124 Proceedings of the Royal Irish Academy^ 



Art. 29. Diferential Equations satisfied hy the Distiirlance which is Stationary 



for the Ghxatest Possible fx. 



Proceeding to a more general investigation, the critical equation for //, 

 whether the fluid be compressible or not, is from (4) : 



- I pu (vd U/dy + wd U/dz) d . vol + f p\duldx + dvjdy + dw/dz) d . vol 



- /i J [2[duldxf + 2[dvldyf + 2 (div/dzy + {dv/dz + dwjdyy + (dw/dx + du/dzY 



+ {du/dy + dv/dxf} d.Yol = 0. (5) 



The variation of u, v, w in this gives, as conditions for a stationary /x, on 

 integrating by parts, 



2fxV~u + 2ixdldx (du/dx + dv/dy + dtv/dz) - p {vd U/dy + wd U/dz) 



= dp/dx + 4/x/3 . {du/dx + dv/dy + dw/dz), (6) 



etc., or, supposing the fluid incompressible,* 



2nVhL - p {vd U/dy + wd U/dz) = dp/dx, 

 2iixV'v - pud U/dy = dp/dy, 



2p.Vhv- pudU/dz = djj/dz. (7) 



If the volume is bounded by fixed surfaces parallel to the direction of flow 

 and by perpendicular planes such that the distance between them is any 

 multiple of a wave-length, the surface terms, which have not been given, 

 vanish; under these conditions also equations (7) with that of continuity 

 satisfy (5), so that (5) need no longer be referred to. 



Art. 30. The uniformly Shearing Stream subject to Boundary- Conditions 

 V = 0, dv/dy = 0. Lorentz' Besidt. 



A stream of uniform vorticity is, of course, the simplest case ; and Eeynolds' 

 method has been applied to it by H. A. Lorentz.f The type of disturbance he 

 selects consists of a species of " Elliptic Whirls " in which each particle of fluid 

 has motion in an elliptic orbit superimposed on its steady motion; these 

 ellipses are similar and similarly situated ; and the angular velocity round the 

 centre is a function of the distance from it ; the orientation and shape of the 

 ellipses and the law of velocity are then determined, so that the value of p. 

 which makes the right-hand member of (4) vanish shall be greatest possible. 

 If the steady velocity be By, and the distance between the bounding-planes i>, 

 his resulting equation is pBD- = 288^. 



* If the fluid be compressible, the variation of p and p in (o) leads to an equation which would 

 determine the scale of the disturbance. 



t " Ueber die Entstehung turbulenter Fliissigkeitsbewegnngen und iiber den Einfluss dieser 

 Bewegungen bei der Stromung durch Rohren." Abliandlungen iiber theoretische Physik, 

 Band 1, s. 43. 



