494 Dr. C. V. Burton on the Motion of a 



from (6). It is further known that the whole energy of the 

 system 



=E + K, (8) 



where E is a function of u . . ., p . . ., only, and K is a function 

 of the momenta /cp only. Suppose, now, that the solid were 

 brought to rest by forces applied to it alone : E would vanish 

 along with u, v, w, p, q, r, while the circulations k, and con- 

 sequently also K, would remain unaltered. The generalized 



velocities %, %'?••• would in general have changed, becoming, 



let us suppose % > %o'> • • • an( i the kinetic energy would 

 accordingly have become 



k=Kto+«W+...) (9) 



Now let 



%=%o + %i, %' = %o' + Xi', (10) 



so that each ^ is that part of the flux of liquid (volume per 

 unit time) which takes place across a barrier-surface owing to 

 the motion of the solid itself. 



Having regard to (8), (9), and (10) our equation (7) for 

 T' becomes 



T'=(E + K)-2K-*pxi -«'/>%/• • • • (11) 

 Let us write for the velocity-potential of the acyclic motion 



®=u$ ¥ +v<l> v + w<l> w +p<l> p + q<l> q + r<l> r , . . (12) 

 and for the value of Xi across the barrier-surface a we have 

 %i= 1 1 A -^ — -\_ul + vm-hivn-\-p(ny — mz) 



+ q{lz-nz) +r{mx-ly)]\do-, . . . (13) 



where x, y, z are the coordinates of the element da and I, m, n 

 are the direction-cosines of its normal v, all referred to the 

 system of axes fixed in the solid. From (12) and (13) sub- 

 stitute in (11) ; thus 



T' = E — K -f uZicp \\il — ^- \da + similar terms in v, w, 



+ptxp ( ny — mz — -^ W + similar terms in q, r, . (14) 



where the summation refers to the m barriers. 



Remembering (8) it will be seen that T' is now expressed 

 in the proper form, namely as a function of u, v, w,p, q, r, and 



