152 



slants, represents a curve or surface derived from the former 

 by means of the following geometrical construction. 



Draw from the origin o any right line meeting the surface 



(1) in the n points n„ Ng, N3, N„ ; and assume on it m 



points Ml, M2, M3, M„i, so as to satisfy the m conditions 



Aflsf ) =Ai2( ) 



Aosf jznAasf ) 



AnSf ) = A32( ) 



V0M1.0M2.0M3/ VoNi.oNa.ONa/ 



U ( ) = Amsf 



\OM1.OMo.OM3 OMot/ \on,. 



.0N2.0N3 oN^y 



Then the points m,, Mg, M3, m^ will lie on the surface (2). 



If we suppose now that the coefficients Aq, Ai, Ag, ... a^ are 

 formed according to any assumed law ; for instance, if they are 

 given functions of m and n, we shall have, as m takes different 

 integer values from 1 up to n, a series of curves or surfaces, 

 derived from the original one, and related in a particular man- 

 ner to it and to each other. 



By making the coefficients Aq, Ai, A2, ... a„„ all equal to 

 unity, we form a series of curves or surfaces most easily de- 

 rived from any given one ; and the consideration of them sug- 

 gests some interesting results. 



The problem of drawing a tangent geometrically at a given 

 point on a curve of the third degree has been elegantly solved 

 by M. Poncelet ; but we are now in a condition to solve it 

 generally for any algebraic curve. 



Having drawn any right line from the given point o, 

 meeting the curve in n— 1 other points Nj Nj N3 . .. . n„_i, let 

 us assume on it a point m such that 



-=A—\ (3) 



OM \0Niy ' 



