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



The same result may also be obtained as follows : viz., observing 

 that a-6a 2 =6 2 + c 2 -5a 2 = — 4a 2 -26c, we have 



-3a 2 v -36 2 z -3c 2 



x — da* y 



i>~2a* + bc' P _ 26 2 + ca' P 2c' 2 + ab' 

 and then by means of the equation 



a 2 b* c 2 



ZcP + bc + 2~F*~+a~c + 2c z + ab ~" 1 = > 



which is identically true in virtue of a + b + c=0 (in fact, mul- 

 tiplying out, this gives 



12a 2 £ 2 c 2 + 4(6 3 c 3 + c 3 a 3 + a% 3 ) + abc{a 3 + i 3 + c 3 ) 



_ 8a «fl8 c S _ 4 ^3 C 3 + c 3 a 3 + a 3^ 3) _ £ a fo( fl 3 + £3 + c 3j _ fl 2 fi « c « _ Q ; 



that is, 



3a 2 6 2 c 2 -0fo(« 3 + 6 3 + c 3 ) = O, or afte(<^ + ^+^— 8o6e)=:(^ 



where the second factor divides by a + b + c), we find the above- 

 mentioned equation, 



x + y+z + 3Y = 0. 

 We then have 



— x + y + z _ x + y + z 2x _ 6a 2 3Ac 



P ~ P "F ~~ 6+ 2a* + bc~~2a ir +Tc'' 



that is, 

 — x+y+z _ —Sbc x—y + z_ — Sea x + y—z_ —Sab 



~~t~ ~ 2a* + bc' P~~ ' ~ 2b* + c ' P ~ %<*+ab 

 And forming the product of these functions, and that of the 

 foregoing values of p, Jj, p, we find as before, 



— ( — x + y + z) (x — y + z) (x + y — z) + xyz — 



for the equation of the locus of the single centre. The equation 

 shows that the locus is a cubic curve which touches the lines 

 x=0, y = Q, z = at the points where these lines are intersected 

 by the lines y—z = 0, z—w=0, x—y = (that is, it touches 

 the lines x=0, y = 0, z = harmonically in respect to the line 

 x + y+z = 0), and besides meets the same lines x = 0, y = 0, 

 z = at the points in which they are respectively met by the line 

 x + y + z = 0. 



2 Stone Buildings, W.C. 

 September 25, 1861. 



