18 
MR. POWER ON THE ABSORPTION OF THE SOLAR RAYS, ETC. 
dence, the cutting planes become parallel to the surface of separation of the two 
media. 
Hence if we take AP=X=BQ, and take a 
section PQ of the incident beam parallel to AB 
(a) the surface of separation, the vis viva of 
PABQ will have for its expression ^ ^ ” cos — 
If instead of X we wish to introduce the period 
of undulation (r),or the number of undulations 
in a unit of time (i v ), since ~=t=^, the expres- 
27r 2 /< 2 2 t i 2 h 3 
sion becomes — .omcqsG, or — .avu cos 6. 
OlT Cl 
13. In the same way*, if h! and h t be the ampli- 
tudes of the reflected and refracted rays, G t the 
angle of refraction, «, the distance of the ethereal particles from each other in 
the second medium, and « / the velocity of undulation in that medium, the vires 
vivce of the same wave after the reflexion and refraction has been completed 
2 ^ h ' 2 27 r 2 /* 2 
will be severally represented by ~ a3 — aw cos G and " ^ 3 ' a t va cos G„ v being taken the 
same as before, inasmuch as phases of any given kind, the nodal points for instance, 
will be transmitted across the surface of separation just as rapidly as they arrive, so 
far as regards the number transmitted in a given time, but with different velocities 
of undulation a and a ; in the two media. Hence, on the supposition that no vis viva 
is lost by the rays, we shall have, omitting common factors, 
h q a cos0 li n a cosfl j_ hfa , , cos0 / 
14. Any particle in the surface of separation will be at one and the same moment 
performing its phase to the incident ray with the transverse velocity cos or 
2 ttIiv cos ( 2 7rvt), and to the reflected and refracted rays with the transverse velocities 
2'r/ft'cos (2 Trvt), 2 t / j ( cos (‘Irvt) ; and since this particle cannot move in more than one 
way at once, it is clear that the two latter must be equivalent to the former, accord- 
ing to the laws of composition of velocities, which is the same as that of forces. 
Hence, omitting the common factors, the amplitudes h!h t of the reflected and 
refracted rays will be statically equivalent to h, the amplitude of the incident ray, 
regard being had to their several directions. If, therefore, we resolve h!h l in the 
direction of li and perpendicular to that direction, the sum of the two former com- 
ponents will equal h, and the two latter components will destroy each other. 
There is no difficulty in pursuing this process, but I prefer the following which 
* It is scarcely necessary to remark, that, for the reflected and refracted waves, at — x should be written in 
the place of at-\-x in the expressions for the displacement, hut the sign of x has no influence on the result of 
the integration. 
