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MR. BOBERT FINLAY ON 



imaginary, so that the origin is a double point of the curve 

 {2) and a conjugate point of the curve (1). Thus we see that 

 a double point on one of the curves is a conjugate point on 

 the supplementary curve; and that the real and imaginary 

 tangents, applied at that point to the two curves, form two 

 supplementary systems of straight lines. 



(c) When m = p, each of the equations (3) and (4) becomes 

 2^* = o ; hence, in this case, the axis of a: is a double tangent 

 to each of the supplementary curves (1) and (2) at the origin. 

 Hence we see that when a curve has a cusp, this point will 

 also he a cusp on the curve which is supplementary to the 

 given one in relation to the double tangent at its cusp. 



Section III. — Of Curves which are Supplementary in relation 

 TO A given Point. 



XI. 



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 which pass through a given point A. 



When the given point A is without the curve, so that two 

 real tangents can be drawn from it to the curve, it is evident 

 that innumerable straight lines can be drawn through the 

 point so that each of them shall cut the curve in two imagi- 

 nary points; but when the given point is on the curve, or 

 within it, it is plain that no straight line can be drawn through 

 A to meet the curve in any imaginary point. Hence it will 

 be sufficient to consider the case in which the given point A 

 is without the given curve, since the locus cannot exist in 

 any other case. 



If the given point A be taken as the origin of rectangular 

 co-ordinates, the equation to the given curve will be of the 

 form 



Ay^ + 2Bry + Cx' + 2I)y + 2Ex + F = o. 



