^O ON THE EQUATION OF CURVES 



iC) When two conjugate diameters are at right angles to 

 each other, it is evident, from the definition in No. iTi., that 

 each of them is an axis of the curve. Hence if a and b denote 

 the semi-axes of the curve (1), we shall have («), 



a'=F'%, h''=F'% (26) 



where «i and z^ are the roots of the equation 



(B^— AC) z^— (A+ C— 2B cos y) z—siu^y=o (27). 



This simple rule determines the magnitude and form of the 

 central curve represented by any equation of the second 

 degree ; and the position of the curve in reference to the 

 original axes of co-ordinates may be found by means of 

 equation (bj, No. iii. 



{f}.) When ¥''=0, it is evident from equations (26) that the 

 semi-axes of the curve (1) are both zero; from which we see, 

 that, if the curve be of the elliptic species, it must vanish in 

 this case into a point ; but, if it be of the hyperbolic species, 

 it must coincide with its asymptotes. Hence, when B^< AC 

 and F"=o, the locus of equation (1) is a point; but when 

 B^>AC and F''=o, the locus breaks up into two straight 

 lines. This remark completes the discussion of central curves 

 given in No. iv. 



IX. 



Let ACA' and BCB' (Fig. 7 and 8) be any system of con- 

 jugate diameters of the curve (1), PP' any line parallel to the 

 latter and meeting the former in Q ; then, by the theorem in 

 No. IV., we shall have 



PaQF : AaQA' : : BC.CB': AC.CA', or 



PQi* : AQ.QA' :: BC^ : AC^ (28) 



(a.) Let CA=a', CB=b', CQ=a?, QP=y/ then, if the 

 curve be an ellipse, we obtain from (28) 



f : (a'+.T) {a'—x) ::b'^: a'^ 

 but if it be a hyperbola we have 



y:(a:+a') (a?— «')::&'': »'S* 



