36 



ON THE GENERAL EQUATION OF CURVES 



B^^ a;=B (y— yi) (x—xi) + Bj/i {x—xi) + Bxi (y—yi) + Bari yi, 



Dy =D (y— yi) + D^i, Ex=E (a;— J7i) + Ea?i, 

 equation (1) may be written in the form 



A (r/—yiy + 2B {y~yi) {x—xi) + C {x—xiY 

 + 2 {Kyi + Bo^i + D) {y—yi) + 2 (B^i + Can + E) {x—x^) 

 + Ayi^ + 2Bxiyi + Cxi^ +2'Dyi + 2 Exi + F=o, 

 and in virtue of equation (2) this becomes 

 ( Aw2 + 2Bm + C) {x—xif + 2 (D'm + E') (x—xi) + F = o . . . (b) ; 

 where, for the sake of brevity, we assume 



Ayi + Bxi + lL>='D',Byi + Cxi + E=E' (c), 



Ayi^ + 2 Ba?i yi + Caii^ + 2Dyi + 2Exi + Y=¥' (d). 



By eliminating x — x from equations (a) and (b), we obtain 

 AmH2Bm + C 2(D'm+E') p .p„_^ ,. 



w2 + 2mcosy+l "^ "^/(wH^mcosy+l) "^ -O.-.^e), 



the roots of which are the segments APi and APg (Fig. 2) of 

 the straight line (2) intercepted between the point a?iyi and the 

 two points in which it cuts the curve (1). 



(a.) By a well-known property of quadratics, we obtain 

 from equation (e) 



AP AP r-(^^+2mcosy+l) ,«. 



Al-i. Ai^,- Am2+2Bm+C ^^^' 



which evidently holds good, whether the points Pi and Pa be 

 real or imaginary. 



(/3.) When the direction-index m of the straight line (2) 

 satisfies the condition 



Am^-\-2 Bw-f-C=o (4), 



equation (e) gives R= 2{D'mi-\<y) ^ ^ ' 



consequently, in this case, the straight line {2) can meet the 

 curve (1) in only one point. 



II. 



When D'ot+E'=o, the middle term of equation (e) will 

 vanish, and the roots of that equation will be equal with 

 opposite signs. Hence, in this case, A must be the middle 



