to the Wave Theory of Light. 25 



08, OM, equal to OQ, ON, OP, OR, respectively ; the angles, 

 at 8 and L being of course right angles. Then it is evident 

 that the point T is on the biaxal surface generated by the ellip- 

 soid abc, because OT is perpendicular to the plane of the ellipse 

 QOq and equal to the semiaxis OQ', and by Theorem III. it 

 appears that 08 is perpendicular to the tangent plane at T. In 

 like manner, the point M is on the biaxal surface generated by 

 the other ellipsoid a'b'c, and OL is perpendicular to the tangent 

 plane at M. Moreover, the rectangles MO 8 and LOT, being 

 equal to the rectangles EOF and NOQ, are each equal to & 2 . 

 Hence the proposition is manifest. 



12. As the ellipsoid whose semiaxes are a, b, c, may be called 

 the ellipsoid abc, so the biaxal surface generated by this ellipsoid 

 may be called the biaxal abc ; and that which is generated by the 

 ellipsoid dlJc' may be called the biaxal a'b'c. 



13. PROPOSITION Y. To find what properties of biaxal sur- 

 faces are indicated by the cases wherein one of the two sections 

 QOq, ROr, in the preceding theorem, is a circle. 



Case 1. "When QOq is a circular section of the ellipsoid abc, 

 the points T and V (9), in the description of the biaxal surface 

 abc, coincide in a single point n. At this point there are an in- 

 finite number of tangent planes; because the semiaxes of the 

 circular section QOq being indeterminate, any two perpendicular 

 radii of the circle may take the place of OQ, Oq, in the general 

 construction. The point n is therefore a point of intersection (3), 

 where the two biaxal sheets cross each other, and it may be called 

 a nodal point, or simply a node. 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 Oq lies in the plane of the circle, we have three right 

 lines OR, Oq, 08, which are at right angles to each other, and 

 of which the first two are confined to given planes. Therefore 

 by Theorem II. the third line 08 describes a cone whose sections 

 parallel to the given planes are circles. Now, T8 or in the 

 present case n8 is parallel to the fixed plane which contains 

 OR, and therefore the point 8 describes a circle ; or, in other 



