136 



PROCEEDINGS OF THE AMERICAN ACADEMY. 



If there is a linear relation between f, e, and d, then these two consec- 

 utive generators intersect in an (« — 2)-flat, i. e., they are coincident 

 and we have a stationary generator of the system. If 



then 



e = 0, 



is the equation of a stationary generator of the system. The equation 

 of the developable S,^_i in this case is 



f, 0, ('■'-nj'n-2) ,^ 



2! 



f, 



0, 



o,d,^^-'\S''-'K, 



2! 



0, 



d, 



0, 



0. 



"We see that / is a factor of the left member of this equation. When 

 this factor is thrown out, the residual or proper developable is of a 

 degree less by one than before. The orders of the multiple loci pre- 

 viously given are also reduced, they only holding when there are no 

 stationary i^„_/s in the system. By means of Veronese's formulae we 

 see that when there are ^ stationary F„_/s the order of the A-way 

 developable is reduced from (m — \ -\- \) (m — n + X) to (n — X -{- 1) 

 {m — n -\- X) — (n — X) p. 



6. Tangent flats to a -^-spread where 2 <; p. 



a. Definitions. 



We Lave treated up to this point the various developables that arise 

 from a curve in n-fold space. We shall show now that similar develop- 

 ables do not arise from the consideration of the tangent flats of spreads 

 of more than one way. 



