294 Proceedings of the Royal Society of Edinburgh. [Sess. 
t ' = RQ 
U 
T-PR T-U/ 
U U 
In the same time the front of the group advances in vacuo a distance Y 0 f, 
represented by OW. Thus 
v/ = ow = v ff 
T-U t 
V~ 
The direction of OW is determined by the ordinary law of refraction, as we 
are in reality now dealing with the emergence of a regular group of waves, 
although their crests are short, and are arranged in echelon pattern. 
Hence, after the rear of the group has reached Q, the wave-crests are all 
parallel to QX and lie along a line QW similar to their former arrange- 
ment along OR. In other words, QW is the locus of points where a 
constant wave-length X is to be observed. By the law of refraction 
OX = ^?SQ=~°(T -Yt). 
The length of the group on emergence is therefore given by 
XW = 0 W - OX 
= V 0 { 
t-u/ 
IT 
T-vn 
i 
p y v 
U'Yf 
Using in this the following equations, ^ = 
finally 
XW I - TX^ 
dX 
— and ^y=/ul 
aX V 
we have 
and the number of wave-lengths in the group after emergence is there- 
fore — T~ , which is in agreement with Lord Rayleigh’s expression. 
§ 7. It may be of interest to show that this value for the number of 
waves of length X in the emergent wave-system can be arrived at without 
making use of the law of refraction. As stated in § 5, at time t the length 
of the group in the prism is (V — U)£, and the number of waves it contains 
is therefore (V — U )//X', where X' is the wave-length in glass. During the 
passage of the group through the prism fresh waves are continually 
entering it, owing to the fact that the individual waves move with greater 
velocity than the group. Thus the number of waves which enter the group 
per second is (V — U)/X / , and the total addition to the group by the time the 
rear reaches Q is (V — U )t'/X' where t' is given in § 6. The total number of 
waves in the group on emergence is therefore given by 
(Y-U )t , (Y-U) (T-U/) 
\~> 1" V/ • fy > 
which reduces to 
( Y — U)T 
X'U 5 
y 
or with Y = v X to 
^ o 
TV 0 f V — U 1 
A 1 UY j 
, which is 
