390 
ME. A. CAYLEY THE COYIC OF EIYE-POIXTIC 
or, what is the same thing, throwing out the factor 2, it is 
= { 3(1 + 4- 6^(1 4- 30Z=*(1 + 
-2Q(1+8P)(x’-2/^); 
or throwing out the factor (1 + 8 Z^) and substituting for Q its value, it is 
= {3^y’+6%2(T’+/) + 30«V2/V-2(yV+2V+^/-3?^^/2’)}(^-’-/)- 
The first factor, reducing by the equation a-'+)/’+t’+6fayz, is 
=3*y-(ai-+y’)(**+y+^*)+(**+3'’+*’)’-2(/^+"'^+*'^) 
= 3xy+«’(3;’+y’+z*)-2(y2’+2'®'+*y) 
= ( 2 ®— 
39. Hence putting for the moment 
it appears that the last of the three equations is M2=0; the fii-st and second are of 
course M ^=:0 and M3/=0, and the required condition is YI-O, that is, 
{f-.z^){z^~x%x^-f)=^, 
the equation which, combined with the equation of the curve 
x^'\‘y^~\~^'\~^^xyz'^^'^ 
.rives the sextactic points. There are consequently twenty-seven such points, and it is 
at once seen that these are the points of contact of the tangents to the cubic irom the 
points of inflexion, or, what is the same thing, that the twenty-seven sextactic points foim 
Le groups of three each, such that the three points of a group have for 
tangeLal one of the nine points of inflexion. In fact, let . be a cube root (real or 
imaginary) of unity, the three sextactic points of one of the gioups wall e giien . 
j" x-—ooy — 0 , 
V' -4 + 2 ® + 6 Ixyz = 0 . 
Now consider the tangential of any one of these points, its coordinates are 
x,—x{y'^~z^\ 
y^z^y{z^-x% 
z,=z{x^—y^)-, 
or, reducing by the equation x—uy—^, x, — ccy{x^—z% yi = —y{x -z ), ^1 — 0, 01 , uhat 
is the same thing, ir.-f.ay,=0, ^, = 0; that is, the point (*,. y„ s,) is one ^ ° 
inflexion. This is the construction of the sextactic points obtained by 1 luclek and 
40 Reverting to the equation of the luie joining the point of contact ^ 
pointic conic with the point of simple intersection, this meets the cubic in a P°^' 
and Mr. Salmon has shown that this third point is in tact the second tangential (tan- 
gential of the tangential) of the point of contact, or, what is the same thing 
