6 Art. 1.— T. Takenouohi: 



= [t]-[t] +i .".^ 



Further it is easy to see that the above also holds good Avhen 

 néaécl. Corresponding to a=l, 2,,.,., i— 1, d, we get cZ 2 =ct— 1, 

 d— 2, ...1, 0, respectively; hence in summation (5), the case di<d 2 

 occurs just (d— 1)— d t times. Therefore 



-[t>*,Dt] + <*- 1 >-* 



=(w+cy-^+(^-i)-^i 



—n-l. Q.E.D. 



Next, to prove (3), we need take notice only of those factors 

 on the left hand side, for which e a '^0. Let p " be the highest 

 power of p in e a ', then as in (4), it can be shewn that 



(l + 7 rf"' = l+[r/ l + ^}. 



Now by supposition no two a's can differ by a multiple of d, bonce 

 no two of the numbers a+dg a , a=l, 2, ...d, can be equal. Also 

 they are all less than », since ß„'<e tt - Therefore, if I be the least 

 of them, then 



//(l + -^'=l + {p'.}*1, (mod. p"). Q.E.D. 



1,4 



Thus we have obtained a system of bases for 21. The in- 

 variants are the orders of these bases, viz. 



| a = 1,2, —d, iîdoi, 

 pL" 3 "", where < a=l, 2, ■••« — ], if \<Cné= 3, 



( a=l, if w=l. 



If we denote by r the number of invariants or, the rank of 21 

 according to Frobenius and Stick elberger, n we get 



1) Crello's Journal, Bd. 86. 



