in certain Problems of Viscous Fluid Motion. 621 



supposed to be of infinite extent, and it seemed to be of 

 interest to solve 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 

 £C=+h } the fluid velocity being parallel to Oy. Let the 



* E. T. Whittaker, Proc. Roy. Soc. A, xciv. p. 367 (1918). 



