62 IRROTATIONAL MOTION. [CHAP. III. 



to infinity and is at rest there, being limited internally by one or 

 more closed surfaces S. Let us suppose a large closed surface 2) 

 described so as to enclose the whole of $. The energy of the fluid 

 included between 8 and 2 is 



.................. (20), 



where the integration in the first term extends over 8, that in the 

 second over S. Since we have by (11) 



(26) may be written 



-C) ( ZS ......... (27), 



where G may be any constant, but is here supposed to be the 

 constant value to which &amp;lt; was shewn in Art. 51 to tend at an 

 infinite distance from 8. Now the whole region occupied by the 

 fluid may be supposed made up of tubes of flow, each of which 

 must pass either from one point of the internal boundary to an 

 other, or from that boundary to infinity. Hence the value of the 

 integral 



taken over any surface, open or closed, finite or infinite, drawn 

 within the region, must be finite. Hence ultimately, when 2 is 

 taken infinitely large and infinitely distant all round from S, the 

 second term of (27) vanishes, and we have 



where the integration extends over the internal boundary only. 

 If the total flux across the internal boundary be zero, we have 



so that (28) becomes 



S*--P//* *&amp;lt;B (29), 



simply. 



