3G2 Mr. T. Muir on an Extension of 



elements of the main diagonal which occupy the same rows be 

 constant, except in the case where the first element of all is 

 concerned, and even then the condition hold when this ele- 

 ment is diminished by the constant referred to, then the con- 

 tinuant is unaltered in value by diminishing the first element 

 by the constant, increasing the last by the same, making the 

 elements of the minor diagonal advance a place, and filling up 

 the vacant last place by the element which would have followed 

 had the continuant been of a higher order." A similar state- 

 ment of (A) would be more complicated, but may without 

 much difficulty be framed on the same model. 



Having thus established a relation (of equality) between 

 two continuants, we should expect to find a relation between 

 the two continued fractions corresponding to two such conti- 

 nuants. To this end let us consider the continued fraction 



-^-g — - + b 2 (b 3 —*) 



13 + 



. b n - x {b n — ct) 

 + — £ . 



which 



=r 6l(6l-")KQ3 ^4-Vff hh-h" . . . bn-Jn-bn-l* /3) _-p T 



K(/3 h 2 J h- b 2 a 3 ^A-^3 a . . . t>n-lb n — l>n-l" Q\ 



Using the elementary theorem that 



K(x x *i x 2 z 2 x 3 H # 4 ...)== x^K.{x 2 z -2 x 3 * 3 .o&i . . . ) 4- ZiK(z 3 H x 4 . . .), 



we have 



(6 2 -/3)K(/3 ft a*4-*.«-jg . . . ?>n-lhn-bn-lK fy 



+ K(/3 b A~ h 2* /3 . . . bn-l-bnbn-l ^) 



= (# 2 - j3)K(j8*A-V fi . . m bn-lb n -bn-i« £) 



+ /3K(/3 hh-\ a j3 . . . 6n-A-6«-i« 0) 



+ (6 2 6 3 -6 2 «)K(/3 ^A-^4« £ . . . bn-xbn-bn-^ ^ 

 = 6 2 K(|3 b A~ b 3 a fi . . . bn-lb n -bn-ix ^ 



+ (b 2 b 3 -b 2 *)K((3 Ma- V . . . bn-xbn-bn-^ ^ 



= b 2 K(p + b s - a ftA-VjS...*"^--*--!* j8) 

 by means of (A). 



