38 Mr. Mac Cuxvacu on the dynamical Theory of 
which, as the variations dé’ and dj, are arbitrary and independent, is equivalent 
to the two equations 
’ 7 
5 ehh Yo i=sOr (21) 
Thus, to find the relations which subsist among the vibrations incident, 
reflected, and refracted, at the common surface of two media, we have four con- 
ditions, expressed by the equations (20) and (21) ; and these conditions are 
sufficient to determine the reflected and refracted vibrations, when the incident 
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 
G = Gs wher Zz, —c- (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 medium, and two sets 
of refracted waves in the second, the resolved displacements, and all the quanti- 
ties which depend upon them, are composed of two parts, due to the coexisting 
waves. Let the point O be, for each set of waves, the origin of a system of 
rectangular coordinates, which we shall call x,, y,, 2, for the incident, and xj, ¥;, 2 
for the reflected wave, the axes of z, and 2; being perpendicular to the respective 
waves, and their positive directions being those of propagation. Let the displace- 
ments in these waves be parallel to y,, 7, and be denoted by 1, 7, respectively. 
Then if the axes of 2, 4) % make with the axis of 7, the angles a, 6, 7, and 
with the axis of 7; the . ai, Bi, yi) we have, by the formule (F), 
/ / Ln, 
sq + Feoa, vy = Heo s+ Fi cos p. (23) 
* In considering the question of reflexion at the common surface of two ordinary media (Mémoires 
de l'Institut, tom. xi., p. 396), Fresnel assumes the conditions (20); but his other suppositions vio- 
late the condition (22). In fact, this last condition 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 Royal Irish Academy, vol. xviii. p. 32. 
