434 Mr. A. Cayley on the Cubic Centres of a Line 



Two of the centres will coincide if the equation for 6 haa equal 

 roots ; and this will be the case if 



or, what is the same thing, if X, p, v = a~ 3 , b~ 3 , c~ 3 , where 

 a + b + c = 0. In fact, if a + b + c=0, then a 3 + b 3 + c 3 = dabc, 

 and the equation in becomes 



that is, 



{abc0) 3 -Z{abc0) -.2 = 0, 

 which is 



{abc0+l)*(abc0-2) = O. 



12 1 



So that the values of are -?-, -r— . First, if 0= j— , 



abc abc abc 



then x, y, z will be the coordinates of the double centre. And 



we have 



or putting for shortness □ = a? 4- £ 2 + c 2 , 



* + 2a 3 6c D ' " abc 6a? 



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



to -f \, + fi, -f v ; and we may therefore write 



?-_°. P - °- ?-_H. 

 *~6a 2 ' y~66 2 ' ^~6c 2; 



whence, in virtue of the equation a + b + c=0, we have for the 

 locus of the double centre, 



V#+ Vy-\- Vz = 0. 

 Or this locus is a conic touching the lines x = 0, y = 0, z = 

 harmonically in respect to the line x + y + z = 0, a result which 

 was obtained somewhat differently in the paper above referred to. 

 2 

 Next, if 0= —r-. x, y, z will be the coordinates of the single 

 abc ° 



centre. And we now have 



3 o-6a 2 

 ~ o6c 6a 2 ' 



