1898 - 99 .] Dr Muir on a Single Term of a Determinant. 459 
and 
^13.13 = ^25.12 “ ^24.12 “ ^23.12 + C 20 .i2 + C 18il2 ~ C 13il2 
= ^23.13 + C' 20(12 + C lg)12 — C 13il2 . 
By proceeding as far as the factor 1 - # 20 it is found that 
V _ p ip i p _p _ p. j_ 
n,n~ 2n- 3,n^ ^2n-6, n-1 T yy 2n-S,n -1 ^2n-13,w-l 2n-l6,n-l T * ' * 
(22) If 8 he less than n, the last term V of the differ- 
ence-equation (§ 17) is of no moment, and the equation to he 
satisfied is 
y =v + v 
ii,8 n,8-l^ n-1, 8’ 
Now it is easily shown that one form of solution for this is an 
aggregate of multiples of combinatorials, that is to say, we may 
have 
n,8 1 4 2 ^w-}-5+0!'2>5+&2 
where A 1? A 2 , . . a v a 2 , b l5 b 2 , ... . are constants with 
regard to n and 8. Thus, on substitution in the right hand side of 
the difference-equation, we have 
» i a-l+ M — 1,5 1 W-f*5 — — 1+&1 2 71+5 — — 1 + &2 
4 i 4 2 - l+tt 2 ,5+&2 
and this by repeated applications of the theorem 
%,q G p-l,q + ^p-l,q-l 
is clearly 
and 
_ a .0 _i_ a .p i 
1 2 
It follows from this that if in the tables of values of Y . the 
n,8 
diagonal 
0, 0, 1, 5, 22, 90, 
which consists of the values of V be expressible as an aggregate 
