On the Homographic Transformation of a Surface. 327 



that the relation contemplated in the conclusion of the theorem 

 (viz. that of two points of a surface of the second order connected 

 by a line touching two surfaces of the second order, each of 

 them intersecting the given surface of the second order in the 

 same four lines) depends upon a proper transformation ; and that 

 the circumstance that an even number of improper transforma- 

 tions required in order to make a proper transformation (that 

 this circumstance, I say), is the reason why the theorem applies 

 to polygons in which an even number of sides pass through fixed 

 points, i. e. to polygons of an odd number of sides. 



Consider, in the first place, two points of a surface of the 

 second order such that the line joining them passes through a 

 given point. Let a?, y, z, w be current coordinates*, and let the 

 equation of the surface be 



and take for the coordinates of the two points on the surface 

 °°\3 Vm z v w i an< ^ x v V<i> z v w & an ^ f° r the coordinates of the 

 fixed point a, /3, 7, 8. Write for shortness 



(a..)(*,/3,y,8)*=p 



(a ..)(«, ft, 7, 8)(x v y v z v «;,)— q» 



then the coordinates x 9f y 2 , z<& w 2 are determined by the very 

 simple formulae 



2/3 



y%-y\-—qi 



27 



28 



* Strictly speaking, it is the ratios of these quantities, e. g. x:w, y.w, 

 z : w, which are the coordinates, and consequently, even when the point is 

 given, the values x, y, z, w are essentially indeterminate to a factor pres. 

 So that in assuming that a point is given, we should write x:y:z: w=a :fi:y:8; 

 and that when a point is obtained as the result of an analytical process, the 

 conclusion is necessarily of the form just mentioned. But when this is 

 once understood, the language of the text may he properly employed. It 

 may be proper to explain here a notation made use of in the text. Taking 

 for greater simplicity the case of forms of two variables, (I, m)(x, y) means 

 Ix+my; (a, b, c)(x, yf means aa?+2bxy+cf ; {a,b,c){%,r)){x,y) means 

 at-x-\-b(%y-\-7)x)-\-cr)y. The system of coefficients may frequently be indi- 

 cated by a single coefficient only : thus in the text {a . .){x, y, z, wf stands 

 for the most general quadratic function of four variables. 



