MR. POWER ON THE ABSORPTION OF THE SOLAR RAYS, ETC. 
15 
communication of vibrations ; indeed the statical condition of two contiguous ethe- 
real media of different densities is impossible, unless we take into account the mutual 
statical actions between the particles of the ether and those of the crystal which it 
is supposed to permeate ; much more are the dynamical conditions likely to be fraught 
with inconsistency, unless we take into account the mutual dynamical actions of 
those particles. I think it is considerably in favour of the present mode of viewing 
the problem, that no difficulties of the kind under consideration are found to present 
themselves. 
10. I confine my attention, as 1 have before stated, to an isotropical singly refract- 
ing medium, like glass or water, though I think, if I had more time at my disposal, I 
could extend the theory to doubly refracting crystals. A very simple integration 
gives me a general expression for the vis viva of an elementary cycloidal wave, in 
terms of the amplitude and the constants of the periodical function. By help of this 
I obtain two equations of vis viva, one for a wave whose vibrations are in the 
plane of incidence, and the other for a wave whose vibrations are perpendicular to 
that plane, both vibrations being transverse to the axis of the ray. By the principle 
of superposition, these two equations will hold true simultaneouely when the above 
waves are regarded as the components of one and the same wave. I obtain three 
other equations between the amplitudes, from the simple consideration that a particle 
situated in the common surface of the two media cannot vibrate in more than one 
way at once. Of these three equations two involve the amplitudes of the first com- 
ponent wave, and the third those of the second. 
The five equations serve to determine, in terms of the angle of incidence and the 
component amplitudes of the incident wave, the five following quantities, namely, 
the angle of refraction, the two component amplitudes of the reflected wave, and those 
of the refracted wave. 
11. By the help of Fourier’s theorem we may decompose any form of undulation, 
extending between given limits, into a series of elementary cycloidal undulations, 
varying in wave-length, amplitude and orientation ; and, again, a wave whose orien- 
tation deviates from the plane of incidence, or a plane perpendicular to this passing 
through the axis of the ray, may be resolved into two, one in each of the above 
planes, which I shall term respectively the primary and secondary planes. 
Let 6 be the angle of incidence of a cylindrical beam or incident ray ; and let 
^7 r 
yz=/isiny ( at-\-x ) 
j • 2 tt 
z = k sin y 0 at-\-x-\-c ) 
represent the displacements due to any one of its cycloidal elementary waves, 
resolved parallel to the primary and secondary plane, x being the distance from the 
point of incidence measured along the axis of the ray, (a) the velocity of undulation, 
and t the time measured from some epoch anterior to incidence. 
