164 AN ESSAY ON THE ROOTS OF INTEGERS, 



Let A = 14 and assumed Root — 3-t-f . 



Then 3~fj[j 2 = 9 + -£-£ -f f$4 = 14 T VV and excess = T W 



Let A — 15 and assumed Root = 3±-f, and in this case r zz 2 a and 

 is a Maximum. 



Then 3~h| 2 = 9 + ff -j- fH = l^V and excess = T ^. 

 And from these examples we may observe — 



72). That each deficiency Z. T V according to Par. 69.) 



73). That each excess Z. i according to Par. 70, for even in the three 

 cases where r is a Maximum, and consequently the excess should, by 

 Par. 68 and 71), be greatest, the excess is 



When a — I, only -/ y . 



When a — 2, it is greater, and becomes ff. 



When a == 3, it is still greater, and becomes -^ w . 



And we may hence also observe, that the excess increases with the 

 increase of a, as by Par. 70). 



74). For more illustration, let z be taken ~ 3, and let other things 



3 r 



remain the same, and then the assumed Root will be a -f ■ and 



6 a -|- l 



the deficiency must be Z. or - 1 \, as by Par. 64.) Then the same 



4r3 2 



