166 AN ESSAY ON THE ROOTS OF INTEGERS, 



Then 3 t V| 2 =9 + ^-J- T | T = 9|f i and deficiency = ^ 

 Let A — 1 1 and assumed Root — 3-^. 



Then 3^ | 2 = 9 -f- ff + t¥* = lOfff- and deficiency T | T . 

 Let A = 12 and assumed Root = 3^. 



Then 3^1 3 = 9 -j- f£ + -&L. = 12^ T and excess = -fJ T . 

 Let A = 13 and assumed Root = 3{-f. 



Then 3ff | 8 - 9 + ff + -B* = 1 W and exce s s = "M- 

 Let A — 14 and assumed Root r= 3-ff. 



Then 3f*- 1 2 = 9 + f£ + -f ff - 14iff and excess ~ ^ 

 Let A = 15 and assumed Root = 3-i-f and r, a Maximum. 



Then 3ff | 2 - 9 -j- VV 8 + iff = 1&H* and excess = fff. 

 And from these examples we may observe- — 



75). That each deficiency Z_ -Jg, as by Par. 75,) and each excess 

 Z_ 1, as by Par. 68,) and also that the deficiencies, though less in value, are 

 yet fewer in number, and the excesses are both greater in number and 

 value than in the examples of Par. 71.) 



76). For still farther illustration, let us take z — a great number 

 = 100, and let us take the three examples in these sets wherein r is a 



