Denoting the cross-section [alpha] of a filament by dS and its mass by dm, the quantity [omega]dS/dm is called the _vorticity_; this is the same at all points of a filament, and it does not change during the motion; and the vorticity is given by [omega] cos[epsilon]dS/dm, if dS is the oblique section of which the normal makes an angle [epsilon] with the filament, while the aggregate vorticity of a mass M inside a surface S is _ / M^(-1) | [omega] cos [epsilon] dS. _/ Employing the equation of continuity when the liquid is homogeneous, / d[zeta] d[eta]\ d² d² d² 2( ------ - ------ ) = [nabla]²u, ... , [nabla]² = - --- - --- - ---, (10) \ dy dz / dx² dy² dz² which is expressed by [nabla]²(u,v,w) = 2 curl ([xi], [eta], [zeta]), ([xi], [eta], [zeta]) = ½ curl (u, v, w). (11) 38. _Moving Axes in Hydrodynamics._--In many problems, such as the motion of a solid in liquid, it is convenient to take coordinate axes fixed to the solid and moving with it as the movable trihedron frame of reference. The components of velocity of the moving origin are denoted by U, V, W, and the components of angular velocity of the frame of reference by P, Q, R; and then if u, v, w denote the components of fluid velocity in space, and u´, v´, w´ the components relative to the axes at a point (x, y, z) fixed to the frame of reference, we have u = U + u´ - yR + zQ, (1) v = V + v´- zP + xR, w = W + w´ - xQ + yP. Now if k denotes the component of absolute velocity in a direction fixed in space whose direction cosines are l, m, n, k = lu + mv + nw; (2) and in the infinitesimal element of time dt, the coordinates of the fluid particle at (x, y, z) will have changed by (u´, v´, w´)dt; so that Dk dl dm dn -- = --u + --v + --w dt dt dt dt / du du du du \ + l( -- + u´-- + v´-- + w´-- ) \ dt dx dy dz / / dv dv dv dv \ + m( -- + u´-- + v´-- + w´-- ) \ dt dx dy dz / / dw dw dw dw \ + n( -- + u´-- + v´-- + w´-- ). (3) \ dt dx dy dz / But as l, m, n are the direction cosines of a line fixed in space, dl dm dn -- = mR - nQ, -- = nP - lR, -- = lQ - mP; (4) dt dt dt so that Dk / du du du du \ -- = l( -- - vR + wQ + u´-- + v´-- + w´-- ) + m(...) + n(...) dt \ dt dx dy dz / / 1 dp \ / 1 dp \ / 1 dp \ = l( X- --- -- ) + m( Y - --- -- ) + n( Z - --- -- ), (5) \ p dx / \ p dy / \ p dz / for all values of l, m, n, leading to the equations of motion with moving axes.