140-141] PRESSURES ON SOLIDS AT REST. 219 



It appears from Art. 137 (10), that under the present circum 

 stances the fluxes through the respective apertures are given by 



dK . dK 



By solving these equations, the circulations K, # ,... can be ex 

 pressed as linear functions of ^ , ; , .... 



If these values of K, K, ...be substituted in K we obtain 

 a homogeneous quadratic function of ^ , ^ , When so ex 

 pressed, the kinetic energy of the cyclic motion may be denoted 

 by T . We have then, exactly as in Art. 119, 



T,+ K=2K = p*x &amp;gt; + pic x;+ ..................... (4), 



so that if, for the moment, the symbol A be used to indicate 

 a perfectly general variation of these functions, we have 



dT () A dT . dT n , dT 



,- . AX O + , . A + ... + y Aft + 

 d%o dxo dq, dq 2 



dK A dK . dK A dK A 



+ A* + ---A* + ...+ i-Aft + ,- A^ + ... 

 d/c d/c dq t dq., 



Omitting terms which cancel by (3), and equating coefficients 

 of the variations A^ , A^ ( /, ..., A^, A^,..., which form an inde 

 pendent system, we find 



and 



Hence the generalized components (2) of the pressures exerted by 

 the fluid on the solids when held at rest may also be expressed in 

 the forms 



dT dT f . 



d qi &amp;gt; 1fc&amp;gt;- 



It will be shewn in Art. 152 that the energy K of the cyclic fluid 

 motion is proportional to the energy of a system of electric current-sheets 

 coincident with the surfaces of the fixed solids, the current-lines being 

 orthogonal to the stream-lines of the fluid. 



