Memoirs presented to the Cambridge philosophical society on the occasion of the jubilee of Sir George Gabriel Stokes, bart., Hon. LL. D., Hon. SC. D., Lucasian professor.

336 MR BERRY, ON QUARTIC SURFACES WHICH ADMIT OF INTEGRALS
second minor, and so on, and if p-pl=a, p1-p2=a..., and 3, ' y...,7,...are the
numbers corresponding similarly to the other factors, s - b, s - c,... of A, then
(S - a)a, (s - a )... ( - ), (s- ).,
are defined as the elementary factors of A.
These factors are shewn by Weierstrass to be invariant for linear transformation of
the variables, and the system of differential equations
dx dy dz dw
= -1, ' -, — --- 04,
dt dlt dt dt
is shewn to be reducible by linear transformation of the dependent variables to a
standard form, in which there are as many distinct sets of equations as there are
elementary divisors of A, the set corresponding to an elementary divisor (x - a)P being of
the form
dx, dx2 dxp
dt a dt = ax2 + x1, dt = a + -.........................(5).
Applying this theory to our equation we see that the possible ways in which A
can be resolved into elementary factors are as follows:
(I) All the roots of A = O equal: (i) (s - a)4,
(ii) (s - a), (s - a),
(iii) (s - a), (s - a),
(iv) (s - a)2, (s - a), (s - a),
(v) (s-a), (s-a), (s - a), (s - a).
(II) Three roots of A = O equal: (i) (s - a)3, (s - b),
(ii) (s - a)2, (s - a), (s - b),
(iii) (s - a), (s - a), (s - a), (s - b).
(III) Two pairs of roots of A = 0 equal: (i) (s - a)2, (s-b)2,
(ii) (- a)2, (s - b), (s - b),
(iii) (s - a), (s - a), (s - b), (s -b).
(IV) One pair of roots of A = 0 equal: (i) (s - a)2, (s-b), (s-c),
(ii) (s - a), (s - a), (s - b), (s- c).
(V) All the roots of A= 0 distinct: (s - a), (s - b), (s - c), (s - d).
Also the equation (2) shews that the sum of the roots of A =0 vanishes, so that
we must have in
CASE I. a = O,
CASE II. b = - 3a J 0, and we may evidently take a = 1, b = - 3,