14 PROFESSOR ANDREW GRAY, 



For simplicity we take the two directions at right angles to one another. If we 

 differentiate the first of these with respect to y and the second with respect to x, and 

 subtract the first result from the second, we obtain 



i +@ * = % + ^ + % = % + V + %T (36) ' 



where £, >/, £ are the components of spin at P about axes perpendicular to the three 

 rectangular planes yz, zx, xy ; u, v, w the component velocities at P ; and the 

 divergence. Two similar equations hold for £, >?, and can be written down by 

 symmetry. 



These equations are equivalent to those given by v. Helmholtz, as generalised 

 for the case of a fluid of non-uniform density. By writing 



©=-±^ 

 p dt 



the equations can be put in the more usual form, and in another, from the second 

 expression on the right of (36). 



Now let £ be the resultant spin at P, that is, let the axis of z be taken along the 

 direction of spin. Then at P, £= n = 0, and we obtain, writing w for £, 



— = - — + - oj . . . . (37). 



dt \dx dyj K ' 



But du/dx + dv/dy may be regarded as implicitly a function of t which changes in 

 value as the particle of fluid followed by the total differentiation moves, and in general 

 will have a positive or negative yjm'te value. If we suppose the axis of z to turn so as 

 to be always in the direction of the axis of spin, then, generally, we shall have at 

 each instant dwjdt = — ko, where k is a finite multiplier. Integrating then for a 

 moving element, we obtain 



log a) = ct + c 



or .... (38), 



o) = e c 'e CT 



where t is the time interval of integration, and c is a proper mean value of 

 — (du/dx + dv/dy) for each successive instant of time, and c' is a constant. If now 

 we suppose that initially w is very small, or zero, c' must be a very large, or infinite, 

 negative constant. Thus w, if at first imperceptible, would not, in any finite time, 

 ac(juire a perceptible value. We obtain again, therefore, the theorem of the per- 

 manence of the non-vorticity of a portion of a perfect fluid. 

 22. Returning now to (36), write in it 



dt dx dy dz dt 



and multiply both sides of (36) by dxdydz, and integrate for the space contained 



