( 342 ) 



isotropic points / ;uhI -/. So the rurvo is ol' oi'dor four. i.e. ralioiial. 

 This I'csiill is nieiilioncd. in I he '■Kiicvklopadic der Malheiuatischeii 

 Wissenschaften" 111. p. KM. II(t\ve\er, if we consnlt in the same 

 woi'lv the theorv of the (piadratic surfaces we find no e^■idence oi' 

 an attempt to determine ihe locus of the principal axes of the sur- 

 faces of a pencil. The preseiU writer makes it his aim in the 

 following' to puhlisii some invesliüations on this locus. 



2. We presu|)pose a simpler special case of the i)encil and we 

 take a pencil (tf conceuti'ic (piadratic cones, of which the locus of 

 the |n-incij>al axes is a cone the order of which can l)e determined. 

 Let US suppose to tlii-^ end the section of one of the cones with the 

 plane at intiuitv : the conic formed in this way determines with the 

 isotropic circle a common autojtolar trianiile and the \ei-lices of that 

 triangle detei-mine the dii-ections of the principal axes of the cone. 

 From this follows: 



The i)rinci|ial axes of all Ihe cones of the pencil cut the j)lane 

 at intinit\ in the ^erlices of ihe common aulopolar triangles of the 

 conies situated in this ])lane and the isotro|Hc circle, 'i'liese lri|>lels 

 of points form Ihe .lacohian curve of the net of conies determinei! 

 l)v two of Ihe conies and Ihe isotropic circde. 



So Ihe cone of the axes is a cone of ordei- three cutting the plane 

 at infinitv in Ihe just mentioned .lacol)ian cur\e. 



To realize the position of the ])i'inci|>al axes of this cnhic cone 

 we choose a generatrix a^. If we assume a jtlane through Ihe vertex 

 normal to o^ this will cut Ihe cone according to three ra\s ^/.j, ^z^, />,; 

 (h and ^/a are normal to eacdi oilier, h^ lielongs to an other Irieder 

 of axes, obtained Iw assnnung through tlu^ vertex a |)lane noi-nial 

 to />! : this i)lane passes through a^ and cuts the cone moi'eover in 

 the two iirincijial axes />,, and />,, normal to each other. 



As a rule this c(me will not have a nodal generatrix, so it will 

 not be rational. 



o. Suppose a pencil of quadratic surfaces be given. Out of a 

 point () in space as vertex we construct the parallel cones of Ihe 

 asymptotic cones of the various surfaces; in this manner a pencil of 

 cones is formed, with respect to which we can consti-uct the cone of 

 the axes. The Irieders of axes of this cone are pai-allel to the Irieders 

 of axes of the surfaces of the pencil. 



Let further a skew cubic fp^ be constructed, which is the locus of 

 the centres of the surfaces of the ])encil : if then out of each centi-e 

 a trieder is constructed parallel to the corresponding trieder of axes 

 of the cone, the surface is formed which is the locus of the principal 

 axes. From this ensues : 



