^Y. Academy ■ 



XIII. 



Oi^ THE HOMOGEAPHIC DTYISION'S OE PLANES, SPHERES, 

 AND SPACE, AND ON THE SYSTEMS OE LINES JOIN- 

 ING CORRESPONDING POINTS. By CHARLES JASPER 

 JOLY, M.A., E.T.C.D. 



[Eead January 11, 1897.] 



1. If OA = a, OB = fS, and OC = y are three given vectors 

 wliose terms A B and C form a triangle ; if a, 3, and c are three 

 given scalars, and x, y, and z three scalar variables, the vector 



^„ aax ■¥ IBy + cy% 



OP = ■S7 = '--- '— 



ax + by -{- c% 



terminates at a point P in the plane of the triangle ABC. 



The variables x, y, and z are called by Hamilton the Anharmonic 

 Coordinates of P, and he uses the equation 



(P) = {x, y, %), or ' (P) = {xy%), 



in order to express that P is determined by x, y, and z, {xy%) he 

 calls the symbol of P. 



The symbols of ^, P, and G are 



(^) = (100), (P) = (010), (C) ^ (001), 



and ^^C is called the unit-triangle. The point U, whose^vector" is 



aa-vbfi ■{- cy 



OU 



a-\-b + c 



and whose symbol is {JJ) = (111), is called the unit point. The 

 arbitrary scalars a, I, and c are introduced in order that ^may occupy 

 an arbitrary position in the plane ABC. 



The coordinates are called anharmonic because the anliarmonics of 

 the pencils ^.PZ76P, B.CUAP and CAUBPixxe 



(A.BUCP) = '^^, B.(CUAP) = -, (CAUBP) = -. 



^ z X y 



E.I. A. PKOC, SEE. III., VOL. IV. 2 O 



