388 Proceedings of Royal Society of Edinburgh. [sess. 
From the original and all these derived results we thus obtain 
by addition 
(l+/3 1 g + fe« 2 + )-aca?(\+p l x + B#?+ . . . ) 
llfo-+ aez 2 “ Po + z Pi x + + 
and therefore 
1 acx 2 
(1 -Aa + OCB 2 ) 2 = + 2 A* + + W 3 + 
whence with the help of the original identity we have 
2 t)x 
(1 - bx + acz 2 ) 2 = + + + + 
Further, from (I) and (II) there results 
( 1 -bx + «CT 2 ) 2 = A + + • • • 
(I) 
(II) 
(HI) 
(2) Again, it is clear that 
a c b - 
+ , + 
a - a 2 x + a s x 2 - a 4 £ 3 + . 
a 
1 + ax 1 + cx 1 -bx + acx 2 
+ c - c 2 »; + c 3 £ 2 - c 4 sc 3 + . 
+ (b - 2aca?) (p o + p-jX + P 2 x 2 + . . . ') 
= (a + c + fy3 0 ) + ( - a 2 - c 2 + bp i - 2 acp o )x 
+ ( a s + c s +bp 2 -2acp i )x 2 
+ 
where the coefficient of x r is 
( - ) r ~ 1 (a r + c r ) + bp^ - '2acp r _ 2 , 
or, since P r = bp r _ x - acp r _ 2 , 
( - ) r ~ 1 (a r + c r ) + p r - acp r _ 2 . 
Now this is exactly equal to the determinant which is con- 
structed by taking p r and putting c in the place (1, r) and a in 
the place (r, 1), viz. : 
b a c 
c b a *. 
c b a 
c b 
b a 
c b 
(IV) 
