In the analytical investigations of line geometry, these six quantities, supposed subject to the relation (4), are used to specify a line, and are called the six "co-ordinates" of the line; they are of course equivalent to only four independent quantities. If a line is a null-line with respect to the wrench (X, Y, Z, L, M, N), the work done in an infinitely small rotation about it is zero, and its co-ordinates are accordingly subject to the further relation L[xi] + M[eta] + N[zeta] + X[lambda] + Y[mu] + Z[nu] = 0, (5) where the coefficients are constant. This is the equation of a "linear complex" (cf. § 8). Two screws are _reciprocal_ when a wrench about one does no work on a body which twists about the other. The condition for this is [lambda][xi]´ + [mu][eta]´ + [nu][zeta]´ + [lambda]´[xi] + [mu]´[eta] + [nu]´[zeta] = 0, (6) if the screws be defined by the ratios [xi] : [eta] : [zeta] : [lambda] : [mu] : [nu] and [xi]´ : [eta]´ : [zeta]´ : [lambda]´ : [mu]´ : [nu]´, respectively. The theory of the screw-systems which are reciprocal to one, two, three, four given screws respectively has been investigated by Sir R. S. Ball. Considering a rigid body in any given position, we may contemplate the whole group of infinitesimal displacements which might be given to it. If the extraneous forces are in equilibrium the total work which they would perform in any such displacement would be zero, since they reduce to a zero force and a zero couple. This is (in part) the celebrated principle of _virtual velocities_, now often described as the principle of _virtual work_, enunciated by John Bernoulli (1667-1748). The word "virtual" is used because the displacements in question are not regarded as actually taking place, the body being in fact at rest. The "velocities" referred to are the velocities of the various points of the body in any imagined motion of the body through the position in question; they obviously bear to one another the same ratios as the corresponding infinitesimal displacements. Conversely, we can show that if the virtual work of the extraneous forces be zero for every infinitesimal displacement of the body as rigid, these forces must be in equilibrium. For by giving the body (in imagination) a displacement of translation we learn that the sum of the resolved parts of the forces in any assigned direction is zero, and by giving it a displacement of pure rotation we learn that the sum of the moments about any assigned axis is zero. The same thing follows of course from the analytical expression (2) for the virtual work. If this vanishes for all values of [lambda], [mu], [nu], [xi], [eta], [zeta] we must have X, Y, Z, L, M, N = 0, which are the conditions of equilibrium.