8 



Robert E. Moritz 



containing- respectively one, two, three, four, etc., cycles of ele- 

 ments. 



We will use the shorter symbols /(o!i, . . .,ai),pi(ai, . . .,ai), 

 P2(ai, . . .,ai),p3(ai, . . .,ai), etc., to represent the above con- 

 tinuants, and generally write /»„(ax, . . .,<?„) for the continuant in 



which the element 2a, recurs 7i times between the initial and final 

 elements a^ and a„. We may then write 



jL^ P(ai, . . .,^0 pijai, . . . ,ai) 



M ~ ~ 





,ai) 



r 



[20] 



p2{(l2, ■ ■ .,^2) * * t)n{^CL2i ■ . .,^2) 



Let us consider the first of the con tinuant quotients [20] for a 

 number whose order is even; in that case its continuant has two 

 equal middle elements, «^^ and by means of [i] we obtain 



p(ai, . . .,ai) =: p{ai, . . .,ak)p(aA, . . .,ai) 



+ p(ai, . . .,ak-i) p(ak-u . . . ,ai) 



=^Vai, . . .,aA)+p^(ci2, . . .,ak-i'), 

 and likewise 



p(.(^2, . . .,a-2) ^^f-i^a-i, . . .,a^)-^p^{m, . . .,a^_i), 

 hence, 



I. Every number zvhose order is even can be expressed as a 

 quotient of sums of squares of positive integers. 



On the other hand, the continuant of a number whose order is 

 odd has a sins^le middle element, ak and in that case 



/>(«!, . . .,rti) = p{ai, . . .,ak)p{,ak-\, . . .,«i) 



+ /(^i, . . .,ak-\)p{ak-t, 

 -= p{a\, . . .,ak)p{ai, . . .,a^_i) 



-\- p{,ai, . . .,ak-{)p{au ■ . 



/>(«!, . . -.^/fe) 



+ 



p{ai, 



akp{.a\s ■ . .,cik-\) ^ p{a.\i • • ■Ak-'i) 



ak 



• .ak-'i) f 

 I 



a^p(ai, . . .,«A_i) — p(ai, 



. .,ai) 

 ,ak-2) 



■ ,ak) J 



— p^{ai, . . .,a^) —p'\ai, . . .,ak-t 



362 



