ADDRESS. 27 
functions, homogeneous of the same degree in the new coordinates ; 
the transformed curve is in general a curve of a different order, with its 
own system of double points; but the deficiency p remains unaltered ; 
and it is on this ground that he unites together and regards as a single 
class the whole system of curves of a given deficiency p. It must not be 
supposed that all such curves admit of rational transformation the one 
into the other: there is the further theorem that any curve of the class 
depends, in the case of a cubic, upon one parameter, but for p>1 upon 
3p — 3 parameters, each such parameter being unaltered by the rational 
transformation ; it is thus only the curves having the same one para- 
meter, or 3p — 3 parameters, which can be rationally transformed the one 
into the other. 
Solid geometry is a far wider subject: there are more theories, and 
each of them is of greater extent. The ratio is not that of the numbers 
of the dimensions of the spaces considered, or, what is the same thing, of 
the elementary figures—point and line in the one case ; point, line and 
plane in the other case—belonging to these spaces respectively, but it is 
avery much higher one. For it is very inadequate to say that in plane 
geometry we have the curve, and in solid geometry the curve and surface: 
@ more complete statement is required for the comparison. In plane 
geometry we have the curve, which may be regarded as a singly infinite 
system of points, and also as a singly infinite system of lines. In solid 
geometry we have, first, that which under one aspect is the curve, and 
under another aspect the developable, and which may be regarded as a 
singly infinite system of points, of lines, or of planes; secondly, the 
surface, which may be regarded as a doubly infinite system of points 
or of planes, and also as a special triply infinite system of lines (viz. the 
tangent-lines of the surface are a special complex): as distinct particular 
cases of the former figure, we have the plane curve and the cone; and 
as a particular case of the latter figure, the raled surface or singly infinite 
system of lines; we have besides the congruence, or doubly infinite system 
of lines, and the complex, or triply infinite system of lines. But, even if 
in solid geometry we attend only to the curve and the surface, there are 
crowds of theories which have scarcely any analogues in plane geometry. 
The relation of a curve to the various surfaces which can be drawn 
through it, or of a surface to the various curves that can be drawn upon 
it, is different in kind from that which in plane geometry most nearly cor- 
responds to it, the relation of a system of points to the curves through 
them, or of a curve to the points upon it. In particular, there is nothing 
in plane geometry corresponding to the theory of the curves of curvature 
of a surface. To the single theorem of plane geometry, a right line is 
the shortest distance between two points, there correspond in solid 
geometry two extensive and difficult theories—that of the geodesic lines 
upon a given surface, and that of the surface of minimum area for 
any given boundary. Again, in solid geometry we have the interesting 
and difficult question of the representation of a curve by means of 
