6 Robert E. Morits 



Moreover, if these conditions prevail, the rational part of [14] 

 vanishes, for in that case, pun-\^=^pn-i,\=p2,n- We have therefore: 



The necessary and sufficient conditions, that [13] may represent 

 a pure quadratic surd, are 



^=0, ax=an, a, =an-i, as=a„^2, • • • ,'^k-\-i=(in-k- 

 With these conditions, the recurring continued fraction becomes 

 x=(o,ai,a2, . . • , ^2,^i>-i'0- The last few terms of this fraction are 



.1 .1 ,1 



^iH 1 ^1 



1 



x+ -^^ ' 1 ^^' ' a2 + 



ai" 



With this equation [14] may be written 



x=(o,ai,a2,. . . ,^2,2^1)==.! ^'^ » ri6l 



* * \ pii *- -' 



or its reciprocal 



where the stars denote the beginning and end of the recurring 

 elements, and for the sake of convenience ^1,1 and ;52,2 have been 

 written iorp^aia-?^. . .,aia^) and/> ( ^2,^3, • • • ,^3,^2 ) respectively. 

 We now need only to take the result just obtained, namely, 

 that every recurring continued fraction which represents a pure 

 quadratic surd takes the form [17], in conjunction with the 

 theorem that every pure quadratic surd can be expressed as a 

 recurring continued fraction, to arrive at the equations. 



, L>M, [18] 



or 



L ^ Pui 



M p2,2 



for every non-quadratic rational number N^^L/M, that is ; 



360 



