= 2/5 (radius of the inscribed circle)², (12) or two-fifths of the k² for the solid triangle. Again, since d[phi]/d[nu] = d[psi]/ds, d[phi]/ds = -d[psi]/d[nu], (13) _ _ / / T = ½[rho] | [phi] d[psi] = -½[rho] | [psi] d[phi]. (14) _/ _/ With the Stokes' function [psi] for motion symmetrical about an axis. _ _ / d[psi] / T = ½[rho] | [phi] ------ 2[pi]y ds = [pi][rho] | [phi] d[psi]. (15) _/ yds _/ 37. _Flow, Circulation, and Vortex Motion._--The line integral of the tangential velocity along a curve from one point to another, defined by _ _ / / dx dy dz \ / | ( u-- + v-- + w-- ) ds = | (u dx + v dy + z dz), (1) _/ \ ds ds ds / _/ is called the "flux" along the curve from the first to the second point; and if the curve closes in on itself the line integral round the curve is called the "circulation" in the curve. With a velocity function [phi], the flow _ / - | d[phi] = [phi]1 - [phi]2, (2) _/ so that the flow is independent of the curve for all curves mutually reconcilable; and the circulation round a closed curve is zero, if the curve can be reduced to a point without leaving a region for which [phi] is single valued. If through every point of a small closed curve the vortex lines are drawn, a tube is obtained, and the fluid contained is called a _vortex filament_. By analogy with the spin of a rigid body, the component spin of the fluid in any plane at a point is defined as the circulation round a small area in the plane enclosing the point, divided by twice the area. For in a rigid body, rotating about Oz with angular velocity [zeta], the circulation round a curve in the plane xy is _ / / dy dx \ | [zeta] ( x -- - y -- ) ds = [zeta] times twice the area. (3) _/ \ ds ds / In a fluid, the circulation round an elementary area dxdy is equal to / dv \ / du \ / dv du \ udx + ( v + --dx )dy - ( u + --dy )dx - vdy = ( -- - -- )dx dy, (4) \ dx / \ dy / \ dx dy / so that the component spin is / dv du \ ½ ( -- - -- ) = [zeta], (5) \ dx dy / in the previous notation of § 24; so also for the other two components [xi] and [eta].