226 
Proceedings of the Poyal Society of Edinburgh. [Sess. 
and 
T = 
(2n - l)a(/5 + 2n-2) 
2 n - l 
y(S + 2n - 2) , (2n - 2) { S(y + 2n - 3) + y } 
3 - 2n 2w - 3 
2{/3(a+2w-3) + a} 
5 - 2n 
(2n - 3){a(/S + 2n - 4) + 2/3} 
2n-5 
(2n - 2){a(/3 + 2n — 3) + (3} 
2n-3 
/3( a + 2 n - 2) 
3 -2n 
(2n-l)8 (y+2w-2) & 
2ra~ 1 
= {ab - a(P + 2 n - 2)8(y + 2n - 2)} - (a . (3 + 2n - 4 + 20)(8 . y + 2n - 4 + 2y) j . . . 
{ah - (3(a + 2 n- 2)y(8 + 2w - 2)}. 
If in T 6 we change the sign of /3 and S, which is equivalent to changing 
the signs of the elements below the principal diagonal, the signs between 
the terms of the binomial factors would be plus instead of minus. 
6. If in T & we put : 
(1) b — a, a = f3, and y = S — 1, it reduces to T a ; 
(2) y = ^ = 0, all the factors are alike and we have 
T b = (ab -a f3) n , 
or if, in addition, b = a and a = 3, 
T,=K-^ r . 
(3) y — — a and S = — (3, two factors become alike and 
T & = (ab - a/3) 2 (ab - 9 af3) . . . (ab -2 n- 3 2 ‘af3 ) ; 
(4) a = y — = y and b = a, then 
T„=(<j2_12)( a 2_ 3 2) . _ . («2_2n-l 2 ). 
This, as far as the factors are concerned, is equivalent to putting 
a = f3 = y = S = 1, 
or « = y = and /3 = S = k, where k is any number, 
or p — a-2n—l, y = S= —2. 
or i3 = a = 2n—l,y = S——l; 
