4^ ANALYTIC GEOMETRY III. 



only those points, are found whose coordinates satisfy the 

 given equation. 



A second conception of the locus of an equation comes 

 directly from this one, for the line or set of lines may U 

 regarded u the /"/'// traced i>\ a point \\hich movrs along 

 it. The path of the moving point is determined by the 

 condition that its coordinates for every position through 

 which it passes must satisfy the given equation. Tims the 

 line EF (the locus of eq. (3), Art. 33) may be regarded 

 as the path traced by the point P, which moves so that 

 its coordinates (*, y) always satisfy the equation 



3:r- 2y + 12 = 0. 



Thus arises a second conception of a locus, viz.: 



(2) The locus of an equation is the path traced by a p'>nt 

 u-hi<'h moves so that its coordinates always satisfy the given 

 equation. 



In either conception of a locus, the essential condition 

 that a point shall lie on the locus of a given equation is 

 that the coordinates of the point when substituted respect i 

 for the variables of the equation, shall satisfy the equation ; 

 and in order that a curve may be the locus of an equa- 

 tion, it is necessary that there be no other points than those 

 of this curve whose coordinates satisfy the equation. 



36. Classification of loci. The form of a locus depends 

 upon the nature of its equation ; the curve may therefore 

 be classified according to its equation, an algebraic curve 

 being one whose equation is algebraic, and a transcendental 

 curve one whose equation is transcendental. In particular, 

 the degree of an algebraic curve is defined to be the same 

 as the degree of its equation. The following pages are 



