ON MATHEMATICAL TABLKS. 



331 



In tho notation of the ' Disquisitiones Arithmeticne,' followed by Eiscnstein 

 and others, 



{"'I ^' ^J denotes ax- + hf + cz' + 2f,jz + 2yzx + 2hxii ; 



the determinant is 



= -{aU-af-hcf-cTi^ + 2fyh), 



a positive form having thus always a negative determinant. And the adjoint 

 form is 



Hence Seeber's determinant is = - 4 into that of Gauss, and his tables really 

 extend between tho values — 1 and — 25 of the Gaussian determinant. 



50. Tables of greater extent, and in the better form just referred to, are 

 given 



Eisenstein, Crelle, t. xli. (1851), pp. 169-190 ; viz. these are 



I. " Tabelle der eigentlich primitiven positiven terniiren Formen fiir alle 

 negativen Determinanten von —1 bis —100" (pp. 169-185). 



A specimen is 



D 



10 



Anzahl. 



Reducirte Formen fiir — D. 



n,l,10>| /1,2,.5\ (2, 2,3\ 

 U 0, 07' Vo, 0, 0/' \p, -1,0 J 



6=8 2=4 a=4 



II. "Tabelle der uneicjenilich primitiven positiven terniiren Formeu fiir 

 alle negativen Determinanten von -2 bis - 100 " (pp. 186-189). 

 A specimen is 



D 



10 



Anzahl. 



Eeducirte Formen fiir — D. 



(2, 2, 4\ 



2=6. 



And there is given (p. 190) a table of the reduced forms for the determinant 

 _385(= - 5 . 7 . 11) (selected merely as a largish number with thi-ee factors); 

 viz. there are in all fifty -nine forms, corresponding to values 1,2,4, 6, 8 of L 



It may be remarked "that 3 denotes for any given fonn the number of ways 

 in which this is linearly transformable into icself, this number being always 

 1, 2, 4, 6, 8, 12, or 24. The theory as to this and other points is explained 

 in the memoir (pp. 141-168), and various subsidiary tables are contained 

 therein and in the ' Anhang ' (pp. 227-242) ; and there is given a small table 

 relating to indctermmate forms, viz. this is 



" C. Versuch einer Tabelle der nicht aquivalenten unbestimmten (indif- 

 ferenten) terniiren quadratischen Formen fiir die Determinanten ohne 

 quadratischcn Theiler unter 20 " (pp. 239, 240). 



A specimen is 



D 



10 



Indifferente ternare quadratische Formen. 



(0, 1, 10\ 



Vo, 0, 1 ; 



/O, 0, 10\ 



\0, 0, l) 



/I, 2, -5\ 



Vo, 0, o; 



