a Theorem in Continuants. 363 



Again, 



(A a -«)K(j3 6 3*4-* 3 * /3 . . . $»-i&»-&»-i* |3) 



+ K(j3 6 A-V |3...6»-i6n-4ii-i« £) 



= (6 2 -«)K(/3 W«-*i« /3 . ,.*it-x»»-«-i« /3) 



+ /3K(/3 J 3 J i-V /3 . . . &»— ift»— 6»- 1* £) 



+ (^3- V)K(/3 *A~V 6 . . . *n-lln-bn-l* ft 



= K(/3 + Z> 2 — « 6 A- 5 2 « |3 6 A-V /3 .. . *»-i&*-'&»-i« 0) 



= K(/3 + ^ 2 -^3 6 3 2 -^ /3 + A 3 -^ 2 -^. . . V-a^ p + b n -*) by (A). 



Now the ratio of the two expressions here transformed becomes 

 on dividing by K(/3 6 A-A« ; £...£), 



h 



and the ratio of the two results is 



0?i{On — x) 



Hence we have an identity of the kind surmised (i. e. connect- 

 ing two continued fractions), viz. 



b 2 (b 2 -a) 6g(6> _ a) _ (b 2 -/3)F + b,{b 2 -x) 



• + b n (b n — u) 

 P + b n -u 



m=%+¥^, = '*+r j - (B) 



F being the continued fraction given above. 



Increasing both sides of this by — - b 2 , we have 



/3 + a j b 2 (b 2 -*) ( _ _*-$ F-b x 



P+.b 3 — b4+. b n (b n — a) 



