128-129] PERFORATED SOLID. 193 



where 



< = n >. 



' 



The functions ^^ </> 2 , </> 3 , % 15 ^ 2 , ^ 3 are determined by the same 

 conditions as in Art. 115. To determine &>, we have the condi- 

 tions : (1) that it must satisfy V 2 o) = at all points of the fluid ; 

 (2) that its derivatives must vanish at infinity; (3) that dco/dn=Q 

 at the surface of the solid ; and (4) that o> must be a cyclic function, 

 diminishing by unity whenever the point to which it refers com- 

 pletes a circuit cutting the first barrier once only in the positive 

 direction, and recovering its original value whenever the point 

 completes a circuit not cutting this barrier. It appears from 

 Art. 52 that these conditions determine o> save as to an additive 

 constant. In like manner the remaining functions o>', a>", . . . are 

 determined. 



By the formula (5) of Art. 55, twice the kinetic energy of the 

 fluid is equal to 



- ...... (2). 



Since the cyclic constants of < are zero, we have, by Art. 54 (4), 



which vanishes, since dfa/dn = at the surface of the solid. 

 Hence (2) reduces to 



"- ...... (3) - 



Substituting the values of <, < from (1), we find that the kinetic 

 energy of the fluid is equal to 



T + # .............................. (4), 



where T is a homogeneous quadratic function of u, v, w, p, q, r of 

 the form defined by Art. 118 (2), (3), and 



13 



