150 AN ESSAY ON THE ROOTS OF INTEGERS, 



direct way therefore of proceeding will be to enquire upon these condi- 



(2 a + I) r — r 1 



tions what is the maximum value of this expression . Then 



(2 a + If 



for this purpose let it be put into Fluxions, and it will become 2 ar -f- r — ■ 

 2rr = 0, and hence 2 a -f- 1 = 2 r and r "zz a -f h Hence it appears that 

 the greatest deficiency is when r — a -+■ §■. Substitute this value of r and 



(2a+il)r.-r* (a + \f (a + $)* 



the expression ■ becomes — — — h Now since 



(2 a + If (2a + l) 2 4 (a + |) 2 



a is an integer, so a -f- \ is evidently a fraction. But r is also an integer, 

 and hence can never be equal to a -f- %. That is the value of r can never 

 be such as to render the deficiency a maximum. In other words, the defi- 

 ciency must always be Z. |. 



57.) As an illustration of this, let us take the following three sets 

 of examples. 



1 



Let A — 2 = l 2 -f 1. Then a = 1 and r ~ 1 and 2 a -J- 1 = 2, and 

 assumed Root = 1§. 



Then 1§| 2 = 1 + •§■ + \ — 1|, and the deficiency = ■§■. 

 Let A = 3 = I s + 2. Then a = 1, r = 2, and assumed Root = 1 



Then Tf] 2 = 1 -f -f- + i = 2£, and deficiency = f . 



Let A = 5 = 2 2 + 1. Then a - 2, r = 1 and 2 a -f- 1 = 5 and 



assumed Root = 2|. 



