248 



b. Let p=rl, q=: co, c = l 



Then x^ _ y2 (l _ z)2 = (1 — z)2 



c. Let p= cc, q^l, c = l 



Then x2 (1 — z)2 — y2 — (1 _ z)2 



Biquadratic Surface with Elliptical Sections. 



Let f (x' y') = x'2 + y"2 _ c = o 



Then f i — — , — — I = j^ -^ + , \. — 0=0 



[p— z q— z J (p — z)2 (q — z)2 



a. Let p = 1, q ^ — 1, c — 1 



Then x2 (1 f X)2 + y2 (l _ z)2 = (1 — z2)2 



Here the volume between rectilinear directrices is exactly that of a 

 sphere of radius one. 



b. Let p ^ aq, c =: 1 



Then — \ ^ = 1 . 



I aq J I q J 



Circular sections are at z ^ o and z ^ 



1+a" 



2 ao 

 The planes z:=o, z=iq, z = :j — - — , z = aq divide every transversal 



harmonically. In particular every element is divided harmonically by the 

 circular sections and the rectilinear directrices. 



c. Combining the last two surfaces and letting p = aq, 



X- , y^ 



\i-^V fi--^! 



= c 



1 i-~ 



I aqJ L q J 



Solve for sections parallel to the xy plane and of the same eccen- 

 tricity : 



ff z 1 ( ^ 7. ^ .... 



ml ±1 which gives 



I aq J I q J '' 



aq(m — 1) , aq (m -f 1) . . .. 



z = and z := ; for similar conic sections. 



m — a m -(- a 



It is then easily seen that the four planes, 



z = q, 



aq ( m — 1 ) 



in — a ■^ 



z = aq, 



_ aq (m + 1) 

 m 4- a ' 



divide any transversal harmonically. 



