ON MATHEMATICAX TABLES. 



327 



composition of forms (although in the mean time established by Gauss), are 

 not therein taken into account ; for this reason the Legendre's Tables I. 

 to VIII. relating to quadratic forms, given after p. 480 (thirty-two pages not 

 numbered), are of comparatively little value, and it is not necessary to refer 

 to them in detail. 



The complete theory was established 

 Gauss, ' Disquisitioncs Arithmeticae,' 1801. 

 It is convenient to refer also to the following memoir 

 Lejeune Dirichlet, "Eecherchos sur diverses applications do 1' Analyse a 

 la theorie des Nombres," Crelle, t. xix. (1839), pp. 338, as giving a succinct 

 statement of the principle of classification, and in particular a table of the 

 characters of the genera of the properly primitive order, according to the 

 four forms D = PS^ P=l or 3 (mod. 4), and D=2PS^ P=l or 3 (mod. 4) 

 of the determinant. 



45. Tables of quadratic forms arranged on the Gaussian principle are given 

 Cayley, Crelle, t. Ix. (1862), pp. 357-372 ; viz. the tables are- 

 Table I. des formes quadratiques binaires ayant pour determinants les 

 nombres negatifs depuis D= — jusqu'a D= —100. (Pp. 360-363.) 

 A specimen is 



where a, /3 denote, as there explained, the characters in regard to the odd 

 prime factors of D ; Z, e, de those in regard to the numbers 4 and 8. The 

 last column shows that the forms in the two genera respectively are 1, (j^, g* 

 and g, g^, g^, where g^ = l, viz. the form g six times compounded, gives the 

 principal form (1, 0, 26). 



Table II. des formes quadratiques binaires ayant pour determinants les 

 nombres positifs non-carres depuis D = 2 jusqu'a I) ==99. (Pp. 364-369.) 



The arrangement is the same, except that there is a column " Periodes " 

 showing in an easily understood abbreviated form the period of each form. 

 Thus D = 7, the period of the imncipal form (1, 0, —7), is given as 

 1, 2 —3, 1, 2, 1, —3, 2, 1, which represents the series of forms (1, 2, —3), 

 (-3, 1,2) (2,1, -3), (-3,2, 1). 



Table III. des formes quadratiques binau-es pour les treize determinants 

 negatifs irreguliers du premier millier. (Pp. 370-372.) 



Arrangement the same as in Table I. It may be mentioned that the 

 thirteen numbers, and the forms for the principal genus for these numbers, 

 respectively are : — 



-D= 



576, 580, 820, 900 



884 



243, 307, 339, 459, 675, 891 



755, 974 



Principal genus. 



(1, r)(l, {\ i\ i') 



(l,d,d'-)(l,d^,d,')(l,ey 



