790] 
GEOMETRY. 
569 
wave surface. Or a surface may be delineated by means of a series of parallel sections, 
or (taking these to be the sections by a series of horizontal planes) say by a series 
of contour lines. Of course, other sections may be drawn or indicated, if necessary. 
For the delineation of a curve, a convenient method is to represent, as above, a series 
of the points P thereof, each point P being accompanied by the ordinate PN, which 
serves to refer the point to the plane of xy; this is in effect a representation of each 
point P of the curve, by means of two points P, N such that the line PN has a 
fixed direction. Both as regards curves and surfaces, the employment of stereographic 
representations is very interesting. 
28. In plane geometry, reckoning the line as a curve of the first order, we have 
only the point and the curve. In solid geometry, reckoning a line as a curve of the 
first order, and the plane as a surface of the first order, we have the point, the curve, 
and the surface; but the increase of complexity is far greater than would hence at 
first sight appear. In plane geometry a curve is considered in connexion with lines 
(its tangents); but in solid geometry the curve is considered in connexion with lines 
and planes (its tangents and osculating planes), and the surface also in connexion with 
lines and planes (its tangent lines and tangent planes); there are surfaces arising out 
of the line—cones, skew surfaces, developables, doubly and triply infinite systems of 
lines, and whole classes of theories which have nothing analogous to them in plane 
geometry: it is thus a very small part indeed of the subject which can be even 
referred to in the present article. 
In the case of a surface, we have between the coordinates {x, y, z) a single, or 
say a onefold relation, which can be represented by a single relation f(x, y, z) = 0; 
or we may consider the coordinates expressed each of them as a given function of 
two variable parameters p, q; the form z = f(x, y) is a particular case of each of these 
modes of representation; in other words, we have in the first mode f (x, y, z)=z —f (x, y), 
and in the second mode x=p, y — q for the expression of two of the coordinates in 
terms of the parameters. 
In the case of a curve, we have bet ween the coordinates (x, y, z) a twofold relation: 
two equations f(x, y, z) = 0, (f> (x, y, z) = 0 give such a relation ; i.e., the curve is here 
considered as the intersection of two surfaces (but the curve is not always the com 
plete intersection of two surfaces, and there are hence difficulties); or, again, the 
coordinates may be given each of them as a function of a single variable parameter. 
The form y = 6x, z = -px, where two of the coordinates are given in terms of the third, 
is a particular case of each of these modes of representation. 
29. The remarks under plane geometry as to descriptive and metrical propositions, 
and as to the non-metrical character of the method of coordinates when used for the 
proof of a descriptive proposition, apply also to solid geometry; and they might be 
illustrated in like manner by the instance of the theorem of the radical centre of four 
spheres. The proof is obtained from the consideration that S and S' being each of 
them a function of the form x 2 + y 2 + z 2 + ax + by + cz + d, the difference S - S' is a 
mere linear function of the coordinates, and consequently that S - S' = 0 is the equation 
of the plane containing the circle of intersection of the two spheres S = 0 and S'=0.