440 ON CONVECTION CURRENTS IN A [412
while here w = 0 at the boundaries requires also dwldx = 0, dw/dy = 0. Hence, at the boundaries, d*u/dxdz, (fivfdydz vanish, and therefore by (5),
Equation (15) remains unaltered : —
/9w + {rc + /e(Z- + m2 + s2)}0 = 0J .. ................ (15)
and (16) becomes
{n + v(l* + m2 + s2)} (P + m2 + s2) w - y (/2 + m2) 0 = 0 ....... (36)
Writing as before a = I2 + m? + s2, we get the equation in n
(n + tea-} (n + va)tr + £7 (Z2 + m2) = 0, ............... (37)
which takes the place of (17).
If 7 = 0 (no expansion with heat), the equations degrade and we have two simple alternatives. In the first n + /or = 0 with w = 0, signifying conduction of heat with no motion. In the second n + va- = 0, when the relation between w and & becomes
(3w + a(fc-v)6 = Q ......................... (38)
In both cases, since n is real and negative, the disturbance is stable.
If we neglect K in (37), the equation takes the same form (20) as that already considered when i/ = 0. Hence the results expressed in (22), (23), (24), (25), (26), (27) are applicable with simple substitution of v for K.
In the general equation (37) if /3 be positive, as 7 is supposed always to be, the values of n may be real or complex. If real they are both negative, and if complex the real part is negative. In either case the disturbance dies down. As was to be expected, when the temperature is higher above, the equilibrium, is stable.
In the contrary case when 0 is negative (- /3') the roots of the quadratic are always real, and one at least is negative. There is a positive root only when
fry (P + m2) > W ............................ (39)
If K. or v, vanish there is instability ; but if K and v are finite and large enough, the equilibrium for this disturbance is stable, although the higher temperature is underneath.
Inequality (39) gives the condition of instability for the particular disturbance (I, m, s). It is of interest to inquire at what point the equilibrium becomes unstable when there is no restriction upon the value of P + m2. In the equation
/3'y (Z2 + m2) - /wo-3 = £'7 (a- - s2) - Kva3 = 0, ............ (40)
we see that the left-hand member is negative when £2 + m2 is small and also when it is large. When the conditions are such that the equation can only just be satisfied with some value of I2 + m2, or cr, the derived equation
/3V - 3/ci/o-2 = 0 .............................. (41)uleur, t. n. p. 172 (1903).