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algebraic forms. Indeed, it is this general correspondence of geometrical and algebraic simplicity which gives to the whole subject its power. It springs from the fact that the connection between geometry and algebra is not casual and artificial, but deep-seated and essential. The equation which corresponds to a locus is called the equation "of" (or "to") the locus. Some examples of equations of straight lines will illustrate the subject. 14

Consider $y-x=0$; here the $a$, $b$, and $c$, of the general form have been replaced by $-1$, $1$, and $0$ respectively. This line passes through the "origin," $O$, in the diagram and bisects the angle $XOY$. It is the line $L^{\prime }OL$ of the diagram. The fact that it passes through the origin, $O$, is easily seen by observing that the equation is satisfied by putting $x=0$ and $y=0$ simultaneously, but $0$ and $0$ are the coordinates of $O$. In fact it is easy to generalize and to see by the same method that the equation of any line through the origin is of the form $ax+by=0$. The locus of the equation $y+x=0$ also passes through the origin and bisects the angle $X^{\prime }OY$: it is the line $L_{1}O{L}_1^{\prime }$ of the diagram.

Consider y − x = 1 : the corresponding locus does not pass through the origin. We therefore seek where it cuts the axes. It must cut the axis of x at some point of coordinates x and 0 . But putting y = 0 in the equation, we get x = − 1 ; so the coordinates of this point ( A ) are 1 and 0 . Similarly the point ( B ) where the line cuts the axis O Y are 0 and 1 . The locus is the line A B in the figure and is parallel to L O L ′ . Similarly y + x = 1 is the equation of line A 1 B of the figure; and the locus is parallel to L 1 O L 1 ′ . It is easy to prove the general theorem that two lines represented by equations of the forms a x + b y = 0 and a x + b y