Absolute Elements We turn now to the case where the absolute elements are coincident. The anharmonic ratio of the pair (z., 22) to the pair (22, 2.), being (2. 22) (2.-24) / (22 2.) (24 zi), is equal to 1, if the elements of a pair, as (21, z.), coincide. Hence if be the absolute ele ments, the distance (angle) z, 24=c log 1=0, if c be finite. The same fact appears from the distance (angle) expression 2 i c cos-1 N12 X' X' 1 where 11 xx=a zit + 2 b x. x. +c x0; for, if the absolute elements are coincident, then (and only then) A a c=0, and X x (pi xi + p. x.)'; whence the distance (angle) expression becomes (p, x'1 + x',) (p. xr, + x",) 2 i c cos V (pi x', + x'02 (p, x", +p,x".)2 =2i if c be finite.
In order to study the metric determination in case of coincident absolute elements, we must view the case as a limiting one, i.e., we must suppose the abiolute elements to differ by an infinitesimal and then examine the law accord ing to which the distance (angle) expression approaches zero as the absolute elements ap proach coincidence. To this end observe that x' x' x' x.' el) 2 A., Now 12 2 i c ..\1 x' x' xr xr 9, ef el) V A 2 i c sin 1 ax'x'axwx' a new expression for distance (angle), admit ting immediate application of the relation lien (sin a:a)=1 as a approaches zero. As the absolute elements approach coincidence, A ap proaches zero, and hence the sine (in the above expression) approaches zero; so that we may replace the sine by its sin'. We thus get for distance (angle) ea. x'sx'i 2i c ( ( ) (p, x', + p. x',) (p. x'. + p, x',) where we choose c so that k 0 as Li ap proaches zero. The last expression gives us the distance (angle) of two elements (x'., x',), (x"i, x".) when the absolute elements (p. x, + coincide. Consider the expression (qi x'i + q2 x'21 / (ti x'2 + p2 +q2x",) / (Pi x"2 Pix It is readily seen that E = k (el x" x' 2 x 2) / + p. x'i) ( p, x", + p. *), provided we make such obvi ously possible choice of the q's that q. p2 q2 k. The form E shows that the addition property of distance (angle) is valid also when the absolute elements are coincident. For let three elements be (x'., x'2), (x"i, x"2), (x"'1, x"'2). Substituting in E, it readily ap pears that distance (angle) between elements one and two + distance (angle) between two and three=distance (angle) between one and three. Also distance (angle) between (x., x2)
and itself is (q2 Xl q2 x2) / xi + P1 X2) (as% q2 x2) / (pi x, + pi xi) = 0. As to the displacement property, it readily appears that E is the difference of two anharmonic ratios, for the form (q.xi + q2 xi) / (p. xi+ p. x,) is the anharmonic ratio of the elements (12 x, + (or 21=q. / (12), pi xi +/02.e=o or .7.2= P2 / (pi qi) (P. q.) x2=0 or 22= (p.q2) / (pi q.) , and 2.=.2% / x,. Now as linear transformation leaves anharmonic ratios invariant, the quantity E will remain unchanged, i.e., the displacement property is preserved also when the absolute elements fall together.
General Metric Determination in the We have hitherto attached the gener alized concept of measurement to one-dimen sional spaces: the range and the pencil. It is easy to carry the concept into spaces of two dimensions, namely, the plane regarded as an ensemble of points (pencils) and of lines (ranges). The pair of absolute elements (of range, pencil) has for analogues in the plane the conic viewed on the one hand as locus and on the other as envelope. Accordingly the Ab solutethe configuration of reference for metric determination in case of the plane will be some chosen conic C. Any line L will cut C in two points, real and distinct, or imaginary, or coincident. These two points will serve for absolute points in case of the range having L as base. Reciprocally, any pencil V of the plane has two lines tangent to C. These tangents, which of course may be real and distinct, or imaginary, or coincident, will be the absolute lines for metric determination in the pencil V.
We may represent C in homogeneous co ordinates by S xx .7_- ax,' + b.r2* + cx,' + 2dxix, + 2exixs + 2fx,x3= 0. Let S denote the result of replacing in Sxx x' for x and let S axix'i + bxzet + cxsx's + d (x, + .r, x'i) + e (xi x's + xs x'i) ± f x's -I- xs x'2). Any two points xi and x'i (i=1, 2, 3) determine a range. This cuts C in two points. The anharmonic ratio of this pair and the given pair is Sxx' + Sxx Sx'x' N .=, R S"xx' Sxx Sx'x' hence the distance between the points is c log R, or 2 i c cos where c is a fixed arbitrary constant, the same for all ranges of the plane.
Reciprocally, if Ziff =0 be the line equation of the Absolute, the angle between two lines G and Vi (i= 1, 2, 3) is £ ; ; , , + / ; - g r 2 [PC/ ; S log or s.