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the locus of the point m will be a conic having a three point 
osculation with the curve ato. 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 
Un + Unit...» $U.4+U,=0 (4) 
be the equation of the given curve or surface, referred to axes 
passing through the given point o, 
U.+uU,=0 (5) 
will be the equation of the curve or surface of the second 
order, constructed in the manner described above. 
Let us suppose = 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 
Uz = 0 
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 vu; = 0, it seems natural to extend this 
conception, and to call the conic uv, + 0, = 0 the generating conic: the curve 
Uy + U, + U,=0 the generating curve of the third degree, and so on. 
o 2 
