152 VORTEX MOTION. [CHAP. VI. 



present themselves; the vortex-lines may be tangential to 2 on 

 both sides, or they may cross the surface, experiencing there an 

 abrupt change of direction. In the first case we have 



% + mi 7l + &amp;lt; = 7 + mi 7a + w? a = &amp;lt;) ............ (13) 



at 2 ; and in the second we have 



^, + wi7 1 + w? 1 = Zf,+-WM7 a + &amp;lt; ......... (14). 



In fact, if d2&amp;lt; be a section of a vortex, taken parallel and infinitely 

 close to 2 on one side of it, the product (l^ + mrj l + wfj d% 

 measures the strength of the vortex, which is (A rt. 127) the same 

 on both sides of S. Now in (12) the region through which the 

 triple integration extends is divided by the surfaces 2 into a 

 certain number of distinct portions. For each of these, taken by 

 itself, the equality of the second and third members of (12) holds ; 

 and if we add the results thus obtained, we see that to make (10) 

 true for the region taken as a whole we must add to -the third 

 member terms of the form 



due to the two sides of each of the surfaces 2. The relations (13) 

 and (14) shew however that these terms all vanish, so that (8) is 

 still satisfied. 



130. Let us examine the -result obtained in Art: 129; and 

 let us suppose first that the fluid is incompressible, so that 6 = 0, 

 and- therefore P = 0. Denoting by Sw&amp;gt;.8v, Biv the portions of u, v, w 

 arising from the element dxdy dz in the integrals (11), we find 



or 



and similarly 





.(15). 



1 / 7 , / rp /v/\ 



C\ -* / O / V V /*X/ t ^ / \ l/T/T/ 



ow = = o V dx dy dz . 



ZTTT \ r r / 



It appears from the form of these expressions that the resultant 



