429 Prof. T. H. Havelock. 
distribution U consists in assuming a solution of (2) of the form 
exp. {i(nr+pE)} fa(n). For any arbitrary real value of p, the corresponding 
possible forms of f,(7) and values of m are found from (2) together with the 
boundary conditions. The distribution U may be said to be thoroughly 
stable if every possible value of 7 has a positive imaginary part, and if this 
holds for all positive values of KR. 
The usual boundary conditions, which we shall assume in the first place, 
are u = 0, v = 0, or 
y=0; apfoy=0: = +1. (3) 
From the work of Kelvin, Rayleigh, Orr, Hopf, and others, it may be taken 
that the simple shearing motion, U = 1+, is thoroughly stable in this sense ; 
and probably a similar conclusion holds for motion under a constant force or 
pressure gradient, namely U = 3(1—7?). 
There are various possible explanations of the well-known divergence 
between these results and the behaviour of actual fluids. In the first place, it 
is obvious that the physical properties, whether of the fluid or of the walls, are 
inadequately specified in the mathematical statement of the problem. But, 
apart from this, the disturbances have been supposed small, and second order 
terms neglected. Again, in a system of this type, a disturbance may be 
small initially and may converge ultimately to zero, but may be very large at 
intermediate times, and may thus give rise to practical instability. 
The energy method of Reynolds is in a different category from these in 
that it takes the mathematical problem as it stands and does not necessarily 
involve the actual magnitude of the disturbance; in fact, it forms a new 
criterion or measure of degree of stability. The energy of the disturbance 
being defined by B= boat? jl (aby. (Sey dé dn, (4) 
we have from (2) and (3), after integrating by parts, 
dk dy d 
= u0?[R((u Se aedn—2 ([(yyyr dean |.) 
Here dE/dt means the rate of increase of E ina region whose end boundaries 
move with the steady velocity U. We may replace this by 0E/ot for a region 
with fixed ends, and we shall then have additional terms on the right of (5) 
denoting flux of energy across these ends. The latter terms may be omitted 
under conditions which cover the usual cases: namely, either the disturbance 
is periodic in &, or it is limited or localised so that y and its derivatives 
converge sufficiently rapidly to zero for = +c. We shall assume such 
conditions to hold in what follows, and references to boundary conditions 
mean those which hold at the planes 7 = +1. 
167 
