202 MOTION OF SOLIDS THROUGH A LIQUID. [CHAP. VI 



If we add to (2) twice the kinetic energy, T 1} of the solids 

 themselves, we get an expression of the same form, with altered 

 coefficients, say 



2T = 4 u 2 1 2 + ^ 22 2 2 2 +... + 2^ 1 ? 2 + (4). 



. It remains to shew that the equations of motion of the solids 

 can be obtained by substituting this value of T in the Lagrangian 

 equations, Art. 133 (17). We cannot assume this without further 

 consideration, for the positions of the various particles of the 

 fluid are evidently not determined by the instantaneous values 

 q l9 q-2,-" of the coordinates of the solids. 



Going back to the general formula 



(AT + 2 (XAf + FAT? + ZAf)} dt 



to 



(5), 



let us suppose that in the varied motion, to which the symbol A 

 refers, the solids undergo no change of size or shape, and that -the 

 fluid remains incompressible, and has, at the boundaries, the same 

 displacement in the direction of the normal as the solids with 

 which it is in contact. It is known that under these conditions 

 the terms due to the internal forces of the solids will disappear 

 from the sum 



The terms due to the mutual pressures of the fluid elements are 

 equivalent to 



or jfp (JAf + mA, + nAf) dS+jffp (^ + ^ + ^/) dxdydt, 



where the former integral extends over the bounding surfaces, 

 I, m, n denoting the direction-cosines of the normal, drawn towards 

 the fluid. The volume-integral vanishes by the condition of 

 incompressibility, 



| 



dx dy dz 

 The surface-integral vanishes at a fixed boundary, where 



