CALIFORNIA ACADEMY OF SCIENCES. [Proc. 3D Ser. 



cient that the three conies <f> = o , yjr = o and % = o have 

 three points, and only three, in common. If these points 

 be all distinct (and of course not collinear) and the equa- 

 tions of the lines joining them in pairs be A, = o , /x = o , 

 v — o , then the equation of any conic through the three 

 points must be of the form 



o 



p/j,v-\-gv\-\- ?' X p, 



and the formulas of transformation may be written in what 

 we shall call the reduced form: 



(2) p y { = pi /x v + q i v X -f 7\ A fx , ( i = 1 , 2 , 3) . 



When the formulas are given in this form, it is clear that 

 the general quadratic transformation is equivalent to the 

 product 6*1 Q S 2 of a linear transformation Si , of what we 

 may call a normal quadratic transformation Q , and of a 

 second linear transformation S 2 . I append the explicit 

 formulas of these transformations: 



p X = l x x\ +- h x 2 -f- h *3 



p jX = m\ Xi -J- M-2 X 2 + m 3 X 3 



p v = 111 xi -\- #2 A-2 -}- n 3 x 3 



a- 7~x = fx v , <r T 2 = v X , <r 2" s — \ ft 



t yx = p x T\ + qi T 2 + r x T 3 

 r y 2 = p 2 Ti -f- ^ 2 ¥1 + ^2 2" 3 

 T JVs = /s 21 + q* 2" 2 + ? 's 2" a 



The transformation can now be readily inverted, for the 

 transformation inverse to S x Q S 2 is S 2 ~* Q~ l Sf 1 , where 



s t 



-1 



Q 



— i 



(6) S 2 



Sr 1 are given by the formulas : 



p T x = Pi y x -f- P 2 J2 + ^3 ^3 

 p 2*2 == Qi ^1 -f Q2 ^2 + Q3 yz 

 P Tz — Ri yi + ^?2 ^2 + 2? 3 ^3 



•A , Qi , etc., denoting as usual the co-factors of pi, qi , 

 etc., in the determinant of 6*2 , viz. : 



(P qr) 



