38 



ON THE GENERAL EQUATION OF CURVES 



centre of a curve represented bj any given equation of the 

 second degree. 



(d.) Since every chord of the curve (1) which passes through 

 its centre is an ordinate to some particular diameter, it is 

 evident that every chord which passes through the centre k 

 bisected at the centre. 



(«.) If B^=AC, while BD is not equal to AE, it is evident 

 from equations (9) that the centre of the curve (1) passes to 

 infinity. In this case it is usual to say that the curve has no 

 centre. Hence arises a division of curves of the second degree 

 into two classes. 1st. The central class, characterized by the 

 condition B^ not equal to AC. 2nd. The non-central class, 

 characterized by the condition B2=AC. 



{(.) By multiplying equation (6) by B, we obtain 

 (Am + B) By + {Wm-[-BC)x + B (D»iH-E)=o. 

 Now when the curve (1) belongs to the non-central class this 

 equation may be written in the form 



(Am + B) (B3/ + C^) + B (Dm+E)=o (10), 



from which we see that all the diameters of a non-central curve 

 of the second degree are parallel to the diameter 



By-{.Cx=o (lO-) 



which passes through the origin of co-ordinates. 



III. 



In a curve of the second degree, any diameter which is per- 

 pendicular to its ordinates is called an axis of the curve. 

 Hence if m' and m denote the direction indices of an axis and 

 its ordinate respectively, we shall have 



l + {m + m') cosy + mm' = o (a), 



y being the angle made by the positive axes of x and y. Now 

 (a) in the case of a non-central curve we have m'= — C : B 

 (by eq. 10), and therefore equation (a) becomes 

 B + (B?w — C) cos y — m C=o, 

 C cosy — B_Bcosy — A 



which gives m-. 



B cos y — C A cos y — B 



