3O2 On the Surfaces of the Second Order. 



angles which the plane of section makes with the directive 

 planes.* 



9. From the centre of the surface expressed by equa- 

 tion (2) let a right line OS be drawn cutting perpendicularly in 

 S the plane which touches the surface at S. Let a denote the 

 length of the perpendicular OS, and a, j3, y the angles which it 

 makes with #, y, z. Then 



a 2 = P COS 2 a + Q COS 2 ]3 + R COS 2 y. (6) 



From this formula it is manifest that, if three planes touching 

 the surface be at right angles to each other, the sum of the 

 squares of their perpendicular distances from the centre will be 

 equal to the constant quantity P + Q + R, and therefore the 

 point of intersection of the planes will lie in the surface of a 

 given sphere. If another surface represented by the equation 



be touched by a plane cutting OS perpendicularly in S () , and if 

 (T be the length of OS , then 



do 2 = Po cos 2 a + Qo cos 2 j3 + -Ro cos 2 7 ; 



and therefore when the two surfaces are confocal, that is, when 

 P - Po = Q - Qo = R - R, = k, 



we have a 2 - cr 2 = k, which is a constant quantity. Hence, if 

 three confocal surfaces be touched by three rectangular planes, 

 the sum of the squares of the perpendiculars dropped on these 

 planes from the centre will be constant, and the locus of the 

 intersection of the planes will be a sphere. 



The focal curves of a given surface are the limits of surfaces 

 confocal with it,f when these surfaces are conceived, by the pro- 



* See Transactions of the Royal Irish Academy, VOL. xxi., as before cited. 

 The formulae (5) were first given, for the case of the ellipsoid, by Fresnel, in his 

 Theory of Double Refraction, Memoir es de VInstitut, torn, vii., p. 155. 



f It was by this consideration, arising out of the theorems given in this and the 

 next section about confocal surfaces, that I was led to perceive the nature of the 



