10 Prof. A. Gray: Notes on Hydrodynamics. 



The first of these is typical of the three equations of 

 v. Helmholtz generalized for the case of a fluid of varying 

 density. The alternative forms here shown are sometimes 

 convenient. 



If we multiply the first of the equations typified by (23) 

 [or (23') ] by £, the second by r), and the third by f, and add, 

 we get, since <o 2 = f 2 + t? 2 + f 2 } 



^+* 0= ^ + Vo- +r o^ • • • (25) 



where "dufda, ...., denote differentiation of u, v, w along 

 the vortex-line, in the direction given by the cosines (£, 77, £)/&>. 

 The equation just found may be written 



^ +6> e=^+^+- ? l^. . . ( 26) 



dt co oo" co oo" co o<r 



12, Now the quantity on the right is the time-rate per 

 unit length at which an element da of the vortex-line is 

 increasing in length in the direction indicated by the cosines 

 (?? Vt ty/ 00 ' For "du/'da.do, 'dvfdo-.da, 'dwj'da.da are the 

 time-rates at which the projections of da on the axes Ox, 

 Oy, Oz are lengthening. It is to be noticed very carefully 

 that this is not the unital rate of elongation of the element 

 da of the vortex-line, for that is clearly 



0O~\C0 CO CO J 



which differs from 



f ~du 7j "dv f "dio 



co oo" co oo" fc> oo" 



by inclusion of the terms 



do- \coJ da\coJ 0(T\coJ 



The expression on the right of (25) is thus the unital rate 

 of elongation in the direction of the tangent to the vortex-line 

 at the point considered. It is thus a component part of 

 the dilatation of the fluid at that point, not following the 

 fluid as it moves. But the dilatation is the volume dila- 

 tation also at a fixed point in space, and might be expressed 

 equally well as the sum of a unital time-rate of elongation 

 along the instantaneous direction of da, and a unital time- 

 rate of expansion of area at right angles to that direction. 

 As will be proved below, that areal expansion will, to the 



