GENERAL DETERMINANTS WITH CONTINUANTS. 9 
(2) will be the corresponding minor of the first determinant on the right-hand 
of (1’) ; and from the two identities we shall have by division 
| @,C,d5¢,| |dod,e,| 0 0 0 
| Dycxlgeq| | bots¢q| | aabse,| 9 0 
0 | Cydge, | | GyC3@4| | UqbsCz | 0 
0 0 | daly, | |Agdzd,| | dob,¢,d; | 
| ayboegdse,| | Cobacy| 9 ° ‘ Haabsta | aabsests| 
| Bocactse, | e: | b,ds¢,| | aab,e,| 0 0 
| coh ges| | AoC | | @ab5% | 0 
O [age] |@adyly| | eadscads | 
0 0 | dgbgeg| | dadaeyes | 
Changing the numerator and denominator on the right hand in accordance with 
the theorem of which 
is an example, we have by SyLvesrer’s fundamental theorem regarding the ap- 
plication of continuants * 
| @b2¢,0 45 | | Cob ge, | | Ul se | | bycofl. 304 | 
| byc3,e | =| 46,0504 eee [bye] | Colye, | LS 
a | dsege, | __ 1eaboeg| aotdses| : 
| cibydl, | —L@aPsute| LaaPore| 
| And 3Cser | 
The corresponding identity for determinants of the next lower order is 
| AydyCal, | | Cotte | 
| byeal, | 
[athe | | Beads | Pe Ae 48) 
Ae | dbz | | Cos | 
| dybq¢,| | dod | 
| aadgtl, | 
~ =| 46,45 |— 
| C3, |— 
The general theorem is readily formulated by attending to the rule given in § 2 
for forming the continuant in (1) and (1’). 
§ 7. If in (4) we write b,c, d,¢ for a,b, c,d respectively, a@ for m, and 5, 1 
* Phil. Mag., 4th ser., vol. v. pp. 446-456. 
