The analytical calculation of the work done by a system of forces in any infinitesimal displacement is as follows. For a two-dimensional system we have, in the notation of §§ 3, 4, [Sigma](X[delta]x + Y[delta]y) = [Sigma]{X([lambda] - y[epsilon]) + Y([mu] + x[epsilon])} = [Sigma](X)·[lambda] + [Sigma](Y)·[mu] + [Sigma](xY - yX)[epsilon] = X[lambda] + Y[mu] + N[epsilon]. (1) Again, for a three-dimensional system, in the notation of §§ 7, 8, [Sigma](X[delta]x + Y[delta]y + Z[delta]z) = [Sigma]{(X([lambda] + [eta]z - [zeta]y) + Y([mu] + [zeta]x - [xi]x) + Z([nu] + [xi]y - [eta]x)} = [Sigma](X)·[lambda] + [Sigma](Y)·[mu] + [Sigma](Z)·[nu] + [Sigma](yZ - zY)·[xi] + [Sigma](zX - xZ)·[eta] + [Sigma](xY - yX)·[zeta] = X[lambda] + Y[mu] + Z[nu] + L[xi] + M[eta] + N[zeta]. (2) This expression gives the work done by a given wrench when the body receives a given infinitely small twist; it must of course be an absolute invariant for all transformations of rectangular axes. The first three terms express the work done by the components of a force (X, Y, Z) acting at O, and the remaining three terms express the work of a couple (L, M, N). [Illustration: FIG. 48.] The work done by a wrench about a given screw, when the body twists about a second given screw, may be calculated directly as follows. In fig. 48 let R, G be the force and couple of the wrench, [epsilon],[tau] the rotation and translation in the twist. Let the axes of the wrench and the twist be inclined at an angle [theta], and let h be the shortest distance between them. The displacement of the point H in the figure, resolved in the direction of R, is [tau] cos [theta] - [epsilon]h sin [theta]. The work is therefore R([tau] cos [theta] - [epsilon]h sin [theta]) + G cos [theta] = R[epsilon]{(p + p´) cos [theta] - h sin [theta]}, (3) if G = pR, [tau] = p´[epsilon], i.e. p, p´ are the pitches of the two screws. The factor (p + p´) cos[theta] - h sin[theta] is called the _virtual coefficient_ of the two screws which define the types of the wrench and twist, respectively. A screw is determined by its axis and its pitch, and therefore involves five Independent elements. These may be, for instance, the five ratios [xi]:[eta]:[zeta]:[lambda]:[mu]:[nu] of the six quantities which specify an infinitesimal twist about the screw. If the twist is a pure rotation, these quantities are subject to the relation [lambda][xi] + [mu][eta] + [nu][zeta] = 0. (4)