382 Proceedings of Royal Society of Edinburgh. [sers. 
^K n + (& + c)K n _j - (b 2 + c 2 )K m _ 2 + . . . 
as was desired. 
Since s = c(l - a)(l - (3) the result (XVIII) may also be written 
in the more symmetrical form 
M(K ) fi ( C T 
( (y-a)( y -/3)(/5-a) 
1 a n+1 a + a 2 + . . . + a n 
1 f3 n+] f3 + /3 2 + . . . + /3 n 
1 y n+1 y + y 2 + ■ • • +y n 
where a, (3, y are the roots of the equation 
ex* - (c - a)x 2 + (b - a)x - b = 0 ; 
and, noting that the coefficients of this equation are the non-unit 
elements of the determinant 
1 
b - a 
-b 
1 
a - c 
b - a 
-b 
1 
c 
a - c 
b - a 
-b 
1 
c 
a - c 
b - a 
1 
c 
a - c 
which is another form of M(K n ) , we have at once suggested the 
problem of evaluating the determinant 
1 
c 
d 
1 
b 
c 
d 
1 
a 
b 
c 
d .... 
1 
a 
b 
c .... 
1 
a 
b .... 
in terms of the roots of the equation 
axP + bx 2 4- cx 4- d — 0 . 
After doing this, however, we should only have reached a simple 
case of a known theorem of wide generality.* 
* Vide my Text-Book of Determinants, p. 173, § 127. 
{Issued separately January 20, 1905.) 
