180 



MB. EOBEBT FINLAI ON 



Section II. — Of Corves which are sdpplemehtary in EELATroN 

 TO A GIVEN Straight Line. 



VI. 



A curve of the second degree being given, to find the locus 

 of the imaginary points in which it is cut by a system of 

 straight lines parallel to a given line. 



If the given line be taken as the axis of y, the equation of 

 the given curve will be of the form 



Ky''+ 2Bxi/ + Ca;'^+ 2D3^+ 2Ej? + F = o. 

 By solving this with respect to y, we get 

 Ay = —{Bx + l))± y^{{Bx+T>T-A{Cx'+2Ex + ¥)}...{l), 



and by changing the algebraic sign of the quantity under the 

 radical sign in this expression, we obtain 



A^ = — (Bx+T» + \/{A. {Cx-'+ 2 Ea; + F) — {Bx + Df] . . .{2}, 



which is the equation of the required locus. 



{a.) Since every value of x which gives real values of y in 

 equation {2) will give imaginary values of y in equation (1), 

 it is evident that the original curve (1) is the locus of the 

 imaginary points in which the curve {2) is cut by a series of 

 straight lines parallel to the axis of y. Hence we see that 

 any straight line, parallel to the axis of y, will meet one of the 

 curves in two real points, tohich may also be considered as 

 imaginary points on the other curve. On account of this 

 remarkable relation, the curves (1) and (2) have been called 

 supplementary conies in relation to the axis of y. But since 

 the property does not hold good with respect to straight 

 lines drawn in any other direction, the curves (1) and (2) 

 are not supplementary, except in relation to the system of 

 parallel straight lines in question. 



(6.) Similarly, any two curves A and B may be said to be 

 supplementary in relation to a given straight line, when any 

 straight line parallel to the given one meets the curve A in 



