330 RSP3IIT— 1875. 



Cayley, Quart. Math. Journ. t. xi. (1871), where the notation &c. is 

 explained, pp. 2.51-261 ; viz. these are : — 



Table I. of the binary cubic forms, the determinants of which are the 

 negative numbers ^ (mod. 4) from — 4 to —400 (pp. 251-258). 

 A specimen is 



Det. Classes. Order. Charact. Comp. 



4-xll. 0,-1, 0,11] on 1, 0,11 1 

 0, -2, -1, 1 \pp pp 3, 1, 4 d 

 0, -2, 1, ij 3, -1, 4 d'. 



Table II. of the binary cubic forms the determinants of which (taken 

 positively) are£sl(mod. 4) from —3 to —99 [the original heading is here 

 corrected] ; and 



Table III. of the binary cubic forms the determinants of which are the 

 negative numbers -972, -1228, -1336, -1836, and -2700; viz. -972 



= 4x -24.3, -2700 = 4x — 675, where -243 -675 are the first 



six irregular numbers for quadric forms]. 



4x —675, = —2700 is beyond the limits of Arndt's tables, and for this 

 number the calculation had to be made anew ; the table gives nine classes 

 (1, d, d') (1, cZj, d^') of the order ip on pp, but it is remarked that there may 

 possibly be other cubic classes based on a non-primitive characteristic ; the 

 point was left unascertained. 



49. The theory of ternary quadratic forms was discussed and partially esta- 

 blished by Gauss in the ' Disquisitiones Arithmeticaj.' It is proper to recall 

 that a ternary quadratic form is either determinate, viz. always positive, such 

 as a^+y'^+z", or always negative, such as —x'—i/- — z' ; or else it is indeter- 

 minate, such as x^+y'^ — z'. But as regards determinate forms, the negative 

 ones are derived from the positive ones by simply reversing the signs of all 

 the coefficients, so that it is sufficient to attend to the positive forms ; and 

 the two cases are practically positive forms (meaning thereby positive deter- 

 minate forms) and indeterminate forms ; but the theory for positive forms 

 was first established completely, and so as to enable the formation of tables, in 

 the work 



Seeber, ' Ueber die Eigenschaften der positiven terniiren quadratischen 

 Formen ' (4to, Freiburg, 1831), 

 which is reviewed by Gauss in the ' Gott. Gelehrten Anzeigen,' 1831, July 9 

 (see Gauss, Werke, t. ii. pp. 188-193). The author gives (pp. 220-243) tables 

 " of the classes of positive ternary forms represented by means of the corre- 

 sponding reduced forms" for the determinants 1 to 100. A specimen is 



-- (J;J;J)(J;J:?> 



igeordnete /S, 8, 3\ / 7, 7, 4\ 

 Formen VO, 0, 8/' V 4, 4, 2/' 



where it is to be observed that Seeber admits odd coefficients for the terms 

 in yz, zx, xy ; \-iz. his 



(^J ^' ^^) denotes ax'+hf- + cz'+fyz+gzx+hxy, 



and his determinant is 



4fflic — af- — hg- — c7r +fgh. 

 Also his adjoint form is 



Zu 



