155 
M,, Mo, My «+++. My, Such that 
Salah a 
OM, n°” \ON, 
OM;.0M, n(n—1) ON}-ONy 
1 _ m(m—1)..... 2.1 <3( l ) 
OM).OMy.....OM,  n(n—1)..(n—m-+1)° \ON}.0NQ...-ONp/’ 
the curve or surface which is the locus of the points M,, M2, 
M;...-++M, will be the m" polar of the point o with relation 
to the given curve or surface. It is plain that, so far as re- 
gards this method of geometrical generation, the polars are 
successive: that is to say, the (m + 1)" polar is derived from 
the m" in the same manner as the m from the (m— 1)". In 
every case the products of the distances OM,, OMy-..OMm, 
taken r by r, have the same harmonic mean as the products of 
ON), ON; ... ON, also taken r by 7: and this for all integer 
values of r from unity to m inclusive. 
It follows from what has been already said, that if 
Un $Unit-->-+UetU+U=0 
be the equation of the given curve or surface, referred to rec- 
tilinear axes ui through o, 
__m(m—}).. m(m—1)...2 
n(n— n(n—1).. aoe rat Te SEND .(n—m-+2) Un—1 + 
m(m — l), m if] 
. “ban 1) 1) Marky oP oe U,;=0 
will be the equation of the m" polar of the point 0, with rela- 
tion to the given curve or surface. By making’m, in this 
formula, successively equal to the integer numbers from 1 up 
to n—1 inclusive, we obtain the equations of all the succes- 
sive polars of the origin. For instance, the equation of the 
first polar is 
U; + 2U, = O 
