OF THE PROPERTIES OF CURVES. 49 



ratio of the lesser axis to tiie greater. For if four quantities are pro- 

 portional, their squares are proportional, and tlie reverse. But the 

 fluxions of the areas are equal to the rectangles contained by these 

 ordinates and the same fluxion of the absciss (190), they are, there- 

 fore, in the constant ratio of the ordinates, and the corresponding 

 areas are also in the same ratio (47). 



204. Definition. If the square of the absciss is equal 

 to the rectangle contained by the ordinate and a given 

 quantity, the curve is a parabola, and the given quantity 

 its parameter. 



Scholium. Thus ABqzrP.BC. 

 If the axes of an ellipsis are sup- 

 posed infinite, it becomes a parabola 



for since — n: — - — , if a becomes 

 a* ax — XX 



infinite,ara: vanishes in comparison with ax, and ~ iz~,— x zry^^ and 



a^ ax a 



12 



is the parameter of the parabola ; and the distance from the focus 



a 



is in a constant ratio to the square of the perpendicular falling on 



the tangent. 



205. Definition. When the ordinate is as any other 

 power of the absciss than the second, the curve is still a 

 parabola of a different order. 



Thus when the ordinate is as the third power of the absciss, the 

 curve is a cubic parabola. 



206. Theorem. If any figure be supposed to roll on 

 another, and any point in its plane to describe a curve, that 

 curve will always be perpendicular to the right line joining 

 the describing point and the point of contact. 



Suppose the figures rectilinear polygons ; then the point of contact 

 will always be the centre of motion, and the figure described will 

 consist of portions of circles meeting each other in finite angles, so 

 that each portion will be always perpendicular to the radius, though 

 no two radii meet in the point of contact. And if the number of 



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