152 Proceedings of Royal Society of Edinburgh. [april 4 , 
and similarly 
r 2 = n.n-l . (l - = ti 2 ^1 - . e log ( 1- ^ 
( 1 \ 1 4 £ 1 8 ? 
1 _ JL V 2 (l + X 2 ) ; X 2 - e 2 »* 3 »»- 
• 3 = n. n - 1 . ^1 - = 7 i 2 (l - . e tlog ( 1- ^) 
_ -L ?i _ L 7 li 
X 3 = e 2 m2- • 3 
— M • 
It is now easy to calculate the successive terms r- r 2 - 2 rr x + r 2 , 
&c., and it is to be observed that, in the parts independent of the 
X’s, we have only terms divided by n } n 2 , or higher powers of n : 
thus in r 4 - 4 r 2 r x + 6r 2 r 2 — 4r 3 r 3 + r 4 we have r 4 into 
1- 
44X-4 
We thus obtain the formula 
1 _ _3\ _ _3_ _ 6 
n 
n 2 n 3 
n' v 
1 
+ 
+ 
+ 
r 
r 2 / 1 
172 \ n 
ty" 
1.2.3 
n 
<r> 
ri 1 
-1X0 
-2X x + (l-- 
1 \ n 
X 
- 3Xj + 3( 1 - — ]X 2 - ( 1 - 
n 
1-- X 
n 
oti( - i i - 4X * + K 1 - ih - 4 ( x - 7X 1 
— )x 3 
n ) 6 
+ 1 - 
71 
1 - 
71 
l-l 
n 
IX 
where r = ne~ iln as above, and X 1? X 2 , • • • have the above-men- 
tioned values, the exponentials being expanded in negative powers 
of 71. 
