284.] PRINCIPAL AXES. 153 



where X is a variable parameter, represents a system of confocal quad- 

 ric surfaces. 



282. As long as X is algebraically greater than <r 2 , the equation 

 (20) represents ellipsoids. For A = c 2 the surface collapses into the 

 interior area of the ellipse in the jry-plane whose vertices are the foci 

 S 2 , -5V and SB, Ss- For as A approaches the limit t 2 , the three semi- 

 axes of (20) approach the limits V# 2 ^ 2 , V^ 2 t 2 , o, respectively. 

 This limiting ellipse is called the focal ellipse. Its foci are the points 

 Sb.Si 1 , since a 2 - c~ - (b 2 c z ) = a 2 - P. 



When X is algebraically < c 2 , but > a 2 , the equation (20) repre- 

 sents hyperboloids ; for values of X < a 2 it is not satisfied by any real 

 points. As long as b 2 < A< c 2 , the surfaces are hyperboloids of one 

 sheet. The limiting surface X = c 2 now represents the exterior area 

 of the focal ellipse in the ary- plane. The limiting hyperboloid of one 

 sheet for X = b 2 is the area in the sac-plane bounded by the hyperbola 

 whose vertices are .Si, /, and whose foci are S 3 , S 8 '. This is called the 

 focal hyperbola. 



Finally, when a 2 < A < b 2 , the surfaces are hyperboloids of two 

 sheets, the limiting hyperboloid X = a 2 collapsing into the jz-plane. 



283. It appears from these geometrical considerations, that there 

 are passing through every point of space three surfaces confocal to the 

 fundamental ellipsoid x 2 /a 2 +y*/b 2 + z 2 /c 2 = i and to each other, viz. : 

 an ellipsoid, a hyperboloid of one sheet, and a hyperboloid of two sheets. 

 This can also be shown analytically, as there is no difficulty in proving 

 that the equation (20) has three real roots, say \ lt A 2 , A 3 , for every set 

 of real values of x, y, z ) and that these roots are confined between such 

 limits as to give the three surfaces just mentioned. 



The quantities A 1; A 2 , A 3 can therefore be taken as co-ordinates of the 

 point (x, y, z) ; and these elliptic co-ordinates of the point are, geomet- 

 rically, the parameters of the three quadric surfaces passing through 

 the point and confocal to the fundamental ellipsoid ; while, analytically, 

 they are the three roots of the cubic (20). To express x, y, z in terms 

 of the elliptic co-ordinates, it is only necessary to solve for x, y, z the 

 three equations obtained by substituting in (20) successively X lf A 2 , A $ 

 for A. 



284. The geometrical meaning of the parameter A will appear by 

 considering two parallel tangent planes TT O and TT A (on the same side of 



