Scientific Papers - Vi

342 ON THE STABILITY OP THE SIMPLE SHEARING [398
positive and indeed greater than v&, inasmuch as this is certainly the case when /? = 0*. The assumption that q = vlcz, by which the real part of the { } in (1) disappears, is indeed a considerable simplification, but my hope that it would lead to an easy solution of the stability problem has been disappointed. Nevertheless, a certain amount of progress, has been made which it may be desirable to record, especially as the preliminary results may have other applications.
If we take a real 77 such that
we obtain . -.•— =
This is the equation discussed by Stokes in several papers f, if 'we take x in his equation (18) to be the pure imaginary itj.
The boundary equation (3) retains the same form, with e^ dy for ekv dy, where
\s=9v/?/P .................... .............. (6)
In (5), (6) t] and \ are non-dimensional.
Stokes exhibits the general solution of the equation
in two forms. In ascending series which are always convergent,
3."576.8. 9+ >"
The alternative semi-convergent form, suitable for calculation when x is large, is
- l'b + 1.5.7.11 1.5.7.11.13.17 1.144»* 1.2.1442^ 1.2.3.144"^
" "
, n -i 2** f, , 1-5 ,1.5.7.11 1.5.7.11.13.17 } ' n
+ Dx *e ]1 + - 8 + - + - J- + ...L ...(9) ( 1.144a;ir 1.2.144V 1.2.3.1443a;- ) -
in which, however, the constants G and D are liable to a discontinuity. When x is real — the case in which Stokes was mainly interested — or a pure imaginary, the calculations are of course simplified.
* Phil. Mag. Vol. xxxiv. p. 69 (1892) ; Scientific Papers, -Vol. HI. p. 583.
t Especially Gamb. Phil Trans. Vol, x. p. 106' (1857) ; Collected Papers, Vol. iv. p. 77.e was there more than the slightest suggestion of an s.