Fior. 3. 



Prof. A. Gray : Notes on Hydrodynamics. 5 



any area of which AB is part of the boundary, is changing 

 in consequence of the fact that each element ds' of AB is 

 being carried towards the right by the motion of the fluid. 



Fig. 3 shows the effect of this 

 motion for a closed path of inte- 

 gration. The area between two 

 stream-line elements ds, and the 

 two positions of the connecting ds, 

 is evidently dsds' sin 6. 



The second term on the right is 

 the rate at which as time passes 

 the flow along ^B~4schanging 

 apart from the mo£ion>Qf the 

 elements ds' . , 



If the path- AB be 

 left-hand side of (8) vanishes and we hAte^ 



where (j) denotes integration round the 



7. Now since the line integral xq'ds' taken rounc 

 closed path is twice the surface-integral of elemental rotation 

 about normals drawn to the elements into which any surface 

 bounded by the path may be divided, if we calculate the 

 whole change of rate of flow along AB due to the various 

 causes, we snail obtain a result which, extended to the whole 

 circuit, will give the exact rate of increase of the surface 

 integral of elemental rotation for any surface of which the 

 path of integration is the bounding edge. 



Now the surface integral is increasing at rate 



2 (J ) co ss ' q sin 6 ds' , 



in consequence of the motion of the elements ds 1 at right 

 an iiles to themselves, and also at rate 



;- q S11T 



closeo 



IncT the 



$16" 



in consequence of the variation of q with time. There are 

 two other causes of variation due to the motion. Each 

 element ds' is being displaced bodily in the direction of its 

 length, and moreover the end of the clement nearer B is 

 undergoing displacement with respect to the end nearer A. 

 It can be proved that each of these displacements gives a 

 rate of increase of flow alono- the element of amount 



