Thus if T is expressed as a quadratic function of U, V, W, P, Q, R, the components of momentum corresponding are dT dT dT (I) = dU + x2=dV, x3 =dW, dT dT dT Yi dp' dQ' y3=dR; but when it is expressed as a quadratic function of xi, 'x2, x3, yi, Y2, Y3, U = d, V= dx, ' w= ax dT Q_ dT dT dy 1 dy2 dy The second system of expression was chosen by Clebsch and adopted by Halphen in his Fonctions elliptiques; and thence the dynamical equations follow X = dt x2 dy +x3 d Y = ..., Z ..., (3) = dt1 -y2?y - '2dx3+x3 ' M =..
If the bubble is in the form of a sphere of radius r this material surface will have an area S = 41rr 2 (I) If T be the energy corresponding to unit of area of the film the surface-energy of the whole bubble will be ST = 41rr 2 T (2) The increment of this energy corresponding to an increase of the radius from r to r-+dr is therefore TdS = 81rrTdr (3) Now this increase of energy was obtained by forcing in air at a pressure greater than the atmospheric pressure, and thus increasing the volume of the bubble.
The great British public would not care if T C Worsley had enjoyed sexual relations with mice.
So if T disappears from where you just slogged your way to get it, well, c'est la vie.
The torque corresponding to this couple is F 1 X z D = 2µWID, and hence follows the fundamental relation, 2T = 2F 1 D = 2µWID, or if W now represents the weight supported by the axle, F will be the tractive force exerted on the frame by the two axle-boxes to propel the vehicle, and the more convenient relation is established, T=2FD=2µWD (3) If T has a greater value than this relation justifies the wheels will slip. F is called the " tractive force " at the rail.
