524 REPORT—1862. 
cannot be represented by the simple formula C’, and we must employ an 
expression of the form C,xC,?xC,?.... To obtain an expression thus 
representing all the classes of the principal genus, we take for C, a class ap- 
pertaining to the greatest exponent 6, to which any class can appertain; and 
in general for C, we take a class appertaining to the greatest exponent 6, 
to which any class can appertain when its period contains no class, except 
the principal class, capable of representation by the formula 0, x C, Sei 
C,_1'#-1, The number a=? x6,xX ... 18s called by Gauss the exponent of 
irregularity ; and similarly we might term &c., the second, 
n n 
0, 6,” 0, 0, 6,” 
third, &c., exponents of irregularity. From the mode in which the formula 
C," x C,” x . . is obtained, it can be inferred that 0, is divisible by 6,, 0, by 
6,, and so on; whence it appears that a determinant cannot be irregular un- 
less n be a divisible by a square; nor can it have r indices of irregularity 
unless ” be divisible by a power of order +1. Moreover, whenever the 
principal genus contains but one ambiguous class, the determinant is either 
regular or has an uneyen exponent of irregularity; if, on the contrary, the 
principal genus contain more than two ambiguous classes, the determinant is 
certainly irregular, and the index of irregularity even; if it contain 2 ambi- 
guous classes, the irregularity is at least of order x, and the « exponents of 
irregularity are all even. 
A few further observations are added by Gauss. Irregularity is of much 
less frequent occurrence for positive than for negative determinants; nor 
had Gauss found any instance of a positive determinant having an uneven 
index of irregularity (though it can hardly be doubted that such determinants 
exist). The negative determinants included in the formule, —D=216k+4 27, 
=1000k+4 75, =1000% 4 675, except —27 and —75, are irregular, and have 
an index of irregularity divisible by 3. In the first thousand there are five 
negative determinants (576, 580, 820, 884, 900) which have 2 for their 
exponent of irregularity, and eight (243, 307, 339, 459, 675, 755, 891, 974) 
which have 3 for that exponent; the numbers of determinants having these 
exponents of irregularity are 13 and 15 for the second thousand, 31 and 
32 for the tenth. Up to 10,000 there are, possibly, no determinants having 
any other exponents of irregularity; but it would seem that beyond that 
limit the exponent of irregularity may have any value. 
118. Arrangement of the other Genera.—In the preceding article we have 
attended to the classes of the principal genus only; to obtain a natural 
arrangement of all the properly primitive classes, we observe that, if the 
number of genera be 2, the terms of the product (1+T,)(1+T,)(1+T,)... 
_ (1+T,,),in which T; represents any genus not already included in the product 
of the i—1 factors preceding 1+4T;, will represent all the genera. If, then, 
A,, A,,... A, represent any classes of the genera I'|,T,,. . I’, respectively, 
and |C| be the formula representing all the classes of the principal genus, the 
expression |K|=|C| x (1+A,)(1+A,)...(1+A,) supplies a type for a simple 
arrangement of all the classes of the given determinant. When every genus 
contains an ambiguous class, it is natural to take for A,,A,,.. A,, the ambi- 
guous classes contained in the genera T,, l’,,.. I’, respectively. When the 
principal genus contains two ambiguous classes (and when, consequently, 
one-half of the genera contain no such classes), let C, be the class taken as 
base (or, if the determinant be irregular, as first of the bases) in the arrange- 
Oe 
a 
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