440 
The equation 
Bn 8 ms ee eee ee bn, 2) = 0 
asl eo wore tees ss n, for roots. If we set 
Sp Sa oe ae et + nt Se vl 
and solve Newton’s formulae* we obtain 
ea wad 0 Oo eer ae 0 | 
S» Si 2 0 <, Sones, eho Le 0 | 
S3 S» Si 3 wusietke, telco 0 
B(k,k) Bien) S4 S3 So Si ENS Cia c 0 k,n — i De oe a 
Sh Sp—-1 Sp—9 S,-3 ey S1 
This determinant vanishes when k > n. 
Inversely, 
Bin) B(O,n) OC Rr Woh ts ee ees 0 
PATEL) — SEX) BOM Pts eek 0 
3B(3,n) B(2,n) UEID\ M kecete Peon eo, 0 
See Me ore) gears) Lacks doce eee ate 
kB(k,n) B(k—1,n) B(k—2,n) ........ B(1,n) 
heii ll 2, coh (CVEMalieh n> 071) 
These sums of the powers of the first » natural numbers are connected 
by the following relations, in which /(//2) signifies the integral part of k/2: 
I(k2) , k 
Sred (anh * ok—la k 
= 2b 1) Sap—1—a= 128 
— 
I(k 2) 
> 2k+1-21 (k k—lo k 
< Y ¢ . Y 
S eS gop 9 = (ot) 2. 
Dye ee A 
whence 
k 
= ee 2k+1-1 os 
= te i| So,—~= 0 where ¢;= = ee when 7 1s even 
1=0 
= —(2n+1) when is odd 
*See, for example, Cajori’s Theory of Equations, pp. 55-S6. 
tStern, Crelle’s Journal, Vol. 84, pp. 216-218. 
