564 
or rather the three reciprocal equations (comp. [ 82. ]), 
(ce AY Riv dg) = (a” BY cv Bi’) = (3 cy’ At 0”) ee its 
correspond simply to the elementary equations, (50), (56), 
(ca'Ba”) = (av/cB”) = (Bc’ac”’) = — 1, 
which may be employed to define the three important points a”, 3”, o”, 
(87), of the first group of second construction [40.], as being the (well 
known) harmonie conjugates of the points a’, 8’, c' of first construction, 
with respect to the three lines of the same first construction, BC, CA, AB, 
on which those points are situated. 
[87.] The equations [ 82. ], which connect the symbols (a) . . (c') of 
the six points, give, by easy eliminations, these other equations of the 
same kind : 
(8) = (6) + 2(¢)5 - (¢) = 2) + (@); 
we have therefore, by (81), the following anharmonic of the group 6, b’, 
¢, e": 
(bb’ce’) = + 4; 
and other easy calculations of the same sort given, in like manner, the 
equal anharmonics, 
(cc'aa’) = + 4; (aa’/bb’) = + 4. 
‘But in general, for any four collinear points, a, b, ¢, d, the definition 
(29) of the symbol (abcd) gives easily the relation, 
(abed) + (acbd) = 1; 
and hence, or immediately by calculations such as those recently used, 
we have this other set of anharmonics, with a new common value : 
(bcb'c') = (cac'a') = (aba'b’) = - 3; 
the negative character of which shows, by the same definition (29), that 
the segment (or interval) aa’, for example, is cut internally by one of the 
two points 4, 6’, or by one of the two points ¢, e’, and externally by the 
other : with similar results for each of the two other segments, 6’, cc’. 
[88.] We may then say that each of the three segments, aa’, b0', cc’, 
overlaps each of the two others, in the sense that any two of them havea 
common part, and also parts not common : whence it immediately follows 
that the cmvolution [82.], to which these three segments belong, has tts 
double points imaginary : whereas it may be proved, on the same plan, 
that each of the three involutions of segments mentioned in [84.], 
namely ad’, be’, cb’; bb’, ea’, ac’; ce’, ab’, ba’, has real* double points; and 
the double points of the three other involutions, determined by the three 
* The determination of these double points gives rise naturally to some new theorems, 
which cannot conveniently be stated here. 
