306 On the Surfaces of the Second Order. 



sections, is equal to the semiparameter of the other principal 

 section. 



11. Let a tangent plane, applied at any point S of a surface 

 of the second order, intersect the plane of one of its f ocals in the 

 right line 0, and let P be the foot of the perpendicular dropped 

 from S upon the latter plane. The pole of the right line 0, with 

 respect to the principal section lying in this plane, is the point 

 P. Let N be its pole with respect to the focal. Then if T be 

 any point of the right line 0, the polar of this point with respect 

 to the section will pass through P, and its polar with respect to 

 the focal will pass through N ; and if the former polar intersect 

 the diligent curve in A, A', and the latter intersect the focal in 

 F, F', the points F, F' will correspond respectively to the points 

 A, A', and the distances A A' and FF' will be similarly di- 

 vided by the points P and N (see Part I. 8). But since the 

 point S is in the plane of the two directrices which pass through 

 A and A', the lengths AP and A'P, which are the perpendicular 

 distances of S from the directrices, are proportional to the lengths 

 FS and F'S. Therefore FN is to FN as FS is to F'S, and the 

 right line NS bisects one of the angles made by the right lines 

 FS and F'S. And as this holds wherever the point T is taken 

 on the right line 0, that is, in whatever direction the right line 

 FF' passes through the point N, it follows that the right line 

 NS is an axis of the . cone which has the point S for its vertex 

 and the focal for its base. Further, if FF' intersect in the 

 point Q, we have FN to F'N as FQ, is to FQ,, because N is the 

 pole of with respect to the focal ; therefore FQ is to F'Q, as 

 FS is to F'S, and hence the right line QS also bisects one of the 

 angles made by FS and F'S. The right lines NS and QS are 

 therefore at right angles to each other ; and as the latter always 

 lies in the tangent plane, the former must be perpendicular to 

 that plane. 



Consequently the normal at any point of a surface of the 

 second order is an axis of the cone which has that point for its 

 vertex and either of the focals for its base. 



It is known that when two confocal surfaces intersect each 



