Theory and applications of finite groups, by G.A. Miller, H. F. Blichfeldt [and] L. E. Dickson.

254 THE LINEAR GROUPS IN THREE VARIABLES [CH. XII

These invariants must evidently be transformed into the
functions f, H, T of the group (E), ~ 105, by the substitution
of -X2X1, X22, -X12 for xi, X2, X3 respectively (cf. Ex. 4, ~ 124).
The relation among the invariants A, B1,... corresponding
to that connecting f, H, T is of degree 30 in the variables xi,
X2, X3, and its determination may be left as an exercise for the
student.*
(I) The fundamental invariants are four in number.t
One is the function
F =A3+XB1,
where (taking for (I) the group given in Ex. 5, ~ 124) A and B1
are the functions listed above under (H), and
-9t3V/- 15. -- 3,V —1-5
20
The three other invariants are obtained by evaluating the
Hessian of F:
F11 F12 F13
X= F21 F22 F23
F31 F32 F33
then the bordered Hessian:
F11 F12 F13 X1
F21 F22 F23 X2
F31 F32 F33 X3
X1 X2 X3 0
and finally the Jacobian of F, X, and Y.
(J) As in the previous case, the fundamental invariants
are four in number; { a function
f= X13X3 +X23X1 +X33X2;
the Hessian of f (say X); the Hessian of f bordered by the
first derivatives of X (as Y above); finally, the Jacobian of
the three functions already determined.
* Cf. Klein, Vorlesungen iiber das Ikosaeder, Leipzig, 1884, p. 219.
t Wiman, Mathematische Annalen, Bd. 47 (1896), pp. 548-556.
1 Klein, Mathematische Annalen, Bd. 14 (1879), pp. 446-448.