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.