194 MR. CHARLES CHAMBERS ON THE 



the equation, so here the circular ordinate must be described 

 (on the plane perpendicular to the tubular ordinate) with 

 its centre at the extremity of the corresponding coordinate, 

 and its radius the function of the latter coordinate given 

 by the equation. Also as the curve, in the Cartesian 

 system, separates the region of greater ordinates than its 

 own at any point from the region of lesser ordinates, so 

 here the curve must be such as will do the same ; in other 

 words, it must be the envelope of the series of circular 

 ordinates. The reason will now appear why we substituted 

 the product of a pair of coincident tubular ordinates for 

 the square of a single ordinate : one of the pair we suppose 

 to be produced by the elongation of a fundamental tube of 

 infinitesimal height and radius, on the positive side of the 

 perpendicular coordinate plane, and the other by elongation 

 of a like tube on the negative side of that plane. The 

 equal dilatation of these tubes will produce, not coincident, 

 but adjacent circular ordinates, and these of opposite sign, 

 of which the product will therefore be a negative square. 

 The curve will be described by the line of intersection of 

 contiguous circular ordinates, that line being of infinitesimal 

 length 2R, half on the positive side and half on the negative 

 side of the plane of the circles. 



Once again we introduce an arbitrary convention that is 

 only additional to, not in contravention of, those of the 

 Cartesian geometry. In whichever manner the pair of 

 fundamental tubes enlarge, whether by elongation or di- 

 latation, we regard the product of the pair of ordinates pro- 

 duced as the counterpart of the square of the correspond- 

 ing variable in the algebraical equation. Thus, if 7 be the 

 magnitude, regardless of sign, of the geometrical ordinate, 

 then, when y* = «*, we may for the algebraical y^ substitute 

 ( + 7) ( + 7) or ( — 7) ( — 7), and we obtain the result 7= +«; 

 and when y'^=—a^, we obtain ( + 7) (—7)= —a% whence 



