1898-99.] Lord Kelvin on Reflexion and Refraction. 
369 
In the wave-system thus expressed the motion of each particle 
of the medium is perpendicular to the wave-front (a, /3, y). For 
purely distortional motion, and wave-front still (a, /3, y) and there- 
fore motion of the medium everywhere perpendicular to (a, /3, y), 
or in the wave-front, we find similarly from (7) and (6) 
where 
j[i __ Vi_ — —f(f — ax y z 
aA /?B yC J \ U 
• (U), 
• (15), 
and so denotes the propagational velocity of the distortional waves ; 
and A, B, C, are arbitrary constants subject to the relation 
a 2 A + /3 2 B + y 2 C = 0 . . . . (16). 
§ 7. To suit the case of solitary waves we shall suppose the 
arbitrary function / ( t ) to have any arbitrarily given value for all 
values of t from 0 to r, and to be zero for all negative values of t 
and all positive values greater than r. Thus r is what we may 
call the transit-time of the wave, that is, the time it takes to pass 
any fixed plane parallel to its front ; or the time during which any 
point of the medium is moved by it. The thicknesses, or, as we 
shall sometimes say, the wave-lengths, of the two kinds of waves 
are ut and m respectively, being for the same transit-times directly 
as the propagational velocities. 
§ 8. And now for our problem of reflexion and refraction. At 
present we need not occupy ourselves with the case of purely 
distortional waves with vibratory motions perpendicular to the 
plane of the incident, reflected, and refracted rays. ' It was fully 
solved by Green * with an arbitrary function to express the char- 
acter of the motion (including therefore the case of a solitary wave 
or of an infinite procession of simple harmonic waves). He showed 
that it gave precisely the “ sine law ” which Fresnel had found for 
the reflexion and refraction of waves “ polarized in the plane of 
incidence.” The same law has been found for light, regarded as 
electro-magnetic waves of one of the two orthogonal polarizations, 
* ‘ ‘ On the Reflection and Refraction of Light at the common Surface of 
two Non-Crystallized Media,” Math. Papers, p. 258. Also Trans. Garnb. 
Phil. Soc., 1838. 
