42 



Ellery W. Davis 



general moves. If it does not move PJ and PJ' are describing 

 the two projective rows determined by P X P 2 and the original P x . 

 Given P 1} P 2 and P s to find the center of the line joining them 

 we generalize the construction in fig. 12. The result is fig. 36. 

 Here PJP S 'R Z S Z are harmonic. Note that, in general, the center 

 varies as z changes, even though z remains a bearer of one of the 

 elements joining the lines P-[P 2 an d P^"P 2 ". Let us call the 

 center we have obtained P ()S , while P 02 is the center obtained when 

 P 2 interchanges its role with P~. The bearer of P 2 we call p 2 . 

 S 2 will lie on p 2 and must therefore be a different point from S z . 

 Thus P 02 will be either at a different point than P oZ on P Z 'R Z or 

 entirely off of that line. The construction gives the point as indi- 

 cated in figure. P is not entirely arbitrary, however, since it is 

 the pole of some bearer p with respect to each of the infinite 

 family of quasi-similar, quasi-concentric conies from some one of 

 which every red element of the line starts in tangency. The 

 center therefore always lies within the conic enveloped by the 

 bearers p l3 p 2 , p 3 , . . . of P 1} P 2 , P a , . . . connecting the blue line 

 with the black line of the figure. When p x is taken for z, 



K = 9 =P =P ' 



and the conies to which the red elements are tangent are all line- 



Y 



pairs, while the imaginary parts of the conjugates of the various 

 red elements vanish. Similarly, when /> 4 is taken for z, 



JV 4 • J 4 L 04 1 4 > 



42 



