247 



Rotate the xy axes through - 4, then the zx axes in the same way, and 

 there results the well known equation, 



x- — z- = 2y. 

 (b) Let f (x^ yO = x' y' — c = o. 



Thenff^^,^^! =-P^ ^l^_c=^^ J__c=:o. 

 Lp — z p — z J p — z q — z p — z z 



q 



Let p = 1 and q become indefinitely great. 



Then xy = c (1 — z). 

 Rotate the zy axes through tt 4, let c = 1 and 



1 — z = Z, 

 Then x^ — y- = 2Z. 

 Compare this operation and result with tlie next. 



The Hyperboloid of One Sheet. 

 Let f (x^ y') ^ x^ y' — c ^ o 



p X q y 

 as above — — -^^^l^ := c. 

 p — z q — z 



let p = 1, q = — 1 



Then xy = c (1 — z^). 



Rotate xy axes througli tt 4, let c ^ 1/2, 



Then x^ — y^ -|- z^ = 1. 



A Cubic Surface with Parabolic Sections. 



Let f (x' y^) = y'^ — x' ^ o. 



rn,, j^rpx qyi q^y^ px 



Then f -^^- , -^^ = 7^^-^ — -- - = o. 

 Ip — z q — zj (q — z)2 p — z 



a. Let p ^ 1 and q = — 1. Then 



y* (1 — z) = X (1 + z)2, one of the cubical warped surfaces. 



b. Let p = 1, q =: 00, then y- (1 — z) ^ x. 



c. Let q ^ 1, p = X, then y^ = x (1 — z)^. 



Biquadratic Surface with Hyperbolic Sections. 

 Let f (x' y') = x'2 — y'2 — c = o 



mi J- f v^ qy 1 p^x2 q^y^ 



Then f | -^— , -^^ | = -^ -„ — , \. — c = o 



[p— z q— ZJ (p— z)2 (q_z)2 



a. Let p^l, qt= — 1, c = l 



Then x2 (1 -| zf — j^ (I — z)2 = (1 — z2)2 



