SUPPLEMENTARY CURVES. 



193 



whereM^={D» — E' — F(A — C)}' + 4(DE— BF)\..(9); 

 and by substituting these expressions in equation (3) we get 

 (D» + E') . PQ* = M . PR. . PR, (10). 



Similarly, if perpendiculars be drawn from any point P in the 

 curve (4) to the straight lines (5) and (6), and if the same 

 notation be adopted, we shall have 



(D* + E') . PQ' = — M . PR, . PR, (U). 



Thus we see that each of the supplementary curves (3) and (4) 

 may he considered as the locus of a point, such that if perpen- 

 diculars be drawn from it to the three straight lines (5) and 

 (6), the product of the perpendiculars drawn to the straight 

 lines (5) shall have a given ratio to the square of the third 

 perpendicular ,' the algebraic sign of this ratio being different 

 for the two curves, but its absolute magnitude being the same 

 for both. From this theorem it is evident (VI, h) that the 

 curves (3) and (4) maj' be considered as branches of the same 

 geometrical curve, although the whole curve cannot be repre- 

 sented by either of these equations unless we take in the 

 imaginary values of the variables. 



(/) ^6 have seen (IV) that any two conic sections des- 

 cribed in a plane have at least one real point of intersection 

 of conjugate common secants. Now when the four points of 

 intersection of the given curves are all imaginary, if the curves 

 be constructed which are supplementary to the given ones in 

 relation to a real point of intersection of conjugate common 

 secants, since each of these curves passes through all the 

 imaginary points of its supplementary curve, it follows that 

 the supplementary curves will cut each other in four real 

 points, which are the imaginary points of intersection of the 

 two given curves. Hence when two conic sections are entirely 

 exterior to each other, or when one of them is entirely within 

 the other, their imaginary points of intersection may be 

 obtained by constructing the curves which are supplementary 



2c 



