OF AIR OBSERVED IN KUNDT’S TUBES. 
3 
and 
so that 
du do 
dy dx ^ 
4 , 1 d 2 U v d V : yfr 
V v»-y Vy = -y— +- -y— 
r at v dx v dy 
(4). 
For the first approximation we neglect the right-hand member of (4), as being of 
the second order in the velocities, and take simply 
V~ 
i//=0 
(5). 
The solution of (5) may be written* 
x P='l J l + 'l'-2 .( 6 )> 
where 
v ^ = 0 , ( v 2 - i |)^=0 .( 7 ). 
We will now introduce the suppositions that the motion is periodic with respect to 
x, and also (to a first approximation) with respect to t. We thus assume that x// 1 and 
v/» 3 are proportional to cos kx, and also to e mt . The wave-length (A) along x is 2ir/k, 
and the period r is 27r/w. The equations (7) now become 
y- r >=°> ($-*- 7)*= 0 .< 8 ), 
by which i/q and x/j. 2 are to be determined as functions of y. If we write 
*'*=**+-.( 9 ), 
V 
we have as the most general solutions of (8) 
i// ) = Ae-^+Be+^.(10), 
xjj . 2 = Cer k 'y +De +X >.(II). 
With respect to the value of k\ we see from (9) that it is complex. If we write 
then 
7r = P 2 cos 2a, -=P 2 sin2a, 
V 
k! = P cos cl -+- i P sin a. 
* Stokes “ On Pendulums,” Camb. Phil. Trans., vol. ix., 1850. 
B 2 
