A Treatise on the Theory of Screws

 448 THE THEORY OF SCREWS. [406- Two linear equations in ajj, x.2, xs, determine a range, and the simul- taneous solution of these equations with f ’ ^3 > *^3 > ^4) “ 0 gives the infinite objects on that range—but there can only be two—and hence we have the following important theorem:— The co-ordinates x2, x2, x3, x4 of the infinite objects in a content satisfy an homogeneous equation of the second degree. We denote this equation by r=o. 407. The Departure. Let the ranges ^,^,0, 0 be formed when the parameter x, 4- x2 has every possible value, then the entire group of ranges produced in this way is called a star. In ordinary geometry the most important function of a pair of rays in a pencil is that which expresses their inclination. We have now to create, for our generalized conceptions, a function of two ranges in a star which shall be homologous with the notion of ordinary angular magnitude. We shall call this function the Departure. Its form is to be determined by the properties that we wish it to possess. In the investigation of the departure between two ranges, we shall follow steps parallel to those which determined the intervene between two objects. If OP, OQ, OR be three rays in an ordinary plane diverging from 0, then < 'P0Q + Z.Q0R = ^P0R. In general the angle between two rays is not zero unless the rays are coin- cident ; but this statement ceases to be true when the vertex of the pencil is at infinity. In this case, however, the angle between every pair of rays in the pencil is zero. Every plane pencil has two rays (i.e. those to the circular points at infinity), which make an infinite angle with every other ray. 408. Second Group of Axioms of the Content. We desire to construct a departure function which shall possess the following properties:— (VI) If three ranges, P, Q, R, in a star, be ordered in ascending parameter, and if the departure between two ranges, for example, P and Q, be expressed by PQ, then PQ + QR = PR.