120-121] EQUATIONS OF MOTION. 177 



If in these we write T T + T l5 and separate the terms due to 

 T and T 1 respectively, we obtain expressions for the forces exerted 

 on the moving solid by the pressure of the surrounding fluid ; thus 

 the total component (X, say) of the fluid pressure parallel to x is 



d dT dT dT 



X = - -T: -T- + r j - Q j- (2), 



dt du dv * dw 



and the moment (L) of the same pressures about x is 



_d_dT_ dT _ dT dT _ JT 

 dt dp dv dw dq dr 



For example, if the solid be constrained to move with a constant 

 velocity (u } v, w), without rotation, we have 



X = 0, Y = 0, Z = 0, 



dT dT _,, dT dT dT dT 



Jj = W . V -r , M = U -j W -j , N = V , U -.- 



dv dw dw du au dv 



where 2T = A*t 2 + Bt&amp;gt; 2 + Cw 2 + 2A vw + 2B W -f 2C uv. 



Hence the fluid pressures reduce to a couple, which moreover 

 vanishes if 



dT dT dT 



-j : u = -j : v = -j- :w, 

 du dv dw 



i.e. provided the velocity (u, v, w) be in the direction of one of the 

 principal axes of the ellipsoid 



A# 2 + &quot;By* + Cz 2 + ZAfyz + VB zx + ZG xy = const.. . .(5). 



Hence, as was first pointed out by Kirchhoff, there are, for any 

 solid, three mutually perpendicular directions of permanent trans 

 lation ; that is to say, if the solid be set in motion parallel to one 

 of these directions, without rotation, and left to itself, it will continue 

 so to move. It is evident that these directions are determined 

 solely by the configuration of the surface of the body. It must be 

 observed however that the impulse necessary to produce one of 



* The forms of these expressions being known, it is not difficult to verify them 

 by direct calculation from the pressure-equation, Art. 21 (4). See a paper &quot; On the 

 Forces experienced by a Solid moving through a Liquid,&quot; Quart. Journ. Math., 

 t. xix. (1883). 



L. 12 



