NOTES ON HYDRODYNAMICS, CHIEFLY ON VORTEX-MOTION. 3 



Multiplying both sides by dm, an element of the fluid mass, and integrating for 

 the whole mass of fluid within the surface, we obtain 



jqqdm = - jq—dm- \q^-d~& . . . (3), 



if dzj be the volume (dm/p) of the element of mass dm. Now usually V is 

 independent of t, and therefore dV/dt = 0. Thus for qdY/ds in the first integrand 

 on the right we can write dV/dt + qdY/ds, which is clearly dV/dt. Thus if T 

 denote the kinetic energy of the fluid which fills the surface at time t, and E 

 denote the potential energy from which the external forces are derived for the 



same fluid mass, we have 



f ., dT fdV, dE 

 qqdm — — , \ — dm = — . 

 J dt' J dt dt 



The equation found above can therefore be written in the form 



*(T + 1>~/4E*, .... (4). 



This gives, in an exceedingly compact form, the value of the total time-rate of 

 variation of the sum of the kinetic and potential energies T and B. 



6. The integral on the right can be transformed by partial integration, without 

 the introduction of Cartesian co-ordinates, by proceeding as follows. The continuity 

 of the motion from point to point in the fluid involves the possibility of dividing 

 the fluid mass up into narrow tubular portions bounded by non-intersecting stream- 

 lines. We shall call these tubes of flow. Their distribution is determinate for any 

 instant (or point of time), but varies in the general case from instant to instant. 

 Each such tube (unless it be endless, and entirely contained within the surface) will 

 enter the closed surface S at one element c/S x of surface, and emerge at another 

 element cZS 2 . We shall denote for any element of one of these tubes, say an element 

 of length ds, the average area of cross-section by a- ; then <r varies along the tube, 

 that is, it is to be regarded as a definite function of s, at each instant of time. 

 The element of volume dzs will now be <rds. 



7. First, then, integrating the expression on the right of (4) by parts, we get as the 

 integrated terms for a single tube —p<>CL2' J 2~' r P\ ( li a '\-> wnere °"i> °"2 are ^ ne cross-sections 

 of the tube at entrance into and emergence from the surface S. If q n he the com- 

 ponent of velocity normal to the surface inward from an element c/S of the surface, 

 this sum of terms may be written (pq n d$) 2 + (pq n d$) v The aggregate result for all 

 the tubes crossing the surface is the integral 



Jpq n dS 



taken over the surface S. Thus we obtain, since dw = <rds, 



^(T + E ) = jpq n dS + jp l - d -{q<r)dZZ 



