NOTES ON HYDRODYNAMICS, CHIEFLY ON VORTEX- MOTION. 



15. Returning now to equation (16), we write instead of \f/, x + \dq/dt . ds, 

 so that 



J p 



Equation (16) may thus be written 



-A = - 2a)„, a sin - -±- 



ds s l dt 



(19). 



(20). 



As in the diagram, let AB be any line drawn in the 

 fluid, and PQ be the element ds', which we suppose to be 

 on this line. 



As before, q is the resultant velocity of the fluid at 

 P, and q' = q cos 6 is the component at P along the tangent 

 to AB at P. Now q sin 6 is the velocity at P at right 

 angles to ds', that is, q sin is the velocity with which 

 each point of the path AB is moving at P towards the 

 right in the plane of the paper. The product q sin 6ds' 

 is therefore, the rate at which any circuit including ds' 



is increasing in area, in consequence of the motion of ds' at right angles to itself, 

 in the plane of the paper, without alteration of length. 



Multiplying (20) by ds', and integrating from A to B, we obtain 



x-x = 



f f W 



I 2o) ss ,q sin 6 ds - I ~rd$ 



(21). 



We see that the first integral on the right is twice the rate at which the surface 

 integral of elemental rotation, taken over any surface of which AB is part of the 

 boundary, is changing in consequence of the fact that each element ds' is being 

 carried at right angles to itself by the motion of the fluid. It is also, it is to be 

 noted, the rate at which the flow of the fluid along AB is changing from this cause. 



16. But the area enclosed is also changing in consequence of the motion of one end 

 of the element ds' relatively to the other. The surface integral is changing from this 

 cause also, and the rate of change can be found most easily by calculating the rate of 

 increase of flow along AB arising from the change of length of the element. 



To do this, consider the new position of ds' after the lapse of time dt. Let the 

 plane of the initial position of ds', and of the displacement qdt of P, be taken as 

 a plane of reference. The other end Q of ds' has been displaced through a distance 

 (q + dq/ds' . ds')dt which is not in this plane, and is not parallel to the displacement 

 qdt of P. The increase of the rate of flow of fluid along ds' arising from the 

 difference (dq/ds' . ds')dt is qdq/ds' . ds'. 



If this is not intuitively perceived, it can be proved as follows : — 



The component of the displacement (q + dq/ds'. ds')dt of Q along ds' is 

 (q + dq/ds' . ds')dt . cos (6 + dd), or (q cos 6-qdd. sin + dq/ds' . ds' cos d)dt. Thus the end 

 TRANS. ROY. SOC. EDIN., VOL XLVII. PART I. (NO. 1). 2 



