348 Mr. G. H. Bryan on the Equations of 



choice of axes it is always possible to reduce this number by 

 six, and a further reduction may be effected when the body 

 is symmetrical. When the motion is cyclic the kinetic 

 energy must be replaced by the modified function which in 

 addition contains the seven terms 



fi* + 7)v + %w + Xp + fJ>q '+ vr — K, 

 of which the last term does not enter into the equations of 

 motion of the solid. The six coefficients (f , . . . X, . . .) are 

 linear functions of the circulations, and they remain con- 

 stant so long as only conservative forces act on the liquid, 

 for the circulations themselves then remain constant. Hence 

 the modified function H may be regarded as a non-homo- 

 geneous quadratic function of the six velocities involving 

 28 constants, of which 27 enter into the six equations of 

 motion of the solid. 



Equations of Motion of a light thin framework of rigid wires. 



12. To take the simplest possible case, let us suppose the 

 solid to consist of a network of infinitely thin rigid massless 

 wires through the meshes of which the liquid is circulating. 

 If the motion of the liquid were acyclic, the wires would 

 simply cut through the liquid without setting it in motion : 

 hence the kinetic energy X' + Si of the acyclic motion 

 vanishes, and the modified function becomes 



TL = %u + 7)v-± Jty-f- Xp + fMq i-vr— K, . . (32) 

 a result otherwise evident from the fact that Si only involves 

 surface integrals taken over the infinitely small surface of the 

 solid, while f, rj. f, X, ju,, v being integrals taken over the 

 finite surfaces of barriers are in general finite. 



If we choose as our axis of x the Poinsot's central axis of 

 the impulse whose six components are f, rj, f, X, /x, v, the 

 modified function will reduce to the form 



H = Ew + Ap-K (33) 



If there is only one aperture, £, rj, ?, \, //,, v are all pro- 

 portional to the circulation k and the central axis of the 

 impulse is fixed in position relative to the solid : if there are 

 several apertures the position of the axis depends on the 

 ratios of the circulations through the various apertures, but 

 throughout the motion it in every case remains fixed rela- 

 tively to the solid. 



The six equations of motion (19) (20) now reduce to 



X = 0, L = 0, ^ 



Y=rB, M.=w& + rA, V . . . (31) 



Z=—qB, l$=—vE — qA.) 



