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.

OF THE FIRST KIND 0F TOTAL DIFFERENTIALS. 337
CASE III. b=-a 0, and we may take a= 1, b =-1,
CASE IV. b + c =- 2a,
CASE V. a + b+c+ d = O.
It follows at once that the Case I. (v) is impossible.
For the purposes of our problem we do not want the general integral of the
equation (1), but only such integrals as are homogeneous quartics; we may also leave
cones out of account, and we must reject solutions giving degenerate (reducible) quartic
surfaces; we find also that in one or two other cases we arrive at surfaces which are
obviously rational and must therefore be rejected.
~ 3. INTEGRATION OF THE DIFFERENTIAL EQUATION, LEADING TO FIVE POSSIBLE SURFACES.
We have in all (after rejecting I. (v)) thirteen cases to consider, which will now
be dealt with seriatim. In each case the transformed variables will still be denoted
by x, y, z, w, and the auxiliary equations will be expressed in the usual Lagrangean
form, the variable t used by Weierstrass being omitted.
I. (i). The auxiliary equations are:
dx dy dz dw
0 x y z'
three integrals of which are:
x = const., y2 - 2zx = const., y3 + 3x2w - 3xyz = const.,
so that the general integral of the equation (1) is
f= (w, y2 - 2zx, y3 + 3x2w - 3xy),
where ~b is an arbitrary function.
The only quartic of this form is a sum of terms
X4 (2 (y2- 2z), (y2 - 2zw)2, x (y3 + 3w2w - 3xyz),
so that w occurs linearly or not at all, and the surface is therefore a cone or rational.
I. (ii). The auxiliary equations are:
dx_ dy_ dz dw
O O y z
three integrals of which are:
x = const., y = const., z2 - 2yw = const.,
so that
fVO- L.,, XVII- 2yI. 4).
VOL. XVIII. 43