158 AN ESSAY ON THE ROOTS OF INTEGERS, 



r 2 z 2 — (2az + \) r r 2 z 2 ?• 



For = • — — — Then- 



(2az + l) 2 (2az + l) 2 2az + 1. 



First. Since r 2 increases faster than r, so by increasing r the ex- 



r z z 1 r 



pression will increase faster than and hence their differ- 



{2az + I) 2 2az+l 



r 2 z 2 r 



ence will increase also. 



(2az+l) 2 2az+l 



Second. Let £ be another value of z greater than the present, and 



r" z" r~ Z; 



let = x and = §. Then — 



(2az + l)* (2a£ + 1/ 



x(2«; + l) 2 %{% aX -f l) 2 r?(2«?+ 1)* 

 = r 2 =: and hence x zz 



? p (a « * + i)* 



4 a* £* " ** -j- 4 a £ s a -f s* 



i x 



4 a* £* * 2 -f 4 a £ 2 « -f £ 2 



Then 4«^f -f 4 a ^r + s 5 2.4 a 2 ^- r + 4 « ^ + £ 2 . For 



4 « 5 ^ 2 f — 4 a 2 £ 2 z", and since by supposition z Z_ £, so 4 a £ £ 2 , or 



4a^: X ^Z.4«^2 X £, or 4 a £ 2 .?, and for the same reason z" Z. £\ 



4 a 2 "Q z % -\- 4 « £ £ 2 -{- s 2 

 Hence a? = i X • or | multiplied by a proper 



4 « 2 ^ 2 + 4«^ + e 



»- 2 £ 2 

 fraction. That is x Z_ |, and consequently the expression 



(2a^-f \y 



r 



increases by the increase of z. And again evidently — which is 



2az -f 1 



the subtracted part of the expression, is diminished by the increase of z, 



that is, as r is divided by a greater number. Then since the increase of z 



