YQ me. a. CAYLEY’S SIXTH MEMOIE UPOX QIJAXTICS. 
171 If the quantic breaks up into rational factors, then the equation of the cune is 
satisfied by equating to zero any one of these factors, or the cui^e breaks up into cun es 
of a lower order, and the order of the entire curve is equal to the sum of the orders o 
the component curves. In particular, a curve of any order may break up mto a sys ein 
of Hnes, the number of lines being of course equal to the order of the cmve, a ^ 
two or more of these hues may coincide with each other. A curve not thus bieakrn^ 
UP into curves of a lower order is said to be a proper curve. 
172. Eeturning to the linear equation and expressing the coefficients, the equation is 
(I, fi, 
or, what is the same thing, 
and we say as a definition, that the coordinates (line-coordinates) of this line aij (|, 0 - 
173. But the same equation, considering (^, y, z) as constant coefficients, and J 
as line-coordinates, is the equation of a point, viz. the point which is the locus (em elope) 
of all those points the coordinates of which satisfy the equation in ques ion, an ^ 
point is precisely the point, the coordinates (point-coordinates) of 
In fact if (I, n, D are considered as variable parameters connected by e q 
+ = then taking (X, Y, Z) as current point -coordinates, the equation 
|X+JjY+^Z = 0 is satisfied by writing (x, y, z) for (X, Y, Z); or the sever nes e 
coordinates whereof are (|, n, ^), all pass through the point {x, y, z). 
174. Hence recapitulating, the equation 
(I, 
^x-\~ny-\-^z, = 0 , 
considering {*, y, z) as current point-coordinates, and (5, », Q as constant coefficients is 
the equation of a line the coordinates (line-coordinates) of which are ( 5 , ,, 0 ; and the 
same equation, considering (S, ,, Q aa current line-coordinates, and (.r, y, z) as constant 
coefficients, is the equation of a point the coordinates (pomt-coordmates) ot which are 
^'"’m'^The expression, the point {a, h, c), means the point whose point-coordmates are 
(u, i, c ) ; and in like manner the expression, the line (u, (3, r), means the Une whose 
line-coordinates are (a, p, r)- The last-mentioned expression may, without any impro- 
priety or risk of ambiguity, be employed when we are dealing with 
but it is of course always allowable, and it is frequently better, to substitu e or ^ ^ 
nition the thing signified, and say the line having for its equation <«-f) 3 y-(-y^- 0 . or 
more briefly, the line c^+l3^+rz = 0. It wUl be proper to do this ni the ^ 
articles. Nos. 176 to 184, which contain some formulse in pomt-coordmates lelatiiig 
the theory of the point and the line. 
176. The condition that the point (a, h, c) may lie in the line 
ax-{-^y-\-'yz=^, 
