56 



ON THE EQUATION OF CURVES 



(9) assume the indeterminate form §. Multiplying equation 

 (7) by B we obtain 



B2 3^+BCb+ BE = o, 

 and in virtue of the preceding conditions this becomes 



Ay + Bx+ D = o (a); 



hence equations (7) and (8) are identical in this case, and any 

 point in the straight line (a) may be considered as the centre 

 of the locus. 



By multiplying equation (1) by A, we obtain, in this case, 



A^y^+2ABxy+'B''x^+2A'Dy-\-2B'Dx+A'E=o, 

 or (Ay+Bar}2 + 2D {Ay+Bx)+AY=o, 



and .-. Ay+Bx=—D±\/(D^'-A¥) (b); 



hence when D^ > AF the locus is two straight lines (b) parallel 

 to the line (a), when D^ = AF the locus is the straight line 

 (a), and when D^ < AF the locus is imaginary. 



The loci which can be represented by the general equation 

 (1) may now be enumerated as follows : — 

 Central Class. 



(a.) If C= A, and B = A cos y, the locus is a circle (iv., 8). 



(/3.) If B^ < AC and F'' < o, the locus is an ellipse (iv., a). 



(y.) If B* < AC and F"=o, the locus is a point (viii., t?). 



(S.) If B'^ < AC and F" > o, the locus is imaginary (iv., a). 



(f.) If B^ > AC and F" not = o, the locus is a hyperbola 

 (IV., )3). 



(C.) If B'^> AC and F''=o, the locus is two straight lines 

 cutting one another (viii., rj). 



(r,.) If B2 = AC, BD= AE, and D^ > AF, the locus is two 

 parallel straight lines (xiv). 



(6.) If B^ = AC, BD=AE, and D^ = AF, the locus is one 

 straight line (xiv). 



(,.) If B* = AC, BD = AE, D^ < AF, the locus is imagi- 

 nary (xiv). 



NONCENTRAL ClASS. 



If B'^ = AC and BD not = AE, the locus is a parabola 

 (iv., y). 



