1898-99.] Dr Muir on a Single Term of a Determinant. 457 
And, since the coefficients of all powers of x higher than the 6th 
must vanish, we have also 
° = C +9 ,a - C r+8 ,3 - C +7>3 + 2C r+43 - C r+li3 - C 3 + C 
for all positive integral values of r : and consequently the identity 
(r + 9 )(r + 8 )(r + 7 )-(r + 8)(r + 7 )(r + 6) - (r + 7 )(r + 6)(r + 5) 
+ 2(r + 4)(r + 3)(r + 2) - (r + l)(r)(r - 1) - r(r - 1 )(r - 2) 
+ (r-l)(r-2)(r-3) = 0. 
For the case of n = 3 the three results similar to these are 
V — P _ pi _ p 
3,5 5+2,2 5+1,2 ^5,2’ 
0 = C r+7,2 ~ C r+6,2 “ C r+5,2 + G r+ 3,2 + C r+2,2 " C r+l,S> 
0 = (r + 7)(r + 6) - (r + 6)(r + 5) - (r + 5)(r + i) 
+ (r + 3 )(r + 2) + (r + 2)(r + 1) - (r + l)r. 
There is little hope, however, of generalisation, the difficulty 
lying in the fact that the expansions of 
1 -x, (1 — x){\ - x 2 ), (l-z)(l-z 2 )(l-a 3 ), 
do not proceed in accordance with a sufficiently simple law. Thus 
after 
v = r _ p — p 
3,5 5+2,2 5+1,2 5 2’ 
^4,5 = ^5+3,3 “ ^5+2,3 ~ ^5+1,3 + ^<$-2,3’ 
we find 
^5,5 = ^5+4,4 “ ^5+3,4 ” ^5+2,4 + ^5 - 1,4 + ^5 - 2,4 
+ ^5 - 3,4 “ ^5-4,4 “ ^5 -5,4 “ ^5-6,4* 
(21) The case where 8 = n is interesting, as then the aggregate 
of the first three terms may be replaced by one term, and there 
is no variation in the expression as we proceed from case to case, 
except through the appearance of an additional term. Thus — 
"^ 3,3 = ^ 5.2 — ^ 4,2 — ^ 3.2 = ^ 4.1 — ^ 3,2 
~ ^3,3 
V 4l4 =C 5>4 
V 5t5 =C 7 , 5 +C 4)4 
^6,6 = C 9.6 +C 6 , 5 
7/4/99 2 G 
VOL. XXII. 
