74 PROCEEDINGS OF THE AMERICAN ACADEMY. 



If we choose the magnitude of j) such that 



p"^ = ml 



f ( ,,-... (30) 



r! {m — r) ! 



where r = 0, 1, . . . . m. 

 Again we have 



and 



[p^-i F-p'] = r [{p""-^ F-p) p'-^]. 

 Hence 



m 

 Solving this for [F ^Z"'] and changing r into r + 1, we have 



^ m[p^-'-Fp^^'], 

 p"^ {m - r - 1) {r -\- 1)' 



a formula holding for r = 0. By continued application of this for- 

 mula we finally get 



Let 



Then 



r! {m -r — l)l{m -1)1 



^ ©. 



(m-1)! 



where r = 0, 1, ,2 , . . m — 1. 



(31) 



24. Space of order 7i = 2 m -\- 1. In this case p"^ is of order 

 n — 1 and hence represents a hyperplane. Since the product p^ is 

 progressive this product must contain p (i. e., p can be expressed 

 as S Xjjt [Ai Akl, the points Ai being contained in <^). Hence, 



[p-jr] = 0. 



