10 Art. 1.— T. Takenouchi: 



-(*-Df])-(Df]-[£l) 



^-«([^•]-W)-M[^r]-0 



=*-l. (18) 



Next, since all the numbers c and fyp' 4 taken together constitute 

 the complete succession of d numbers, fc, fc+1, ..., &+d— 1, we 

 have, supposing n>fc, 



i[ ^ ]+i[ ^ ]=i l <i[ w ] 



=i' r w ~ a/ i + v r "~ ( <? + g ' / ) '] 



».d L d J l.T-i L et J 



hence by (5), —n — Ti. (19) 



From (18), (19), we get (10). Also the proof of (17) can be 

 obtained from (18), by replacing k by n. 



To prove (15), let p 0i and p<>° be the highest powers of p in 

 e b and e ' respectively (supposing e b '=k0, e c '=f=0.). Then, as in (7), 

 (8), (9), we get 



(l + ; rf'=l + {p^], if^<;„ j 



= l + {p^' 6+ ^-^}, if^y,, I (20) 



(H-^) e/ =l+{p c+ ^3, - ) 



and here bp 9!> <zi, while bp 3b -\-{g h —j b )cl^'k, c+g c d^Jc. 



As remarked before, the numbers bpfo are all different from one 

 another. And, since the numbers &/*"> and c taken together 



