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. 341
of which the general integral is
f- f(z3x, x2zw, y/x),
leading to
3 (x, y) + zw (, y)2= 0,
a degenerate surface.
Similar reasoning shews that no term of the type (x, y) w3 can exist.
If a term (x, y) z2w, or (x, y) zw2 exists, then a =b or a = c, contrary to hypothesis;
and if a term z3W or zw3 exists, then also a = b or a = c.
Thus the only possible terms are of the type (x, y)2 zw and the surface consequently
degenerates.
V. Since no two of a, b, c, d are equal, one of them at most can vanish. We may
therefore distinguish two sub-cases:
(a) d=O, a+b+c=O,
(3) a+0, b+0, c+O, d$O.
SUB-CASE (a). Proceeding as in IV. (ii) (y) we see that the terms x4, y4, z4 cannot
exist in f, but a term w4 may exist.
If a term sxy exists, then b =-3a, c = 2a, so that the differential equation
reduces to
xf, - 3yfy + 2zfz = 0,
whence f -b (xy, Z/X2, w),
so that the only possible terms are x3y, xyzw, w4. The quartic is therefore rational,
since y only occurs linearly, if at all.
If a term x2y2 exists, then a + b= O, and therefore c = 0, contrary to hypothesis.
For the same reasons no terms of the types (y, z)4, (z, x)4, (x, y)4 can exist.
If a term x2yz exists, then 2a + b + c = O, whence a = 0, contrary to hypothesis.
Thus no term of the type (x, y, )4 can exist, so that f contains w as a factor
and is degenerate.
SUB-CASE (/3). Under the conditions assunied it is evident that no terms such
as x4, or x2yz can exist; and there cannot be more than one term belonging to a
group of the type (x, y)4.
Let the term xc3y exist, then b =-3a, c+ d = 2a, and no other term involving