162 AN ESSAY ON THE ROOTS OF INTEGERS, 



8 ar 8 a 2 



and consequently also Z. or J. That is to say, the 



16 a 2 + 8 a -J- 1 16 a 2 



2r 



error of excess committed by assuming as the Square Root of 



4a-f 1 



a f -j- r continually increases as a is greater and greater, but can never 



exceed the limit J. This is also proved by the same consideration of 



Infinity, as in Par. 68. For if a be infinitely great, then, in the expression 



8 a" — 2a 



, 2 a vanishes before 8 a", and 8a f 1 vanishes before 16 a 2 , 



16a 2 -f-8a + 1 



8 a 2 



and hence it will be reduced to z= J as before. 



16 a 2 



71). As an illustration of all this, let us resume the former 3 sets of 



examples of Par. 57,) and suppose z zz 2, so that the assumed Root will 



2 r 



be a -\ and then — 



4a-f 1 



Let A zz 2 and assumed Root - If 



Then l-~-| 2 — 1 -f- f- -f sV = If-f an d deficiency — ^_. 



Let A = 3 and assumed Root — If, and in this case r = 2 a, and is 



a maximum. 



Then 1|- | 2 = 1 -j- f- -f 4f = 3 inr> anc ^ tne excess is -^ 



o 



Let A — 5 and assumed Root = 2-f. 



Then 2f j 2 = 4.+ -§- -J- -^ — 4$-f and deficiency = T 5 T , 



Let A — 6 and assumed Root — 2 f, 



