A MEMOIR ON QUARTIC SURFACES. 
135 
445] 
and with either arrangement we may form one and the same determinant, the Jacobian 
determinant J (P, Q, P, S), or, equating it to zero, the Jacobian surface J(P, Q, R, £) = 0, 
of the four surfaces. 
2. In the case of more than four surfaces, adopting the arrangement 
p, Q, n, s, t, .. 
8 y 
8 Z 
&w 
and considering the several determinants which can be formed with any four columns 
of the matrix, these equated to zero establish a more than one-fold relation between 
the coordinates ; viz., in the case of five surfaces, we have J(P, Q, R, S, T) = 0, a 
twofold relation representing a curve ; and in the case of six surfaces, J(P, Q, R, S, T, U)=0, 
a threefold relation representing a point-system; and (since with four coordinates a 
relation is at most threefold) these are the only cases to be considered. 
3. In the case of fewer than four surfaces, adopting the arrangement 
8x> $y> $z> ^w 
p 
Q 
and considering the several determinants which can be formed with any 3 or 2 columns 
of the matrix, and equating these to zero, we have in like manner a more than one 
fold relation between the coordinates; viz., in the case of three surfaces, we have 
J (P, Q, R) = 0, a twofold relation representing a curve; and in the case of two 
surfaces J (P, Q) = 0, a threefold equation representing a point-system, (viz., this 
denotes the points 8 X P : 8 y P : 8 Z P : 8 W P = 8 X Q : 8 y Q : 8 Z Q : 8 W Q); for a single surface 
we should have a fourfold relation, and the case is not considered. But observe that 
if the notation were used, J (P) = 0 would denote 8 X P = 0, 8 y P = 0, 8 Z P = 0, 8 W P = 0, 
equations which are satisfied simultaneously by the coordinates (x, y, z, w) of any 
node of the surface P = 0. Although in what precedes I have used the sign =, there 
is no objection to using, and I shall in the sequel use, the ordinary sign =, it being 
understood that while J (P, Q, R, S) = 0 denotes a single equation or onefold relation, 
J (P, Q, R, S, T) — 0 or J (P, Q, R) = 0 will each denote a twofold relation, and 
J(P, Q, R, S, T, U) = 0 or J (P, Q) = 0 each of them a threefold relation. 
4. It is not asserted that ... J(P, Q, R) = 0, J(P, Q, R, S) = 0, J(P, Q, R, S, T) = 0,... 
form a continuous series of analogous relations; and there might even be a propriety 
in using, in regard to four or more surfaces, J, and in regard to four or fewer surfaces 
an inverted J (viz., in regard to four surfaces, either symbol indifferently); but there 
is no ambiguity in, and I have preferred to adopt, the use of the single symbol J.