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. 335
where 01, 02,,0, 04 are quantics of order n - 3, which satisfy the equation
ao, ao2 a,03 4
+ + w =..................................... (2).
ax ay az azf
These equations being satisfied, the differential
dx, dy, dz Of, ~y z aw
01 02, 03
satisfies the condition of integrability, and its integral is finite everywhere with the
possible exception of certain singular points and lines.
When n =4 the quantics 0 are linear and the equation (1) becomes a familiar
partial differential equation.
If we write
0i - aix + biy + ciz + diw, (i = 1, 2, 3, 4),
then, in accordance with the usual elementary theory, the integration of the equation
(1) depends upon the roots of the algebraic equation
/x — a1-s, b1, c1, d1 =0.(3).
1a2 b2 - s c, c,
\ac, b3, C3-s, dS
a4, b4, c4, d4-s ]
If the roots of this equation are all distinct we can at once obtain three independent integrals of the auxiliary system
dx dy dz dw
= - Y....................................... (4**
01 02 03 044)
and deduce the general integral of the partial differential equation. But if two or
more roots of A=0 are equal the integration of the system (4) is less simple, and one
or more of the integrals is in general logarithmic, though these integrals may again
become algebraic if the coefficients of the 0's satisfy certain further conditions. Although
the complete discussion of these cases by quite elementary methods presents no serious
difficulty it is rather long and tedious, and the work can be considerably abbreviated
by reducing the equations (4) to a standard form by means of the method which was
given by Weierstrass as an application of his theory of bilinear forms*. This method,
stated in a forin applicable to our particular problem, depends on the resolution of the
determinant A into "elementary factors" (Elementartheiler). If s - a occurs p-tuply as
a factor of A, p p-tuply as a factor of each first minor of A, p2-tuply as a factor of each
* "Bemerkuugen zur Integration eines Systems liîlearer Differentialgleichungen mit constanten Coefficienten,"
lIathenmatische Werke, I. pp. 75-6.