78 



It may be obsen-ed that the velocity is tlie same as that acquired by a 

 body falling tlirough the lieight b, and is independent of the radial dis- 

 tance, r. The time of descent is directly proportional to r; and botli are 

 independent of the weiglits. That is, we have tlie tlieorem : 



Motion on till' lii'li.r snrfacp is cijiiii-dJinl tn thnl on tin- inclinr ploni', irften r is 

 constant. 



A Generalization of Fermat's Theorem. 



Jacob Westlund. 



Consider tlie function 



n(A) n(A ) 



n, V n(A) n(Px) n(PO\ 



_n(A)_ d(A) 



/ n(P.Po) \ i n(P,P2...Pi) 



+ ^« + ...J- ....+( -1) « 



where « is any algebraic integer and A any ideal in a given algebraic 

 number field, Pi, ... Pi are the distinct prime factors of A, and n(A) de- 

 notes the norm of A. The theorem whicli we shall prove is that F( « , A) 

 is always divisible by A. 



For the case when « and A are rational integers several proofs of the 

 divisibility of F(«, A) by A liave been given*. 



Wlien A is a prime ideal the function F(«, A) reduces to a"^-'^) — a, 

 which, as we know, is divisible by A. 



Let us first consider the case when A=zP/, where Pi is a prime ideal 

 of degree f, and pi the rational prime divisible by Pi. Tlien 



fSi ffSi — 1). 



Yi'u p^)=« - u 



But 



ffai — 1) 



hence 



(2) F(", P^)=o, modP^ 



Now, suppose A = B.Pj' where B is any ideal not divisible by Pi. 

 Then we can easily derive the following relation : 



tsi ftsi-lj 



¥(a\ B)-r(", B.P^) = F(a'" , B), 



■•' Dickson, Annals of Mathem.atics, 2d Series, Vol. 1, 1899, p. 31. 



