METRICS The question arises: Has a range two poem equidistant and infinitely distant from all r. other points, and has a pencil two lines eqt inclined to and making an as angle with a: other lines of the pencil, in case of the gener, ized system of measurement? The answer .s affirmative. In this case, the fixed elements e the scale-making transformation have tam taken for origins 02 and Os. Their equatim are z= 0 and z= co . Now be any point line. The distance (angle) from to s= is c log (0 : z')=oo. In like manner, the tance (angle) from z z' to z = co , beic c to ( 00 : a'), is oo. Hence the distance (angle) between either of the absolute pol, (lines) of a range (pencil) and any other tow: (line) of the range (pencil) is infinite (lel* rithmically).

A Restriction We will rev abandon the supposition that the origins 0, z:;:. 0, coincide with the absolute elements. The absolute elements will, of course, be given Is writing z' = z in the transformation z'=(a.: b) : (cz d); i.e., they are given by the eq-.2 tion (d—a) z — b= 0, or, as we may wit it, changing the values of our constant letteri az' 2bz c= 0; or, as .r2 Li ax,' 2bx + ex Now let z',) dIlG x",) any two points (lines) of a raw (pencil), then the quantities (x", Ax",) for varying 1 are the co-ordinates a variable point (line) of the range This variable element will coincide with a: absolute element for such and only such valuo of 2- as satisfy the equation a (x', + A 2b A- X ei) +( ± el) 2 = 0, or These values are — x'x' z' z' m'e As= conjugate of Xi.

The anharmonic ratio of the element pair (x',, x',) and I's.) to the pair of absolute= is & : At Hence, if we denote x : by z' and (z",:z"%) by z", we have the proposition Distance (angle) z' z"= - ax'xdaie c log x' x' I c log K.

Such is Klein's expression for the metric num ber of distance (angle) measured in accordance with the generalized system or scale. From it Cayley's expression is obtainable as follows: ti n be any number, then log n = 2i cos +I 2 Vn), a fact verifiable thus: —in+ 1 n=e i2cos =cos (2 cos i sea' (2 cos = 1 + 2 V m 4x i 1 (x+ 2 Applying that fact to the Kleinian expression, we readily obtain c log K =2ic cos I NI fix' De el whence, on letting c = —i:2, we obtain x' cos —1 x' Uzi x' which is Cayley's expression for distance (angle) of the elements and s"= :e2.

Application to Real Elements, Absolute Elements Two cases arise accord ing as the absolute elements are real or are imaginary. Consider first the case where they are real. For the sake of convenience we shall first consider the range. Denote the absolute points by F. and Two points 2 and z' ma be situated (1) both of them between, (2) neither of them between, or (3) only one of them between F1 and F,. In situations (1) and (2) the anharmonic ratio of the pairs and (F,, F,) is positive; in the remaining situa tion, negative. Accordingly in the former case the logarithm of the anharmonic ratio is im aginary, while in the latter case, the logarithm is real except for the imaginary period before indicated. Hence in order that the distance be tween two points not separated by Fi or F, shall be real, we must assign c for our distance func tion a real value. This being done, we have the theorem: The absolute points being real and distinct, the distance between two real points z and z' is real or imaginary according as the points are not or are separated by one of the absolute points. If, with Cayley, we assigned to c a pure imaginary value, then in the foregoing theorem we should have to inter change the words "real° and "imaginary° or else the phrase "not separated° and "separated.° The Cayleyan choice is logically allowable. It is rejected as being inexpedient. It gives a gen eralized scale needlessly unlike the ordinary scale. For example, if 1, 2, 3 denote successive divisions of the Cayleyan scale such that the successive divisions shall be a unit apart, then, if division 1 falls between F. and division 2 must fall without, division 3 within, etc.