with respect to Three Lines and a Line. 419 



tions with Y, Z, be £ ; then the locus of the point f is a conic 

 passing through the points YI, ZI, YZ. 



If, in like manner, through the point of intersection of the 

 lines Y, L, there is drawn any line meeting the lines I, Z, X, 

 and the harmonic of the intersection with I, in relation to the 

 intersections with Z, X, is called tj, the locus of the point ?) is a 

 conic passing through the points ZI, XI, ZX. 



And so, if through the point of intersection of the lines Z, L 

 there is drawn any line meeting the lines I, X, Y, and the har- 

 monic of the intersection with I, in relation to the intersections 

 with X, Y, is called £, then the locus of £ is a conic passing 

 through the points XI, YI, XY. 



The pairs of conies, viz. the second and third, third and first, 

 first and second conies, have obviously in common the points 

 XI, YI, ZI respectively. They besides intersect all three of 

 them in three points, which may be termed the cubic centres of 

 the line L in relation to the lines X, Y, Z and the line I. 



The line L may be such that two of the three cubic centres 

 coincide ; the locus of the coincident centres is in this case a 

 conic which touches the lines X, Y, Z harmonically in regard to 

 the line I ; that is, it touches each of the three lines in the point 

 which is the harmonic of its intersection with I in relation to its 

 intersections with the other two lines. 



Except that the line I is there taken to be infinity, the fore- 

 going theorems occur in Pliicker's System der analytischen Geo- 

 metrie (Berlin, 1835), p. 177 et seq.; and they play an import- 

 ant part in his classification of curves of the third order (see 

 p. 220 etseq.). It is, I think, an omission that he has not sought 

 for the curve which is the envelope of the line L in the above- 

 mentioned case of the two coincident centres : I find that the 

 envelope is a curve of the fourth order, having four-pointic con- 

 tact with the lines X, Y, Z harmonically in regard to the line I ; 

 viz., if the equations of the lines X, Y, Z are x—0, y=0, z—0 

 respectively, and the equation of the line I is x-\-y-\-z — 0, then 

 the equation of the envelope in question is 



a result which is also interesting as exhibiting a geometrical con- 

 struction of the curve represented by this equation. 



The investigation of the series of theorems is as follows ; taking 

 a7 = for the equation of X, 



y=o „ Y, 



x+y + z=Q „ I, 



\x + fiy + vz=0 „ h, 



