353 
on one common plane of homology, of which the equation and the symbol 
may be thus written, 
et+y+r+w=0,[e]=[1, 1, 1,1], 
and which may be called (comp. 4) the Unit-Plane. 
17. Any four collinear points, Pp, Pi, Pe, Ps, have their anharmonic 
symbols connected by two equations of the forms, 
(P:) =¢ (€) + & (B:), Ps =U (Po) + (2) 
each including four ordinary linear equations between the co-ordinates of 
the four points, such as 
a= tx + UXo, Yi = tYo + UY2, &e. > 
and the anharmonic of their group is then found to be expressed by the 
formula, 
ul! 
(Po, Fiy Pay Ps) Ree 
And similarly, if any four planes 11,.. I; be collinear (that is, if they 
have any one right line common to them all), their symbols satisfy two 
linear equations of the corresponding forms, 
[11, | = t [To] + U [TI], [13] = v [Tp | = u [rast 
and the anharmonic of the pencil is, 
T1,11,1,0,) = ad 
( (Vaasa tu!” 
18. If p(ayzw) be any rational fraction, the numerator and denomi- 
nator of which are any two given homogeneous and linear functions of 
the co-ordinates of a variable point ; and if we determine a line A, and 
three planes 1, 11, 11, through that line, by the four local equations, 
0 
$= b= we H=1, b= 05. 
then I find that the function @ may be expressed as the anharmonic of 
a pencil of planes, as follows :— 
(aye) = (THAD); 
where II is the variable plane Ap, which passes through the fixed line 
A, and through the variable point Pp = (xyzw). 
19. And in like manner, as the geometrical dual (9) of this last the- _ 
orem (18), if &(/mnr) be any rational fraction, of which the numerator 
and denominator are any two given functions, homogeneous and linear, of 
the co-ordinates of a variable plane ; and if we determine a line A, and 
three points Po, P1, Pz on that line, by the four tangential equations, 
0 
Pies ®=1, ®=0; 
I find that the proposed function ® may then be thus expressed as the 
anharmonic of a group of points. 
ae &(Imnr) = (PoP P2P) ; 
