6 Prof. A. Gray: Notes on Hydrodynamics. 



q'dq/'ds' .ds'. Thus the whole rate of diminution of flow 

 along the unclosed path AB is 



- U j 1 ®,.' q sin dds'+2 (V|| <**' + f ^' <fcj , 



AB AB 



and so from (8) we get 



AB AB AB AB 



. . . (10) 



The left-hand side is thus the whole rate of diminution of 

 flow along AB, and, since when extended to a complete 

 circuital path, it gives the whole rate of change of the surface 

 integral of spin about the normals, may be regarded also as 

 twice the rate of diminution of this surface integral as 

 depending on AB. Writing 



AB 



ws have from (10) when the value of ^ is inserted 



B 



or 9 <n 



z dt 



= 2(co ss 'qsm6ds / + q 2 -g 2 +('^ds'. . (12) 



AB AB 



Here the expressions 



Cdp Cdp 

 J P* J P' 



denote the integrals indicated, taken from some chosen 

 initial point on the line of integration up to the point A and 

 the point B respectively. 



If the circuit be closed dljdt vanishes by (11), that is the 

 time-rate of change of the surface integral of spin is zero. 

 Equation (12) then gives 



2(ba„,qsin0ds , + {§'&ds'=O, . . (13) 



which of course is the direct consequence of (7) obtained 

 above as (9). 



