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ON THE CENTEO-SUKFACE OF AN ELLIPSOID. 
[520 
through each of which, in the two different planes through the axis respectively, there 
passes an ellipse and an evolute. In the assumed case a? + c 2 > 26 2 , the disposition of 
the points is as shown in the figure. 
z 
Plane of xz, evolute is outside ellipse, 
yz, „ inside 
xy, „ cuts „ ; 
hut in the contrary case a? 4- c 2 < 2b 2 , the disposition is 
Plane of xz, evolute is outside ellipse, 
yz, „ cuts 
xy, „ is inside „ ; 
there is no real difference, and to fix the ideas I attend exclusively to the first-mentioned 
case 
a? + c 2 > 2b 2 . 
18. In each of the principal planes, the evolute and ellipse, qua curves of the 
orders 6 and 2 respectively, intersect in twelve points, 3 in each quadrant; viz. of 
the 3 points two unite together into a twofold point or point of contact, and the 
third is a point of simple intersection; assuming for the moment that this is so, the 
figure at once shows that in the plane of xz or umbilicar plane the contact is real, 
the intersection imaginary; in the plane of xy, or major-mean plane, the contact is 
imaginary, the intersection real; but in the plane of yz or minor-mean plane the 
contact and intersection are each imaginary. The contacts arise, as will appear, from 
the umbilici of the ellipsoid, and may be termed “umbilicar centres,” or “omphaloi;” 
the simple intersections “points of outcrop,” or simply “outcrops.” By what precedes 
there are in the umbilicar plane, four real umbilicar centres (in each quadrant one); 
and in the major-mean plane four real outcrops (in each quadrant one); the other 
umbilicar centres and outcrops are respectively imaginary.