876 ILYTIC GKOMrriiY (Cii IV. 



(1) The traces on th<> .///- and rr-planes are 

 with semi-axes a and 6, c and a, resp.-, i ivly, and \\ith the 

 transverse axis along the z-nxis, while the traces on tho 

 planes parallel to the yz-plane are imaginary if x lies 

 between a and a, real ellipses if x is beyond those li; 

 and poinU if x = a. 



(L') The traces on planes parallel to any coordinate plane 

 are similar (Art. 225). 



(3) The elliptical sections parallel to the yz-plane increase 

 continuously and indefinitely as x varies from + a to -f oo, 

 or from a to oo. 



(4) The surface is symmetrical with respect to each 

 coordinate plane. 



This quadric surface, whose equation is [3fi], is called a 

 bi-parted hyperboloid, or hyperboloid of two sheets. It : 

 !> conceived as generated by a variable ellipse which has 

 its vertices upon, and moves always perpendicular to, two 

 fixed hyperbolas which in turn are perpendicular to each 

 other, and have a common transverse axis. This definition 

 leads readily to the equation [M], 



Every equation of the form Ax 2 By 2 Cfe 2 K = rep- 

 resents a bi-parted hyperboloid. If the coefficients of the 

 two negative variable terms are equal, i.e., if b = <?, the sur 

 face is the double hyperboloid of revolution (cf. Art. 213, 

 eq. (13)). 



231. The paraboloids: equation ^i^"** A discussion 

 of the equation ^i+xi 32 * * * P?] 



similar to that of the preceding articles shows that its locus 



