246 Geometrical Propositions 
OT describes the curve of contact. Therefore the curve of coptact is a cirele J 
0 Lat VecleT 
passing (8) through the point S. pty 
16. We have examined the two cases of circular section with peewee only to the 
biaxal abc. If we examine the same cases with regard to the second biaxal wb'c’, we 
shall find that their indications are reversed ; the supposition which gives a node 
upon one biaxal, giving a circle of contact on the other: and that the node and the 
circle, thus corresponding, are so related, that a line drawn from O to the node 
passes through the circumference of the circle, cutting the plane of the circle perpen- 
dicularly ; whilst every line drawn from O through the circumference of the circle is 
perpendicular to some nodal tangent plage. 
These things are evident on looking at the figure. For when Or is a circle, 
it is plain that the point M is a node of the biaxal ab’c', since OM is perpen- 
dicular to the plane of the circle # Or and equal to its radius O#. But we have 
already seen (15) that when ROr is a circle, the other biaxal abe has a circle of con- 
tact, whose plane is perpendicular to OM at the point S of its circumference. The 
line OTL is perpendicular, in general (11), to a tangent plane at WM, and therefore 
perpendicular, in the present case, to a nodal tangent plane; whilst the point 7’, 
through which it passes, is on the circle of contact. It is also evident that OT’ x OL 
sire 
We have here an example of the general remark in the corollary of Theorem I. 
17. The section made in a biaxal surface abe, by any of the principal planes of its 
generating ellipsoid, consists of an ellipse and a circle. 
For let the plane Q Og pass through one of the semiaxes a, and let it revolve round 
this semiaxis, while the right line OZV (9), perpendicular to the plane QQq, re- 
volves about O in the plane of the semiaxes b,c. Then the semiaxis a of the ellipsoid 
will always be one of the semiaxes of the ellipse QQq ; and if O 7’ be equal to this 
semiaxis, the point 7’ will describe a circle with the radius a about the centre O. 
The other semiaxis of the ellipse QOq is that semidiameter of the principal 
ellipse be which lies in the intersection of the plane bc with the plane @Oq ; and as 
OV is equal and perpendicular to this semidiameter, the point V describes an ellipse 
equal to be, but turned round through a right angle, so that the greater axis of the 
ellipse described by V coincides in direction with the less axis of the ellipse be. As 
the radius a of the circle is greater (4) than both the semiaxes b,c, of the ellipse, the 
circle will lie wholly without the ellipse. 
In like manner, the section made in the biaxal surface by the plane aé consists of 
a circle with the radius c, and an ellipse with the semiaxes a,b ; and as the radius of 
the circle is less than both the semiaxes of the ellipse, the circle lies wholly within the 
ellipse. 
18. But when the section lies in the plane of the greatest and least semiaxes a,c, _ 
the circle and ellipse, of which it is composed, intersect each other. For the radius 
6 of the circle is less than one semiaxis of the ellipse ac and greater than the other. 
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