NUMBERS 621 The necessary and sufficient condition in order that it may be possible to compound together two given forms (a, b, c), (a, b , c), is that their determinants shall be to each other in the proportion of two square numbers ; in particular, the two forms may have the same determinant D ; and when this is so the compound form (A, B, C) will also have the same determinant D. The rules for this composition of two forms of the same determinant have been (as part of the general theory) investigated and established. The forms compounded of equivalent forms are equivalent to each other ; we thus in effect compound classes, viz., considering any two classes, the composition of their representative forms gives a form which is the representative of a new class, and the composition of any two forms belonging to the two classes respectively gives a form belonging to the new class. But, this once under stood, it is more simple to speak of the composition of forms, that is, of the forms belonging to the finite system of representative forms for a given determinant ; and it will be enough to consider the properly primitive forms. 30. The principal form (1, 0, D) compounded with any other form (a, b, c) gives rise to this same form (or, b, c) ; the principal form is on this account denoted by 1, viz., denoting the other form by </>, and expressing composition in like manner with multiplication, we have 1 . < = </>. The form (f> may be compounded with itself, giving a form denoted by < 2 ; compounding this again with <, we have a form denoted by </> 3 ; and so on. Since the whole number of forms is finite we must in this manner arrive at the principal form, say Ave have <f> n = 1, n being the least expo nent for which this equation is satisfied. In particular, if the form <f> belong to the principal genus, then the forms </> 2 , < 3 , . . . </> n - l will all belong to the principal genus, or the principal genus will include the forms 1, </>, < 2 , . . . </>"-~ 1 , the powers of a form </> having the exponent n. 31. Regular and irregular determinants. The principal .-^enus may consist of such a series of forms, and the deter minant is then said to be regular ; in particular, for a nega tive determinant D, = - 1 to - 1000, the determinant is always regular except in the thirteen cases - D = 243, 307, 339, 459, 576, 580, 675, 755, 820, 884, 891, 900, 974 (and, Perott, in Crelle, vol. xcv., 1883, except also for -Z> = 468, 931); the determinant is here said to be ir regular. Thus for each of the values-/) = 576,580,820,900, the principal genus consists of four forms, not 1, (f>, (f> (/> 3 , where < 4 = 1, but 1, <, <J> V <</>!, where < 2 = 1, fa 2 = 1, and therefore also (^></> 1 ) 2 = 1. Compounding together any two forms, we have a form with the characters compounded of the characters of the two forms ; and in particular, combining a form with itself, we have a form with the characters of the principal form. Or, what is the same thing, any two genera compounded together give rise to a determinate genus, viz., the genus having the characters compounded of the characters of the two genera ; and any genus compounded with itself gives rise to the principal genus. Considering any regular determinant, suppose that there is more than one genus, and that the number of forms in each genus is = n ; then, except in the case n = 2, it can be shown that there are always forms having the exponent In. For instance, in the case D - 35 we have two genera each of three forms ; there will be a form g having the exponent G, </ = 1 ; and the forms are 1, g, g 2 , g g*, g 5 , where 1, g 2 , c? 4 , belong to the principal genus, and g, g 3 , g 5 , to the other genus. The characters refer to (-? j , (4 j , and the forms are + +, (1, 0, 35) 1 (4, 1, 9) r/ 2 (4,-l, 9) ^ -, (3,-l, 12) <7 (5, 0, 7) g 3 (3, 1, 12) g 5 . An instance of the case n=l is D = -21, there are here four genera each of a single form 1, c, c v cc v where c 2 = 1, Cj 2 = 1 ; an instance of the case n = 2 is D = - 88, there are here two genera each of two forms 1, c, and c l} cc v where c 2 = 1, c L 2 = 1, thus there is here no form having the exponent 2n. (See Cayley, Tables, dc., in Crelle, vol. lx., 1862, pp. 357-372.) We may have 2 k + l genera each of n forms, viz., such a system may be represented by where </> 2 l =l, c 2 =l, c 1 2 =l, . . . c._ 1 = l; there is no peculiarity in the form </>, we may instead of it take any form such as c<f>, cc^, <fcc., for each of these is like </>, a form belonging to the exponent 2n, and such that the even powers give the principal genus. 32. Ternary and higher quadratic forms cubic forms, lire. The theory of the ternary quadratic forms (a, b, c, a , b , c )(.v, y, z)-,=ax- + by- + cz* + 2a y~ + ?. or when only the coefficients are attended to ( , ,, ,
- , b, c
has been studied in a very complete manner ; and those of the quaternary and higher quadratic forms have also been studied ; in particular the forms x 2 + y 2 + z 2 , x 2 + y 2 + z 2 + ic 2 composed of three or four squares ; and the like forms with five, six, seven, and eight squares. The binary cubic forms (a, b, c, d)(x, y) 3 , = ax 3 + 3bx 2 y + 3cxy 2 + dy*, or when only the coefficients are attended to (a, b, c, d), have also been considered, though the higher binary forms have been scarcely considered at all. The special ternary cubic forms ax 3 + by 3 + CZ 3 + Qlxyz have been considered. Special forms of the degree n with n variables, the products of linear factors, present themselves in the theory of the division of the circle (the Kreistheilung) and of the complex numbers connected therewith ; but it can hardly be said that these have been studied as a part of the general theory of forms. Complex Theories. 33. The complex theory which first presented itself is that of the numbers a + bi composed with the imaginary i, = /^l ; here if a and b are ordinary, or say simplex posi tive or negative integers, including zero, we regard a + bi as an integer number, or say simply as a number in this complex theory. We have here a zero (a = 0, b = 0) and the units 1, i, - 1, - i, or as these may be written 1, i, i 2 , P (i 4 =l) ; the numbers a + bi, a - bi, are said to be conjugate numbers, and their product (a + bi)(a - bi), = a 2 + b 2 , is the norm of each of them. And so the norm of the real number a is = a 2 , and that of the pure imaginary number bi is = b 2 . Denoting the norm by the letter N, A"(a bi) =* a 2 + b 2 . Any simplex prime number, = 1 (mod. 4), is the sum of two squares a 2 + b 2 , for instance 13 = 9 + 4, and it is thus a product (a = bi)(a - bi), that is, it is not a prime number in the present theory, but each of these factors (or say any number a + bi, where a 2 + b 2 is a prime number in the simplex theory) is a prime; and any simplex prime number, = 3 (mod. 4), is also a prime in the present theory. The number 2, = (1 + i)(l-), is not a prime, but the factors l+i, l-i are each of them prime; these last differ only by a unit factor i 1 + 1 = i(l-i)so that 2, = - t(l + i) 2 , contains a square factor. In the simplex theory we have numbers, for instance 5, - 5, differing from each other only by a unit factor, but we can out of these select one, say the positive number, and attend by preference to this number of the pair. It is in this way viz., by restricting a, b, c, . . . to denote terms of the series 2, 3, 5, 7, ... of positive primes other than unity that we are enabled to make the definite statement, a posi tive number N is, and that in one way only, = atfrtf . . . ; if N be a positive or negative number, then the theorem of course is, Nia, and that in one way only, = ( - l) w a a 6/V/. . .
