A Treatise on the Theory of Screws

450 THE THEORY OF SCREWS. [410- An object L will be defined by an equation of the form, where Llt L2, La are numerical, L1x1 + 4* L3x3 = 0, for any two sets of values, xlt x.2. x3, which satisfy this equation, will deter- mine a pair of ranges which have the required object in common. Let the relation between the co-ordinates of an infinite range be /(«„ x2, «s) = 0; then the infinite ranges in the star, whose vertex is the object L, will be defined by co-ordinates obtained from the simultaneous solution of L-^Xi -I- L<jX2 4" L3x3 — 0, f (^1 > ^2 > ®S) ” 0. But there can only be two such ranges; and, accordingly, the latter of these equations must be of the second degree. We hence deduce the following important result:— The infinite ranges in an extent may be represented by the different groups of values of the co-ordinates xlt x2, x3 which satisfy one homogeneous equation of the second degree. Remembering that the existence of zero intervene between every pair of objects on a range is a consequence of the coincidence of the two objects of infinite intervene on that range, we have the following result:— The range through two consecutive objects of infinite intervene is a range of zero intervene. And, similarly, we have the following :— The object common to two consecutive ranges of infinite departure is the vertex of a star of zero departure. 411. Relations between Departure and Intervene. With reference to the theory of Departure, we thus see that there is a system of critical ranges, and a system of critical objects in each extent. Every critical range has an infinite departure from every other range. Every critical object is the vertex of a star of which the departure between every pair of ranges is zero. It will be remembered that in the theory of the intervene we were conducted to the knowledge that every extent contained a system of objects and ranges, critical with regard to the intervene. Each critical object had an infinite intervene with every other object in the extent. Each critical range possessed the property that the intervene between any two objects thereon is zero. There are, thus, objects and ranges critical with repect to the intervene. There are also objects and ranges critical with respect to the departure. But nothing that we have