DE. J. CASEY ON CYCLIDES AND SPIIEKO-QTJAETICS. 
G69 
Hence in this section we shall discuss the curve WU by regarding it as the intersection 
of V and U. 
For the purpose of classification I shall, following Cayley and Salmon, consider the 
curve (UV) as made up of points ; then the points of UV will he the points of the 
system, the line joining two consecutive points will be a line of the system, and the 
plane of two consecutive lines will be a plane of the system. If a plane of the system 
contains four consecutive points it will be a stationary plane ; and reciprocally, if four 
consecutive planes of the system intersect in a point of the system, it will be a stationary 
point. Again, if a line join two non-consecutive points it will be a line through two 
points ; reciprocally, if a line be the intersection of two non-consecutive planes, it will 
he a line in two planes ; finally, if two non-consecutive lines intersect, their point of 
intersection will be a point on two lines, and their plane a plane through two lines. 
For the purpose of denoting these singularities the following notation will be used. 
Thus we shall denote by 
r, the number of lines of the system which meet an arbitrary line. 
771, 
99 
points of the system which lie in any plane. 
a, 
99 
stationary planes of the system. 
X, 
99 
points on two lines which lie in 
a given plar 
( Ji 
99 
lines in two planes which lie in 
a given plar 
The reciprocals of m, a, x , </will be denoted by the letters respectively consecutive to 
them, namely, n, (3, y , h. 
m is called the degree of the system. 
n ,, class „ 
r „ rank ,, 
219. The complete surface formed by the lines of the system (UV) is the developable 
A, whose properties we have discussed in Chapter VIII., the curve (UV) being the cus- 
pidal edge. Again, the developable 2 of Chapter VIII. formed by the tangent planes 
to U along the curve UV is the reciprocal of A, and the point, lines, and planes of A are 
respectively the reciprocals of the planes, lines, and points of 2, with respect to U ; so 
that when we have the characteristics of A, by reciprocation we shall have the charac- 
teristics of 2. 
220. Let us consider the cone whose vertex is at any point and which stands on the 
curve (UV). If, for the sake of distinction, we denote by Greek letters the characteristics 
of a plane curve (that is, if v, e), r, z, i denote the degree, class, double points, double 
tangents, cusps, points of inflection of the curve), then, for a cone standing on that plane 
curve, it is evident that the same letters will denote the degree, class, double edges, 
double tangent planes, cuspidal edges, and stationary tangent planes ; then for the cone 
on (UV) ( jj = m , v = r , l = h , r = y , z = f 3 , i = n . (See Salmon’s ‘Geometry of Three 
Dimensions,’ art. 321.) 
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