MR. POWER ON THE ABSORPTION OF THE SOLAR RAYS, ETC. 
17 
each component wave separately. Let us first take the primary component whose 
displacement is in the plane of incidence and determined by the equation 
27 T 
y—h sin-^--(«^+' :l ')- 
In order to determine the vis viva due to one undulation, let a be the distance of 
the particles of the ether from each other; the medium being isotropical, ~ will be 
the number of particles contained in a unit of line, and the number contained in 
a unit of surface, in whatever direction the line or surface may be turned : the 
interval a being supposed extremely small compared with X the length of an nndula- 
dx 
tion, — will be the average number of particles contained in a portion dx of X, which 
. . , . 27 rha 271 - 
may be regarded as vibrating with the common velocity cos — ( at—x ) found by 
differentiating the expression for y with respect to t. The vis viva of a single line of 
vibrating particles at the given instant ( t ), therefore 
27T 2 // 2 ff 2 
«A 2 
K 2 2tt 
-?cos . — (at-j-x)dx fi 
om 
x=x to x=x-\-X. This is easily found to be 
A « 
; and it is worthy of remark, 
though this is no more than we ought naturally to expect, that the result is inde- 
pendent of the phase at the beginning and end of the integral. The same will be 
true of every length X of particles which constitute the incident cylindrical beam, 
whatever may be the nature of the phase at the two extremities. Let u be the ob- 
lique section of this beam made by the common surface of the two media ; then 
co cos 0 is the transverse section of the incident beam made by a plane perpendicular 
to its axis, and w ^° S - is the number of ethereal particles in this section. Hence to 
obtain the vis viva of one undulation of the incident beam we have only to multiply 
the vis viva of each line of particles 
27 cVdd 1 
by 
co cos 9 
and the result is 
27r 2 /i 2 « 2 w cos 9 
A « a 2 5 Aa 3 
which holds true whether the particles be in the same phase for the whole extent of 
each transverse section, as, I think, is commonly supposed, or whether the phase be 
supposed to vary from particle to particle in such transverse section, according to 
some continuous law depending on the original vibrations at the surface of the sun or 
other origin of the beam ; it being understood of course that at a given instant the 
same type or system of phases will recur for sections separated from each other by 
the interval X. There will be the same recurrence of type for sections made by planes 
inclined to the beam at any given angle 0, and separated from each other by the per- 
pendicular interval X cos 0 ; and, further, the vis viva contained between two sucb 
planes will be the same as before, since every line of particles will have as much vis 
viva added at one extremity as is cut off at the other, when the cutting planes are 
turned from the transverse position through the angle 0. If 0 be the angle of inci- 
MDCCCLIV. D 
