492 Dr. C. Y. Burton on the Motion of a 



to determine the entire configuration of the system (including 

 the positions of all the particles of liquid) ; hut we shall see 

 immediately how, in virtue of the proposition (A), Lagrange's 

 equations may be written down. 



6. Since an increment &% in one of the coordinates % is the 

 volume of liquid which flows across a barrier- surface (i. e., 

 which flows through an aperture relatively to the solid), the 

 generalized force corresponding to % must be conceived of as 

 a uniform pressure exerted over the said geometrical surface, 

 by means of some immaterial mechanism attached to the solid; 

 while the impulse corresponding to ^ is of course a uniform 

 impulsive pressure applied in the same manner. From hydro- 

 dynamical considerations we know that the measure of such an 

 impulsive pressure is pS/c, where p is the density of the fluid, 

 and 8k the change produced in the circulation through the 

 corresponding aperture. 



Hence the impulses corresponding to %, %', . . . are 



K P> K 'P> ( 6 ) 



where /e, */, . . . are the circulations through the various 

 apertures. 



7. Now when the motion of the liquid is irrotational, we 

 have 



T =a homogeneous quadratic function of 0, r , . . . k, k' . . . 

 only ; coefficients functions of 0, 0' . . . only; ^ 



Y, -%'j • • • = homogeneous linear functions of 0, 0', . . . 

 K, k' , . . . only ; coefficients functions of #, r , . . . only. 



Since the %'s are equal in number to the /e's, let us suppose 

 the last-written system of linear equations to be solved for 

 the ks in terms of the %'s ; we then have 



k,/c',... = homogeneous linear functions of 0, d',... %, %', . . . only ; 

 coefficients functions of 6, 0', . . . only. 



Substituting in the expression for T we get 



T = a homogeneous quadratic function of 0, 0',... %,%', ... only ; 

 coefficients functions of 0, 0' , . . . only. 



This, then, remains true so long as the motion of the liquid is 

 irrotational; in other words,, so long as the only forces and 

 impulses acting are of types corresponding to 0, r , . . . (since 

 these are applied to the solid) , %, y/, . . . (since these are 

 uniform over the barriers, by § 6). 



If we identify ^, 0, . . . with the coordinates 0, 0',... % y/, • • • 

 of the present example, we see that the proposition (A) of § 4 

 is immediately applicable to this case. We may therefore 



