248 Geometrical Propositions 
common tangent di’ to be drawn, in a quadrant adjacent to that which contains the 
node ”, and let it touch the circle in d’ and the ellipse acin7’. A radius Od", drawn 
through the point d’ to meet the ellipsoid a’d’c’ in the point d”, will be reciprocal to the 
radius O07’, because it is perpendicular toa tangent ut 2’, and it will be equal in length to 
b', because Od" x Od' =k’ = bb’, and Od'=6 ; whence Od'=0'. Therefore Od" isin a cir- 
cular section of the ellipsoid ab’c’ Two planes perpendicular to the plane of the 
figure, and passing through the radlikocal radi Od’, O7’, are (6) reciprocal planes 
we have seen that the first of them makes a circular section in the ellipsoid a‘b’c’. They 
are therefore (15) the fixed planes in the second case of Prop, V. 
22. Now draw di a common tangent to the circle b and ellipse ac composing the 
biaxal section, and let it touch the circle in d and the ellipse m 7. The lines Od, Oi, are 
of course perpendicular to the lines Od’, Oz’, and therefore perpendicular to the fixed 
planes just mentioned. Hence the me Od and the point d are the same as the fixed 
line OS and the point S in the second case of Prop. V. The plane of the circle of 
contact is therefore perpendicular to Od at the point d (15); and the points d and 7, 
where its plane intersects the right lines Od, O7, perpendicular to the fixed planes, are 
(8) the extremities of a diameter. 
These things agree with the obvious remark, that the points of contact d and 7 
must be points of the circle of contact; and that d? must be a diameter, because the 
plane of the circle is perpendicular to the plane of the figure, and this latter plane 
divides the biaxal surface symmetrically. 
As the circle and ellipse may have a common tangent opposite to each node, there 
are four circles of contact in planes perpendicular to the plane of the nodes.* 
23. The biaxal surface belongs to a class that may be called apsidal surfaces, from 
the manner in which they are conceived to be generated. 
Let G be agiven surface, and O a fixed origin or pole. If a plane passing through. 
O cut the surface G, the curve of intersection will in general have several apsides 
A, A’, A’, &c., where the lines OA, OA’, OA", &c. are perpendicular to the curve. 
Through the point O conceive a right line perpendicular to the plane of the curve, and 
on this perpendicular take from O the distances Oa, Oa’, Oa’, &c. respectively equal 
to the apsidal distances, O04, OA’, OA”, &c. Imagine a similar construction to be 
made in every possible position of the intersecting plane passing through O, and the 
points a, a’, a’, &c. will describe the different sheets of an apsidal surface. 
*The curves of contact on biaxal surfaces, and the conical intersections or nodes, were lately disco- 
yered by Professor Hamilton, who deduced from these properties a theory of conical refraction, which 
has been confirmed by the experiments of Professor Lloyd. See Transactions of the Royal Irish Aca- 
demy, Vol. XVII. Part. I, pp. 132, 145; and the present paper, Art. 55—58. 
The indeterminate cases of circular section—at least the case of the nodes—had cogurred to me long 
ago; but haying neglected to examine the matter attentively, I did not perceive the properties involved 
in it (13). April 2, 1834. 
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