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 OF TOTAL DIFFERENTIALS. 339
III. (i). The auxiliary equations are:
dx dy dz dw
x x+y -z z-w'
three integrals of which are:
y/x - log = const., w/z + log z = const., zx= const.,
so that the only possible quartic is the degenerate surface
z2X2 = 0.
III. (ii). The auxiliary equations are:
dx dy dz dw
x x+y -z -w
three integrals of which are:
y/x - log x = c onst., /x = c ons t.,w = costt,
leading to f=- (zx, z/w), which gives a cone.
III. (iii). The auxiliary equations are:
dx _dy dz dw
x y -z -w'
leading to
y/x = const.,t., x= costt, const.,
whence f/_ (xz, xw, y/x), so that the quartic is quadratic in x, y and in z, w, viz.
of the form
2 2
(x, y= z, w ).......................................... (8).
IV. (i). The auxiliary equations are:
dx dy cdz dw
ax x + ay bz cw
where 2a + b + c = 0.
If a 0, three integrals are:
ay/ - log x = const., z nt., zw/xb = c on s t., onst.,
of which the first is essentially logarithmic, so that we have f ()(za/xb, wa/Sx), leading
to a cone.
If a = 0, so that c =-b 0, three integrals are:
x = const., x log z- by = const., zw = const.,
so that f - (zw, x), leading again to a cone.
43-2