200 ON THE EQUATION OF THE SECOND DEGREE. 



forms of expression. He became strongly impressed with 

 the notion that fruitful speculation would lie in the direction 

 indicated, of regarding a plane and a straight line as limits 

 of some kind, and that kind such as would, if possible, 

 cause them, at infinity of distance from the parts considered, 

 to deviate symmetrically from the positions due to the 

 abstract conceptions of a plane and a straight line. The 

 rotation of the system of circles shown in figure 2 about 

 the straight line AB as an axis will yield examples o£ a 

 straight line and a plane of the symmetrical kind required, 

 provided that the opposite pairs of equal circles are at the 

 infinitesimal distance R from the central point of the figure, 

 and that their radii are infinities of a second order in com- 

 parison with R ; for, with these conditions, finite quantities, 

 which are infinite in comparison with R, will be infinitesimal 

 in comparison with the radii of the circles. Let a sphere 

 of finite radius be described with its centre at the central 

 point of the figure, then the part of the tube of which AB 

 is the axis that is cut ofi' by this sphere has the character- 

 istics of our tubular ordinates, and the part of the same 

 sphere cut off" by the space between the upper and lower 

 circles forms circles of the nature of our circular ordinates. 

 When the sphere becomes infinite and of radius greater 

 than the diameter of the circles of the figure, the infinite 

 ordinates (tubular and circular, positive and negative) will 

 all go out into space altogether without the system, in ob- 

 vious analogy with the sudden passage, in certain otherwise 

 continuous variations, of -i-c» and — c» into each other. 



