of the Unitates of Powers and Roots. 119 



Example II. — Find some of the values of 8 in which \/3 lias 

 an integral imitate. 



The values obtained from the formula 8 = n m — q, by substitu- 

 tion of integral numbers for n, are 6, 13, 22, 33, 46, &c. 



Example III. — v^35 has integral unitates in the following 

 systems — 14, 29, 47, 65, 86, 109, &c, of which the first two 

 refer to the expression U 5V /n 35. 



Example IV. — Integral unitates to the square root of 499 or 

 of the unitate of 499 are found when the base of the system of 

 imitation is one of the following numbers — 30, 77, 126, 177, 

 230, 285, 342, 401, 462, 525, 590, 657, 726, &c. 



Example V. — The imitation bases which give integral results 

 in the case of ^47 are 17, 78, 169, 296, 475, 682, 953, 

 1284, &c. 



4. Other values of 8 may be found, which furnish integral values 

 for the unitates of a given surd, in those systems which are exact 

 divisors of the systems found by means of the formula 8 = n m — q. 

 For instance v^3 has an integral unitate to the base 11, and 11 

 is an exact divisor of 22 and 33, both of which are found in the 

 values of 8 obtained from the formula 8 = n <2 — 3. Again, 23 is 

 an exact divisor of 46, and accordingly the unitate V^s/'S is in- 

 tegral. This plan can always be relied upon; for it is evident that 



7l8 



— will give the same unitate to a given number as 8, 



5. Unitation squares may be used to obtain collateral proofs 

 of the imitates of the roots of numbers. For instance, in the 

 unitation square of the powers to the unitation base 7 in the 

 order a 3 , a*, a?, a), a* 2 , a 4 , a s , &c, the unitates that continue the 

 law of series are 1*, 2*, 4% and 7", It would appear at first 

 sight that, in the case of square roots, unitation squares give only 

 one of the two roots ; but it is not so ; for in general U 5 a* = U 5 ( + ah} 

 or V s { — a?); that is to say, in the above instance, U 7 v/2 = 4 or 

 — 4 = 4 or 3*. 



6. The unitation square for the base 11, in the order a*, a?, 

 a?, a} , a 2 , a 4 , a s , a 16 , &c, is given below; in the case of the square 

 roots the whole-number unitates are comprised in the num- 

 bers 1, 3, 4, 5, 9, 11, and — 1, -3, —4, —5, -9, -11, or 10, 

 8, 7, 6, 2, 11, the corresponding natural numbers being 1, 9, 5, 

 3, 4, 11. For instance, U 11 v / 5=4 or —4 = 4 or 7. 



* Phil. Mag. S. 4. vol. xlvi. p. 38. 



