155 

 M|, Mj, M3 iM„„ such that 



KOMyJ n VON]/ 



/ 1 \ m(m—\) ^ ( 1 



VoMi.OMoy ?«(?/—!) \ON,.ON2 



1 



OM1.OM2 OM,„ w(«— !)..(« — ?W + 1) \0N,.0N2....0N„ty' 



the curve or surface which is the locus of the points m,, 1VI2, 



M3 Mot will be the w"' 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 ?«"' in the same manner as the m"' from the {in — 1 )"'. In 

 every case the products of the distances OMi, 0M2 • . . OM„j, 

 taken r by /•, have the same harmonic mean as the products of 

 oNi, 0N2 . . . ON„, also taken r by ?• : and this for all integer 

 values of /' from unity to m inclusive. 



It follows from what has been already said, that if 

 u„ + u„_i + . . . + U2 + Ui + Uo = o 



be the equation of the given curve or surface, referred to rec- 

 tilinear axes passing through o, 



?«(m— 1)...2.1 m(m — \)...2 



Uot + —, r^ -r r^ u™-, + 



n{n— \)...{n — m-\-\) n{n—\)...{n—m-\-2) 



mhn— 1) in 



...+-7 rrUo-h— u+ u,= o 



n{ii — 1) n 



will be the equation of the m''' polar of the point o, with rela- 

 tion to the given curve or surface. By making »i, in this 

 formula, successively equal to the integer numbers from 1 up 

 to w — 1 inclusive, we obtain the equations of all the succes- 

 sive polars of the origin. For instance, the equation of the 

 first polar is 



Ui -j- HUo =■ o 



