38 Ellery W. Davis 



of all the I's, while the / intersection is the collectivity of all the 

 J's. The particular pair I and J used for a symbol of all of them 

 is then what depends upon the choice of axes. The blue points 

 /", J" form, in fact, an elliptic involution 1 of points upon the line 

 z. Von Staudt calls this involution of points regarded as having 

 the / direction, one intersection ; while regarded as having the 

 J direction he calls it the conjugate intersection. It will be 

 noted that any /' and Y are a pair belonging to the involutions of 

 points on z. Thus the involutions can be called /" /' /" Y and 

 ] r /" /' /". Similarly there are the projective involutions of 

 center O determined by and therefore properly denoted by any 

 two pairs of axes. Every real line that bears a red vector, 

 bears an involution of points determined by that red vector. 

 Every two red vectors not borne by the same real line determine 

 an involution of lines whose center is the center of the imagi- 

 nary line joining the two red vectors and which cuts out on the 

 bearer of each red vector the very involution of which the bearer 

 may be taken as a symbol. 2 I say taken as the symbol, for a 

 vector P has a definite conjugate vector P and its bearer meets 

 z in a definite point K, so that if P'" is the blue point of P the 

 involution is P'"P'P"K. Similarly P"P'P'"K is symbolized 

 by P. 



Again OP is the line of center of which P is an element, 

 while OP is the conjugate line. Suppose that Q is the real point 

 of the /-ray through P, while R is the real point of the /-ray. 

 Then Q — ~JP"'IP" is a harmonic set as is also R—JP"'IP". If 

 on the bearer of P we associate all the pairs of points S'", S" such 

 that Q — JS'"IS" is a harmonic set we merely reproduce the 

 involution P. In particular, if axes become rectangular and z 

 actually goes to infinity, S'"Q" is a right angle whose revolving 

 arms sweep out the involutions J'"XV'Y on z, P'"P'P" oo on 



1 Reye, Geometrie der Lage. 



2 If the involution be taken on the .r-axis, say, with the origin as center 

 and foci at x = ± i, then any real point pair belonging to the involution 

 is x = ± ch 9" ~\- i sh 9". An imaginary point pair is gotten by multiply- 

 ing the coordinates of a real pair, the one by cos &' -f- i sin 9' t the other by 

 cos 6' — i sin 9' . This is the same as to say that any point pair is given 

 by x = ± cos 9 -)- j'sin 9. 



38 



