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given transversal A’/B'C’. Also, let the variable point P be situated 
upon the variable dine LMN; and let Q, R, S be the intersections of 
AP, BP, CP with BC, CA, AB. Then, because 
(BA‘CA”) = (CB'AB”) = (AC'BC’) =-1, 
we have 
_ (LBa’C), - = @LCB’A),— 4 = (NAC'B) 
n q m 
n " l Ud m ee Ue . 
l —7, = (LCA B), — 7 = (MAB C),-F =(NBC"A); 
as well as 
Y = Ut i, eS " z a "7 
[ z= (QCA"B), == (RAB'O), | = (SBO"A), 
ca 
7 7 (QBA'C) - =(RCB’A), - =(SAC’B); 
and therefore, 
m. 
sae =(MARC); —-~ =(LBQC). 
NZ 
But, by the pencil through P, 
(MARC) = (LQBC); 
and by the definition of the symbol (ABCD), for any four collinear 
points, 
AB CD 
BC’ DA’ 
which is here throughout adopted, we have the ¢dentity, 
(ABCD) + (ACBD) =1; 
(MARC) + (LBQC) = 1, 
(ABCD) = 
therefore 
or, 
la + my +nz=0. 
6. We arrive then at the following Theorem, which is of fundamental 
importance in the present system of Anharmonic Co-ordinates :— 
‘‘Tf the unit-point O be the pole of the unit-line A/B/C’, with re- 
spect to the unit-triangle ABC, and if a variable point P, or (a, y, 2), be 
situated anywhere on a variable right line LMN, or [/, m,n], then the 
sum of the products of the corresponding co-ordinates of point and line 
is zero.” 
7. It may already be considered as an evident consequence of this 
Theorem, that any homogeneous equation of the yp” dimension, 
Sn (&, Y, %) = 9, 
