Oer — Btability or Instability of Motions of a Viscous Liquid. 123 



On integrating by parts all the terms on the right, except the first, the 

 right-hand member may be written ... 



pu(vdU'/dy+ivdU/dz)d. vol-- plU{ii? + v^ + vf) dS+ u{l;p^^^'nip^y+npx^)dS 



+ two terms similar to the last 



[iJxxditldx + fyydvjdf + p~~divldz 



, + py., (dv/dz + divjdy) + p.^^ {diojdx + duldz) + p^y {duldy + d^ldx) j d . vol, (3) 

 dB denoting an element of the bounding surface, and /, m, n the direction- 

 cosines of the outward drawn normal. The term involving the first surface- 

 integral represents the rate at which kinetic energy of disturbance is convected 

 into the volume considered, and the other three surface-terms denote the rate 

 at which the additional stresses pncxs Pxy, etc., called into existence by the 

 disturbance, would do work in the additional motion ii, v, %o on the fluid 

 contained in the surface. In many cases the joint effect of the surface-terms 

 is nil ; this happens, for instance, when the disturbance has a definite wave- 

 length in the direction of flow, if the volume is bounded by surfaces parallel 

 to the direction of flow, such that u, v, w vanish at them and by perpendicular 

 planes, such that the distance between them is any multiple of a wave- 

 length. In any such case, by substituting in the last integral in (3), the 

 values of the stresses, viz., 



Pxx = - i^ - I A* (du/dx + dvjdy + diojdz) + 2f.idiildx, p^y = fx (dtc/dy + dv/dx), etc., 

 the right-hand member of (2) becomes 



- J pu (vd U/dy + vjd Uldz) d . vol 



- fxj [2{duldxf + 2{d.vldy)~ + 2 (d-w/dzy + (dv/dz + dw/dyV + {d'w/dx + du/dzY 

 + (dujd.y + dv/dx)-} d , vol + \p'{du/dx + d.v/dy + dtv/dz) d . vol, (4) 

 where p)' =p ^2fx/?> .{dii/dx + dv/dy ^ div/dz). 



The second member is essentially negative ; the first may be either positive or 

 negative ; the third is, of course, zero, though it is convenient to retain it for 

 the present,* thus not assuming the fluid to be incompressible ; and whether 

 the disturbance increases or decreases, depends on the sign of the whole. If 

 then, for a given steady motion we could find the lowest value of p. for which 

 it is possible to choose u, v, lo, so that the expression (4) may be zero, there 

 would be no possibility of the motion being unstable for a greater value of p. 

 In the applications of the method by Eeynolds, Sharpe, and H. A. Lorentz, 

 the character of the disturbance is to a certain extent assumed, and apparently 

 somewhat arbitrarily ; and I proceed in the present chapter to conduct similar 

 investigations, while endeavouring to avoid any such arbitrary choice. 



* For the purpose of variation. 



