145 
1903 - 4 .] Dr Muir on the Theory of Continuants. 
and thence, doubtless, divined the generalisation 
U n = H w - B n x • (x -n + + B n>r (x -n + 1)(^ - n + 3)-H M _ 4 - . . • • 
where 
H w = (0 + n - 1 ){0 + n- 3)(d + n - 5) . . . . to n factors 
and 
n(n - l)(n - 2 ) • • • • (n - 2s + 1 
^■ s= ~ 2 s • 1 • 2f~3 s 
The establishment of the truth of this is all that the paper is occu- 
pied with, the procedure being to expand U n in terms of the elements 
of its last row and their complementary minors, thus obtaining 
U n = 6\J n _ x - (n-l)(x-n + 2)JJ n _ 2 
and thence 
U w + | (n - l)(cr - u + 2) + (n - 2)(x - n + 3) - 0 2 J U n _ 2 
+ (n - 2)(n - 3)(x -n + 3)(x- n + 4)U n _ 4 = 0, 
and showing that the above conjectural expression for ~U n satisfies 
the latter equation. The process of verification is troublesome, and 
was not viewed with satisfaction by Cayley himself. 
As a preliminary the coefficients of the H’s in the value of U n 
are for shortness’ sake denoted by A n>0 , — A n>1 , . . . , and for 
the same and an additional reason the coefficient of U n _ 2 in the 
difference-equation is denoted by 
M„, s - j 0 2 - (n - 2s - l) 2 } , 
which is equivalent to putting 
M n>s = (n-\){x-n + 2) + {n - 2)(x - n + 3) - (n-2s-iy. 
The operation to be performed being thus the substitution of 
- A Mj 1 H w _ 2 + . . . . + ( — ) s A M S H n _ 2s + . . . . 
for U n in the expression 
M. 
|e 2 -(»-2s-l) 2 | 
U w _ 2 + (n- 2 )(n - 3)(x -n + 3)(x -n + 4)U n . 
PROC. ROY. SOC. EDIN. — VOL. XXV. 
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