The Stability of Fluid Motion. 432 
equation (5), without any surface integrals expressing transfer of energy across 
the boundaries, with the following combinations : 
(i) w= 0, v= 0; (i) w= 0, py, = 0; (ail) v=0, pr, = 0; (iv) Pry =0, Py = 0. 
We may also verify that, under these conditions, the variation of (6) leads 
to the same differential equation (7). 
5. Most of the fluid motions whose stability has been examined, come under 
case (i) of the above. A different case of special interest is a stream with a 
free upper surface, the conditions at the upper surface being as in (iv). These 
conditions, however, do not lead to simple expressions in terms of the 
disturbing function, yr; moreover it is not permissible to regard the upper 
free surface as rigorously plane. We therefore, following Kelvin,* replace 
the problem by one which is very nearly the same but is more easily 
specified; it may be described as a broad river flowing over a perfectly 
smooth inclined plane bed, the upper surface being fitted by a parallel plane 
cover moving with the water in contact with it. The conditions at the 
upper surface then come under case (ili) of the previous section. 
We take the origin in the upper surface in this case, so that a is the depth 
of the stream and R is aU/v. The steady state is given by 
U =$3(1—-7’). (10) 
Using this in (7) and assuming y to be proportional to ¢’?‘, the differential 
equation becomes 
(4-1) ¥-1 (20a +) =0, (11) 
where « = pn, and k = 31R/2p*. 
The boundary conditions are w= 0, v= 0 at the bed of the stream, and 
v = 0, Pr = 0 at the upper surface; these reduce to 
ay = 0, Ayr /da? = 0; C—O 
wv =0, dy/de = 0; Cs = 70. (12) 
Equation (11) was solved by Orr for flow between two fixed planes with w 
and v zero at both boundaries, and it was found necessary to consider only 
solutions in even powers of « We shall require here the corresponding 
solutions in odd powers. Writing a solution in the form 
v= >; A,a"/n! 
we have the sequence relation 
An+a— 2 Angot {1—(2n+1)k} A, = 0. (13) 
* Kelvin, ‘Math. and Phys. Papers,’ vol. 4, p. 380. 
