334 Sir William Thomson on the Motion of 



3. The kinetic energy T is, of course, necessarily a quadratic 

 function of the generalized momentum-components f, ?;,... k } k'...- 

 with coefficients generally functions of ^, 0, ... , but necessarily 



independent of x , %', In consequence of this peculiarity it 



is convenient to put 



T==Q(f-«yc-^V-&c.^-^/ C -^V-&c.,...)+ ( ®(^^.„) j (5) 



where Q, <3 denote two quadratic functions. This we may clearly 

 do, because, if i be the number of the variables f , 97, . . . , and,; 

 the number of k, k 1 ,..., the whole number of coefficients in the 

 single quadratic function expressing t is (? + ^)( z+ i+ 1 )^ ^[^ 



is equal to the whole number of the coefficients i^±D 4. <lLLt}) 



2 2 



of the two quadratic functions, together with the «,/ available 

 quantities a, (3, . . . a! , /3', . . . , ... 



4. The meaning of the quantities «,&...*',..■ thus intro- 

 duced is evident when we remember that 



3F"* ST*"/ 5-?* 37=^... (6) 



For, differentiating (5) and using these, we find 



and using these latter, 



*=S-«+-0#-&°" #=^-«'*-^-&e.,... (8) 



Equations (8) show that -«^,-^ _«f^ fecare the contri- 

 butions to the flux across 12, &', &c. given by the separate 

 velocity-components of the solids. And (7) show that to pre- 

 vent the solids from being set in motion when impulses k k' 

 are applied to the liquid at the barrier surfaces, we must apply 

 to them impulses expressed by the equations 



£ =: a/c + u'j + &c, 7j =J3k + j3 r /c ! + &c (9) 



5. To form the equations of motion, we have in the first place 

 fcT *T 



and therefore, by (1), 



* =K > *- =K '--' • • • (») 



