PROFESSOR W. K. CLIFFORD OFT THE CLASSIFICATION OF LOCI. 
665 
To particularise still further, a system of conics having the characteristic /x= 2 must 
always have quadruple contact with a quartic curve ; and the different species may be 
enumerated by studying the successive degeneration of the curve, ending with the 
fundamental system v= 1, when it breaks up into four straight lines. 
So again, there is no quadric scroll, in any number of dimensions, except the ordi- 
nary quadric surface which is in flat space of three dimensions. 
A curve of the third order must be either the known skew cubic in three dimensions, 
or a plane cubic. Hence, if a system of curves be such that three of them can be 
drawn through an arbitrary point, the equation of any curve of the system is of one of 
the two forms — - 
U+3V7 +3W* 3 +X* 3 =0, 
IJ + Ysrru + 2 Ws n u cn udnu= 0, 
where t, u are parameters. Hence it is easy to write down the equations to the 
envelopes in the two cases, and to enumerate the distinct species. 
one element in the 7c- flat in the form a 1 tq + a 2 'W 2 + . . . +a.£ +1 Wi +1 = 0, and take the It ratios of the a as the 
coordinates of a variable cnrve. 
“ For It — 2 we get a net as the flat space of two dimensions or as the plane in this space, and for h— 1 
a pencil corresponding to the line. 
“ If, on the other hand, we assume in the It - flat It— 1 equations between the coordinates a, there remains 
a singly infinite number of curves, that is according to Professor Clifford a curve (with curves as 
elements), according to the usual nomenclature a series of curves. 
“ To determine the order of this curve we have to find the number of elements on it which satisfy a 
linear relation between the coordinates. In our case the condition that a curve shall pass tkrough a given 
point gives such a relation, and the number of curves through a point is the order in question. 
“ Hence, if we wish to extend a theorem relating to a curve (in the ordinary sense with points as 
elements, but in any number of dimensions) to a proposition relating to a series of curves, or if we wish 
to illustrate in a plane a theorem relating to a curve in more dimensions than three, we have instead of a 
point on the curve to take a curve in the series, and to replace the order of the curve by the index of 
the series. 
“ The theorem that every curve of order two is a plane curve becomes thus — the curves in a series of 
index 2 belong to a net. 
“ Further, the coordinates of a point on a conic may be represented as expressions of the second degree 
in a variable parameter, say by x\ + 26yi + 6 2 zi ; where i— 1, 2, 3, if the coordinates are taken in the plane 
of the conic, but if they are taken in space we have to take i— 1, 2, 3, 4, and so on for more dimensions. 
The locus of these points, that is, the conic, is. then given by an equation of the form 
yu+yy+ v/W =0 
where U, Y, W are three of the points. 
“ If we apply this to our series we obtain the results stated in the text, viz., the coordinates of any 
cnrve of a series such that two curves pass through a given point are of the form quoted, and the 
equation of the envelope is of the form 
v/u+ yy+yw^o, 
IJ, Y, W being three of the curves. 
“ Similarly, if the series is such that three pass through any point, then the series may be considered as 
a ‘ curve’ of order three, and the statements made in the text follow at once from the known properties 
about cubic curves, which are either unicursal (twisted, or plane nodal, cubics) or they are plane curves 
of deficiency one.”— January, 1879.] 
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