156 
which involves in it the celebrated theorem of Cotes, so 
successfully used by Maclaurin. And the equation of the 
(n— 1)" polar is 
Un) + 2Un_9 + 3Un3 +... + (n— 1)U, + NUy = 0, 
an equation which we recognize as belonging to the curve or 
surface of the (n— 1)" order, which passes through the points 
of contact of all the tangents drawn from the origin to the 
given curve or surface. It is by this geometrical property 
that M. Bobillier, who first directed attention to these suc- 
cessive polars,* has chosen to characterise them. 
, ee ie : 
By substituting "7 4, 7? for x, y, z, in the general equa- 
tion of the surface, and in the equations of its polars, Profes- 
sor Graves shews that, when the point o recedes to an infinite 
distance, the whole series of successive polars become diame- 
tral lines or surfaces of the different orders belonging to the 
given curve or surface. This had, in fact, been observed, in 
the case of the first polar, by M. Poncelet, who has shewn 
that the theorem of Cotes is an extension of Newton’s propo- 
sition relative to the rectilinear diameters of plane curves. 
This theory of polars enables us to give a geometrical 
construction of the problem, ‘* From a given point in its 
plane to draw all the possible tangents toa curve of the third 
order.” 
We have only to construct the second polar of the given 
point, which will be a conic section, and its intersection with 
the curve will give the points of contact. Here we see the 
advantage of adopting the geometrical definition of polars 
employed in this paper. 
In connexion with the present subject, Professor Graves 
announced an extension of a theorem deduced by Maclaurin 
* See a Series of Articles contained in the cighteenth and nineteenth 
volumes of Gergonne’s Annales de Mathématique. 
{ See Chasles’ Histoire de Geometric, p. 147. 
