with respect to Three Lines and a Line. 435 



with similar values for 6 + fi, 6 + v. But -, -, - are proportional 



to 9 -f- \, 6 + fi, d + v, and we may therefore write 



P _ D-6a 2 P_ p-66 2 P_ p-6c 2 

 x ~ 6a 2 ' y ~ 66 2 ' z~ 6c 2 ' 

 from which equations, and the equation a + b + c = 0, the quanti- 

 ties P, a, b, c have to be eliminated. I at first effected the eli- 

 mination as follows : viz., writing the equations under the form 



_^_6a 2 _])___§&_ z _ 6c2 

 x + Y~ d' y+P — □' z + F~ a' 

 we obtain 



_*_ + _*_ + _£_ =6 



# + P y + P * + P J 



which are easily transformed into 

 *+P + y+P + *+P J 



( y +P)(^+P) + (^+p)( a7+ p) + (^+p)( y+ P) - y > 



or, what is the same thing, 



6(P + *) (P + y) (P + z) -*(P + y) (P + *) -y(P + z) (P + a:) 



-*(P + *)(P + y)=0, 

 9(P + <z)(P + y)(P + *)-y*(P + #) -**(P + y) 



-tfy(P + *)=0, 

 which give 



6P 3 + 5P 2 (# + y + *) +4P(#+y + ;?) +3^=0, 

 9P 8 + 9P 2 (# + y + z) + 8P (a + y + z) + 6xyz = 0. 

 Or, multiplying the first equation by 2, and subtracting the 

 second, 



3P + # + y + 2- = 0; 



and we thus obtain for the locus of the single centre the equation 



x | V , g = o 



—2x + y + z — 2y + z + x^ —2z + x + y ' 



or, what is the same thing, 



x 3 + y 3 + r 3 - (yz 2 + zx* + xy* + y*z + z*x + x*y) + Zxyz = 0, 

 which may also be written, 



— (— x + y-t z){x—y + z)(x + y— z)+xyz = 0. 



