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the locus of the point m will be a conic having a three point 

 osculation with the curve at o. The tangent and osculating 

 circle of this conic belong therefore likewise to the given curve. 

 So also, in the case of a point o given on any algebraic sur- 

 face, the surface of the second order, which is the locus of a 

 point M determined in the same way, will have its lines of 

 greatest and least curvature coincident with those of the given 

 surface at o. All this is obvious, since if 



U» + Vn-i + + Ua + Ui = o (4) 



be the equation of the given curve or surface, referred to axes 

 passing through the given point o, 



Ua + Ui = o (5) 



will be the equation of the curve or surface of the second 

 order, constructed in the manner described above. 



Let us suppose nz=:3, and (4) to be the equation of a 

 plane curve of the third degree; its intersections with the 

 conic (5)* determine the three right lines represented by the 

 equation 



U3 = o 



which are obviously parallel to the asymptots. The directions 

 of the asymptots being thus ascertained, we may readily deter- 

 mine their actual position. For this purpose draw tangents 

 to the curve of the third order at the extremities of any one of 

 the chords common to it and the conic ; they will meet the 

 curve in two points ; and the line joining these points will cut 

 the curve in the point through which the asymptot parallel to 

 that chord passes. 



Of course the results here stated are subject to modifica- 

 tions when o is a singular point. 



* As we are accustomed to regard the curve as generated by the motion 

 of the tangent, whose equation is Ci = o, it seems natural to extend this 

 conception, and to call the conic Cj + tJ, = o the generating conic : the curve 

 u^ -f. Uj + Ui=: o the generating curve of the third degree, and so on. 



o 2 



