482 
NCI OD aaa 
[ Marcu 25,.1897 
results it is shown that all equations of the ninth degree 
with a linear congruence group are metacyclic, and a 
complete account is given of metacyclic equations of the 
degrees 4 and 8 respectively. 
The next two chapters illustrate the power of group- 
theory in dealing with a certain class of problems in 
analytical geometry. The configuration of the inflexional 
tangents of a plane cubic, and the much more compli- 
cated configuration of the twenty-eight double tangents 
of a quartic, are here reduced to the scheme of a group. 
The advantage thus gained is twofold: a clear compre- 
hension of the structure of the configuration, and the 
appropriate engine for attacking the algebraic problems 
which the geometry suggests. Thus (p. 389) the fact 
that the Galoisian group of the equation of the twenty- 
eight double tangents of a quartic is simple and doubly 
transitive, is intimately connected with the existence of 
Steiner's sets of six associated pairs of double tangents ; 
and the structure of the group shows the exact nature of 
the algebraical problem which consists in the separate 
determination of these sets of lines. 
Chapter xiii. deals with the solution of the general 
quintic equation. It is now well known that the general 
quintic cannot be solved by radicals, and that it has no 
resolvent of lower degree than the sixth. By the solution 
of the quintic is now understood either the expression 
of its roots by means of transcendental functions, such 
as elliptic or modular functions; or else the expression 
of its roots in terms of a definite algebraical irrationality, 
such as that furnished by the icosahedral equation. The 
chapter before us is chiefly concerned with the second 
method ; it is shown that the equation 
yP+5a+5oy+c=o 
where @, 2, ¢ are any constants whatever, may be identified 
with one of the principal resolvents (Hauptresolventen) 
of the icosahedral equation of the sixtieth degree, usually 
written in the form j 
1S Fe So 
The process of identification requires the determination 
of sand of two other auxiliary parameters A, p, which 
fix the particular resolvent to be chosen. If A is the 
discriminant of the given quintic, the three auxiliary 
parameters are expressible as rational functions of a, 4, ¢ 
and ,/5A. Ultimately, then, the roots depend in a quite 
simple way upon those of the icosahedral equation ; this 
latter, although of a high degree, is very convenient of 
application, because its Galoisian group is known, and 
its roots are algebraical functions of a single parameter 
(z). Moreover, one of its roots may be simply expressed 
by means of the hypergeometric series (see p. 432). 
From this point of view then, if the solution of numerical 
quintics were a matter of practical importance, we should 
construct a single-entry table of the values of the icos- 
ahedral irrationality for different values of 2, and then 
make use, in each particular case, of the formule of 
identification above referred to. 
Chapters xiv. and xv. contain a theory of ternary 
groups of substitutions, and deal in particular with a 
group isomorphous with the G,,,, which may be otherwise 
represented as a congruence-group, mod. 7. This admits 
of a very interesting application to a special class of 
NO. 1430, VOL. 55] 
equations of the seventh order, analogous to the use of 
the icosahedral equation in solving the quintic. 
The fourth, and concluding, Book is on algebraical 
numbers ; and to those whose predilections are arith- 
metical this will probably prove the most interesting 
of all. When Kummer generalised Gauss’s theory of 
complex integers by introducing complex roots of unity 
of any order, he was at first baffled by the perplexing 
fact that in certain cases complex integers presented 
themselves which were incapable of resolution into 
factors, and yet did not possess all the essential qualities 
of prime factors ; thus, for instance, one and the same 
number might be expressible both as a8 and as 76, where 
aand # were integers essentially distinct from y and 6, 
and yet a, 8, y, 6 were all indecomposible. By a stroke 
of unsurpassed genius, Kummer devised a theory of 
ideal primes, which at once removed the difficulty, and 
enlarged the province of arithmetic indefinitely. The 
divisibility of one real complex integer by another may 
be expressed by a series of linear congruences : Kummer 
succeeded in showing that, associated with every cyclo- 
tomic corpus, there are certain sets of congruential 
conditions which are precisely analogous in the general 
theory to divisibility by different primes in ordinary 
rational arithmetic. The satisfaction of one of these 
sets of congruences way denote divisibility by an actual 
(complex) prime; but whether this is so or not, the 
nature of the limitation thus imposed is just the same, 
and so, when the actual prime divisor does not exist, we 
say that the satisfaction of the congruential conditions 
expresses the existence of an zdea/ prime factor. As an 
example of how ordinary divisibility may be expressed 
by congruential conditions, we may take 
ax + by =o, dx—ay =0 (mod. a? + 67) 
which, if satisfied simultaneously, are equivalent to the 
divisibility of « +47 by a+ 62. Here, of course, when 
the congruences are satisfied, the complex factor a + dz 
actually exists ; but the congruences may be discussed, 
and their arithmetical significance developed, quite 
independently of this fact. 
Kummer actually succeeded in showing how to con- 
struct, for any given cyclotomic corpus, the congruential 
conditions associated with the actual or ideal primes. 
contained in it; but when his theory is extended to 
general algebraic corpora, it becomes impracticable to 
carry out the investigation precisely on Kummer’s lines. 
The fundamental idea remains the same; and by an 
appropriate modification at the outset, Dedekind and 
Kronecker each succeeded in constructing an arithmetical 
theory capable of application to any corpus of algebraicali 
integers whatever. 
Their methods are not so different as at first sigh 
they may appear; this may be shown by an example 
which illustrates a fundamental point of the theory. 
Suppose that a and f are two ordinary rational integers ; 
then the linear form +a + _y8,in which x, y assume all 
rational integral values, comprises a certain set of rational 
integers, and these are, in fact, the multiples of the 
greatest common measure of a and 8. Thus, since a 
rational integer is given when all its multiples are given» 
we may say that the greatest common measure of a and 
B is represented by the linear form +a + 78,.or by the: 
