448 ON A NE\Y ANALYTICAL REPRESENTATION OE CURVES IN SPACE. [284 
function of p, q, r, s, t, u, through which quantities it is a function of a, /3, y, 8; the 
last-mentioned four equations become therefore 
0 = . — d r V. y + d q V. 2 — d s V. w, 
0= d r V. x . — d p V.z — d t V. w, 
0 = — d q V. x + dpV .y . —d u V.w, 
0= d s V. x + d t V. y + d u V. z . , 
and from the first, second, and third equation, or any other combination of three 
equations, we obtain at once the condition 
d p V. d s V + d q V. d t V 4- d r V. d u V = 0, 
so that if this condition is satisfied, the four equations do in fact reduce themselves 
to two independent equations; the condition in question is thus shown to be necessary. 
But the condition only implies that all the cones having their vertices in the neigh 
bourhood of the point (a, yS, y, 8) pass through one and the same curve ; this will be 
the case for instance for a series of cones all of them circumscribed about one and 
the same surface, those having their vertices in the neighbourhood of the point 
(a, /3, y, 8) will all pass through the curve of contact with the surface of the cone 
having for its vertex the point in question. But the curve of contact is not a fixed 
curve for all positions of the vertex, and the condition before referred to is consequently 
insufficient. 
It may be noticed that the systems p, q, r and s, t, u are not similar to each 
other and that the six coordinates cannot be in any way divided into two systems 
which are similar to each other: the symmetry of the coordinates is in fact that of 
the vertices (or sides) of a complete quadrilateral (or quadrangle); thus we may divide 
the coordinates into two sets in a fourfold manner as follows: 
u,t,p\ r, q, s, 
s, q, u\ p, t, r, 
r, t, s ; u, q, p, 
p, q, r; s, t, u, 
where each left-hand set corresponds to three vertices forming a triangle and each 
right-hand set to the remaining three vertices in lineo. It may be noticed also that 
if in the equation V = 0 of any curve in space we substitute for p, q, r, s, t, u, their 
values, and equate to zero the coefficients of the different powers and products of 
a, ¡3, y, 8, each of the equations so obtained will belong to a surface passing through 
the curve, and the entire system of these equations will be equivalent to two relations 
only between the coordinates x, y, z, w. But any two of these surfaces will not in 
general intersect only in the curve, i.e. the curve will not be the complete intersection 
of any tAvo of the surfaces. It may be added that the equation of any other surface 
whatever through the curve will be obtained by equating to zero a syzygetic function 
of the functions which equated to zero give the surfaces first referred to.