234 VI. SYSTEMS OF VECTORS. DISTRIBUT. OF MASS. INSTANT. MOTION. 



gyration about that axis are large, so that the inverse ellipsoid has 

 a large dimension in the direction of that axis, while the direct 

 ellipsoid, like the body, has a small one. 



73. Ellipsoidal Coordinates. The equation of a central 

 quadric referred to its axes may be written, 



'82) L' + JL'+! = i, 



a, ^ a z ^ a s 



where a lt a%, a 3 , may be positive or negative. If they are all negative, 

 the surface is imaginary, for the equation is not satisfied by any 

 real values of x,y, 2. 



1. Suppose one is negative, say 



% = - c 2 , 

 while 



a^ = a 2 , a. 2 = b 2 . 

 Let 



a > I > c. 

 The equation now is 



/y.2 ,,.2 ,.,2 



fL i y ___ \ 



a? + 5 2 c 2 ~ 

 The surface is cut by the XY- plane in the ellipse 



whose semi -axes are a and 5, and whose foci are at distances from 

 the center 



/a 2 b 2 = /% a 2 

 on the X-axis. 



The section by the ZX~ plane is the hyperbola 



with semi- axes a, c, and foci at distances "j/a 2 -}- c 2 = y^ a B on 

 the X-axis. The section by the YZ- plane is the hyperbola, 



with semi -axes fc, c, and foci at distances "J/6 2 -}- c 2 = j/a 2 a 3 on 

 . the F-axis. The surface is an hyperboloid of one sheet. 



2. Let two of the constants a 19 a 2 , a 3 , be negative, say 

 The equation is 



