328 REPOKT — 1875. 



^liexe d' = (y^-[, e'=e^'=l, i' = l, viz. (], r) (1, e^-) denotes four forms, 

 1, e\ e^'', e'e^ ; and so in the other cases. 



46. Gauss must have computed quadratic forms to an enormous extent; 

 but, for the reasons (rather amusing ones) mentioned in a letter of May 17, 

 1841, to Schumacher (quoted in Prof. Smith's Report on "The Theory of 

 JN'umbers," Brit. Assoc. Eeport for 1862, p. 526), he did not preserve his re- 

 sults in detail, but only in the form appearing in the 



" Tafel der Anzahl der Classen biniircr quadratischer Formen," Wevlce, t. ii. 

 pp. 451-475; see editor's remarks, pp. 497-49'J. 



This relates almost entirely to negative determinants, only three quarters 

 of p. 475 and p. 476 to positive ones ; viz. for negative determinants it gives 

 the number of genera and classes, as also the index of irregularity for the 

 determinants of the hundreds 1 to 30, 43, 51, 61, 62, 63, 91 to 100, 117 to 

 120 ; then in a different arrangement for the thousands 1, 3 and 10, for the 

 first 800 numbers of the forms — (15n + 7) and -(15n-f-13) ; also for some 

 very large numbers, and for positive determinants of tlie hundreds 1, 2, 3, 9, 

 10, and some others. 



A specimen is Centas I. 



GIL (58) (280) 



1. 5, 6, 8 

 9, 10, 12 



13, 15, 16 

 18, 22, 25 

 28, 37, 58 



2. 14, 17, 20 

 32, &c. 



Summa233 477 



Irreg. Imp. 74 



viz. this shows (as regards the negative determinants 1 to 100) that the 

 determinants belonging to G II. 1, viz. those Avhich have two genera each of 

 one class, are 5, 6, 8, 9, &c. (in all fifteen determinants) ; those belonging to 

 G II. 2, viz. those which have two genera each of two classes, are 14, 17, 20, 

 &c., and so on. The head numbers (58) . . (280) show the number of deter- 

 minants, each having two genera, and the number of classes ; thus, 



GIL 1x15= 15 

 2x17= 34 

 3x17= 51 

 4x 6= 24 

 5x 2= 10 

 6x 1= 6 



58 140 

 Xll 



= 280 

 and the bottom numbers show the total number of genera and of classes, 

 thus. 



G L 17x1= 17 61 



IL 58x2 = 116 280 

 IV. 25x4=100 136 



l'';0 233 477 



