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



the functions fa, fa, ..., being determined by the same conditions 

 as in Art. 134, and 



fa = ica) + K'G> f + ........................ (2), 



a, a/, ... , being cyclic velocity-potentials determined as in Art. 129. 

 Let us imagine barrier-surfaces to be drawn across the several 

 channels. In the case of channels in a containing vessel we shall 

 suppose these ideal surfaces to be fixed in space, and in the case of 

 channels in a moving solid we shall suppose them to be fixed 

 relatively to the solid. Let us denote by % > %</> -> the portions 

 of the fluxes across these barriers which are due to the cyclic motion 

 alone, and which would therefore remain if the solids were held at 

 rest in their instantaneous positions, so that, for example, 



where So; 3cr', ... are elements of the several barriers. The total 

 fluxes across the respective barriers will be denoted by % 4- % , 

 X + %<>'> ' so that %> X> wou ld b e the surface-integrals of the 

 normal velocity of the fluid relative to the barriers, if the motion of 

 the fluid were entirely due to that of the solids, and therefore 

 acyclic. 



The expression of Art. 55 for twice the kinetic energy of the 

 fluid becomes, in our present notation, 



This reduces, exactly as in Art. 129, to the sum of two homogene- 

 ous quadratic functions of q lt q 2} ..., and of tc, K', ..., respectively*. 

 Thus the kinetic energy of the fluid is equal to 



T + K .............................. (5), 



with 2T = Aufc" + A^ 2 + ... + 2A 12 & + ......... (6), 



and 2K = (K, K) K* + (K, K) K 2 + . . . + 2 (*, K') KK + ...... (7), 



where, for example, 



* An example of this reduction is furnished by the calculation of Art. 99. 



