84 



Mr. H. S. Warner on Conjugate Points. 



1. Prof. Young says (p. 92), that in the extreme case of 

 the ellipse (the minor axis diminishing till it vanishes), the 

 curve degenerates into a finite straight line, the equation in 



this ultimate state being y — — VcP—x 2 . He thus conti- 

 nues : " Between the limits x— —a and #= +a, the evanescent 

 quantity on the right is real, implying that every point on the 

 axis of x, which does not lie without these limits, is a real 

 point of the locus. But if x exceed a, in either the positive 

 or negative direction, the evanescent quantity is imaginary, 

 and consequently in accordance with the ordinary interpreta- 

 tion, the corresponding points on the axis of x are beyond 

 the bounds of the locus." Now for any point in the axis of 

 x, at which x is greater than a, positively or negatively, the 

 equation is of the form y = V — 1 ; and as he says that this 

 point is beyond the bounds of the locus, he must consider 

 0*/ — 1 as unassignable: in this consists his error, for this ex- 

 pression is equal to neither more nor less than 0; thus 



OV — 1=V(— 1 x0 2 )=\/ — 0=\/0 (since +0=— 0) = 0: 



and hence for any point in the axis of x (whether at that 

 point x is greater or less than a)y = 0, so that the locus is an 

 infinite straight line coinciding with the axis of x, and not, as 

 he asserts, a finite straight line of the length of the major 

 axis, or 2 a. 



2. Perhaps however the geometrical consideration of the 

 case will render this more evident. If A B C be a section of 

 a cone by a plane passing through the axis A H, and if an- 

 other plane at right angles to this one intersect it in the line 

 D E, then the line of intersection of the conic superficies and 

 this latter plane will be an ellipse, 

 of which the major axis a will be 

 = DE; and if D F, EJ be lines 

 drawn across the triangle ABC, 

 perpendicular to A H, it is well 

 known that the minor axis b of the 

 ellipse will be = ^DF.EJ. If 

 now the points D and E be sup- 

 posed to move along the lines B A, 

 A C (D approaching A, and the di- 

 stance D E remaining unchanged, 

 which is equivalent to supposing 

 that in the equation to the ellipse 

 b diminishes while a remains con- 

 stant) till b = 0, this will be the case only 'when D coincides 

 with A ; and then the conic section will coincide with the side 



