or 
447 
( 18 ); 
of Edinburgh, Session 1879 - 80 . 
10 = Cl, (err) + di (err) 
where C and (£ denote arbitrary constants, to be determined in the 
present case by the equations of condition (12). These are equi- 
valent to g = c when r = a, and g = t when r = a, and, when (16) is 
used for w in (13), give two simple equations to determine C and 
The problem thus solved is the finding of the periodic disturb- 
ance in the motion of rotating liquid in a space between two bound- 
aries which are concentric circular cylindric when undisturbed, pro- 
duced by infinitely small simple harmonic normal motion of these 
boundaries distributed over them according to the simple harmonic law 
in respect to the co-ordinates z , 6. The most interesting Sub-case is 
had by supposing the inner boundary evanescent (a = 0), and the 
liquid continuous and undisturbed throughout the space contained 
by the outer cylindric boundary of radius a. This, as is easily seen, 
makes w = 0 whenr = 0, except for the case i= 1, and essentially, 
without exception, requires that c be zero. Thus the solution for w 
becomes 
w = C (vr) . . . (19) 
or w = Cl (or) . . . (20) ; 
and the condition g = c when r — a gives, by (13), 
,2/i 
c = 
vJ'i (yd) — — —————— — ^ (vd) 
— ioi)a 
( 21 ). 
or the corresponding I formula. 
By summation after the manner of Fourier we find the solution 
for any arbitrary distribution of the generative disturbance over the 
cylindric surface (or over each of the two if we do not confine our- 
selves to the Sub-case), and for any arbitrary periodic function of 
the time. It is to be remarked that (6) represents an undulation 
travelling round the cylinder with linear velocity na/i at the sur- 
face, or angular velocity nji throughout. To find the interior effect 
of a standing vibration produced at the surface we must add to the 
solution (6), or any sum of solutions of the same type, a solution, or 
a sum of solutions in all respects the same, except with -n in 
place of n. 
It is also to be remarked that great enough values of i make v 2 
