1888.] viscous rotating cylinder. 251 
5. We may now show how to determine the forced oscillations 
or waves produced when, in addition to surface tension, small 
variable disturbing forces act on the surface of the liquid. Let 
the normal and tangential components of these surface tractions 
at any point be F, G, and let the radial and transversal component 
velocities of the fluid relative to the system of moving axes be 
U, V, so that 
OS ya CO 
U= "aa: ree nouiaadasbiers seuldeag sles (19) 
For the component rates of distortion of the liquid we have* 
é,, (rate of elongation along 7) = 
: oV U 
é,, (rate of elongation perpendicular to r) = Siias Fe (20), 
OO Ol 
s, (the rate of shear) Sa ae 
and for the stresses we get 
ue aif 2ve, 
Pia RERICS a ascies sateen! (21) 
ue | 
ry = VS j 
Substituting we find 
Te ais monet “i 
orp p av ag tote reece ee eeeee (22), 
Ce SLICES Gio laa: 
Pp = (5 ae? a Or? ae + sh charm atosrdaral( 2rd) 
in which r is. put =a after the ditterentiations have been per- 
formed. 
Whatever be the expressions for J’, G, they may be expanded 
in circular functions of multiples of 0, and the coefficients which 
are functions of the time may be expanded in terms of the form 
e-* where the a’s are certain, real, imaginary or complex constants. 
Thus, f’, G are expressible as series of terms of the form fe~“e”®, 
ge-“e™ where f, g are constants, and the effect of each term must 
be found separately. 
The corresponding expression for yr will have the same a whilst 
h will be determined by (9). 
* R. R. Webb, “On Stress and Strain in Cylindrical and Polar Coordinates.” 
Messenger of Mathematics, Feb. 1882, p. 147. 
VOL VI. PT. IV. 18 
