Vk. 1. 



projections on the plane tlirougli the four vertices wiien one keeps 

 in the same direction. To each quadruplet of corresponding rays 

 belongs a hvperholoid of revolution with OZ for axis and circle 

 OJ\OtO, as minimal circle ("cerclc de gorge"), whilst the hyper- 

 boloids of revolution belonging to the various values of y, touching 

 each other according to that circle, form a tangential pencil as well 

 as an ordinary one. Each of those surfaces presents itself twice as 

 bearer of two reguli corresponding to two supplementary values 

 of <f. In this case is a rule what was an exception above ; here 

 the number to be found is infinite, as two lines satisfying the con- 

 ditions pass through each point of/°, the two generators of the surface 

 of this peculiar pencil passing through this point. Indeed, the case 

 of an infinite number of solutions makes its appearance even as 

 soon as there is only one hyperboloidic quadruplet and 1° is at the 

 same time director ray of the regulus determined by this quadruplet 

 as director rays ; then through each point of I' passes only one 

 line satisfying the question. 



To simplify the representation the preceding particular case has 

 been talcen on purpose as regularly as possible. Tiie principal thing 

 is what the figure retains after a projective transformation, that 

 namely the vertices 0^, 0,, 0„ O^ lie in the same plane, that the 

 planes «j, «„ «,, «^ pass through the same point and that all quadratic 

 surfaces touch those planes in the vertices mentioned above ; the 

 regular situation of the four vertices on the common minimal 

 circle is of secondary impoi'tance. 



This leads us to a new ([uestion, viz. whether it is impossible to find 

 four pi-ojecti\ely related pencils of rays where each quadruplet of 



