536 
rent), or that the fourth passes through the point of intersection of the 
other three. 
[8.] It is easy to conceive how problems respecting intersections of 
lines and planes can be resolved, on the foregoing principles. And if we 
define that a point p, or plane II, is a Rational Point, or a Rational 
Plane, of the System determined by the five given Points a..=, or that 
it is rationally related to those five points, when its coordinates are equal 
(or proportional) to whole numbers, it is obvious, from the nature of the 
eliminations employed, that a plane which is determined as containing 
three rational points, or a point which is determined as the intersection 
of three rational planes, is itself, in the above sense, rational. We may 
also say that a right line A is a Rational Line, when it is the line p-P 
which connects two rational points, or the intersection IL- IL of two 
rational planes: and then the intersection of a rational line with a ra- 
tional plane, or of two complanar and rational lines with each other, will 
be a rational point. 
[9.] When any two points, P, P’, or any two planes, I, Il’, have 
symbols which differ only by the arrangement (or order) of the five co- 
efficients or coordinates in each, those points, or those planes, may then 
be said to have one common type; or briefly, to be syntypical. For 
example, the five given points are thus syntypical, because (omitting 
commas, as in [ 5. ]) their symbols are, 
4 = (10000), B=(01000), c=(00100), n=(00010), r=(00001). (27) 
In general, any two syntypical points, or planes, admit of being derived 
from the five given points, by precisely similar processes of construction, 
the order only of the data being varied ; and in the most general case, a 
single type includes 120 distinct points, or distinct planes, although this 
number may happen to be diminished, even when the coordinates are all 
unequal: for example, the type (12345) includes only seaty distinct 
points, because, by (17), we have in this case the congruence, 
(12845) = (54821). (28) 
[10. | The anharmonic function of any group of four collinear points 
aBcp being denoted by the symbol (azcp), and defined by the equation, 
(aBcD) =— .— = (29) 
it will be found that if P).. P, be thus any four collinear points, of which 
therefore, by (18), the quinary symbols @.. @3 are connected by two 
linear relations, of the forms, 
@1 = by Qy + bz Qo + UU, Q3 = Uy Qo + t/2Q+ WU, (30) 
then the anharmonie of this group of points is given by the formula, 
i, U4 
(PoP P2P3) =——, (31) 
of which the applications are numerous and important. 
