626 Prof. T. H. Havelock on an Integral Equation 
In the same way, the solving function has a limiting form 
which follows directly from (15) and (19), namely 
2pve( ? axe @lt=7) 
oT { a+ 40*y/o? ae 
Using this value as before, we obtain the same result (24). 
6. It is clear that the same procedure is sufficient when 
the applied foree is any assigned function of the time. 
For example, if the accelerative force is acospt and the 
motion starts from rest, we have 
dV ; —vA2(t=7)/h2 
TE 2 Os pt OCR EAC dt, . (25) 
where the summation extends over the roots of the same 
equation (15), and the coefficients are given by (19) The 
solution follows on completing the integrations ; it consists 
of a periodic motion in different phase from the applied 
force, together with the disturbance due to taking into 
account the initial conditions. 
7. A final example may be taken from cylindrical motion 
when there is no limiting steady velocity. Suppose the 
motion to be symmetrical round an axis; then if ris dis- 
tanee from the axis and v is the fluid velocity, supposed 
perpendicular to the radius vector, we have 
CO fOr, LO o 
ot ras Or oF ror =*) 
Consider the motion of a hollow cylinder, of radius a, filled 
with the liquid. Suppose the motion to start from rest and 
let the velocity of the cylinder be Q(¢). Then it may be 
shown that the angular velocity of the fluid at any time is 
given by 
ad (pr/a) —yp2(t=7)/a2 
=|) Or Dj aN Wd e—Et—7)/a har 
© \ (x) { 1+ aap) , 
where the summation extends over the positive roots of 
J, (p)=0. 
Let the cylindrical shell start from rest under the action 
of a constant couple N, and let I be its moment of inertia, 
both quantities being for unit length along the axis. The 
retarding couple due to fluid friction is the value of 
2rpurOo/dr when r==a. Hence the equation of motion 
of the cylinder is 
t 
Ql(t) + Umut) O! (rt) Se--e dg = NI, (28) 
0 
(26) 
182 
