10 Prof. Powell on the Demotistration of FresnePs Formulas 



if n be the number of vibrations in a unit of time^ v=n\; and 

 for two media, 



—=fx, -- = -^ =n for homogeneous light. 



V. A. A,, 



84. If we consider, first, a single line of vibrating molecules, 

 we may investigate the vis viva of that line ; but from this pro- 

 ceeding to the vibrating mass, we must multiply the previous ex- 

 pression by the density (or equivalent retarding property), which 

 we may express generally by 8 and S, for the external and in- 

 ternal ffither; and by the rectangular breadths of the oblique 

 rays on the same base or section of the surface, which will be 

 respectively proportional to cos i and cos r : thus we shall have 

 in general for the multipliers, B cos i and 8^ cos r ; the former for 

 the incident and reflected, the latter for the refracted ray. 



35. Mr. Power's investigation is restricted to a particular 

 hypothesis as to the density, but may be more simply and gene- 

 rally followed out thus : — 



For a length dx, in which the molecules have a common 

 velocity, we may take for that velocity, 



imd for a single line of vibrating molecules, the vis viva 



I v*ar = — Tj — I C0B^—{vt—iC)ax, 



For a portion from a? to a7+X, the integral is easily found to 

 reduce to ^ ; and thus for those limits, 



\^dx = — r — = 27r^nh^v = (p) ; 



and for the vibrating mass, as before explained (34), 



For the incident ray p=27r^nh^vB cos i 



. . . reflected ray p' = 27r^nh!^vB cos i 



. . • refracted ray j9^ = 2'7T%h^vPi cos r . 



But since the principle of vis viva gives p = j^' +/>y, we have the 

 equation connecting the vires viva, 



fiB cos i{h'^'-h'^) = 8 1 cos rh^^, 

 or generally 



m(A«-.A'2) = m^«; . (M) 



where, on substituting the values of m and m^ on the respective 

 hypotheses, we can express the vires viva equally on the respect- 

 ive views of Fresnel, Maccullagh, and others. 



r 



