431 Prof. 'T. H. Havelock. 
Carrying out the variation, and using the boundary conditions (3), we 
obtain 
OW py Oh 
4viy+ 2RU aEon nD eT (7) 
To find the critical value of R, we assume first that & occurs in y as a 
factor exp. ip£, and then solve (7); using the boundary conditions, we have 
an equation from which we can find the least value of I for a given value 
of p, and finally we take the minimum value of R with respect to p. 
The process has been expressed in a different form by Hamel.* Using the 
corresponding Green’s function for the equation y*y = 0, the equation (7) 
may be replaced by a linear integral equation for y, of which the required 
value of R is the lowest characteristic number. 
Returning to equation (5), if dE/dt is positive for any assigned initial 
disturbance, it does not follow that the motion is unstable in the ordinary 
sense. But, if there exists an absolute minimum for R in the manner 
explained above, it follows that, when R is less than this value, dE/d¢ is 
negative for every initial disturbance, and must always remain negative. 
Thus the system has at least a much higher degree of stability for such 
values of R compared with those greater than the critical minimum. 
Obviously, this method does not produce any new information which is 
not implicit in the ordinary equations, such as equations (2) aud (3); but it 
presents part of that information in a different form, so that the critical 
minimum of R may be used as a measure of the degree of stability of various 
distributions of velocity under different boundary conditions. 
4. It is convenient to classify the boundary conditions under which the 
energy equation (5) is valid. For this purpose we use an alternative form 
derived directly from equations (1), with the ordinary notation 
0 Ou , oO 0: 
Pz = —pt 2a; Pay = (SESE) Py = —P+2u5, (8) 
On, 
We have 
di 
a= | {u (lprr+ Mpry) +0 (pry + mpyy)} ds—p {{ wv a dx dy 
i [[{e= 2 + Pyy x + Pry (S42) b dxdy, (9) 
where ds is a line element of the boundary and (/, m) the normal. 
We have specified the conditions at the end boundaries, and we are con- 
cerned now with the planesy= +a. It follows that we get the energy 
* G. Hamel, ‘Gott. Nachr., Math. Phys. Klasse,’ 1911, p. 261. 
169 
