applied to the Wave Theory of Light. Q45 
point there are an infinite number of tangent planes; because the semi-axes of the 
circular section QOq being indeterminate, any two perpendicular radii of the circle 
may take the place of OQ, Og, in the general construction. The point 7 is there- 
fore a point of intersection (3), where the two biaxal sheets cross each other, and it 
may be called a nodal point, or simply anode. As OQ always lies in the plane of the 
circle QOq, the line OR, which is reciprocal to OQ, must lie (6) in a given plane 
reciprocal to the plane of the circle. And as Qg lies in the plane of the circle, we 
have three right lines OR, Og, OS, which are at right angles to each other, and of 
which the first two are confined to given planes. Therefore by Theorem If. the third 
line OS describes a cone whose sections parallel to the given planes are circles. Now 
T'S—or in the present case »S—is parallel to the fixed plane which contains OR, 
and therefore the point § describes a circle; or, in other words, the feet of the per- 
pendiculars OS, let fall from O on the.nodal tangent planes, eg) the ee 
ence of a circle passing (8) through the nodal point. 
14, Parallel to the plane of the circle and to its reciprocal plane, conceive two 
planes passing through the node, and call them the principal tangent planes at n. 
The plane of the circle and its reciprocal plane are intersected in the right lines Og, 
OR, by the plane gO which is parallel to a tangent plane at m. | Consequently this 
tangent plane at ” intersects the two principal tangent planes in lines that are paral- 
lel to Og, OR ; and as Og, OR are perpendicular to each other, it follows that every 
nodal tangent plane intersects the two principal tangent planes in lines that are at 
right angles. 
Hence again, the nodal tangent planes touch (7) the surface of a cone whose sec- 
tions, parallel to the principal tangent planes, are parabolas. As this cone touches 
the biaxal surface all round the point 7, it may be called the modal tangent cone. 
15. Case 2. When ROr is a circular section of the ellipsoid a’‘b’c’, any two per- 
pendicular radii of the circle may be taken for OR, Or: and because OR =’, and 
OR x OP =Kk?=b1'", we have OP or OS equal to b, the mean semiaxis of the ellip- 
_ soid abe. Hence OS is given both in position and length; for it is perpendicular 
_ to the fixed plane ROr, and it is equal to 6. Now a plane cutting OS perpendicularly 
at S,is a tangent plane to the biaxal abc ; and we have just seen that this tangent 
plane remains the same, whatever pair of rectangular radii are taken for OR, Or. 
But the point of contact J’ is variable, for the plane ROS in which it lies changes 
with OR. Therefore as OR revolves, the point J’ describes a curve of contact on 
. a the tangent plane of the biaxal abc. 
The lines OR, Or, are in the fixed plane ROr ; and as OQ is reciprocal to OR, 
it lies in a fixed plane reciprocal to the plane Ror (6). Therefore the first two of 
the three perpendicular right lines Or, OQ, OT, are confined to fixed planes. Hence 
the third line OT describes a cone, whose sections parallel to these planes are circles. 
But the tangent plane is parallel to the fixed plane ROr, and its intersection with 
