ON MATHEMATICAL TABLES, 



3S9 



viz. seventeen determinants, each of one genus, and together of sixty-one 

 classes; lifty-eight determinants, each of two genera, and together of 280 

 cUisscs ; and twenty-iive determinants, each of four genera, and together 13G 

 classes, give in all 233 genera and 477 classes : these are exclusive of 74 

 classes helonging to the improperly primitive order ; and the number of 

 irregular determinants (iu the first hundred) is =0. 

 The irregular determinants arc indicated thus : — 



243(*3*) 



307(*3*) 



459(*) 



576(*2*) 



675(*3*) 



755(*3*) 



891(*3*) 



339(*3*) 



580(*2*) 



820(*2*) 



900(*2*) 884(*2*) 974(*3*) 



*3* 243, 307, 339, 459?, 675, 755, 891, 

 *2* 576, 589, 820, 884, 900, 974, 



■which is a notation not easily understood. 



As regards the positive determinants, a specimen is 



Centas I. 



Excedunt determinantis 



quadrati 10. 



G I (12) 



1. 2, 5, 13 

 17, 29, 41 

 53, 61, 73 

 89, 97 

 3. 37 



viz. in the first hundred the positive determinants having one genus of one 

 class are 2, 5, 13, &c. . . (eleven in number) : that having one genus of three 

 classes is 37 (one in number), 11 + 1 = 12. The irregular determinants, if 

 any, are not distinguished. 



47. Binary cubic forms. — The earliest table is 



Arndt, " Tabelle dcr reducirten biniiren kubischeu Formen und Klassen 

 fiir alle negativen Determinanten — D von D = 3 bis D=2000." GriinerVs 

 Archiv, t. xxxi. 1858, pp. 369-448. 



The memoir is a sequel to one in t. xvii. (1851). The binary cubic form 

 (a, h, c, d) of determinant — D( = (6c— ac?)'^ — 4(6"— «c)(c" — hcl)) is said to be 

 reduced when its characteristic ^, =(A,B,C), =(2(6- — oc),6c — oc?,2(c"— 6c?)) 

 is a reduced quadratic form, that is, when in regard to absolute values B is 

 not > gA, C not < A. 



A specimen is 



D Eeduced forms with characters. 



Classes. 



Two subsidiary tables are given, pp. 351, 352, and 353-368. 



48. Tt appeared suitable to remodel a part of this table in the manner made 

 use of for quadratic forms in my tables above referred to, and it is accordingly 

 divided into the three tables given 



