86 
ay? + b2a? = v2 2 
ea’? —h” = f 
There are four cases to be considered, according to the different 
signs the arbitrary constants a* and b® may assume. There are 
however two of these cases in which one of the loci is impossible. 
If a? < 0 and b” > 0, the second equation is impossible, for it would 
equate the sum of two quantities essentially negative with a quantity 
essentially positive. _As therefore the equation of condition between 
the constants a? and b” cannot under these circumstances be fulfilled, 
there can be no corresponding lines of curvature. If a? and b” be 
both negative the first equation is impossible, and therefore there is 
no corresponding lines of curvature. The only cases therefore 
which remain to be investigated are where a” and b’ are both posi- 
tive, or b’ alone negative. 
If a? and b” be both positive the first equation represents an el- 
lipse, and the second an hyperbola. If b? < 0 the first represents 
an hyperbola, and the second an ellipse. The semiaxes of the el- 
lipse and hyperbola represented by the second equation are 
Joe 
A = a, —S=——— 
V/a—c 
; J/ a—b? 
Bo) eS 
JS P—e 
With these common axes let an ellipse and hyperbola be described 
upon the plane xy. With the coordinates of any point of the ellipse as 
semiaxes, let an hyperbola be described upon the same plane, having 
its axes coincident with those of the surface, and with the coordi- 
nates of any point in the hyperbola as axes, let an ellipse be de- 
