152 AN ESSAY ON THE ROOTS OF INTEGERS, 



Let A — 14 = 3 2 -f 5. Then a == 3 r =. 5, and assumed Root = 3-f . 

 Then 3f] " — 9 -f- 3 T n +11= 13-f-f-, and deficiency — if. 



Let A =; 15 — 3 2 -}- 6. Then «= 3r-0, and assumed Root — 3-f- . 

 Then 3f~| 2 = 9 -f V 6 + if == I44fi and deficiency — T V 

 From these examples we may observe— 



58). That the deficiencies are in every case Z_-*- according to Par. 57). 



59). That when the remainder is very great or very small, the defi- 

 ciency is small, but when the remainder is a medium, that is, as it ap- 

 proaches to be equal to a -\- h, the deficiency becomes great, and is great- 

 est when the deficiency is — a, and == a -f 1. That is, it is greatest in 

 the 3d set of cases when /• — 3 and — 4. In the 2d set of cases when 

 r — 2 and — 3. And in the 1st set of cases, of course when r — 1 and — 2. 

 This observation is confirmed by the examples of Par. 44). For in the 

 1st and 4th examples where r — 1 and — 728 that is very small and very 

 great, the deficiency is small, and in the 2d and 3d examples when r =z 

 332 and — 333, that is, a medium, the deficiency is great. 



60). And that when A is equally distant from a" below, and (« -j- l) e 

 above the deficiency is equal. That is the deficiency is equal when A is 

 equal to a" -f- 1 and {a -\- l) c — 1, and the deficiency is equal when A is 

 equal to a 2 -j- 2 and (a -j- I) 2 — 2, and the deficiency is equal when A is 

 equal to a 2 -J- 3 and (a -j- l) 2 — 3, and so on. Thus — ■ 



. In the 1st set of Gases. ■ 



When A — 2 — l 2 + 1, and when A = 3 = 2 2 — 1, the deficiency is 

 the same, viz. f . 



