Oke — Stahility or Instability of Motions of a Viscous Liquid. 131 



Art. 33. The more General Investigation. 



Proceeding to a more general investigation, if the axis of x be taken 

 midway between the planes, and the steady velocity be ?7 = C{a^ - y"^), and 

 keeping to the two-dimensioned case, equations (7) are replaced by 



2fiV~u + Wpyv ^ dp/dx, 



2fi^H + 2Cpyu = dpidy. (54) 



Eliminating 23, and substituting for U, we obtain 



2fxV^{duldy - dvjdai) + 2Cp[y{dvldy - dujdx) + vj = 0, (55) 



or, introducing the stream function, i//, 



fxV'xl^ - Cp{2ydmdxdy + d^/dx} = 0. (56) 



If we now further suppose that \P varies as e*'^ where I is definite, but 

 undetermined, this is reduced to 



fx {d^dy^ - IJ4j - Cpli{2yd-^ldy + ^) = 0. (57) 



It seems convenient to substitute ly = a, Cpi/juP = k, and doing so this 

 equation becomes 



{d'Ida' -1)^- k {2ad^lda + ^) = 0. (58) 



This can be solved in series preceding in ascending powers of a. Writing 



^ = S^„a7|_7i, (59) 



the coefficient law is 



An,, - 2A+2 + [l-{2n + l)k]An = 0. (60) 



There are, therefore, series whose first terms are respectively 1, a, a^ al 



If ti, V, or ^, dxp/dy are to vanish at the boundaries y = ± a, there is evidently 



one solution of the problem in which xfj is an even function of y, and another in 



which it is odd. And there are various reasons for supposing that the former, 



i.e., that in which v is an even, and u an odd, function, will give the narrower 



limit of stability. This view is in conformity with the fact that Sharpe 



obtained a lower value for I Vp/fi than Eeynolds did ; I understand Sharpe 



to state that it seems more in accordance with experiments that v should 



have a maximum midway between the planes than that tc should ; and I 



obtained this result when la is very small. 



When la is sufficiently small, we may replace the coefficient law (60) by 



the simpler one 



An,, - {2n + 1) kAn = 0. (61) 



