Further Note on Factorizable Continuants. 



185 



be resolvable into linear factors by means of the set of column- 

 multipliers (S), then 



Pi + 7i = ft + Ts = Ps + 7s = 



/3 2 + y 2 = A + 74 = 06 + 76 = 



a-) 



Of the truth of this there is no doubt, but a really short proof of 

 it is much to be desired. 



Calling the constant in the case of the odd suffixes s, and in the 

 other case t, we may consequently write the continuant in the form 



a 



ft 



s-ft a+p 



t - /3, a + q 



ft 



and when we perform upon this the first operation 



col x + col 2 -f col 3 + 



we find, by reason of the removability of the factor « + /3 T , that 



a =p- x -t+p a -P 3 , 



and that the resulting determinant can be lowered in order, being 

 in fact 



t ' 3 



a + pD I - s - j3 2 



ft 

 ft 

 ft 



n 



a + (j 

 s - p 3 



a + ) 

 t 



l\ 



n-l. 



By the performance on this of the second operation, viz. 

 col r - col 2 + 2 coL - 2 col 4 + 3 col 5 - 3 col 6 + 



and the use of the fact that another factor is thus made removable 

 we obtain a set of n - 2 equations, which turn out to be sufficient 

 for the complete solution of our problem. The equations involve 

 all the /3's and s and t, that is, in all n + 1 quantities ; and the key 

 to the proper mode of treating the set lies in the selection of the 

 three quantities in terms of which the others are most conveniently 

 expressible. These three are s, t, /3 T ; but even with this fact in 

 possession the solution merits some attention in detail, 



