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FACTOEIZABLE CONTINUANTS. 



By Thomas Mum, LL.D. 



(Eead December, 1903.) 



1. The first to note the existence of a continuant resolvable into 

 linear factors appears to have been Sylvester," the continuant in 

 question having for the elements of its main diagonal a constant 

 quantity, for the elements of one minor diagonal the integers 

 1, 2, 3, ... in order, and for the elements of the other minor 

 diagonal the same integers in reverse order : for example 



a 1 . ■ . I 



3 2 I 3 =-(^^l»)(a»--3-). 



. . 1 a\ 



The next was Painvin,t whose continuant had not only the elements 

 of its minor diagonals in equidifferent progression, but those of its 

 main diagonal as well : for example 



(a + 36-3)(a + &-2) 

 (a-b-l)(a-3b), 



— an identity which, though involving an additional variable, does 

 not include the identity preceding it. 



The initial object of the present note is to establish a theorem 

 much more general than either and including both. This done 

 there is then added one or two allied propositions of greater 

 analytical interest. 



2. The ia.-line continuant whose right-hand minor diagonal is 

 b, 26, 3b, ..., whose left-hand minor diagonal reversed is -c, -2c, 



* [Sylvester, J. J.] Theoreme sur les determinants de M. Sylvester. — Nouv. 

 Ann. de Math., xiii., p. 305 (Aug., 1854). 



f Painvin, [L.] Sur un certain systeme d'equations lineaires. — Journ. de 

 Liouville, 2 e ser., iii., pp. 41-46. 



a 



b 



3(6 - 1) 



a-1 2b 

 2(6-1) a -2 3b 

 b-1 a-3 



