248 



b. Let p=^l, q^ oo, c = l 



Then x^ _ y2 (1 _ 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' yO = x^2 + y'2 _ c = o 

 Then f (-P^.-^^l =,-Pi^,+ ^'y' --- 



[p-z' q-z J (p-z)2 ' (q-z)2 



a. Let p = 1, q = — 1, c — 1 



Then X^ (1 f X)2 -f yZ (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 ^^ = 1 . 



L aq J 



aq J I q J 



2 aq 



Circular sections are at z ^ o and z 



1+a- 



2 ao 

 The planes z ^ o, z =: q, z = - — ■ — , z = aq divide every transversal 



i — p a 



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- , y2 



I aqJ L q J 



= C 



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

 tricity : 



( 1 z "1 Ti z ^ 1 ■ , • 



ml ±1 which gives 



I aq J L q J 



z = — ^— and z rz= — ; for similar conic sections. 



m — a m -)- a 



It is then easily seen that the four planes, 



z = q, 



aq (m — 1) 



z = —2: 



m — a 



z = aq, 



_ aq (m + 1) 



m 4- a * 



divide any transversal liarmonically. 



