119, 120.] PERFORATED SOLID. 137 



(rf) that o)j must be a monocyclic function, the cyclic 

 constant being unity; viz. the increment of the function must 

 be unity when the point to which it refers describes a circuit 

 cutting the first barrier once and once only, and zero when the 

 point describes a circuit not cutting this barrier. 



It appears from Art. 62 that these conditions completely de 

 termine a) lt save as to an additive constant. 



The energy of motion of the fluid is given by Art. 67, viz. 

 we have 



Substituting the values of &amp;lt;, j~ from (47) we obtain a homo 



geneous expression of the second degree in M, v, w, ... , K I} K V ... . 

 This expression consists of three parts. The first is a homogeneous 

 quadratic function of u, v, w, p, q, r, the coefficients in which are 

 given by the same formulae as in Art. 110 ; the second part consists 

 of products of u, v, w,... into x lt /r g ...; whilst the third part is a 

 quadratic function of the coefficients K. The coefficients of the 

 second part all vanish. Thus the coefficient of UK^ is 



and to see that the value of this expression is in fact zero, we 

 have only to compare (30) and (31) of Art. 66, writing &amp;lt;f&amp;gt;=&amp;lt;f&amp;gt; lf 

 -\Jr = a) l , and therefore K V = K Z . . . = 0, A:/ = 1, * 2 = K^ = . . . = 0. 

 The coefficients of the third part are found as follows. We have 



by another simple application of Thomson s extension of Green s 

 theorem. 



