40 Mr. Mac Cuutacu on the dynamical Theory of 
Hence, 
dn, am ys ; dn, an ys ; 
P= —rs cosa, + ——;7's cosa Q@=——"7s cos B, + —>r’'s' cos B;. (24 
dz,’ are: Ler ee os dz, Bsr dz; 28) 
The quantities s, s’ are, as appears by the last section, the normal velocities 
with which the two sets of refracted waves are propagated. The velocity with 
which the incident and reflected waves are propagated is taken as unity. There- 
fore, if 7, 7; be the transversals, or amplitudes of vibration in the incident and 
reflected waves, and 7,, 7, the transversals of the refracted waves, the lengths of 
the latter waves being denoted by A,, A;, and the length of an incident or reflected 
wave by A,, and if we put 
d= x7 —atn) & => (- 2+); 
oe 1 (25) 
Qa Oe ere oe ‘ Z 
$. = x (86 — 2+ 2), $2 = x7 (St— a + %)s 
2 2 
where v,, vj, v,, v, are constants, and z is the ratio of the circumference to the 
diameter of a circle, we may write 
M=7,COSH, 1 = 7 COS Hi, fy = 7 COS Hy 7, = 7 COS H). (26) 
By means of these values the formule (23) and (24) become 
iF Qr . 1 riya ‘ 
X= (7, cos a, sin @, + 7; cos a; sin ¢;), 
L 
2 . ' Qecbe nh 
Y= x (7, cos B, sin @, + 7; cos B; sin ¢;), 
- Jp f (27 
rs : tas 3 
P= 27 = T, COS a, SIN @, + yr 7? Cos a, sin $%)> 
A ui 
Ted. 
Q=— 27 = 7, cos B, sin @, ++ so 7, Cos By sin $4). 
The angles ¢, ${, . ¢, are the phases of vibration in the different waves at 
the time ¢. To see how they depend on the coordinates x,, Y» 2, conceive the 
axis of z, to be directed from O towards the interior of the second medium, and 
the axis of x, to lie in the plane of incidence, so that the positive directions of 
2 2» 2, may lie within the angle made by the positive directions of 2, and z,, 
