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 0, 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 likewise. 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, 0 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 n order. The geometric mode of generating 
them is as follows: 
Draw from any point 0 a right line meeting the surface in 
nm points, Nj, Ny, N3..-++.Nn, and assume on it m points, 
