and a couple G whose components are L = [Sigma](L_r), M = [Sigma](M_r), N = [Sigma](N_r), (7) where r= 1, 2, 3 ... Since R² = X² + Y² + Z², G² = L² + M² + N², it is necessary and sufficient for equilibrium that the six quantities X, Y, Z, L, M, N, should all vanish. In words: the sum of the projections of the forces on each of the co-ordinate axes must vanish; and, the sum of the moments of the forces about each of these axes must vanish. If any other point O´, whose co-ordinates are x, y, z, be chosen in place of O, as the point to which the forces are transferred, we have to write x1 - x, y1 - y, z1 - z for x1, y1, z1, and so on, in the preceding process. The components of the resultant force R are unaltered, but the new components of couple are found to be L´ = L - yZ + zY, \ M´ = M - zX + xZ, > (8) N´ = N - xY + yX. / By properly choosing O´ we can make the plane of the couple perpendicular to the resultant force. The conditions for this are L´ : M´ : N´ = X : Y : Z, or L - yZ + zY M - zX + xZ N - xY + yX ----------- = ----------- = ----------- (9) X Y Z These are the equations of the central axis. Since the moment of the resultant couple is now X Y Z LX + MY + NZ G´ = --- L´ + --- M´ + --- N´ = ------------, (10) R R R R the pitch of the equivalent wrench is (LX + MY + NZ)/(X² + Y² + Z²). It appears that X² + Y² + Z² and LX + MY + NZ are absolute invariants (cf. § 7). When the latter invariant, but not the former, vanishes, the system reduces to a single force. The analogy between the mathematical relations of infinitely small displacements on the one hand and those of force-systems on the other enables us immediately to convert any theorem in the one subject into a theorem in the other. For example, we can assert without further proof that any infinitely small displacement may be resolved into two rotations, and that the axis of one of these can be chosen arbitrarily. Again, that wrenches of arbitrary amounts about two given screws compound into a wrench the locus of whose axis is a cylindroid. The mathematical properties of a twist or of a wrench have been the subject of many remarkable investigations, which are, however, of secondary importance from a physical point of view. In the "Null-System" of A. F. Möbius (1790-1868), a line such that the moment of a given wrench about it is zero is called a _null-line_. The triply infinite system of null-lines form what is called in line-geometry a "complex." As regards the configuration of this complex, consider a line whose shortest distance from the central axis is r, and whose inclination to the central axis is [theta]. The moment of the resultant force R of the wrench about this line is - Rr sin [theta], and that of the couple G is G cos [theta]. Hence the line will be a null-line provided tan [theta] = k/r, (11)