491 



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

 tion (2) let a right line OS be drawn cutting perpendicu- 

 larly in S the plane which touches the surface at S. Let cr 

 denote the length of the perpendicular OS, and a, j3, y the 

 angles which it makes with x, y, z. Then 



o-^ = p cos ^a + Q cos '^jS + R cos ^7. (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 4- 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 



Pq Qo Rq 



be touched by a plane cutting OS perpendicularly in So, and 

 if o-Q be the length of OSo, then 



gq = Po cos ^a + Qo cos ^/3 + Ro cos "^y ; 



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

 when 



P — Pq =: Q — Qo = R — Ro = '^j 



we have (P' — gq = 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 sur- 

 faces confocal with it,* when these surfaces are conceived, 



* 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 na- 

 ture of the focal curves, and the analogy between their points and the foci of 



