Quotients of Sums and Differences of Perfect Squares 3 



Pr,i /Pr+ti = ar+ r 7-7 , 



Pr+l,l/pr+2,t 



Pr+l,i /pr+2,i = ^r+l"!" T J-7 



Pr+2,i / Pr+3,t 



• • 



Pi-U/Pt =«/!-l+ —7 7-7 . 



pi / pi+l,t 



Now Pt = ^i and A+i,^ = i by [2] , consequently 



pr,t 1 

 ■~^-= ar-\ ^ 



«r+2 + [5] 



, 1 



at-\- 



at 



Dirichlet^ has introduced the symbol (^^/!;,.fi, . . . , ^/) to represent 

 the fraction [5] ; we may therefore write 



—p^ — = ( a,-,a,-+i, . . . , a^ ), r</. [6] 



Similarly we obtain from [4] 



~f^ — = ( at^at-x, . . . ,ar). r<t. [7] 



Pr,t-\ 



Finally, since by the definition of a continuant 



Pr^t=pi,r [8] 



we have also 



iDirichlet, Werke, Bd. 2, S. 141. 



357 



