OF THE SECOND DEGREE. 



57 



r^ XV. 



Hitherto our attention has been chiefly directed to the most 

 general form (1) of the equation of the second degree, but in 

 many cases the equation becomes simplified in form by the 

 evanescence of one or more of its co-efficients A, B, C, &c. 

 Some of the simplest of these forms have been given in Nos. 

 VII., IX., XI,, and we now proceed to point out a few others. 



(a.) When the curve (1) passes through the origin of co- 

 ordinates its equation must be satisfied by the simultaneous 

 equations x=o and t/=o; hence we shall have F=o, and 

 equation (1) takes the form 



Af + 2Bxy+ Cx^ + 2D2/ + 2Ex-o (a). 



(^.) When E=o the diameter (7) which bisects chords 

 parallel to the axis of x passes through the origin ; hence 

 when the origin is on the diameter which bisects chords 

 parallel to the axis of a; equation (1) takes the form 



A^^ + 2Bxy+Cx^ 4- 2I>y + Y=o (b). 



Similarly, when the origin is on the diameter which bisects 

 chords parallel to the axis of y, equation (1) becomes 



Af + 2Bxij+ Ca?2 + 2 Ecc + r=o (c). 



When the origin is at the centre we have D=o, E=o, and 

 the equation becomes 



Af -{- 2 Bxy + Cx^ + F=o (d). 



(y.) When B=o equations (7) and (8) become 

 Cx-\-'E=o and Ay-{-D=o, 

 and therefore the diameters which bisect chords parallel to 

 the axes of x and y are respectively parallel to the axes of y 

 and X. Hence, when the axes are parallel to a system of 

 conjugate diameters, equation (1) takes the form 



Ay^-{-Ca^ + 2I>y-t2 Ea?-i- ¥=o (e). 



When the curve (1) is a parabola the condition B=o gives 

 A=o or C=o. 



(8.) When C=o equation (13) gives m=^o, and therefore 

 the curve (1) has an asymptote parallel to the axis of x. 



I 



