OF THE SECOND DEGBEE. 



41 



and consequently the denominator of the fraction under the 

 radical sign in equation (12) has the same sign as — A; but 

 for all other values of m the factors have like signs, and there- 

 fore the denominator has the same sign as + A. Hence we 

 see, that when F' has the same sign as A, the central radius 

 vector CP is real for all values of m between m' and m" and 

 imaginary for all others, or that the curve (1) is included 

 within the angles S C T and S' C T ; and that when — F' has 

 the same sign as A, CP is imaginary for all values of m, 

 between m! and iri' and real for all others, or the curve is 

 included in the angles S C T and S' C T. In both cases the 

 curve is called a hyperbola, and therefore the equation (1) 

 always represents a hyperbola when B'^>AC, provided that 

 F" be finite. 



(y.) When the curve (1) belongs to the non-central class it 

 is called a parabola; hence (ii) the conditions in order that 

 the equation (1) may represent a parabola are B^= AC and BD 

 not equal to AE. 



(8.) When C = A and B = A cos y equation (li^) becomes 

 CP= ^—F'TX, 

 and consequently when F" and A have unlike signs, the curve 

 (1) is a circle, but when F" and A have like signs the locus 

 is imaginary. Hence equation (1) will represent a circle 

 when C=A and B = A cos y, provided that F" be negative. 



V. 



We have seen (No. 1) that the equation 



Aw^-1-2 Bm+C=o (a) 



expresses the condition that the straight line (2) may meet 

 the curve (1) in only one point. Now (a) when B^ > AC the 

 curve is a hyperbola, and the roots of this equation are the 

 direction-indices of its asymptotes (iv) ; hence, if from any 

 point two straight lines be drawn parallel to the asymptotes of 

 a hyperbola, each of these lines will cut the curve in only one 

 point. 



