198 ON THE STABILITY OF THE [377
where p, q, a. @ are real. Substituting in (3) and equating separately to zero the real and imaginary parts, we get
d*a d*U (p
^ = £2a +
dif
whence if we multiply the first by $ and the second by a and subtract,
d f da dp\__ffiU q(tf + /3*} dy{Pdy • dy)~'dif (p+Uf + f ................ ( >
At the limits, corresponding to finite or infinite values of y, we suppose that v, and therefore both a and ft, vanish. Hence when (4) is integrated with respect to y between these limits, the left-hand member vanishes and we infer that q also must vanish unless d2U/dyz changes sign. TJ^us in the motion between walls if the velocity curve, in which V is ordinate and y abscissa, be of one curvature throughout, n must be wholly real ; otherwise, so far as this argument shows, n may be complex and the disturbance exponentially unstable.
Two special cases at once suggest themselves. If the motion be that which is possible to a viscous fluid moving steadily between two fixed walls \mder external pressure or impressed force, so that for example U=yz — fr, d2 Ujdyz is a finite constant, and complex values of n are clearly excluded. In the case of a simple shearing motion, exemplified by U = y, d2 U/dy* = 0, and no inference can be drawn from (4). But referring back to (3), we see that in this case if n be complex,
would have to be satisfied over the whole range between the limits where v=0. Since such satisfaction is not possible, we infer that here too a complex n is excluded.
It may appear at first sight as if real, as well as complex, values of n were excluded by this argument. But if n be such that n/k + U vanishes anywhere within the range, (5) need not there be satisfied. In other words, the arbitrary constants which enter into the solution of (5) may there change values, subject only to the condition of making v continuous. The terminal conditions can then be satisfied. Thus any value of — n/k is admissible which coincides with a value of U to be found within the range. But other real values of n are excluded. ;
Let us now examine how far the above argument applies to real values of n, when d^Ufdy2 in (3) does not vanish throughout. It is easy to recognizeal velocities, the motion