170 On a Dynamical Theory of 



vibration is given. But though, by the nature of the question, 

 four conditions only are required for its solution, there remains 

 another condition which ought to be satisfied; for we ought 

 evidently to have 



= 0. (22) 



This condition is apparently independent of the rest ; but it 

 cannot really be so, if the preceding theory is consistent with 

 itself. We shall accordingly see, in what follows, that the last 

 condition is included in the other four ; which is a remarkable 

 circumstance, and a singular confirmation of the theory.* 



As the incident and reflected waves coexist in the first me- 

 dium, and two sets of refracted waves in the second, the resolved 

 displacements, and all the quantities which depend upon them, 

 are composed of two parts, due to the coexisting waves. Let 

 the point be, for each set of waves, the origin of a system of 

 rectangular co-ordinates, which we shall call x^ y^ z l for the in- 

 cident, and o/i, /'i, s'i for the reflected wave, the axes of z l and z\ 

 being perpendicular to the respective waves, and their positive 

 directions being those of propagation. Let the displacements in 

 these waves be parallel to y\, y\, and be denoted by iji, 1/1, re- 

 spectively. Then if the axes of # , / , z make with the axis of 

 #! the angles ai, |3i, 71, and with the axis of x\ the angles 

 a'i, /3'i, y'i, we have, by the formulae (F), 



J'. = P cos , + C ^ cos ' r o = pi cos ft + J < /3',. (23) 

 azi az i uZi az i 



Again, let the co-ordinates #2,2/2, s 2 have reference to one set of 

 refracted waves, and #' 2 , y' 2 , s' 2 to the other, the axes of z 2 

 and z' z being perpendicular to the respective waves, and their 

 positive directions being those in which the waves are propa- 



* In considering the question of reflexion at the common surface of two ordi- 

 nary media (Memoires de VInstitut, torn. ix. p. 396), Fresnel assumes the conditions 

 (20) ; but his other suppositions violate the condition (22). In fact, this last con- 

 dition is inconsistent with the supposition that, in a polarized ray, the direction of 

 the vibrations is perpendicular to the plane of polarization. See the Transactions of 

 the Eoyal Irish Academy, VOL. xvui. p. 32 (supra, p. 88). 



