( 5 ) 



For tp = rp ± 180° this equation transforms itself into that of /a'. 



5. We signalize here some particularities of the found surface. 



a. The intersection of the cone a-- -f y^ — s^ = with the plane 



at infinity is a fourfoM curve of the surface. For the substition of 



X := p -\- £ COS A , y ::zL q -\- z sin X 



into the equation 1) yields an equation of the fourtli degree for z. 

 It is reduced to a cubic equation under the condition 



( p cox k -\- (] sin A)^ = a^ cos" X -{- b^ sin" X 



and to a quadratic equation for 



p cos X -{- fj nin A ^ 0. 



So the four tangent planes in the point x ^ z cosX, y ^= z sin X at 

 infinity are represented by 



;r cos X -{- y sin A — z := ± \/a^ cos^ A -j- 6^ siti^ X , 

 rf' cos A -\- 1/ sin X — c = , 



the last of these counting twice. The deduction of the envelopes of 

 these planes for various values of A shows that the surface 1) is 

 touched at infinity by the developable surface or torse circumscribed 

 to the tangential pencil of quadrics, to the four quadrics flattened to 

 conies of which belong the fourfold conic of 1) and the ellipse e; 

 moreover it is osculated at infinity by the cone x^ -\- y^—z^ = 0. 

 So this cone intersects the surface 1) in a curve in space of the 

 sixteenth order to which the fourfold conic of 1) belongs six times. 



The completing curve in space lies on the cylinder—--] — —=1, 



in which 1) is transformed for m^ = ; this cylinder meets the 

 surface 1) in another curve of the twelfth order, etc. 



b. The intersection of the surface 1) with each of the planes 

 Z X, ZOY consists of four straight Hues and a rectangular hyper- 

 bola counting twice. So ?/ = yields the four lines x ± z ^ ± a and 

 the hyperbola x"—z'^ -{- a" = 0, and likewise .^ =; yields the four 

 lines y ±1 z ^n ±b and the hyperbola y"- — z'^ -\- b~ =i^. So 1) con- 

 tains besides the fourfold conic at infinity still two double conies; 

 moreover it bears eight right lines, viz. the four pairs of lines into which 

 the rectangular hyperbolae of the vertices of « are degenerated. 



