92 
Let a” and c’ be the semiaxes of the elliptic directrix 
Since a’—c’ is greater than either a*—b? or b’—c? it follows that 
a” and c” are greater than a and c respectively. On the axes of x 
and z assume portions equal to a” and c’, and with these as axes let 
an ellipse be described. This ellipse will determine the several el- 
lipses which are the projections of the lines of curvature in the same 
manner as the hyperbolic directrix determined the projections upon 
the plane of xy. The limits of these ellipses are those which ul- 
timately coincide with the axes of the elliptic directrix. Asc” di- 
minishes a’ approaches to equality with a’, and the corresponding 
ellipse continually becomes more eccentric until it flattens into a right 
line when c” vanishes. Similar observations apply to the other axis 
when a’ vanishes. It is obvious that all the ellipses which are pro- 
jections of the lines of curvature are included within the elliptic di- 
rectrix. 
The elliptical section by the plane wz is itself one of the ellipses 
given by the equation; for if a’ = a c”’= cc. Those ellipses in which 
e’<c divide the elliptic section into zones in the direction of the 
axis 2a, and those in which ¢”> ¢ divide into zones in the direction 
of the axis 2c. These are therefore the projections of the lines of 
curvature perpendicular to the former. 
If the inscribed lozenge be formed in the elliptic directrix by 
connecting the extremities of its axes, the four sides of this will be 
