MATHEMATICS: H. B. FINE 549 
Similarly 
k2Pj^(k2py\ (11) 
Let r denote any positive number less than 1, and suppose that ki <po< r. 
Then since, by (5), p^np.<})Q, we also have kipKr, and therefore, by 
(8) and (9), 
k2<pj<r^\ k2Pjr^\ (12) 
Again pj+i^ ^2P/ Pj ^nd therefore 
Pj+i<r^'pj, (13) 
Also, ^2 Pj ^ ^2 p (^2 p)^~ ^ and therefore 
Pi</-'p. (14) 
Therefore, since /*< 1, the sum P1 + P2 • • • p?- increases withy to a 
limit which is less than pr / (1 — r^). 
But pr / (1 - r^) < p when r ^ 1/2. 
Therefore when r = 1/2, that is when the condition knpo< r becomes 
2^1 n^'^yTv 
the point whose coordinates are x^i^ = (4°^ + ) + + . . . + 
(^ = 1, 2, . . . , w), will remain in ^S" asj increases and will approach a 
definite limiting position (ci, C2j • • • ? O in 5 as 7 = 00 . Moreover 
by (12), limy«co^i = 0, and therefore, since the functions are con- 
tinuous,/,- (fi, ^2, . . . Cn) = 0, (7* = 1, 2, . . . w). Therefore the equa- 
tions /; = 0 have the solution {c^,C2^. . . c„) in S. 
Observe that it follows from (5) that when (15) is satisfied, 
2^2 n^^ fip 
The equations /,• = 0 have no other solution than (ci, C2, . . . Cn) in S. 
For suppose that (ci -\- hi, . . . , Cn + hn) represents another such solu- 
tion. Then by developing as in (l) and applying (6) 
Sg7^* + f'^''2^*' (i=1.2.. . .,«), (17) 
where | ^/ 1 ^ 1. 
Solving these equations for the numbers in terms of hi, we 
obtain, by the method used in deriving (5) from (2), 
