressed, in short, with the idea " that the 

 next step in physical optics would lead to those higher and 

 more elementary principles by which the laws of reflexion 

 and the laws of propagation are linked together as parts of 

 the same system." 



This step the author has now made ; and the present 

 paper realises the anticipations scattered through the for- 

 mer. Setting out with the general dynamical theorem 

 expressed by the equation 



SSSrf^<^2^rf«(585+08.) + 5 2"') =llldxdyd^Sv, (1) 



where S, Vt ?» are the displacements at the time t of a par- 

 ticle whose co-ordinates are x, y, ^, and where the density 

 of the ether is supposed to be unity, as being constant for 

 all media, the author determines the form of the function v, 

 for the particular case of luminiferous vibrations, by means 

 of the property which may be regarded as distinguishing 

 them from all others — namely, that ihey take place entirely 

 in the surface of the wave. From this property he shows, 

 in the first place, that v is a function of the three dif- 

 ferences 



^^^ ^^^ ^-.^. 

 dz dy^ dx dz^ dy dx' 



2k2 



