I 



OF LUMINOUS VIBRATIONS. 593 



15. Hitherto we have reasoned on the supposition that no extraneous /orce acted on the aether. 

 It is quite possible tliat a ray, after taking its original form and direction, may be modified 

 subsequently in both these respects, by the action of forces, and retain the new form and direction 

 after the action of the forces has ceased. For instance, in the case of the ordinary reflexion of a 

 ray, forces act upon it for a short time and through a short space at the surface of the reflecting 

 medium, which, as they do not act symmetrically with reference to the axis of the ray, alter the 

 form of /. The analytical fact that this function is given generally by the integration of a partial 

 differential equation, and therefore not necessarily always of the same form, is quite consistent with 

 such an alteration. But on the principle that the transverse motion in the modified ray, so far 

 as it corresponds to pha'uomena of light, is still defined by an exact expression, the new form 

 of/ will be consistent with equation (18). Consequently, as A and A' in that equation are 

 arbitrary, the new ray will either be completely polarized, or will consist partly of a common ray 

 and partly of a polarized ray. We cannot however suppose any alteration of the function <p, 

 unless the forces be such as to destroy the luminous character of the ray ; for on the particular 

 form of (p which we found in Art. 4, depends the uniformity of propagation, a property which a 

 ray of light is supposed to retain under the modifications here contemplated. It is unnecessary 

 to point out the accordance of the above theoretical inferences with observed facts. 



16. A ray may also be modified by forces which act upon it continuously, as is the case on its 

 intromittence into a transparent medium, the modifying forces being the retardations which the 

 vibrations suflTer by encountering the atoms of the medium. This kind of modification I have 

 considered in my Paper on the " Transmission of Light through Transparent Media, and on 

 Double Refraction." (Cambridge Philosophical Tra7tsactions, Vol. viii. Part iv. p. 524.) I 

 have seen no reason to correct the Theory therein contained, and have only to remark, that the 

 approximate equation in p. 529, which determines /, may be arrived at by reasoning similar to that 

 in Arts. 2 and 3 of this Paper, as follows. We have, as in Art. 7 of the Paper cited, 



,, ds dw 

 (-■- . -r +-, — ^ 0. 

 dz dt 



Hence, 



d . fsdt 



doc dy dz 



no arbitrary function of co-ordinates being added for the reasons heretofore given in .Art. 1. 



Jl V w 



Consequently —^dx + —-dy+ —, dz must be an exact differential. This will be the case if 

 ^ ^ a'- h- c' 



s=fcp', f being a function of x and y, and ch' a function of z and t. For then, 



/.,<//=/r^'d.=/^;sothat, i^ = ./,g, ~ = <P%y 



w ,dd) , u V w idf df \ dd) . , 



- =/ /^, and —dx + T7-,rfy + — dz = d>[f^ dw + —dy] +/ ^ dz = (</./f). 

 c dz a- h' r- ^ > dx dy j dz ' 



ds „d''<p du ,,, df dv .,„ d-f dw ,„ „ rf'0 



Also — =f~~, -r = «'*-- , — = b''(h •' , — = c'f -[ . 



dt ■' df' dx *^ dx= dy ^ dy' dz ' dx' 



Hence by equation (4), 



dt' dx' \c' fda:' c' fdy'j ^ 



a' , b'-' 



or, if -;— = A and -t-„ == I, 



c' c^ 



