447 
284] ON A NEW ANALYTICAL REPRESENTATION OF CURVES IN SPACE. 
or what is the same thing we have identically 
d p V. d s V + d q V. d t V + d r V. d u V =LV + M(ps + qt + ru), 
where L and M are functions of p, q, r, s, t, u. But the converse proposition, viz. any 
equation whatever V= 0, where V satisfies the condition just referred to represents 
a curve in space, is not true; it would obviously be an important point in the theory 
to ascertain what further conditions must be satisfied by the function V. 
The establishment of the foregoing results is very easy; in fact if x, y, z, w are 
current coordinates of the ordinary kind (point-coordinates) and a, /3, y, 8 the coordi 
nates of an arbitrary point; the equation of any cone whatever having for its vertex 
the point (a, /3, y, 8) may be represented by a homogeneous equation between the six 
determinants of the matrix 
( x, y, z, w ) 
i ¡3, y, 8 \ 
or if we write 
p = yy — /3z, s = 8x — aw, 
q = az — yx, t = By — ¡3iu, 
r = fix — ay, u = 8z — yw, 
values which it is well known give identically 
ps + qt-1- ru = 0, 
then the cone will be represented by a homogeneous equation 
F = 0 
between the six coordinates (p, q, r, s, t, u). It remains to find the conditions in order 
that all the cones so represented, viz. the cones obtained by giving any values whatever 
to the arbitrary quantities a, fi, y, 8 which enter implicitly into the coordinates 
p, q, r, s, t, u, pass through one and the same curve; for when this is the case, the 
equation V — 0 may be properly considered as the equation of the curve. 
Assume then that all the cones pass through the same curve; if we give to one 
of the arbitrary quantities a, fi, y, 8, say a, the infinitesimal variation da, then the 
function V becomes V + d a V. da, and each of the equations V = 0, V+d*V.da = 0 
belongs to a cone passing through the curve; the equation d a V = 0 is therefore the 
equation of a surface passing through the curve; and in like manner the four equations 
d a V = 0, dpV = 0, d y V= 0, d s V= 0 
are each of them the equation of a surface passing through the curve, or these 
equations must be simultaneously satisfied for all the points of the curve, they must 
consequently reduce themselves to two independent relations. But V is given as a