156 



which involves in it the celebrated theorem of Cotes, so 

 successfully used by Maclaurin. And the equation of the 

 («— l)"* polar is 



u„_, + 2u„_2 + 3u„_3 + ... + {n- l)Ui + nvo - o, 

 an equation which we recognize as belonging to the curve or 

 surface of the (n— I)*'' 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. 



By substituting -, -, -, 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.t 



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 to a 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 eighteenth and nineteenth 

 volumes of Gergonne's Annales de Matkcmatique. 

 t See Chasles' Histoire de Geometrie, p. 147. 



