704 Dr. C. V. Burton on the Kinetic 



apertures in the solids. We shall be able to treat the energy 

 of the circulation-momenta as potential if certain conditions 

 are satisfied which are equivalent to (9a) and (19). The 

 condition (9 a) (that the coordinates ^ shall not appear in the 

 coefficients of the expression for the total kinetic energy of 

 the system) is obviously realized. But (19) will only be 

 consistently fulfilled when the sum of the C^'s for each solid 

 is always zero ; the momentum C corresponding to the coor- 

 dinate % being Kp where k is the cyclic constant of circu- 

 lation for the aperture in question, and p is the density of 

 the liquid. 



32. For each solid therefore there is a condition to be 

 satisfied of the form 



t*px=% ...... (29) 



the fas being homogeneous linear functions of x, y 3 z, w ly co 2 ,(i) 6 , 

 where o) 1} co 2 , g>3 are the angular velocities of the solid about 

 axes instantaneously coincident with a set of rectangular axes 

 moving with the solid, and x, y, z are the Cartesian coordinates 

 of the origin of those moving axes. For the particular solid 

 under consideration, let 



Xi = W * + W i) + [ «Q k + M »i + M « 2 + M <*>z 1 / 30) 



Then (19) or (19 b) for that solid is equivalent to the six 

 conditions 



= 2*p|>] = 2*/>[y] =Xrcp[z], ~\ 



= %/cp[(o 1 ~] =2#/o[a> 3 ] — 2/c/o[a> 3 ]. J 



33. Let S, S', . . . be geometrical surfaces invariably re- 

 lated to the solid with which w T e are dealing, and sufficing 

 to close all its apertures. Then since [x]x, [x]'x, . . . are the 

 volumes of liquid flowing per unit time past the surfaces 

 S, S', . . j ow r ing to the velocity-component x of the body, we 

 easily see that 



\x] = -J cos vdS, H'=-j'cosj/^S', . . (32) 



where v is the angle which the positively drawn normal at 

 any point of the surface S makes with the axis of x, and so on. 

 Thus the first of the conditions (31) may be written 



Z. K p$cosvdS = (33) 



34. Remembering that the '/cp's measure the impulsive 

 pressures which must be applied over the surfaces S, !S', . . . to 



