Criterion of Potential Energy. 705 



produce the circulations /c, we see that (33) amounts to this : 

 that when all the impulsive pressures Kp are so applied, there 

 must, on the whole, be no component o£ impulse parallel to 

 the axis of x (or of course to the axis of y or o£ z). 



35. Similarly 2tf/o[o>i] =0, which is one of the conditions 

 (31), may be put in the form 



2.*/ojVcos&dS = 0, (34) 



where r is the perpendicular distance of any point on the 

 surface S from the axis about which co 1 is measured, and £ is 

 the angle which the normal to S at the point in question 

 makes with a line perpendicular both to r and to the axis 

 of ft> l5 the positive direction of this latter line corresponding 

 (let us suppose for definiteness) to the positive sense of the 

 angular velocity co v Now (34) expresses the condition that, 

 when all the impulsive pressures Kp act over the surfaces S, 

 there shall be no resultant impulsive moment about the axis 

 of <tii (or of course about the axis of <w 2 or of e» 3 ). 



36. The results obtained in §§ 31-35 may be summarized 

 as follows : If in a Motionless liquid free from vortex motion 

 a number of solids are immersed, each consisting of a rigid 

 framework of thin wires, then the energy of the circulation- 

 momenta may be treated as potential energy, provided that 

 for each single solid the impulses required to initiate all the 

 circulations of that solid are such as, being applied to a rigid 

 body, would be in equilibrium. 



37. A dynamical system of the kind just considered may 

 -also be made to furnish an example of the fulfilment of the 

 conditions (21). With our previous stipulation as to the 

 thinness of the wires of which the solids are built up, it is 

 evident that (21) will be satisfied, provided only that each 

 body of the system is limited to translational freedom, without 

 the possibility of rotation. In this case, no matter what may 

 be the values of the circulation-momenta, the energy due to 

 those momenta may be treated as potential energy, although 

 in general the translational movement of any solid will 

 involve reactions against the constraints arising from what 

 maybe called " want of balance "o£ the circulation -momenta. 



38. The case of a mass of gas whose pressure is varied 

 adiabatically may serve as a final example ; isothermal con- 

 ditions are excluded from consideration, as in such case 

 the system is not properly speaking conservative, although 

 simulating a conservative system in its general dynamical 

 behaviour. For simplicity let the gas be monatomic and be 

 contained in a fixed cylinder, in which works a gas-tight 



Phil. Mag. IS. 6. Vol. 17. No. 101. May 1909. 3 B 



