398 



If the triangle touches the conic in A, B, C, then A x , A 2 coincide 

 with A, &c., and the above equation becomes 



2. 2^.0.1 = 1 OT -1. 

 2 C 3 A 0,8 



The first case shows that the three points lie on a straight line ; 

 the second, that the three lines C^A, O 2 B, O 3 C meet in a point*. 

 If the point C were to lie at an infinite distance from O l and O 2 , 



then since 0^=0^ + 0,0, and 2 = 1 -{- 2i, where the last 



O C 



term is infinitesimal, we have merely to consider i = 1, or omit it 



from the product f. Thus if we take as two sides of the triangle 

 the asymptotes of an hyperbola meeting at O t , and suppose the 

 third side O 2 O 3 to cut the curve in A, and A 2 , then since both 

 OjO 2 and OjO 3 cut the curve at an infinite distance, the equation 

 reduces to 



9A Q a A a= i 



OA'OA 



whence we readily find 



3 A 2 =O 2 A 2 , or=-O a A 1 - 



The first value is impossible, because A 2 lies between O 2 and O 3 ; the 

 second gives the well-known property of the hyperbola J. 



11. As regards the clinant roots, or the intersections of the sides 

 of the polygon with the clinant radical loci, the wording of the pro- 

 position (9) requires to be changed to adapt it to the more compli- 

 cated form of the product ; and perhaps it may be considered suffi- 

 cient to refer to the forms of the products in the final equation of (8), 



* Pliicker interchanges the two cases, and makes the positive result show the 

 intersection of the three lines in a point, which is shown to be wrong in the last 

 note. He rejects the first case, " because a conic can only be cut in two points by 

 a straight line " (p. 45). But in this case both the conic and the triangle reduce 

 to a straight line (that is, the points O^ 2 , O 3 , A, B, C all lie in the same straight 

 line), and the points A, B, C do not indicate contacts, but double-intersections, 

 which have the same analytical expression, owing to the rejection of infinitesimals 

 in the above calculation of contact. 



t Pliicker says that " in every such case there must be a fresh change of sign " 

 (p. 46). This apparently arises from his having neglected to particularize the 

 directed unit, or to note the primary relation O 1 C = 1 2 4-O 2 C. 



t Plucker makes O iZ A 1 =O 3 A 2 (p. 46); that is, he neglects the relation of 

 direction, which is all-important in such investigations. 



