.Equations in certain Cases of Fluid-Motion. 357 



bodies momentarily' fixed. We have then to show that, for 

 such a displacement, 



^dm(xhx + ySy + zSz) = 0. 



This is evident ; for the work done by the momenta of the 

 particles of the system must for any possible displacement be 

 equal to that done by the impulse ; and the latter in this case 

 vanishes, as the bodies remain fixed. 



The same thing may be shown analytically thus: — 



%dm(ocSa + ySy + zdz) 



= \\d&.p.4>.hn 



The first term vanishes, since the motion of the fluid in con- 

 tact with the bodies is tangential; the second, since the fluid is 

 incompressible. 



Addendum, 



Sir William Thomson has shown, in his paper on Vortex- 

 Motion (Trans. E. S. E. vol. xxv.), that if one or more solid 

 bodies are moving in an infinitely extended frictionless incom- 

 pressible fluid, the motion of the fluid being supposed at one 

 instant and therefore always irrotational, the impulse (i. e. the 

 system of forces which would at any instant, if applied to the 

 solids, generate the motion of the system of solids and fluids) 

 would, if applied to a rigid body, represent a constant motive. It 

 may be interesting to show that this conclusion follows directly 

 from the Lagrangian equations. 



First, let there be only one rigid body. Take two systems 

 of coordinate axes — the first (OX, OY, OZ) fixed in space, 

 the second (O'X', O'Y', O'Z') attached to the body. 



Let u } Vj w be the components of the velocity of 0' esti- 

 mated along the moving axes ; 

 p, q, r the rotations of the body round these axes ; 

 <f),yjr 7 6 the usual angles denoting the position of the 



moving axes with respect to the fixed axes ; 

 x, y, z the coordinates of ; with respect to the fixed 

 axes. 



Then, taking as generalized coordinates x, y, z, <£, ^r, 0, we 



