a Horizontal Layer of Fluid. 533 



touched. It would seem to demand the inclusion of the 

 squares of quantities here treated as small. But the size of 

 the hexagons (under the boundary conditions postulated) is 

 determinate, at any rate when they assert themselves early 

 enough. 



An appendix deals with arelated analytical problem having 

 various physical interpretations, such as the symmetrical 

 vibration in two dimensions of a layer of air enclosed by 

 a nearly circular wall. 



The general Eulerian equations of fluid motion are in the 

 usuai notation : — 



Du _^ 1 dp Dv _ v 1 dp Dw _ 7 1 dp .... 



Wt~ pda' Bt" pdy' Ut~ pTz" ' W 



where 



D d , d d d /ON 



T>t = dt +u Tx + % +w dz> • ■ ■ (2) 



and X, Y, Z are the components of extraneous force reckoned 

 per unit of mass. If, neglecting viscosity, wo suppose that 

 gravity is the only impressed force, 



X=0, Y=0, Z=-g, ... (3) 



z being measured upwards. In equations (1) p is variable 

 in consequence of variable temperature and variable pres- 

 sure. But, as Boussinesq * has shown, in the class of 

 problems under consideration the influence of pressure is 

 unimportant and even the variation with temperature may 

 be disregarded except in so far as it modifies the operation 

 of gravity. If we write p = p + 8p, we have 



9P =Wo(l + Sp/po) =gpQ—9Po"0, 



where is the temperature reckoned from the point where 

 p = Po and a is the coefficient of expansion. We may now 

 identify p in (1) with p 0i and our equations become 



T)u _ _ 1 dP Bv_ 1 d? Dw_ _ldP 



\)t " p dx ' Dt~ p dy> Bt~ p dz + 7 ' ' ' W 



where p is a constant, y is written for got, and P for p+gpz. 

 * Theorie Analytique de la Chaleur, t. ii. p. 172 (1903). 



