247 



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

 there results the well known equation, 



X- — z- = 2y. 

 (b) Let f (x^ y') =x'y' — c = o. 



Thenff^^, AZ.1 

 Ip — z p — z J 



Let p = 1 and q become indefinitely great. 



Then xy = c (1 — z). 

 Rotate the zy axes througli - 4, let c = 1 and 



1 — z = Z, 

 Then x^ — y-^ = 2Z. 

 Compare tliis operation and result with the next. 



The Hyperboloid of One Sheet. 

 Let f (x^ y') ^ x' y^ — c ^ o 



p X q y 

 as above — — ^-^^— :;= c. 

 p — z q — z 



let p = 1, q = — 1 



Then xy = c (1 — z^). 



Rotate xy axes through n- 4, let c = 12. 



Then x^ — y2 -j- z^ = 1. 



A Cubic Surface w^ith Parabolic Sections. 

 Let f (x' yO = y"'' — x" = o. 



Thenf f^^ , A^l = ,q^y' - -P^ = 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 =r 1, q ^ oc, 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 



rr., ^ (^ P5^ qy I p^x^ q^y^ 



Then f 1 -^^^ , -^^ I = —^ -„ — , ,, — c = o 



[p— z q— ZJ (p — z)2 (q— z)2 



a. Let p=:l, qrrr — 1, c^l 



Then x2 (1 -+ Z)2 _ y2 (l _ z)2 = (1 — z2)2 



