289 



By the dual transformatioii ?ii {((,^) passes into a ^i* (a,« f ^), 

 wliich has 2 {(( -\- i-i) singuhir points more than ^"ï. 



6. The harmonical transfoimalioii may be replaced by a more 

 general transformation in the following way. 



The |)olar curve rr of a point N with regard to a given curve 

 ipm-\-i intersects the null-ray n in m points N*, which we shall 

 consider as new null-points of n. In the new null-system ^1* each 

 straight line has then imi null-points N*. 



As N* lies on the polar curve jt'" of ;V, xV belongs to the polar 

 line p of N^ with regard to «/^"' + i. Now^ {<^-\-?) tangents of the 

 curve {p) pass through iV* ; they are the null-rays of xV* for ^"t*. 

 I. e. ^"ï (a, ^) is transformed into a ^1* [it -j- ^, nij^) by the new 

 transformation. 



In opposition to the harmonical transformation this transformation 

 produces no new singular straight lines. 



7. If we write <tz:=:\, {i=zl, m=::2, we tind from a bilinear 

 null-system a ?il^{2, 2) for which the three singular straight lines of 

 5)'i(J,l) are also singular. 



We may indicate the bilinear null-system by 



and the curve 0" by 



The polar curve of (?/) is then expressed by 



For the null-system ^'i (2, 2) we have therefore 



el -^'i f §, •^', + §, ■«'•3 = 



In order to find the equation of the curve (P)'^ we have to com- 

 bine these two equations with 



Pi ll + P, k -f Ih ^3 = 0. 



Elimination of §k then produces for (Py 



(.!;,' + .1-1^3) - 2 (p,,r, - ;?,A-,) (p,.*-, —p,x,) (.f,' f .t-,.r,) - 0. 

 The equations (1) determine the two null-points of the straight 

 line (^) as intersections of (l) with a conic. As a condition for the 

 coincidence of tlie two null-points we tind after some reduction the 

 equation 



bi-,^3 — Ui -T-^i )i^t ^^1 irj b, — '^Si^ji-ai.si f-s, ; — 4rs, b, — "• 



0) 



