54 AXAI.YIH' QEOME'fliY (c... III. 



<jr=3* + 2, ft I 



4. { 



4. 



r.TJ ~7* . "' u-*=3. 



7. " = !/<r ' 



-*- 



1 yr. 12. 



-x = 0. 



13. Trace carefully the above loci; by measurement, find the coordi- 

 nates of the points in which each pair intersect; and compare these 

 results with those already obtained by computation. 



40. Product of two or more equations. Given two or more 

 equations with their second members zero ; * the product of their 

 members, equated to zero, has for its locus the combined 

 of the given equations. 



This follows at once from the fundamental relation be- 

 tween an equation and its locus (see Art. 35 (1)), for the 

 new equation is satisfied by the coordinates of those points 

 which make one of its factors zero, but it is satisfied by 

 the coordinates of no other points ; i.e., this new equation 

 is satisfied by the coordinates of points that lie on one or 

 another of the loci of the given equations. 



The following example illustrates this principle in the 

 case of two given equations. 



Let the given equations be : 



. . . (l)and*-y = ... (2) 



If equations whose second members are not zero are multiplied togetlir r. 

 member by member, the resulting equation is not satisfied by any points 

 of the loci of the given equations except those in which they intersect each 

 other ; the new equation therefore represents a locus through the points of 

 intersection of the loci of the given equations. 



