PEOFESSOE CAT LEY ON CUEVATUEE AND OETHOGONAL SUBEACES. 
231 
we have an equation between the cordinates (x, y , z) and a parameter r (r being there- 
fore in general an irrational function of x, y, z ), which, when>=r 15 reduces itself to U = 0 : 
we have then r—)\ as the equation of the surface ; and the corresponding equation of the 
chief cone is ; this is not the same as the cone ??d^+£cL) 2 U==(), 
although of course it intersects the tangent plane in the same two lines, viz. the chief 
lines ; and so in general there is a distinct chief cone corresponding to each form of the 
equation of the surface. But adopting a definite form of equation, we have a definite 
chief cone intersecting the tangent plane in the chief tangents. 
4. Observe that the equations U = 0, r=r„ although each relating to one and the same 
surface, serve to represent this surface, and that in different ways, as belonging to a 
family of surfaces, viz. one of these is the family U= const., and the other the family 
r— const. In order to represent a given surface as belonging to a certain family, we 
need the irrational form of equation ; thus r denoting the irrational function of x, y, z 
x 1 y- z 2 
determined by the equation ^q; + ^y.+gij^;=l 5 we have r=0 as the equation of the 
yQ *,2 
ellipsoid — +^ + : g=l, considered as belonging to a family of confocal quadrics. 
5. Although at first sight presenting some difficulty, it is convenient to use the same 
letter r to denote the parameter considered as a function of the coordinates, and the 
special value of the parameter ; thus in general the equation of a surface may be written 
r(x, y, z)—r= 0 (in which form the first r may be regarded as a functional symbol), or 
simply r— r=0, viz. the first r here denotes the given function of (x, y, z), and the 
second r the particular value of the parameter. 
6. By what precedes we have through the point and in the tangent plane two circular 
lines, the intersections of the tangent plane by the circular cone having the point for its 
vertex. 
We have also through the point and in the tangent plane two other lines, termed 
the principal tangents, viz. the definition of these is that they are the double (or sibi- 
conjugate) lines of the involution formed by the circular lines and the chief tangents, 
or, what is the same thing, they are the bisectors (and as such at right angles to each 
other) of the angles formed by the chief tangents. 
7. The principal tangents may also be considered as the intersections of the tangent 
plane by a quadric cone, called the principal cone ; this being a cone constructed by 
means of the circular cone and the chief cone, and thus depending on the particular 
chief cone, that is, on the form of the equation of the surface. The definition is that 
the principal cone is the locus of a line (through the point), such that the line itself, the 
perpendicular (or harmonic in regard to the circular cone) of the polar plane of the line 
in regard to the chief cone, and the normal of the surface are in piano. 
8. Analytically, taking, as before, (x, y, z) for the coordinates of the point, and u, v, w 
as current coordinates measured from the point as origin, then the equation of the cir- 
cular cone is v? + v 2 -}- w 1 — 0 ; and taking Xw -j-Y-y+ Z w — 0 for the equation of the tangent 
2 i 2 
