1887.] Professor A. Cayley on a Formula for A w 0 i /n\ 151 
whence the formula becomes 
1 - 1 - i' 
A”0*’ r f , 1 + At r 
-~ r J 1 + r — /- 1 
n 2 
= e 
x 
n c 
1.2 
n 
+ . . 
It seems to me that the correct result up to this order of approxi- 
mation is 
A w 0* 
n~ 
= e 
1 + r 
w- 
1.2 
1 
+ 
n n c 
My investigation is as follows : we have 
A"0*’ 
= 1 - 
n 
n 
1- 
1 
n 
+ 
n. 7 i — 1 
T72~ 
1 - 
n 
+ . . . 
the series being a finite one, but the number of terms is very large. 
But observe that, however large n is, we can take i so large that the 
second term n(\ — — ^ may be as small as we please ; taking this term 
to be of moderate amount, say = r 1 , the subsequent terms will be not 
ry* 2 ry* 3 
very different from -A— , — i — , . . . and the approximate value 
1 . 2 1.2.3 
is 1 - - &c., which is a convergent series having its sum 
= e~ r \ To work this properly out, I represent the successive terms by 
v r, 2 ? 3 go that the series is 
19 1.2’ 1.2.3’ “ * 
= 1 - r-L + 
To 
~x + • • 
1.2 1.2.3 
Taking r a value at pleasure not very different from r 1} and multi- 
plying by 
(l = )e~ r . e r = e- r .(l+r + ^ 
the sum is 
= e" r . | 1 + (r - jq) + -t- r 2 ) 
+ T^r^^’ 3 “ 3? ‘ 2r i + 3?t 2 ~ r s) + • • • 
1.2.3 
Assume nowr = ne~' ln , we have 
^1_J_^ =ne X ° s ^ «) = r(l+X 1 ); X 1 = e 2 »s 3 «s 
i\ = n 
