541 
A; = EA‘ BCD, B, = EB‘ CAD, C; = EC* ABD, D,; =ED~* ABC, (64) 
Ao=DA‘* BCE, B,= DB‘ CAE, C2 = DC“ ABE, (65) 
and the seven harmonic relations, 
(BA,44";) = (EB, BB’,) = (RC,C0’;) = (ED\DD’;) = - 1, (66) 
(DA,AA’s) = (DB2BB’2) = (DC,Co’,) = — 1, (67) 
by means of which 14 last relations these 14 new points can all be 
geometrically constructed; we shall then be able to interpret, on the 
recent plan [13.], the three new quotients, /: 7, m:r, m: 1, as anhar- 
monics of groups, as follows: 
-b m ! nN ; 
a (Da',AQ) ; 7 (BBE) nate (DC’2Cs) ; (68) 
with the analogous interpretations, 
Uy ‘ , n , r 
ia (BA’|AX) ; — = (EB BY) ; a0 (EC’,0Z) ; som (zD/,;DW), (69) 
if x, ¥, z, w denote the intersections FA ° II, nB* II, uc‘ II, ep: II, so that 
xX = (80007), y=(0500m), z= (00802), w=(0008r), Where 3 =—s. (70) 
[16.] As regards the notations employed, it may be observed that 
although we have often, as in (9) or (27), &c., equated a point, or rather 
its (eteral symbol, a or P, &e., to the corresponding quinary symbol (10000) 
or (zyzwv), &e., of that point, yet in some formulz, such as (17) (18) (19), 
in which we had occasion to treat of linear combinations of such quinary 
symbols, we substituted new letters, such as @, Q', for Pp, Pp’, &c., in order 
to avoid the apparent strangeness of writing such expressions* as ¢P+¢¥’, 
&e. To economise symbols, however, we may agree to retain the literal 
symbols first used, for any system of given or derived points, but to en- 
close them in parentheses, when we wish to employ them as denoting 
quinary symbols in combination with each other ; writing, at the same time, 
for the sake of uniformity, (v) instead of 7, as the guinary unit symbol 
(16). And thus, if we agree also that an equation between two unen- 
elosed and literal symbols of points, » and ¥’, shall be understood as ex- 
pressing that the two points so denoted coincide, we may write anew 
those formule (17) (18) (19) as follows: 
Pr =p, if @’) =¢(2) +4); (71) 
P” on line vv’, if (P”) =t(e) +f (2’) + u(U); (72) 
Pe” in plane pr’e”, if (P"”) = t(e) + U(e’) + t'(e”) + U0). (73) 
* Expressions of this form occur continually in the Barycentric Calculus of Moebius, 
but with significations entirely different from those here proposed. 
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