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Mathematics. — ^' General considerations on the carves of contact of 

 surfaces with cones, löitJi application to the lines of saturation 

 and binodal lines in ternary systems.'* (Ooinmunicated by 

 Prof. D. J. KoRTEWEG and Prof. F. A. H. Schreinemakers). 



Introduction. 



It is a known fact that in the study of the ternary solutions 

 which for given temperature and pressure can be in equilibrium 

 with a solid substance a great part is played by the curve of contact 

 of the tangential cone of the ^-surface with a given point as vertex. 



If namely we project the vertex of the cone and its curve of 

 contact on the horizontal plane, then the projection of tiie curve of 

 contact represents a ternary line of saturation, namely the series of 

 the solutions, which for assumed temperature and pressure are 

 saturated with the solid substance indicated by the projection of the 

 vertex of the cone. 



The form of the line of saturation of a solid substance being 

 thus determined by the form of the curve of contact of a cone, it 

 was our aim to investigate which peculiarities this curve of contact 

 could display in some points of a given surface and in particular 

 of the ^-surface. 



We choose as origin of the system of coordinates a point of 

 the surface. We assume the A"- and F-axis in the tangential plane 

 of the surface in point 0. 



For the equation (►f the surface in the vicinity of point we can 

 then write : 



The equation of a tangential plane in a point ,f, y, 2 of this surface 

 becomes : 



d^ dz 



If we wish to let this tangential plane pass through a point 

 P[p.q) of the A^. 3^-plane, then we must have 



0^ bz 



bz bz 



If in this equation we substitute the values of z,— and —out of 

 ^ ox oy 



'Vj we get ; 



