Geometric Interpretation of Homogene ous Co-ordinates.— In case of the range assume two origins 0, and 0, instead of one and let them be 6 apart. These divide the range into two parts, the short segment be tween and the long one (including not between 0, and 02. Strictly, any point of the range other than 0, or 02 is between these points, for the range is a closed figure, but the meaning of the preceding sentence is sufficiently clear. Let it be agreed that a point in the shorter segment is on the positive side of both O's, whence, naturally, a point in the longer segment will be on the positive side of the more remote, and on the negative side of the nearer, 0. Denote by x, and x, respectively the distances of any point P from 0, and 02. For any P, x, + x2=6. To any pair of x's satisfying that relation there corresponds a point, and conversely. The homogeneous co-ordinates exi, axi of a point of a range are the dis tances (multiplied by any finite constant) of the point from two chosen fixed points. Analogously for the pencil, where, however, distances are replaced not by the angles but by the sines of the angles made by the variable line p with two fixed lines o, and oi, and where it is understood that an angle and its vertical angle are one and the same angle.
Anharmonic Ratio.— The ruling notion in the doctrine of the range (pencil) is the anhar monic (double or cross) ratio of four elemer.ts.
If xi, x,, xa, x, be any four numbers (say any four values of a continuous variable x) the expression -x8-4) is called the anharmonic ratio of the four values taken in the order xi, .vs, xi, x,, and is con veniently denoted by the symbol (xix,x,x4). If a one-to-one correspondence be established between the continuum of x-values and the elements of a geometric continuum, the notion of the anharmonic ratio of any four x-values may be and is associated with the correspond ing four geometric elements, as the points of a range, the lines of a pencil, the planes of an axial pencil (assemblage of all planes containing a same line), and so on. The order of ele ments is essential. The 24 possible per mutations of 4 elements yield six (in general distinct) values of their anharmonic • ratio. The exchange of two alternate elements, as x, and x,, inverts the ratio. Thus, if (xixix,x4) then (xix,x8xi) =1:r. To exchange two consecutive elements, as x, and xi, takes the complement of the ratio to 1. Thus (xixix.x.) 1—r. The six values are r, 1 :r, 1 —r, 1:(1— r), (r-1) :r, r:(r —1).
Geometric Interpretation of Anharmonic Ratio in Range and Pencil.— Let xi, x,, x,, xa be the distances of the points P,, P,, Pa, P. of a range from the origin. Then xi—xi, xi,