(4 ) 



(x — a cos (p)" -f- (y — b sin (f)'- 



ab^ cos tp \- { a% sin (p 



+ u — 



\ b~ cos^ cp -\- a^ sirfi (pJ V 6^ cos^ (f -\- o? sin^ (p 



So these equations represent together the rectangular hyperbola 

 projecting itself in /,,' . Putting for simplicity's sake u^ for :t^ + «/^ — ^^ 

 these e(]uations can be reduced to 



u- -\- b^ -\- c^ cos'^ (p — 2 ax cos rp =: 2 6?/ sin (p ^ 



u^ («3 — (;2 cggi fp^ — 2 ab^ X cos (p -\- a-b'^ =^2 a^ by sin <p J 



So elimination of sin (p gives for cos cp the relation 



cos (p \(ifi -\- a^) cos (p — 2 aj] =: , 



which breaks up into 



2ajc 

 cos rp := , cos (p =: ■— ^ . 



M- -|- d" 



As (fi is variable, the first condition cannot serve here. It cor- 

 responds in fact to this, that the two cones with the vertices A^ 

 and 0„, coincide instead of determining together a rectangular 

 hyperbola, when A' is one of the extremities of the minor axis 

 of t ; whilst a similar treatment, in which the parts of cos rp and 

 sin q> are inverted, leads to the relation sin (p ^ 0, corresponding in 

 the same manner to the coincidence of these cones, if A' is one of 

 the extremities of the major axis off. Substitution of the other value 

 of cos (p in the first of the second pair of cone e(|uations gives the 

 result in the form 



4a2;i;2 4 6^ / 



which really represents a surfiice of the eighth order. 



Inversely this simple equation shows that the surface represented 

 by it may be generated by rectangular hyperbolae, by considering 

 it as the result of the elimination of ipi between 



2 ax =. {u~ -\- a") cos tp j 

 2by — {ifi + 62) sin ip \ ' 



which represent for any constant value of ip a rectangular liyper- 

 bola lying in the plane 



2 ax 2by _ ., 



cos ip sin (p 



