352 PROCEEDINGS OF THE AMERICAN ACADEMY. 



the cones of the system Fj passing through the vertices of the pencils. 

 Then 



Through a line p of V^passsio'teen cones of V^ which have inflexional 

 planes, and sixteen curms of V^ lijing in plaiies passing through p have 

 the line p for inflexional tangent. 



The tangent ttj to V^ at x cuts V^ in a curve which has in .^-a multiple 

 point of order sixteen; hence The -j^^ linear series C^, tangent Vr^ in a 

 line p cut I "5 in a ruled' surface having p for multiple generator of order 

 sixteen. 



6. The locus of all osculating AS'4's of the curves traced on V^ and 

 passing through x and having the same osculating *% is a hyperplane 

 /Sg- To show this let us consider the S^ as defined by the points 



(.r), {dx), (d'-x), (d^x), (d^x), 



which in hyperplanar coordinates are 



(4) f=0, 



(5) ^fdUi = 0, 



(6) %fijduiduj + Ifdhii = 0, 



(15) IfjkdUidUjdUk + 3 If/Puiduj + ^fd^Ui = 0, 



(16) 'Efij^ckidUidUjduj^dui + 6 %fij]flUidUjd'^Uk + 3 'Lfjdhiid-Uj^ + 



4.^fflhi,duj + ^fd'ui = (). 



If d'^ui are allowed to vary and all the other differentials are held fixed 

 by reasoning similar to that previously used, it will be seen that the 

 osculating >S4 will generate an /S's determined by the nine points 



/= 0, /i = ... /5 - 0, ^fjduSuj = 0, 

 ^fijkduidujdu„ 4- 3 X/'ijdrUiduj = 0, 

 ^fijkiduidUjdu^dui + 6 ^fij^duidujd'hik + 3 lAjd-Uid^U/, + 'SfijdUid^Uj = 0. 



We see that this *% contains the hyperosculating /% and the tangent 

 S5. In a similar manner it is seen that the osculating aSVs which have 

 an osculating plane fixed generate an ^S^ which is contained in the Sg and 

 which contains the hyperosculating Sq. If dui defines one of the sixteen 

 principal directions, the above Sg becomes an S-;, etc. 



We saw that a hyperosculating *Sc defined by any direction through 

 a point X hyperosculates along a whole cone of directions passing 

 through X. If, however, the given direction is one of the sixteen prin- 

 cipal directions, the hyperosculating Sq becomes indeterminate ; that is, 



