oo, y, 



% 



i, 1, 







a, a', 



1 



PROFESSOR CAYLEY ON POLYZOMAL CURVES. 31 



line through / or J is = 0. And in particular, if z = 0, then a? + y 1 = ; that is, 

 the distance of the point (a, a', 1) from lor J is in each case = 0. 



62. Consider for a moment any three points P, Q, A , the perpendicular dis- 

 tance of P from QA is = 2 triangle PQA -s- distance $/4 ; if # be any point on 

 the line through A to either of the points J, J, and in particular if Q be either of 

 the points 7, J, then the triangle PQA is finite, but the distance QA is = : that 

 is, the perpendicular distance of P from the line through A to either of the points 

 JT, J, that is, from any line through either of these points, is = co . But, as just 

 stated, the triangle PQA is finite, or say the triangles PI A, PJA are each finite; 

 viz., the co-ordinates (rectangular) of P, A being (x, y,z = 1), (a, a\ 1) or (circular) 

 (£, >7, z = 1), (a, a', 1), the expressions for the doubles of these triangles respec- 

 tively are 



- i, 1, o 



that is, they are (rectangular co-ordinates) #— «£ + i{y—a! z),x — az — i [y — a's), 

 or (circular co-ordinates) £ — a#, >; —a' 2;. 



Representing the double areas by PI A, PJA, respectively, and the squared 

 distance of the points A, P by A, we have — 



A = (x — az) 2 + (y — a'zf 



- (| _ az ) („ _ a 'z), = PIA, PJA. 



Antipoints ; Definition and Fundamental Properties — Art. No. 63. 



63. Two pairs of points (A,B) and (A^BJ which are such that the lines 

 AB,A 1 B 1 bisect each other at right angles in a point O in such wise that 

 OA = OB = i OA 1 —iOB v are said to be antipoints, each of the other. In 

 rectangular co-ordinates, taking the co-ordinates of (A B,) to be {a, 0, 1) and 

 (—a, 0, 1), those of {A 1 ,B 1 ) will be (0, ai, 1) and (0, — ai, 1) respectively, whence 

 joining the points (A,B) with the points (I, J), the points A 1 ,B 1 are given as the 

 intersections of the lines AI and BJ, and of the lines A J and BI respectively. 

 Or, what is the same thing, in any quadrilateral wherein I, J are opposite angles, 

 the remaining pairs (A,B) and (A^BJ are antipoints each of the other. 



64. In circular co-ordinates, if the co-ordinates of A are (a, a', 1), and those of B 

 are (/3, /3', 1), then the equations of 



Al, A J are | — az = 0, ij — az = 

 PI, PJ „ I- fiz =0, n - jS'z = 



whence the equations of 



Aj I, A 1 J are i; — az = 0, % — fi'z — 

 ^ J, ^ 7 „ I - j8.j = 0, 4 - «z = 0.. 



