to the Wave Theory of Light. 27 



ROr, and its intersection with T describes the curve of contact. 

 Therefore the curve of contact is a circle passing (8) through the 

 point 8. 



16. "We have examined the two cases of circular section with 

 reference only to the biaxal abc. If we examine the same cases 

 with regard to the second biaxal ab'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 to the node passes through the circum- 

 ference of the circle, cutting the plane of the circle perpendicu- 

 larly ; whilst every line drawn from through the circumference 

 of the circle is perpendicular to some nodal tangent plane. 



These things are evident on looking at the figure. For when 

 ROr is a circle, it is plain that the point M is a node of the 

 biaxal ab V, since OM is perpendicular to the plane of the circle 

 ROr and equal to its radius OR. But we have already seen (15) 

 that when ROr is a circle, the other biaxal abc has a circle of 

 contact, whose plane is perpendicular to OM at the point 8 of its 

 circumference. The line OTL is perpendicular, in general (11), 

 to a tangent plane at Jf, and therefore perpendicular, in the 

 present case, to a nodal tangent plane ; whilst the point T, 

 through which it passes, is on the circle of contact. It is also 

 evident that OT x OL = k\ 



We have here an example of the general remark in the corol- 

 lary of Theorem I. 



17. The section made in the biaxal surface abc, by any of 

 the principal planes of its generating ellipsoid, consists of an 

 ellipse and a circle. 



For, let the plane QOq pass through one of the semiaxes, a, 

 and let it revolve round this semiaxis, while the right line OTV 

 (9), perpendicular to the plane QOq, revolves about in the 

 plane of the semiaxes , c. Then the semiaxis a of the ellopsoid 

 will always be one of the semiaxes of the ellipse QOq] and if 

 T be equal to this semiaxis, the point T will describe a circle 

 with the radius a about the centre 0. The other semiaxis of the 



