MR. A. B. KEMPE ON THE THEORY OF MATHEMATICAL FORM. 
63 
(1.) If the three points in each of the nine triads of points 
a , b, c; a, d, e; a, g, f; b, d, li; c, e, h; d, f, l; e, g, l; b. f, k; c, g, k. 
are collinear, the points of the triad h, k, l are also collinear (fig. 67). 
(2.) If the three points in each of the eight triads of points 
a, b, c; a, h, , cl, e, J', ci, g, &, b, g, d, c, h, d, b, k, J', c, k, c 
are collinear, the points in the triad g, h, k are also collinear (fig. 68). 
a 
Coplanctr Points, Lines, and Conics. 
358.. In the case of the treble system composed of coplanar points, straight lines, 
and conics, each line touches, cuts, or does not touch or cut each conic, and each point 
lies on, in, or outside each conic ; we thus have three sorts of connecting pairs in each 
case. We may conveniently connect the graphical unit representing a conic by links 
to the graphical units representing points lying on the conic, and to the graphical 
units representing lines touching the conic; no links being drawn in the other 
cases ; so that the unlinked pairs are a double system. 
359. If a be any point, C any conic, of the system, there is one line and one line 
only of the system which is unique with respect to the pair a, C, viz., the polar X of a 
with respect to C. Likewise a is the only point unique with respect to X, C. 
360. It will be convenient to approach the consideration of Formal Logic from the 
standpoint of the algebraist. Classes are denoted by terms A, B, C, D, . . ; The 
class which just contains, and the class which is just contained in, all the classes 
A, B, C, are denoted respectively by the sum A + B+C, and the product ABC of the 
terms denoting the several classes. Whatever class A may be, we have AA= A, and 
A+A=A. Addition and multiplication are each commutative and associative ; and 
are also distributive as regards each other, so that we have 
(A+B)(C+D)=AC+AD+BC+BD 
AB+CD=(A+C)(A+D)(B+C)(B+D) 
If AB = A, which implies also A~bB=B, then A is contained in B. 
