BKMKTKY 



[_('... IV. 



the semi-axes a and b in the ^//-plane, I and c in the yz- 

 plane, and e and a in the zx-plane. 



i The traces on planes parallel to any coordinate plane 



nnilar ellipses (Art. 225). 



) The equation may be written 



I" ~~ > / ) n v "^ * f 



a* a a 



hence for a plane section parallel to the y-plane, the semi- 

 axes are real if the value of x lies between a and +</, 

 imaginary if beyond those limits, and zero if x= a. More- 

 over, the length of the axes diminish continuously fmni tin- 

 values b and c, respectively, when x = to the value zero, 

 when x = a. 



Similarly for sections parallel to either of the other 

 e.H.rdinate planes. 



(4) The surface is symmetrical with respect to each co- 

 ordinate plane. 



This quadric surface, the locus of equation [34], is called 

 an ellipsoid. It may be conceived as generated by a varial >! 

 ellipse, which has its vertices upon, and moves always per- 

 pendicular to, two fixed ellipses, which in turn are perpen- 

 dieular to each other and have one axis in common. 



From this definition equation [34] can be easily derived. Let 

 CRA and A SB be fixed ellipses perpendicular to each other, and I 



the semi-axis OA in conn 

 and the second axes OC and 

 OB, respectively ; and 1-t s/'li 

 be the variable el! 

 semi-axes MS and Ml!. If 

 "1, OB, OC be taken 

 x, y, z axes, respectively : 

 P be any point on the moving 

 ellipse, with coordinates Oi 

 Fw.157 MM', .1/7', then (hy Art. 112), 



