( 209 ) 



XL — The Differentiation of a Continuant. 

 By Thomas Mum, LL.D. 



(Read January 21, 1901.) 



(1) The continuant 



a l 



h 





• 



-1 



°2 



h 





• 



-1 



«3 



h 







-1 



a i 



is conveniently denoted by 



K ( b i & 2 h \ or ( b l b 2 h V 

 yftj a 2 a 3 a 4 / \a 1 a 2 a 3 a J ' 



and, when the elements b v b 2 , b 3 of the variable minor diagonal are each equal to 1, a 

 further simplification is obtained by leaving them out ; — thus, 



K(a,b,c,d) or (a,b,c,d) 

 abed + ab + ad + cd + 1. 



stands for 



(2) The general continuant, K, of the n th order is a function of not more than 2n—l 

 independent variables, — n in the principal diagonal and n—1 in the variable minor 

 diagonal. We thus may be expected to know the differential-quotient of the continuant 

 with respect to an element in either of these diagonals. 



Taking first the case where the element is in the minor diagonal, b h say, we use the 

 known identity 



c 



*i h-i b h &„_! 



a 1 a 2 . . . %_! a h a h+l . . . a n _ t a, 



_ ( &1 &A-1 \ . / ^A+1 • • • K-l \ 



V^ a 2 . . . a h _y aj \a k+1 . . . a n _-^a n ) 



+ b h ( 6 i ••• )■( ••■ b ->) 



\a x a 2 .. . a h _J \a h+2 ... a n _ aj 



to separate out the element in question, and thus see at a glance that 



9K = / b x . . . \ / b n _, \ 



db h \«i « 2 • ■ • 0>h-\) W+2 • • • a n-\ a n) ■ 



In other words, the differential-quotient of K with respect to b h is obtained 

 mechanically by taking K, deleting 6 A _ l5 a h , b h , a h+1 , b h+1 from it, and inserting the 

 brackets )( instead. 



Secondly, taking the case where the element is in the principal diagonal, a h say, we 

 use the same identity at the outset, but go a step further and alter 



/ b x b h _ x \ . / 6 X \ / 6j \ 



\tt-L a. 2 ... a h _ x aj in ° ^ a 2 . . . a h _J h ~ l + \a x a. 2 . . . a h .J ak 



VOL. XL. PART I. (NO. 11). 2 1 



