Factorizable Continuants. 31 



the determinant in which reduces to 



a-36 + 3c 46 + 12c 

 - c a-4:b - 4c 



Finally the operation 



roWj + 4 row 2 



enables us to remove the factor a-Sb - c and to disclose the final 

 factor a-Ab. The result thus is 



(a - 4c) (a - 3c - 6) (a - 2c ~ 2b) (a-c- 3b) (a - 46). 



3. It is important to take notice of the multiples in the successive 

 row-operations, especially as they appear when placed in the form 



11111 



12 3 4 



13 6 

 1 4 



1 



— a form resembling what used- to be known as " Pascal's triangle." 

 In the second place it should be noted that the sum of the factors 

 appearing in the final result is the same as the sum of the elements 

 ■of the main diagonal of the continuant ; the fact being, indeed, that 

 if from the elements of the main diagonal we remove the terms in b, 

 thus obtaining 



a, a-c, a- 2c, a - 3c, a - 4c, 



a;nd then in reverse order re- annex the said terms we arrive at the 

 factors. Lastly, note should be taken that before deciding whether 

 a given continuant is of the form of § 2, and therefore resolvable 

 into linear factors, it is necessary to bear in mind that a factor in the 

 place (r, s) may be transferred to the place (s, r) without affecting 

 the value of the continuant. 



4. Painvin's result is obtained by putting c = - b + 1, and 

 Sylvester's by putting c= -b — 1. 



Curiously enough, however, there are more continuants than one 

 which resolve into exactly the same factors as Painvin's ; thus for 

 the fourth order we have 



a + 3b b 



-6 a+b-1 2b 



-4 a-b-2 3b 



-2 a -3b -3 



\ ( a + 3b-3)(a+b-2) 

 ( (a- 6.-1) (a- 3b), 



so that the continuant here and that instanced in 8 1 are identical. 



