O'SuLi.iVAN — Points on the Curve of Intersection of two Quadrics. 139 



axe?, /'(^ meets <i", a", and also tlie tangents to Wat A A' A" A'", so that the 

 planes joining FQ to these points are tlie tangent planes to ^7 F which pass 

 through the chord P'Q'. The tangent planes through P'Q' touch at the same 

 four points, while those through PQ', P'Q touch at the points A-^A-^'A^'A^". 

 If A, A' be connected by the relation 



waa' = (/3 + 7 - « - S) XX.' - (/jy - ag) (A + A') + /S^Ca + g) - «cX|3 + 7) = 0. 



7<A'-«) =(«-/^)(«-7)(X-g). 



Z;(A'-fJ) = -(/3-«)0-S)(A-7). ■ (3) 



/<A' - y) = - (y - a) (7 - g) (A -\i). 7^ = (/3 + 7 - a - 8) A - (/By - ag). 



7<A'-g) =(S-i3)(g-7)(A-a). 



Hence if the coordinates of the points AA' ... be (sj, y, z, w), {x, - y, - z, iv), 

 &c., and those of A^, A^' . . . be (ajj, y-^, %, «'), (0;^, - y.^, - Sj, Wj), &c., 



(S-;8)(8-7)A'« _ 



IV^' fa 



A'a 



' A'S 





A73 



Ay 



(» - ^) (« - y) A'g 



(y - g) (y - S) A'/3 



" (|3 - «) (/3 - S) \'y 



AS 



(S- 



/3) (g - 



7) 



w^ 



Aa 



(«- 



/3)(«- 



t) 



o;"- 



Ay 



_ T 

 "/3 



- ri . y 



-^ 



s^ 



A/3 



-«./3 



-g 



2/^ 



(4) 



Thus the points A-^ and their collinears are connected with the points A and 

 their collinears by relations of the type x^ : y-^: z-^: w^ 



= Jg-/3.g-y w : Jy - « . y - g .^^ : JjS - a . /3 - g . y : j., - j3 . a - g . .'.;. 



(5) 

 We may call the points .-/j the inverses of the points A in the first correspon- 

 dence. 



If v\\" = 0, ivkk" = 0, we get two other sets of inverse points related 

 similarly to the second and tliiixl correspondences. These sets of points will 

 be denoted by A^ . . . A^ . . ., and these " inversions " may be regarded as three 

 linear transformations by which the curve is transformed into itself. By the 

 first transformation A is transformed into A', by the second, A' into A'", by the 

 third \"' into A ; so that if the three transformations be applied successively 

 the point A returns to its original position. 



From Fig. 3 it is clear that two sets of inverse points A . . ., A^ . . . are 

 the extremities of the four axes of a system of quadrangles (AA'X, and the 

 coUinear points A, . . . A-^^. . . those of the axes of the collinear system (AA')2- 

 The whole set of 32 points which can be derived from A by linear transfer-^ 

 mation, viz. four cotangential points and their collinears, and the inverses of 

 these eight points in each of the three correspondences, are called by Harnack 

 the 32 coUineatious of the curve. 



