74 
ME. A. CAYLEY’S SIXTH MEMOIE TJPOX QrAXTICS. 
The equation of a curve in point-coordinates, or as it may be tenned the point-equa- 
tion of the curve, is the relation which exists between the pomt-coordmates of any 
ineunt of the curve. • 
The equation of a curve in line-coordinates, or line-equation of the curi e, is 
tion which exists between the line-coordinates of any tangent of the cun e. 
186. It has been mentioned, that the order of a curve is the degi-ee of its pomt^ua- 
tion; in like manner the class of a curve is the degree of its line-equation ; and in the 
same way that a curve, as represented by a point-equation, may break up mto curves 
havinff the order of the entire curve for the sum of their orders, so a cuiwe as repre- 
sented by a line-equation may break up into curves having the class of the entae curve 
for the sum of their classes. And, in particular, a curve may break up into a system of 
points, the number of points being equal to the class of the cuiwe, and two or more o 
these points may coincide together. _ ^ ^ ^ 
187. A line is a curve of the order one and class zero ; a pomt is a cmwe of the order 
zero and class one. A proper conic is a curve of the order two and class two ; but w en 
the conic breaks up into a pair of lines, the class sinks to zero ; and when tke come 
breaks up into a pair of points, the order sinks to zero. It is to be noticed a a 
point, or system of points, cannot be represented by an equation in pomt-cooi ina es, 
nor a line or system of lines by an equation in line-coordinates. We may say, m gene^ . 
that a curve is both a point-curve and a line-curve, but a point or system ot pom s is a 
line-curve only, and a line or system of lines is a pomt-emwe only. • x 
188 The points of intersection (common ineunts) of two cuiwes are the points e 
coordinates of which satisfy simultaneously the point-equations of the two curias. 
Hence the number of common ineunts is equal to the product of the oi eis o ® ® 
curves; and, in particular if one of the curves be a Une, the number of pomts of inter- 
section (common ineunts) is equal to the order of the curie. In i^e ^ 
common tangents of the two curves are the lines the coordinates of which satisfy simu - 
taneously the line-equations of the two curves. Hence the number of common tangents 
is equal to the product of the classes of the two curves; and, in particular, if one ot me 
curves be a point, the number of common tangents (tangents to the curve throug t e 
point) is equal to the class of the curve. Since the tangent is the line t u-oug mo 
consecutive ineunts, it besides meets the curve, assumed to be of the order m m (m ) 
points; and in like manner we may from any ineunt of a curve of the class n draw (w ) 
tangents to the curve. _ / < i i\ a r -"'i 
189. The point-equation of a line passing through the points (a- , 3 / , 2 : ) an [x , if ) 
Is, as already noticed, 
X , 
y ^ 
z 
x\ 
y'^ 
z' 
x\ 
y\ 
z" 
: 0 . 
Suppose that {x, y, z) are the coordinates of a point (ineunt) of the curve U 0, the 
