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



(as the case may be) of the normal to a barrier, drawn in the 

 direction in which the circulation is estimated. 



Let us now suppose that the slightly varied motion, to which 

 A refers, is arbitrary as regards the solids, except only that the 

 initial and final configurations are to be the same as in the actual 

 motion, whilst the fluid is free to take its own course in accordance 

 with the motion of the solids. On this supposition we shall have, 

 both at time t and at time t lt 



for the fluid in contact with an element SS of the surface of a 

 solid, and, at the barriers, 



The right-hand side of (14) therefore reduces, under the present 



suppositions, to 



r ~\t t 



L*4 (x + xo) + PIC A ( x f + XQ ) + .. .J , 



and the equation may be put into the form 



{ A T - p* A ( X + %o) - PK A ( x f + ft ) - ... 



+ QiA ?1 + Q a A ?a +...}d* = ...... (16)*, 



which now takes the place of Art. 134 (7). 



Since the variation A does not affect the cyclic constants 

 K, K, ..., we have by (11), 



and therefore, by (12), 



...}&amp;lt;fc = ...... (17). 



It is easily seen that x , x , ... are linear functions of q lf q. 2 , ..., 

 say 



%= MI + O-&+ -..} 



^ / = 1 / g 1 + a L /g,+ ... I .................. (18), 



where the coefficients are in general functions of q lt q. 2 , If we write 



* Cf. Larmor, &quot; On the Direct Application of the Principle of Least Action to 

 the Dynamics of Solid and Fluid Systems,&quot; Proc. Lond. Math. Soc., t. xv. (1884). 



