12 Robert E. Morits 



The values of the constantly recurring continuants /(2<2i, . . .,«^), 

 p{2ai, . . .,ax;_i), /(as, . . .,ak), p{az, . . .,a^-i), p(a2, . . .,^-t). 

 /'(a2, . . .,«>6-i) are computed once for all, and then [23] and [24] 

 are applied alternately in the computation of successive quotients. 

 A different set of recurrence formulae similarly obtained can be 

 emploj^ed in the computation of successive quotients when the 

 number JV is of odd order. For example, 



p( 10,1,1 1,2,2,1,1,1,10) 



11-5 = - r ■ 



/'(i,i,i,2,2,i,r,i) 



/>(«!, . . .,ak) = 85, p(a2, . . .,ak') — 8, 



pias, . . .,«^) = 5, p(2ai, . . .,ak) = 165, 



p(ai, . . .,a^_i) = 32, p(a2, . . .,ak-i) = 3, 



pias, . . .,ak-i) = 2, /(2<2i, . . .,ak-i) = 62, 



and now by successive substitution in [23] and [24] we readily 

 obtain 



^^ ^ 85-4-32^ ^ 16 009^—489^ ^ 131 952'+49 579^ 

 8-^+ 3"^ I 506-^— 46-' I2 4i3»+ 4664^ 



24 845 978^ — 758 918^ 204 789 589'+76 946 640^ 



~ 23373132—713932" 19264984^+72385312 



^ 38 560 973 865^—1 177 841 225^ 

 3627511282^ — 1 10 801 982"^ 



^ 317 833574080^+119421 2348592 

 298992675812+ II 234 204 776"'' 



59 846656 2844532 — I 828 010340 118'^ 



:=: etc. 



5629899846977-— 171964747457^ 



Of special interest is the inquiry under what conditions 

 p(ai,...,a\)/p(a2,...,a-2) represents an integer. Let us suppose all 

 the elements but one of the concinuant given. Let the variable 

 element .r occupy the >('th place. The problem then is, for what 

 integral values of ,v will 



P(ai, . . .,aji-.i,x,ak->ri, . . .,«^+i,:i-,<r^_i, . . . ,ai) 



P((^-2 a^-i,x,aij^i, . . . ,ai-^i,x,aji-i, . . . ,a-2) 



represent integers? 



366 



[25I 



