Criterion of Potential Energy. 703 



mitre-wheels T, T', fixed upon the spindles A A, A' A' re- 

 spectively and gearing with a mitre-wheel S, would effect 

 this result ; a fourth mitre-wheel S being introduced for tho 

 sake of symmetry and balance. Though the omission is not 

 essential to the case now considered, it will be assumed for 

 simplicity that the rotational energy of the mitre-wheels S, S 

 is relatively small enough to be left out of account. 



30. The expression for the whole kinetic energy is obtained 

 from (22) by substituting — x ^ or %'' while the momentum 

 corresponding to the ^-coordinate is now 



3T/3x = 21fX = « (say); .... (27) 

 and the modified Lagrangian function is accordingly 

 T' = T-wX 



-W-IS (28) 



If the momentum u remains constant, the last term on the 

 right-hand of (28) is the energy of rotation of the governors 

 about their axes, with sign reversed. When u is given, this 

 term is a function of 6 only, and T', given by (28), may 

 accordingly be taken as made up of the difference of the 

 kinetic and potential energies of the system. As regards 

 the equations of motion corresponding to the working coor- 

 dinates, the present example differs from that of § 2S in that 

 the moment of inertia Ti of the frame B about the axis AA' 

 is now effectively increased by 2lf, the sum of the moments of 

 inertia of the two governors about their axes A A, AAA 



31. An interesting example of potential energy is furnished 

 by a system of perforated solids, immersed in a frictionless 

 incompressible fluid which is circulating irrotationally through 

 their various apertures. Let each of the solids be in the form 

 of a thin rigid wire or wires, forming a closed loop or a 

 framework. Then, provided no two solids approach one 

 another very closely, the component of fluid motion contributed 

 by any one of the solids is due almost exclusively to the cyclic 

 constants of circulation associated with that solid, and is appre- 

 ciably the same as if the remaining solids were non-existent. 

 In the present case the working coordinates are any such as 

 serve to define at each instant the position and orientation of 

 every one of the solids, and each of the coordinates (%,%', ...) 

 to bo subsequently ignored is the volume of liquid which, 

 starting from a definite configuration, has flowed across one 

 of the ideal geometrical surfaces required to close the various 



