91 
of curvature upon the plane 2z. The quantity a” is variable, and 
for each value of it there is a distinct line of curvature. Since it has 
been already proved that a’ can never exceed a’, it follows that a@—a® 
> 0, and therefore that all these projections are ellipses. Let a’c” be 
the semiaxes of any of them corresponding to any proposed value of 
U 
2 se 
G7, ies 
ea’a? 
12 — 
_ec*(a’ — a”) 
hh? 
ec” 
Eliminating by these equations the variable quantity a”, the result is 
fea’? + a°b’c”? =ea'c? 
or c*(a’—b’)a” + a? (b°—c?) c’”? = a®c*(a’—c’) 
Considering a’c’ as variable coordinates this is the equation of an 
ellipse, since a’ —b*>0 and b*’—c*>0. This is the elliptic 
directrix for determining the projections of the lines of curvature 
upon the plane of the greatest and least axes of the ellipsoid. 
This equation might have been derived at once from the equation 
(page 86) for the projections of the lines of curvature upon the plane 
xy-by changing b into cand v.v. The result here however is some- 
what different from the former. In the present case the lines of cur- 
vature are all projected into ellipses, and there is only an elliptic di- 
rectrix to determine them. The projections of the lines of curvature 
are therefore all determined by the equations 
aa? +624? = Q’2¢"2 
c?(a?—b?) a” + a? (b> —c2) c”? = ac? (a?—c?). 
VOL. XIV. Q 
