422 On the Cubic Centres of a Line. 



that is 



x* + y* + z*-2yz- 2zx-2vy=0; 

 or what is the same thing, 



s/~x + \/y + i/J=0, 



for the equation of the locus of the coincident centres : such 

 locus is therefore a conic touching the lines x = 0, y=0, z=(), in 

 the points of intersection with y— z=0, z—x = 0, x—y = 

 respectively ; it is a conic touching the lines X, Y, Z harmoni- 

 cally in regard to the line I. 



To find the envelope of the line L, the most convenient course 

 is to take the equation in 6 in the reduced form 



^~0(/*]/ + v\ + \//,)-2Vv = O; 



this will have a pair of equal roots if 



{fiv + vX + \/x) 3 -27AV" 2 =0 j 

 that is, if 



or if 



or finally, if 



fiv + v\ -f \/ul — 3 (X/av)* = ; 



which is the relation between X, fj,, v in order that the line 



\x-\-fjby-\-vz=Q 



may have two coincident centres ; this gives at once for the equa- 

 tion of the envelope 



\/x + y~y + V~2 = 0, 

 which is the equation of a curve of the fourth order having four- 

 pointic contact with the lines x=0, y=0, z=0, at the points of 

 intersection with the lines y—z = 0, z—x=0, %—y=0 respect- 

 ively, i. e. it has four-pointic contact with the lines X, Y, Z 

 harmonically in regard to the line I. 



It may be noticed that the rationalized form of the equation 

 \/x + \/y + \/z=0i8 



x 4 + 2/ 4 + z 4 — 4(ys 8 + 'fz + zx 3 + z 3 x + xy 8 + a?y) 



+ 6{y*z* + z*x* + xy)-124(x*yz + y' i zx + z' 2 xy)=0. 



If, to fix the ideas, the signs of the coordinates x, y, z are so 

 determined that a point within the triangle x=0, y=0, z=0 

 has its coordinates positive (in which case the line x + y + z = 

 will cut the three sides produced), the curve X/lc + \/y + %/ z = 

 will lie wholly within the triangle, and will be of the form shown 



