790 PROFESSOR G. G. STOKES ON THE COEFFICIENT OF VISCOSITY OF AIR, 
become infinite at the centre is replaced by the condition that it shall not be infinite 
at an infinite distance. 
In the present case the motion is necessarily symmetrical about the axis, so that it 
is alike all round any circle that has the axis for its axis ; it is, moreover, tangential 
to the circle. Let the fluid be referred to polar coordinates r, 6, ro-; r being the 
distance from the centre, 0 the inclination of the radius vector to the axis, and ct as 
usual. Then, taking p, p, to denote the density and coefficient of viscosity, and 
observing that v = q cos where q is the velocity, we easily get from the second 
equation of motion, by putting, as we may, ttt=0 after differentiation, 
A + ? l( sin 0 ‘k\ j_ _p k = o . 
dr^r drr 2 sin 6 dd\ ddj r 2 sin 2 6 p dt 
( 1 ) 
and we have the condition at the surface — 
q=a)ct sin 0 when r=a, .(2) 
where oj is the angular velocity of the sphere, and a its radius. 
The motion with which we have to deal is periodic, subject to a secular diminution. 
The latter being actually very slow, it will suffice, in calculating the force of the air 
on the sphere, to take the motion as periodic, and expressed, so far as the time is 
concerned, by the sine or cosine of nt. It will be more convenient, however, to use 
the symbolical expression e int , where 7=^/( — 1). The general equation (1) and the 
equation of condition (2) can both be satisfied by taking q to be expressed, so far as 0 
is concerned, by sin 6. Assuming, then— 
q — e int sin 0f(r), . 
...... (3) 
and writing 
ipn iirp 0 
p, pr 
.(4) 
we have 
/»+J:/'(^)-“- 2 /(^)-^/( r )- • • • 
.( 5 ) 
Taking fl-m for that root of the imaginary m 3 which has its real part positive, we 
have for the integral of (5), subject to the condition of not becoming infinite at an 
infinite distance— 
J\r) = A C -(i+V)..(6) 
Omitting the pressure in equilibrium, we shall have for the force of the fluid on an 
element of the sphere a tangential pressure (say T, referred to a unit of surface) 
