73, 74] COORDINATE SURFACES ORTHOGONAL. 237 



Similarly for the normals to the surface p = const,, 



cos 



89) cos (n^y) = 



cos (n u z) = 



The angle between the normals to A and /A is given by 



f # 2 w 2 



^ ^ \( a2 ~F ^) l^ 2 -}- ^) (^ 2 4~ ^) (^ 2 ~F f 1 -) 



i__ 



Now by subtracting from the equation 



_x< 



the equation 



,2 W 2 



-f 



we obtain 



/ ^1 2 J_ 1 2J_ f I y \ 13 I 1 7,2 I ., I I 



or 



2/ 2 



f,v,2 2 2 \ 



(a + ao(a* + fO + (& 2 + ^)(& 2 + ^) + ( C 2 + ^)(c 2 + ")J ^ 



Accordingly, unless h = ii, cos (n^n/^) = 0, and the two normals .are 

 at right angles. Similarly for the other pairs of surfaces. Accord- 

 ingly the three surfaces of the confocal system passing through any 

 point cut each other at right angles. 



If we give the values of A, ^, v we determine completely the 

 ellipsoid and two hyperboloids, and hence the point of intersection 

 x, y, 8. To be sure there are the seven symmetrical points in the 

 other quadrants which have the same values of A, ft, v, but if we 

 specify which quadrant is to be considered this will cause no 

 ambiguity. Thus the point is specified by the three quantities >L, ^, v, 

 which are called the ellipsoidal or elliptic coordinates of the point. 



74. Axes of Inertia at Various Points. Let K = MW be 



the moment of inertia about an axis whose direction cosines are 

 , ft y, at a point whose coordinates with respect to the principal 

 axes at the center of mass G- are xyz. Let p be the distance of the 

 axis at from a parallel axis through 6r, and q the distance of the 



