72, 73] ELLIPSOIDAL COORDINATES. 235 



The sections by the coordinate planes and their focal distances are 



XY - = 1 Hyperbola, i/o^+T 2 = V^~^ on X-axis, 

 ^ - - ~ = 1 Hyperbola, ]/a 2 T7 2 = Va^a, on X-axis, 

 * -f = - 1 Imaginary Ellipse, /-(6 2 _ c 2 ) = Va^a*. 



The surface is an hyperboloid of two sheets. 



3. If a l9 a 2 , a s are all positive, the sections are all ellipses, and 

 the surface is an ellipsoid. In all three cases, the squares of the 

 focal distances are the differences of the constants %, & 2 , a 3 . Con- 

 sequently if we add to the three the same number, we get a surface 

 whose principal sections have the same foci as before, or a surface 

 confocal with the original. Accordingly 



^ 2 _L y* + g2 _ i 



- 



represents a quadric confocal with the ellipsoid 



tf + V + ^ = 1? 

 for any real value of Q. 



If a > 6 > c and p > c 2 , the surface is an ellipsoid. If 

 c 2 > Q > & 2 , the surface is an hyperboloid of one sheet, and if 

 - Z> 2 > Q > a?j an hyperboloid of two sheets. If Q < a 2 , the 

 surface is imaginary. 



Suppose we attempt to pass through a given point x, y, s, a 

 quadric confocal with the ellipsoid 



Its equation is 83), where the parameter Q is to be determined. 

 Clearing of fractions, the equation is 



84) ft?) EH (o + 9 ) (6* + 9) (o 2 + 9) - a? (V + 9 ) (c 2 + 



a cubic in p. But this is easily shown to have three real roots. 

 Putting successively p equal to oo, c 2 , V, a 2 and observing signs 



of f(9), 



P= oo, f(9) = 



