550 
MATHEMATICS: H. B. FINE 
n 
and therefore, if h\ ^ 0, 
> 2 
(18) 
Hence (ci -r hi, . . . , Cn + hn) Kes outside of S. Therefore the exist- 
ence of a second solution in S is impossible. 
The equations {b) will also yield a sequence of approximations to 
{cy, C2, . . . , c„) similar to the sequence (xi \ Xi \ . . . xi^^) if instead of 
{xi^\ xf\ . . . x^^^) any otlter point in 6* be taken as the point of de- 
parture. For let {af\ af \ . . . , ) be any point in and ap + S 
4^^, (^ = 1, 2, . . . , w), the numbers determined by the equations 
h{a'^\ai^\ . . . ,aF) + ^^fi^' =0, (» = 1, 2, . . . , »), 
(7+1) 
+ d^^=c,-a^\ (^ = 1,2, 
fory = 0, 1, 2, . . . successively. 
Then since 
we shall have, on developing as in (1) and taking into account the 
equations which determine the numbers 
where each x^^'^ lies between a[^^ and c^, (^ = 1- 2, . 
From (19) it follows, as in the proof of (5), that 
oy+i ^ ^2 
where % = ] [^^F^l^ ^^^^^ above, ^2 = 
n). 
n), (19) 
(20) 
From (20) in turn it follows, as in (10), that 
therefore, if ^2 %< 1, that lim;=co 5; = 0 and lim^^oo (^P\ • 
But 60 is the distance of the point {aT ^ • • • ? from the 
point (ci, C2, . . . , Cn), and ^25o<l if 60 < 1/^2; therefore, by (16), if 
(21) 
