Motion of a Perforated Solid in Liquid. 341 



We also notice that 



as may be at once seen by differentiating (2) with respect 

 to u and p respectively. 



5. Substituting for p 1 in (5) in terms of the velocities, we 

 have 



+ 1 \< (u—yr-\-zq)^-+ (two similar) >ldS 



-«JJ{(!^H1*)> «> 



The first line of this expression is, from (6), equal to 



-JJ ^ sT /S 



JJJ l da 3* oj/ ^ 3e ds J J 



by Green's transformation. Remembering that <j> u is inde- 

 pendent of the time, this integral, taken throughout the liquid, 

 becomes 



mm<Mhm>* 



== l^3X = l^BXi 

 pdt~du p~dt ~du 



By Green's transformation the second line is equal to 

 "" 1 1 1 7T \ ( u —y r + Z( l) ^r + ( two similar) I dxdy dz 



-'-JK'U-fg)*** 



which by a second application of Green's transformation 

 becomes 



