154 



If we draw from o two right lines parallel to the asymptots 

 of the generating conic, in virtue of the equation (3), they 

 will each meet the curve of the third degree in two points 

 equidistant from o. It follows, therefore, that the two recti- 

 linear diameters of the curve, conjugate to these right lines, 

 must pass through o. Hence we know how to construct 

 these diameters. Again, as there are plainly but two diame- 

 ters passing through o, the curve enveloped by all the diame- 

 ters of the curve must be of the second class, that is to say, 

 it must be a conic section. 



If o be a conjugate point, the conic and its asymptots are 

 imaginary ; consequently o must lie within the envelope. 



If o be a cusp, the conic degenerates into two coincident 

 right lines. The diameters of the curve conjugate to them 

 must, therefore, be coincident Hkewise. Hence o lies on the 

 envelope. 



If o be a double point, the conic is replaced by two inter- 

 secting right lines ; and it is easy to see that, as we can draw 

 from o two diameters of the curve, conjugate to the two systems 

 of respectively parallel chords, o must lie outside the envelope. 



The preceding theorems, relative to the envelope of the 

 rectilinear diameters of a curve of the third order, and to the 

 positions of singular points of the curve with respect to it, are 

 due to Professor Pliicker of Bonn, who obtained them by ana- 

 lytical methods. They are here presented as consequences, 

 derived by purely geometrical considerations from the proper- 

 ties of the generating conic ; in order to illustrate the impor- 

 tance of the place which it holds in the theory of curves of the 

 third order. 



Perhaps the most interesting class of derived curves or 

 surfaces is that of the Successive Polars of a given curve or 

 surface of the w"* order. The geometric mode of generating 

 them is as follows : 



Draw from any point o a right line meeting the surface in 

 n points, Nj, Ng, N3 n„, and assume on it 7n points, 



