4 ' Robert E. Morits 



J^ — == («;.«,.+i, . . . , at), r<_t, [9] 



and 



£^ — = («/,«/_!, ...,«;,), ;-</. [jo] 



pt-\r 



The formulae just deduced lend themselves admirably to the 

 study of continued fractions, in fact many known results have 

 thus been repeated by Sylvester, Giinther, Muir, and others. I 

 \wish to add one application, which opens the way to some inter- 

 esting results in the theory of numbers. 

 Let 



X =^ (^0,<^i-^2, • • • , ^n,y) [11] 



where the ^'s are positive integers, and y arbitrary. Expressed as 

 a quotient of continuants, by [6] we have 



p (aoau^2, • ■ , an,y) 



and then by [4] 



If X is a pure recurring continued fraction, y is equal to x, 

 and [12] gives us the quadratic in x, 



from which 



Pl,n-\^p0.n _[- V ( Phn-l—pO,ny-\-4-PuipO,n-\ . [13] 



that is, 



Every pure recurring continued fraction is equal to some 

 quadratic surd number. 



Similarly, for the mixed recurring continued fractions 

 y = (^1,^2, • • • , ^^> x) 



where ;i: is a pure recurring continued fraction we have 



__ p{.bx,bz^...,bkx') __ xp{bi,b2, ...bk )+/'(<^i,<^2. ■•• t>k-\) 

 ^~~ pib2,...,bkx) ~ xp{bt^...bk)^-p(^b2,...bk-x) 



and therefore, since x is a quadratic surd number, we have 



358 



