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enumeration of the reciprocal properties of curves and curved surfaces. 

 Hence given a series of points, lines, and planes, we may construct a 

 series of as many planes, lines, and points, according to a fixed and 

 simple law. 



Now we know that in the application of algebra to geometry by 

 the method of coordinates, a point is determined in position by its 

 projections on three coordinate planes, or by three equations, that is 

 by three conditions. A right line may in like manner be determined 

 when we are given the positions of two points in it ; and a plane is 

 determined by one condition, which is called its equation. But 

 in the inverse method, a point should be determined by one con- 

 dition, a right line by two, and a plane by three. Again, a right 

 line may be determined by considering it as joining two fixed points, 

 or as the common intersection of two fixed planes. Now all these 

 conditions may be expressed by taking as a new system of coordi- 

 nates the segments of the common axes of coordinates between the 

 origin and the points in which they are met by a moveable plane. 

 Thus if these segments be designated by the symbols X, Y, Z, the 

 three equations which determine a plane are 



X= constant, Y= constant, Z = constant. " 



Again, the equation in (#, y, z) of a plane passing through a point 

 of which the coordinates are xyz, and which cuts off from the axes 



of coordinates the segments X, Y, Z, is -j- $- 4. = 1 . Now this 



X Y Z 



is the protective, or common equation of the plane, if we make x, y, 

 and z vary, and consider X, Y, Z as constant. But we may invert 

 these conditions, and consider x, y, z constant, while X, Y, and Z 

 vary. And the equation now, instead of being the common equation 

 of a fixed plane, becomes the inverse or tangential equation of a fixed 

 point. In this latter case let a, /3, and y be put for x, y, and z, and 



, , for X, Y, Z ; then the equation may be written 



which may be called the tangential equation of a point. 



Moreover, as the continuous motion of a point, in a plane suppose, 

 subjected to move in accordance with certain fixed conditions ex- 

 pressed by a certain relation between x and y may be conceived to 

 describe a curve, so the successive positions of a straight line cutting 



o 2 



