14 TRANSFORMATION OF A ‘DETERMINANT INTO A CONTINUANT. 
Solving now the second set of equations, we get 
* Ej F a sf 
| Dos as "> scuimepla Oyliee Ey | 
D, E, F; ) 0: De | 
‘ : | ase 
D,, 1 E, 1 ra | 4 
SOT ty Te, Me RNG ey a ee 
=(-1)9,F,... Fak; 
but from the first set 
2A =(-1)'F,. : 
Hence 
ad eo FP Pe 
- 2 pe 2 aie reba oy ‘ ; 5 5 : (3) ; 
AD, Dy D,, | ee: 
whete it is to be noticed that every row of the continuant except the first contains an index susceptible 
of 7-1 different values. ; a a 
By giving these indices the proper values, we get, as particular cases, Mr Murr’s formule 
(1), (1’) and (5). 
By solving for «,, we get at once a result like that of Mr Murr in § 6. 
II. The above may be generalised as follows :— 
Noticing that every one of the equations in set (2), except the first, is susceptible of »—-1 different 
forms, and multiplying each of these by one of the arbitrary quantities m,, m,.... ™,,-,, we get, by 
addition, in each case a new equation. Hence (writing also for uniformity’s sake ¢, /, for E, and F,), 
we get the new set : 
ety +f =0 
d,2, 1 + aie +f Lrsy = 0 . a GQ 2 : (4). 
A, lp -l + Cytin =f, f 
Whence, as before, 
i Yoinee Ba 1 
€ \ 3 : A 4 (5) ; 
\d,d, 4, 
hoSs se Wis Ti 
1 2 ! 
where e, f, are the same as E, F,; but d, e, f, now omit only the 7" row, and have each an additional 
first column 77, , M7, 266 Myy_ - 
It is to be observed that in each of the rows of the continuant we have a different set of 7-1 
arbitrary quantities. The identity (5) is therefore one of considerable generality. ' It gives Mr Muir’s 
identity (2) as a particular case. 
