EQUATION OF THE SECOND DEGREE. 



199 



Years ago the writer became acquainted with the method 

 in analytical geometry which regards a plane and a straight 

 line, not as the pure abstractions of the mathematician, but 

 as small portions of the surface of a sphere and of the pe- 

 riphery of a circle respectively in the limits in which the 

 radii of the sphere and circle become infinite ; and, whilst 

 noting that these definitions were open to serious objections 

 (chiefly as giving sides to a plane and a straight line of an 

 unsymmetrical character), he was much struck by the extent 



Fig. 2. 



to which they enlarged the field in which geometrical in- 

 terpretation might be consistently applied to algebraical 



pairs of impossible values are given for the coordinates of the points of in- 

 tersection, then as many pairs of real values vrill be given by elimination 

 betvFeen the equations of the supplementary pair of conies which lie in the 

 plane that contains the two centres ; but he has yet to apply the analytical 

 test of the correctness of this impression. Tlie nature of this test will be to 

 find whether for each case in which the primitive pair of equations give the 

 squares of the ordinates negative, the s\ipplempnt:iry pair give the squares of 

 the ordinates positive, and flee ver.td. 



