ON THE THEORY OP NUMBERS. 



125 



what follows, that the units of which we speak have been thus reduced to a 

 real form. 



For all values of X greater than 5, the number of systems of fundamental 

 units is infinite. For if u v u 2 , . . . w^_i still represent a system of fundamental 

 units, it is evident that the system E„ E 2 , . .. E^_i, defined by the equations 



E(i» i) 



E 2 =«<*•» 



,(1,2) 

 .(2, 2) 



X.U 



(1, (*-D 

 /x-1 ' 



(a, M-D 



>• 



(A.) 



E^^-'-'Vr 1 '^ 





is also a fundamental system, if the indices (1, 1), &c. be integral numbers, 



and if the determinant 2±(1, 1)(2, 2) (ju— 1, jx— 1) be equal to unity. 



And conversely, every system of fundamental units will be represented by 

 the equations (A.), if in them we assign to the indices (1, 1), (2, 2), &c. 

 all systems of integral values in succession consistent with the condition 

 2 + (l,l)(2,2)(3, 3)...(/i — l,/i— 1)= + 1; so that a single system of fun- 

 damental units represents to us all possible systems. 



We shall also have occasion to allude to independent systems of units. A 

 system of ^—1 units, u lf u 2 , .. w^_i, is said to be independent when it is 

 impossible to satisfy the equation 



2 3 



"3 



.11 



»(X_1_ 



= 1, 



whatever integral values are assigned to the indices «j, n 2 , n 3 , ... w^-i. 

 The equations (A.) will represent all possible systems of independent units, 

 if we suppose that in them the indices (1,1), (2, 2), (3, 3) . . . receive all 

 positive and negative integral values, subject only to the condition that the 

 determinant A = 2 + (l, 1) (2, 2) . . . (/*— 1, ft— I) must not vanish. Every 

 system of fundamental units is also independent; but not conversely. Every 

 unit can be represented as a product of the powers of the units of an inde- 

 pendent system ; but if the system be not also fundamental, the indices of the 

 powers are not in general integral, but are fractions having denominators 

 which divide A. Lastly, if c^a), c 2 (a), .... c^^a) be a system of inde- 

 pendent units, the logarithmic determinant 



L.c,(a), L.c 2 (a), L.^_j(a), 



L. Cl (*y), L. c.£x<), L. <v_,(V), 



Wa^ ), 



-2. 



L.c.,(a v ), L.c M _,(a v ), 



in which y denotes a primitive root of \, is different from zero; and con- 

 versely, if the determinant be different from zero, the system of units is inde- 

 pendent. For all systems of fundamental units, the absolute value of the 

 logarithmic determinant is the same; for any other independent system, its 

 value is A times that least value. The quantities denoted by the symbols 

 L. Cj(a), L. c 2 (a), &c, are the arithmetical logarithms of the real units ^(a), 

 &c, taken positively. 



43. Gauss's Equations of the Periods. — In Gauss's theory of the division 

 of the circle, it is shown that if \ be a prime number, and if ef=\ — 1, the e 

 periods of/ roots each, that is the quantities t] , ij x , jj 8 .... »/,,_,, defined by 

 the equations 



