line. IU direction coincides at every point with that of the axis of rotation 

 of the fluid. 



We may now prove the theorem of Helmholtz, that the points of the 

 fluid which at any instant lie in the same vortex line continue to lie in the 

 ^m vortex line during the whole motion of the fluid. 



The equations of motion of a fluid are of the form 



dV 



when p is the density, which in the case of our homogeneous incompressible 



g 

 fluid we may assume to be unity, the operator *- represents the rate of varia- 



tion of the symbol to which it is prefixed at a point which is carried forward 

 with the fluid, so that 



Sw du du du , du . . 



K7=j7 + u-T--fv-7- + M> -j- ......................... (4), 



8t at ax ay dz 



p is the pressure, and V is the potential of external forces. There are two 

 other equations of similar form in y and z. Differentiating the equation in // 

 with respect to z, and that in z with respect to y, and subtracting the second 

 from the first, we find 



d 8v d 810 . 



Performing the differentiations and remembering equations (1) and also the 

 condition of incompressibility, 



du dv dw 



we find 



Now, let us suppose a vortex line drawn in the fluid so as always to 

 begin at the same particle of the fluid. The components of the velocity of this 

 point are , v, u>. Let us find those of a point on the moving vortex line at 

 a distance ds from this point where 



ds = otda- .................................... (8). 



