28 VARIABLES AND FUNCTIONS. [INT. II. 



2. Let two of the constants a lt a 2 , a s be negative, say 

 The equation is 



/y?2 /i>2 ^-2 



a*~P~& =1 ' 



The sections by the coordinate planes and their focal distances 

 are 



XT sr W3L H yp erbola 



^^ ^-^ =1 - Hyperbola Va' + c 2 = V ai -a 3 , 



Ct (j 



^A rg. 



YZ - + -^ = 1. Imaginary Ellipse V- (6 2 c 2 ) = Va 2 - a s . 



t? C 



The surface is an hyperboloid of two sheets. 



3. If a l} a a , 3 are all positive, the sections are ellipses, and the 

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

 distances of the principal sections are differences of the three 

 constants a 1} a 2 , a 3 . Accordingly 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 



/T\ _i_ y 



V*/ ~2i,*~k2l. 



represents a quadric confocal with the ellipsoid 



for any real value of p. 



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



c 2 > p > b 2 , the surface is an hyperboloid of one sheet, and if 



6 2 > p > a 2 an hyperboloid of two sheets. If a 2 > p, the surface 

 is imaginary. 



Suppose we attempt to pass through a given point cc, y, z a 

 quadric confocal with the ellipsoid 



