1903 - 4 .] Dr Muir on the Theory of Continuants. 
143 
He then recalls the further fact that if y 0 , y Y , y 2 , . . . , y n he the 
numerators of the convergents of the continued fraction 
\ \ K 
a i + a 2 + • • • + a n 
there exists the set of equations 
Vo = a o 1 
- «i2/ 0 + Vi =\ 
~ ^ 2^0 “ a 2^1 + 2/2 = 0 
- \Vi - + Vs = 0 
- Ky n - 2 - = 0 , 
and he thereupon draws the natural conclusion that the previous 
result can be applied to the determination of y 0 , y 1 , y 2 , . . . , y n . 
Making the necessary substitution for the u’s and for he of 
course obtains 
y n = a 0 A n ° + J^Af, 
A n ° , being now determinants which for want of Cayley’s 
notation he cannot accurately specify, but which he persists in 
writing in the form 
a n n aAa c 
L!» - 2±«„v 
. . a 
From this result he calculates in succession the values of y lt y 2 , 
y 3 , y 4 ; hut it will readily be understood that the process is neither 
elegant nor short. 
In the remainder of the paper (§§ 4-9) no further use of the 
properties of determinants is made, the contents of the last ten 
pages being such as might appear in any ordinary exposition of 
continued fractions. First there is established the old “rule” for 
writing out the value of above referred to as being given by 
Stern. This is followed by the results 
factor ( - 1)*+*, he takes the further step of moving the row with the index k 
over K-i + l rows, thus arriving at 
^ = - 2 
± a 0 ° af 
' X a i+1 n K ~ X n K n K+X 
-1 a i + 1 • • • « K+1 
Of course there is at the second step the option of moving the column with 
the index i over ic-i + 1 columns, and this Ramus does. 
