for Reflected and Refracted Light. ^ 



Sucli is the principle of equivalent vibrations, according to 

 Prof. Maccullagh; in other words, it is founded on the pro- 

 position that " the incident and reflected vibrations are equivalent 

 to the refracted,'' 



30. It does not appear to what extent Fresnel had adopted 

 any generalized view of this kind ; but in the deduction of his 

 formula, he introduces an equation which is equivalent to 



, cos r , 

 cos I ' 



for vibrations parallel to the plane of incidence ; and if we sup- 

 pose this to be the expression for the law of equivalence, in that 

 case we must also take for that oi perpendicular vibrations, 



h'-h'=h,. 



In other words, the law of equivalent vibration, according to 

 FresnePs view, will differ from that of Maccullagh in that it 

 affirms the proposition that " the refracted and reflected vibrations 

 are equivalent to the incident.'' 



These values of h^, which we will call (L'), may be combined 

 in the same expression as before (L). 



Equation o/vis viva. 



31. That the amplitudes of the vibrations are the measures of 

 the velocities of those vibrations whose time is constant, is obvious; 

 and further, that the square of the amplitude or velocity, multi- 

 plied by the vibrating mass, is the true measure of the intensity ; 

 and consequently that the principle of vis viva, of which it is a 

 simple application, is true, undoubtedly receives ample proof 

 a posteriori, inasmuch as all the calculations founded on this 

 principle agree to such extreme accuracy with the experimental 

 results, whatever question may have existed as to its establish- 

 ment a priori, 



32. Mr. Power, assuming the general principle, but following 

 an original analytical method, deduces directly expressions for 

 the vis viva of the incident, reflected, and refracted ray respect- 

 ively ; and applies them by introducing the same expressions for 

 the amplitudes as those of Maccullagh, and on the same suppo- 

 sition as to the directions of the planes of vibration, though not 

 on the same hypothesis as to the constitution of the sether, with 

 respect to which he is led to conclusions of a peculiar kind, and 

 which have been considered by some as questionable, but the 

 consideration of which is foreign to the present inquiry. 



33. Assuming the ordinary formula for vibrations in ang plane, 



u-=:hmn-:—(vt''x); 



