36 Mr. Mac Cutracn on the dynamical Theory of 
when the particle is in the first medium, and by &/, mj, ¢{/ when it is in the 
second, the equation (1), adapted to the present case, will be 
0 dng 0 aes wa Pn, 0 sidst (au 
(Sdrady dey Teves + Man, + = az ) +4 SSdndydz, (< 8 + ey mi! +E 8E,! ) 
= WWGdaydy,dz,dV’ + \\Sda,dy,dz,dV" ; (17) 
wherein 6V’ and éV” are the respective values of €V for the two media, which 
are conceived to extend indefinitely on each side of the plane of x, y,; that plane 
being an upper limit of the integrations relative to one medium, and a lower 
limit of the integrations relative to the other. Each medium is conceived to be 
occupied by systems of plane waves, the first by incident and reflected waves, the 
second by refracted waves; and, except where they are bounded by the plane of 
Ty) Yo these waves are regarded as unlimited in extent. 
For the ordinary medium, if we put 
XA Oty Eo Yo kG ae Li 
Fa Ua * at Ga ieeae te Pi ens aren, 
and suppose the velocity of propagation to be unity, we have* 
Save a(R de a ee =) i (Ge ony, 
dz,  dy,/ dx, dx, 
For the crystallized medium, if its principal axes be those of 2, y, z, the 
value of 6V” will be the same as that of 6V in formula (3); but instead of the 
variations of é, 7, ¢, we must use those of &’, 7/’, ¢;’. Denoting the cosines of the 
angles which the principal axes respectively make with the axis of x, by /, m, 
with the axis of y, by l’,m’,n’; with the axis of z, by /’, m”, n”’; and putting 
Sr OG Oe a een, ee 
x, = 5 = ) = ’ 
a dz, dy 0 dx, dz, dy, dx, 
, deni’ dae pon! iG paBE S205, na Oba dono, 
x 2 
d dz, dy,” pees Y dy, dx, 
? 
* It is assumed here, and in what follows, that when there are two or more coexisting waves in 
a given medium, the form of the function V is the same as for a single wave, provided the displace- 
ments which enter into the function be the resultants of the displacements due to each wave sepa- 
rately. This, however, ought evidently to be the case, in order that the principle of the superposi- 
tion of vibrations may hold good. 
