

7 //A / <>CU8 OF AS 



n 



paiOOO through every Othn n \vhi h the 1<. 



equations (1) ai> itersect each other. 



precisely the same way it may be proved generally 

 the low* of the turn of two equation! pattet throuyh all 



ft in trhi,-h the loci of the two given equation* intersect 

 each other. 



If i-ithrr of the given equations (1) or (2) had been inulti- 

 I by any constant factor before iul-liu-. the above reason- 

 ing \\niihl still have led to the same conelu>i>n ; in fact, 

 rem may be generally, staU<l t i. 



v are any function* of the two variable* x and y, and 

 k if any const ant, then the locu* of 



patift through every point, of intersection of the loci of 

 u = and v 0. 



he locus of the equation u = be the curve 



he locus of v = be the curve DBF. and let 



P } be any on 



the points in \vhi.-h these 



fs intersect each <>t!. 

 Then the equat 



is satisfied by the n.onli- 



nates of the point />, = 



V,), because if these 



coordinates be substituted for x and y in the functions M and 



make both these functions separately equal to 



zero. Therefore the locus of u + kv passes through 



every point in \\hich the loci of u and r = intersect 



each other. 



