150 Proceedings of Royal Society of Edinburgh. [apeil 4, 
Tait, it is certainly not applicable to fclie case wliere i is of the order 
n, for taking i — A n, A a given number however large, then q is in- 
definitely near to the very small value e~ A , but nevertheless the last 
term - \(n + i+ 2)q 2 , by taking n sufficiently large may be made as 
large as we please, and the value would thus come out negative. It 
is thus necessary that i should be at least of the order n log n ; but 
it may be of any higher order.. 
Writing for greater convenience r = ne~ i,n (where r is not very 
large) then nq = re~ 1,n — r( 1 -X) if X = l-e~ 1/w , and the formula 
becomes 
A «o< 
n 
e -r(l-X) j~ J 
i+1 —%n y a r n -f i + 2 
»- 2 /« 
:n 
n 
n 
1 11 11 
Here X = if + \ 2 ~3 if an( ^ ex P onen li a l 
= l+rX+ 
n 
r 2 X 2 
1.2 
.. . is thus also expansible in negative powers 
of n } and the formula becomes 
A ? Q S " / r 2 X 2 
e~ r 1 +rX + 
n l 
1.2 
'l . e -Vn_ 
r 7i + i+2__ 2ln r 2 
n 
n- 
viz„, putting for X its value. 
= e (l 
, % 4- 1 — 2 7i . 
+ r . ( — — e~ lfn + 1 - e~ lln 
+ r 
2n 2 
- n - % — 2 
,-2/w 
2 n 2 
i + 1 - 2% 
2?^ 2 
1 _ e -V n ^ e -Vn + _ g-1 fn'J'j 
+ &C.} 
or finally expanding the and taking the whole result as far as 
1 
— : coefficient of r is 
n l 
V 7i 2 n“ )\ n ) n 2n 2 ’ n 2 7 
coefficient of r 2 is 
i + Ji\i + J_ I , -i-h . 
n 2 n 2 ) n 2 v? ’ ~ n 7 i 2 
