

Prof. A. Gray: Notes on Hydrodynamics. 11 



first order at least of small quantities, be unaltered by asso- 

 ciation with the fluid as it moves, since, to the degree of 

 approximation stated, the effect of the displacement of the 

 fluid will be simply to turn the area round through a small 

 angle. 



Supposing then ® to be expressed as has been suggested, 

 we subtract the time-rate of unital elongation in the direction 

 of the tangent to the vortex- filament from both sides of (25), 

 and obtain 



Ir+^=0, (27) 



where 2 is the unital time-rate of increase of area. If S be 

 the cross-sectional area of the vortex-filament at the point 

 considered, we have S = S2, and therefore also 



&>S-|-a>S = 0, 



that is o)S = constant. 



The moment of the vortex-filament thus remains unaltered 

 as the fluid moves. 



I have not hitherto seen this important result derived 

 directly from equation (22). Moreover, the proof seems 

 free from the objections brought by Stokes to the proof 

 given by Cauchy, objections to which the proof given by 

 v. Helmholtz is also open. 



13. As to the point referred to above regarding the areal 

 expansion at right angles to the tangent at P to the vortex- 

 filament, let rectangular axes be so chosen at P that the axis 

 of the filament is along Fz, and consider the expansion in 

 the plane which at the beginning of an interval dt is at righr 

 angles to P-. Take a distance APB = J.i- extending from 

 — \d.r, to 4-Jfito, and another CPD extending from — \dy to 

 + \dy. At A and B the component velocities of the fluid 

 are 



_ U» 7 _1^, _ 1 ~dio . 



2 ox 2 0<v 2 ox 



where the upper signs apply to A and the lower to B. 

 Initially, then, the coordinates of A and B are — \dx, 0, 0, 

 and +\dx 3 0, 0, and after dt has elapsed these have become 



+ l i dx+(u-\£d*)dt ) (v + l^ds)dt, (u, + \^dx)dt. 



