Art. 1.— T. Takenouchi : 



e a =p\- d , if a SÄ', 



=^" + L— S - J, ifo<X-, Ä-<n, ( 10 ) 



^PofcHr] if a <Ä, ig„. 



This premised, we shall shew that the classes represented by 

 the numbers 



1 + n*, a=l, 2, 3, — , d + k — 1, excluding the multiples of p, (11) 



form a system of bases for 3Ï. Here, as a preliminary for the 

 proof, let us divide these numbers into two sets : 



1+a*, 6=1,2,-4-1, 6*0, (mod. p), (12) 



and 1 + 7T"', c=k, Jc + 1, ■■■h + d—l, c*0, (mod. p), (13) 



and make a few observations. 



The number of multiples of p between 1 and fe— 1 (here as 



well as hereafter, including the both ends) is evidently — — 1 ; 

 hence there are fe— — numbers in (12). The number of 

 multiples of p between fe and k + cl—1 is I — — J — I —J > which 

 is equal to fe— I— J ; for if we put d=k(p — 1)— q (o^q<cp — 1), 

 then d+k=kp— q and consequently I — ■ — - J =fe. Hence there are 

 d—k + — numbers in (13). Therefore there are d numbers 

 in (11). 



Since all b's are not divisible by p, no two of the k— I —J 



numbers of the form hpi* can be equal. Also, they are all less 

 than h+d—l, for 



bp jb =bp jb ~\p-l) + bp il >~ l ^(k-])(j)-l) + (k-l)<:d + k-l. 



It follows therefore that the fe- I —J multiples of p between fe 



and h+d—l are completely exhausted by the numbers bpi*> (6=1,2, 

 ..., Ic-1, b^O, mod. p). 



