CHAP. Hi] SYMMETRIC FUNCTIONS MODULO p. 103 powers of a primitive root h of p in a rectangular table of t rows and r columns, where tr=p — 1. For p = l3, h = 2, 2=4, the table is shown here. Let R range over the numbers in any 124 column. Then 2R and Sl/fi are divisible by p. If t is 8 36 even, Sl/JB is divisible by p2, as 1/1+1/8+1/12+1/5= 12 11 9 132/120. For t = p — 1, the theorem becomes the first one 5 10 7 due to Wblstenholme.267 Generalizations are given at the end of the paper. N. Nielsen311 proved his286a theorem and the final results of Glaisher.294 Nielsen312 proceeded as had Aubry301 and then proved (p-D/2 tt —3 s2n+1=0 (mod p2), S j*sO (mod p), lgn:S ^~. y-i & Then by Newton's identities we get Wilson's theorem and Nielsen's306 last result. E. Cahen313 stated Nielsen's286" theorem. F. Irwin stated and E. B. Escott314 proved that if S,- is the sum of the products j at a time of 1,1/2,1/3,.., 1/J, where t= (p-l)/2, then 2£2-/Si2, etc., are divisible by the odd prime p. '"Oversigt Danske Vidensk. Selsk. Forhandlinger, 1915, 171-180, 521, "Wd., 1916, 194-5. 813Comptes Rendus Stances Soc. Math. France, 1916, 29. »"Amer. Math. Monthly, 24, 1917, 471-2.