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, K',... can be ex- 

 pressed as linear functions of % > %<>'> 



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

 a homogeneous quadratic function of %o> %</> When so ex- 

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

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



T +K=2K = P Kx* + pK'Xo' + ..................... (4), 



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

 a perfectly general variation of these functions, we have 



dT . dTn . . , dTn . dT tt A 



dK . dK . dK . dK 



,- A* + -J-; A/c' + ... + -j- Aft + , Ag 2 + 

 d/c d/c dq : dq. 2 



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

 of the variations A^ , A^ ', ..., A^, Ag 2 ,..., which form an inde- 

 pendent system, we find 







and = --- - = _ ............ (7). 



dq 1 dq l dq. 2 dq 2 



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 



, . 



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. 



