EQUATION OF THE SECOND DEGREE. 193 



of the two coincident ordinates as tubes instead of mathe- 

 matical straight lines^ provided the tubes be of infinitesi- 

 mal transverse section^ which may be of circular form and, 

 say, of radius E,, and provided further that the nume- 

 rical values of the tubes are the quantities of curved surface 

 which they possess, the convention as to signs being un- 

 altered. The unit of curved surface must obviously be 27rR. 

 Instead, however, of the curve being described by the 

 motion of the central point of the extremity of the or- 

 dinate, let it be described by a diameter, through that 

 point, perpendicular to the plane of the coordinates ; it will 

 still be of infinitesimal depth. Further, let us take as the 

 fundamental ideal of the tubular ordinate the one whose 

 height is R. It may be varied in value by elongation in a 

 positive or negative direction, its diameter remaining 

 constant ; or it may be varied by dilatation into larger and 

 larger circles, its height remaining constant : in the latter 

 case the ordinate, which may be called circular, to distin- 

 guish it from the tubular ordinate, Avill be of positive sign 

 if it be on the positive side of the coordinate plane, which 

 is perpendicular to the tubular ordinate, and of negative 

 sign if on the opposite side of that plane. Elongation to 

 unit length of the tube, and dilatation to unit radius of the 

 circle, alike make the quantity of curved surface 27rR, which 

 is the unit of this magnitude. Adopting this latter unit 

 as the analogue of the algebraical unit, we shall thus have 

 two distinct but connected systems of variation of the mag- 

 nitude — that of elongation yielding tubular ordinates ana- 

 logous to and coextensive with the ordinates of the Cartesian 

 geometry, and that of dilatation yielding the circular or- 

 dinates, the nature of which we now proceed to consider. 

 Just as in our modified Cartesian system the ordinate 

 is erected with the axis of the tube at the extremity of the 

 corresponding coordinate, and with a length determined by 



SER. III. VOL. VI. o 



