On Lagrange's Equations of Motion. 491 



the other, so that initially and finally Byfr, Bcj>, . . . are all zero. 

 It does not follow that all the Bx, By, Bz's are zero ; but 



%n(xBx + yBy + zBz) 



is the so-called " virtual moment " of the actual momenta in 

 the hypothetical displacement ox, By, Bz ; that is, the virtual 

 moment, in the same displacement, of the impulse necessary 

 to produce the actual motion from rest. In virtue of (3), 

 therefore, and of the initial and final vanishing of B-yfr, B<p, . . . 

 we see that the bracketed terms of (4) must both be zero ; 

 hence 



The increment BA vanishes and A has a stationary value for 

 all ivorklessly effected variations of path which leave the 

 initial and final values of yjr, <£, . . . . unaltered. . (5) 



4. Lagrange's equations for the coordinates i|r, (/>,... may 

 now be written down at once^ since the investigation of 

 Thomson and Tait* becomes applicable to the present case 

 without modification. It will be noticed that in their equa- 

 tions (10) v and (10) vi , § 327, the sign of dV/d^ should be 

 reversed. 



We have thus a perfectly general proof of the proposition : 

 If the kinetic energy of a material system can be expressed as a 

 homogeneous quadratic function of certain generalized velocities 



yfr, $>, . . . only, the coefficients being functions of yjr, <£>,... only, 

 and if this remains always true so long as the only forces and 

 impulses acting are of types corresponding to tjr, <f>, . . , , the 

 equations of motion for the coordinates yjr, <£, . . . may be written 

 down from this expression for the energy, in accordance with 

 the Lagrangian rule. Provided only that the stated conditions 

 are satisfied, we need not consider ichether the whole configura- 

 tion is determined by the values of yjr, <£, . . . , or what is the 

 nature of the ignored coordinates (A) 



5. Passing over the known application of this result to the 

 motion of solids through an irrotationally and acyclically 

 moving liquid, we come to the more general case of a perfo- 

 rated solid, with liquid irrotationally circulating through the 

 apertures. Take as coordinates any six 6, 6 r , . . . which deter- 

 mine the position of the solid, together with ^, %', . . . equal 

 in number (in) to the apertures ; each ^ being the volume 

 of liquid which, starting from a given configuration, has flowed 

 across some one of the m geometrical surfaces, required to 

 close the apertures, these surfaces being supposed to move 

 along with the solid. 



Of course the coordinates 6, 6' , . . . %, %', . • • are insufficient 

 * Loc. cit. 

 2 M 2 



