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stants, 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 2 points Nj, No, Ng-- +++ N,3 and assume on it m 
points M,, My, M3,...+++Mms SO as to satisfy the m conditions 
- mela eed 
Pe kon)  * "Nom 
a2 (4 +a Grom —) 
r 1 
OM).0M9.0M; ON}-ON».0ON3 
( ot 2( 
Ay (| ———_-—_——_ : 
OM).OMo.0M3......-OMm 6 ON,.ON5.ON3..0-6 se) 
Then the points Mj, My, M3, «--+«- M, Will lie on the surface (2). 
If we suppose now that the coefficients Ao, Aj, Ag, «+» Am 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 , 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 A, 1, Ao, --- Amy 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 ona 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 N, Ny N3...+Np—1, let 
us assume on it a point m such that 
See ) 
