SUPPLEMENTAEY CURVES. 



183 



To obtain the most general solution of this question, it 

 would seem that the locus must include not only all the points 

 between the straight lines (6) which satisfy the proposed con- 

 dition, but also all points external to these lines on either side 

 of them which satisfy the condition ; and in a geometrical 

 point of view the exterior and interior points in question seem 

 to have perfectly equal claims to be considered as points of 

 the locus. Now it is evident from what has been advanced 

 above (</), that, when the given ratio is considered as a positive 

 quantity, the equation to the required locus is (4), which 

 represents the internal points; but when the ratio is con- 

 sidered as negative, the equation to the required locus is (5), 

 which represents the external points. Thus we see that 

 neither of the equations will represent the whole locus of 

 external and internal points, unless we include the imaginary 

 values of the variables ; but that, with this assumption, either 

 of the equations will represent the entire locus. Thus we 

 have a remarkable instance, in which the science of algebra 

 seems to be at fault, in point of generality, as compared with 

 the kindred science of geometry ; and it is not possible to 

 remedy the defect except by admitting imaginary values of 

 the algebraic symbols. This seems to afford a strong argu- 

 ment for considering the two supplementary curves (4) and 

 (5) as branches of the same general curve ; but I must admit 

 that the authority even of Chasles himself is opposed to this 

 view of the subject.* 



[i.) If the direction of the straight lines (6) be supposed to 

 change, while the curve (4) remains invariable, the magnitude 

 and position of the curve (5) will be changed ; and conversely, 

 if the straight lines (6 change their direction, while the curve 

 (5) remains unchanged, the magnitude and position of the 

 curve (4) will vary. In fact, we have seen {g) that the two 

 supplementary conies (4) and (5) can be viewed as intimately 

 related to the three straight lines (3j and (6), and of course 

 * See his History qf Qeomelry, Note XXVI., page 396 of the German edition. 



