TRANSACTIONS OF THE SECTIONS. 13 



P 



Suppose also that the continued fraction (i) is equal to -=-, and let Ri, R2 . . . -Ru . . . 



be such that 



Ri = «,P-/3,Q, 



R2=«2Rl+^2P, 



R3 = eesRa + ^sR I , 



Rn — ofnRn— J +/3nR„_2, 



then 'R^=q^—p Q, as can be shown by induction ; so that 



Now 



P Pn _ Rn /• ^ 



therefore 



P _Pn . fPn+l Pn\ fPn+i _Pn+l \ 

 Q~9« U«+l 9„/ Un + 2 %+!/ ' 



^_Pn _/-_-)« ^l • • -^n+l I (_-)n+i ^' • • ■ ^n + 2 _^ _ 

 Q 9n ?«7„ + l 9n+l9'n + 2 



from (iii). By equating this value of -=. _-^ to that in (iv), we obtain 



(-)»+iR«=Q ^'---^"+' -Qgn ^'---^"t ^+ (v) 



9n+l 9n+l9»+2 



If P and Q be integers and «i ...««..- /3i ••• /3n ••• he also all integers, then 

 from the equations by which Ri . . . R„ . . . are determined, we see that they also 

 are integers. 



Now in the case of the continued fraction for tan—, 



I3n=-X-, 



and we notice that if x and y be integers, then <«i ...«»... ^1 ... /3n .. . are so too, 

 and consequently (if P and Q are integers) Ri . . . R„ . . . are integers. 



The factor by which ^'•••'^'" is multiplied to obtain ^' • ' ■^'•+ ' is 

 9r qr+l 



= + ^' 



(2r+l)y-x-^-r=}' 



which can be made as small as we please by increasing r. 



We can therefore from (v), Q being finite, make Rn as small as we please by 

 taking n sufficiently large ; but if P and Q be both integers, Rn must remain an 



integer whatever value n may have ; thus if - be rational, 7; ( = tan - ) must be ir- 



rational ; but tan^ =1, so that - cannot be rational. 

 ' 4 4 



The above is in substance Lambert's demonstration ; alterations have been made 

 in points of detail &c., and the notation has been changed. 



It may be noticed that the proof does not (as of course it should not) hold good if 

 P and (^ be infinite integers ; for we cannot make R» as small as we please in (v) 

 if Q be infinite. 



