152 KINETICS OF A RIGID BODY. [279. 



Cartesian co-ordinates, that is, to express x and y in terms of Xj and X 2r 

 we have only to solve for x and y the two equations 



279. The two confocal conies that pass through the same point P 

 intersect at right angles. For the tangent to the ellipse at P bisects the 

 exterior angle at P in the triangle SPS', while the tangent to the hyper- 

 bola bisects the interior angle at the same point ; in other words, the 

 tangent to one curve is normal to the other, and vice versa. The elliptic 

 system of co-ordinates is, therefore, an orthogonal system ; the infinitesi- 

 mal elements d\ l d\ 2 into which the two series of confocal conies (19) 

 divide the plane are rectangular, though curvilinear. 



280. These considerations are easily extended to space of three 

 dimensions. 



An ellipsoid 



-i+^+ Z ^ = i, where a >*><:, 

 a o c 



has six real foci in its principal planes ; two, .Si, .Si', in the dry-plane, on> 

 the axis of x, at a distance (2Si = V0 2 ft 2 from the centre O ; two, 

 S 2 , S 2 ', in the jyz-plane, on the axis of y, at the distance OS 2 = V^ 2 t 3 ' 

 from the centre ; and two, S 3 , S 3) in the &#-plane, on the axis of x, at 

 the distance OS 3 vV c 2 from the centre. It should be noticed 

 that, since b > c, we have OS 3 > 0.Si ; i.e. S lf SJ lie between S 3) S 3 ' on 

 the axis of x. 



The same holds for hyperboloids. 



281. Two quadric surfaces are said to be confocal when their princi- 

 pal sections are confocal conies. Now this will be the case for two- 

 quadric surfaces whose semi-axes are a lf b^ c^ and a z , 2 , c z , if the 

 directions of their axes coincide and if 



Writing these conditions in the form 



,,2 -2 Z2 Z2 -2 -2 cav \ 



a 1 a \ = &2 Pi ^2 C \ ) Sa Y A > 



we find 2 2 = a} + X, 2 2 = <V + X, r 2 2 = c? + X. Hence the equation 



x 2 



