§14] 
INFLEXION POINTS OF CUBIC 
29 
where w is an imaginary cube root of unity. Their left members are 
constant multiples of 3h+rf, where r=3/3 2 , — (/a—/3) 2 are the four roots 
of (1), with 
b = p(a 3 -p 3 ), 4o = a 6 —20a 3 /3 3 — 8/3 6 , 
3. The Jacobian of ffxi,. . ., x n ),. . . , f n (x h . . , x n ) is 
9/i dfx 
9/i 
dxi 9*2 
’ 9 x n 
9fn dfn 
dfn 
9*i 9*2 ' ' 
' 9*?. 
Show that it is a covariant of index unity of f x , . . . , /„. 
4. Hence the resultant of three ternary linear forms is an invariant of 
index unity. 
5. If/i,. . .,/« are dependent functions, the Jacobian is zero.