Prof. T. H. Havelock 621 
supposed to be of infinite extent, and it seemed to be of 
interest to solye similar cases of motion when the fluid has 
a fixed outer boundary. In the following paper considera- 
tion has been limited to the motion of a plane between fixed 
parallel planes and to similar problems with cylinders, the 
ordinary hydrodynamical equations for non-turbulent motion 
not involving terms of the second order in such conditions. 
The results are perhaps not of practical importance, but, 
apart from the particular problems, the method of solution 
may be of interest. Stating the problem as in the cases to 
which reference has been made, we are led to an integral 
equation of Poisson’s type in which the nucleus is an infinite 
series of exponentials. This equation can be solved by fol- 
lowing a method suggested by Whittaker*; the solving 
function is obtained as an infinite series of exponentials, the 
exponents being the roots of a certain equation. It seems 
that examples of this method have not been given hitherto, 
though equations of this type should arise naturally in 
various physical problems. The particular cases worked 
out in detail are the fall of a thin material plane in a liquid 
bounded by two fixed parallel walls, and the motion of a 
cylindrical shell filled with liquid and acted on by a constant 
couple. The same method gives the solution when the force 
is an assigned function of the time, for instance an alter- 
nating force which is suddenly applied. Motion in an 
infinite fluid may be included in the scheme by replacing 
the infinite series of exponentials by corresponding infinite 
integrals. The case of systems with a natural period of 
oscillation will be considered in a subsequent paper. 
It will be clear, from the examples, that the method of 
solution could be formulated in general rules for obtaining 
the solving function. This has not been attempted here, as 
an examination of convergence would be necessary to estab- 
lish any general theorem. A knowledge of the differential 
equations and the boundary and initial conditions enables us 
to verify the results which are given ; in these circumstances, 
of course, they can be obtained by other methods without 
difficulty. However, there are probably other physical 
problems, in which the conditions are not so completely 
known, whose statement leads to an integral equation of 
the same type, and its solution can be obtained in the same 
manner. 
2. Consider laminar fluid motion between two fixed planes 
w= +h, the fluid velocity being parallel to Oy. Let the 
*E.T. Whittaker, Proc. Royal Socy. A, vol. 94 (1918), p. 367. 
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