( 51J ) 



{2c,p + c.,q)x -f- ('-^p 4- 26-3^)?/ \- {M,p + d,q —c,).7r + 

 + (2c?, p + 2^3^ — c,yvy + {d,p -f- 80^,7 — c,)f + 

 + (4., p-^e,q- 2d,)w' + (3.,2> + 2^3^? — 2c/J .^^^ + 

 -f. {2e,p + 3.,7 - 2d,)xy' + (., p 4 Ae,q - 2d,)f + . . . . rr: O . (2) 



The above form (2) is therefore the equation of the curve of 

 contact of a cone touching the surface and having point P[p.q)a.s 

 vertex. 



We shall now distinguish three cases : 

 I. is not a parabolic point. 

 II. is a parabolic point. 



111. (9 is a point of osculation. 



1. Point is not a parabolic point. 

 As is an elliptic or a hyperbolic point, it follows that 



c,c, c,^ ^ 0" We now assume the line OP as X-axis, so that 



' ' 4 ' < 



q=:zO. We can now distinguish two cases according to OP being 



an asymptote of the indicatrix or not. 



Ia' The line OP is not an asymptote of the indicatrix. 



We assume OP as X-axis and the conjugate diameter of the 

 indicatrix as T-axis ; so q^O and c, = 0. From (2) follows then : 



2c,px + {M,p—c,)x" + 2d^pxy + {d^p — C3)?/- + . . . = . (3) 



The curve of contact touches therefore the 1^-axis in point 0. 

 As the A-axis (the line OP) and the F-axis are conjugate diameters 

 of the indicatrix, it follows that the line OP, connecting the vertex 

 P of a cone with a point of its curve of contact, and the tangent 

 in point to this curve of contact are conjugate diameters of the 

 indicatrix of point 0. 



In general the curve of contact in the vicinity of point is of 

 finite curvature and determined by: 



2c,px-^{d,p—c,)y' = (4) 



If p is chosen in such a way that d^p — Cj = then the equa- 

 tion is 



2c,px-\-{e,p-2d,)y' = (5) 



so that the curve of contact has a point of inflection in point 0. 



►Several ternary lines of saturation with one or more points of 

 inflection are known. We find e.g. on the line of saturation of the 



