. Gordan’s series. 
NovEMBER 9, 1916] 
NATURE 
187 
preparation from verdigris, from spirit vinegar, 
and from pyroligneous acid, and there is a short 
reference to a few of the processes which have 
been described (and patented) for the synthetic 
production of acetic acid by inorganic methods. 
But perhaps the most interesting section of the 
book to the lay reader is the description of the 
modern methods of making vinegar from an in- 
fusion of malt or malt and grain, as practised in 
England, and it is here that the author’s technical 
knowledge and experience give his work a special 
value. The processes are described in detail, and 
are well illustrated, showing the most approved 
forms of mash-tuns, mashing machines, sparges, 
refrigerators, fermenting tuns, acetifiers, steri- 
lisers, etc. The remaining chapters deal with 
the chemical methods of examining vinegar, and 
with the characteristics of different vinegars. 
The author as a chemist concerned with the 
manufacture of vinegar has naturally something 
to say on the relations of that substance to the 
Food and Drugs Acts, and on what is known as 
the malt vinegar question. At present both the 
law and the practice are admittedly in a somewhat 
chaotic condition. What is held to be legitimate 
trading in one county renders a dealer liable to a 
criminal prosecution in another. The Local 
Government Board has no power to fix legal de- 
finitions of food substances, but in response to 
the appeal of the Association of Vinegar Brewers 
it has suggested certain definitions. These 
definitions have not been universally accepted by 
public analysts, nor, when accepted, have they 
been regarded as obligatory by certain stipendiaries 
who are more concerned to dispense law - than 
justice. The consequence is there is a consider- 
able amount of confusion in administration, and 
malpractices tend to be perpetuated which might 
readily be put an end to by the exercise of a little 
common sense on the part of judicial authorities. 
INVARIANT THEORY. 
A Treatise on the Theory of Invariants. By Prof. 
OQ. E. Glenn. Pp. x+245. (London: Ginn and 
Co., i915.) Price 10s. 6d. net. 
iB: many other branches of mathematics, the 
theory of invariants has gone through 
stages similar to those of gold-mining. We may 
reckon Gauss, Lagrange, and Eisenstein among 
the pioneers; Boole, Cayley, Sylvester, and 
Salmon found the first big nuggets; and Aron- 
hold’s symbolic method may be compared to the 
rocker which extracted gold-dust from alluvial 
deposits. Finally, the refractory problem of find- 
ing complete systems led Gordan to invent his 
transvectant formule, corresponding to the 
stamps and cyanide tanks now used in South 
Africa. ; 
The present book illustrates very well the state 
of the subject at present. The author gives all 
the important methods, both for binary and 
ternary forms, including annihilators of sources, 
polar theory, Aronhold’s symbolic method, and 
Complete systems are given for 
NO. 2454, VOL. 98] 
| binary forms up to the quintic inclusive, and for 
certain pairs of forms; tables are also given for 
two ternary quadratics, and for the ternary cubic. 
Gordan’s theorem is proved with the help of 
Hilbert’s theorem and a lemma by Jordan, which 
simplifies the analysis. In the ternary theory an 
account is given of Clebsch’s translation (Ueber- 
tragung) principle; nothing is said, however, 
about connexes. 
The main novelty of the work is the account of 
modular invariants, the invention of Prof. Dickson. 
This remarkable theory illustrates once more the 
striking difference thére is between umbral and 
arithmetical analysis. Fundamental problems, 
such as finding a complete system, assume an 
entirely new aspect, and lead to quite different 
results. Whether this new theory will have wide 
applications is uncertain; but there is no doubt of. 
its theoretical interest and of the new turn it has 
given to a somewhat stereotyped part of analysis. 
In connection with Gordan’s theorem and the 
‘use made of Hilbert’s theorem we may, add a 
few remarks. To us, at any rate, there is some 
vagueness both in the statement and the proof of 
Hilbert’s theorem; fortunately, however, so much 
of it as is wanted for the proof of Gordan’s 
theorem practically amounts to the fact that linear 
diophantine equations of a certain type must have 
solutions that form what Dedekind calls a finite 
modulus. Hilbert’s theorem is that any definite 
set of polynomials forms the whole or part of what 
Kronecker calls a modulus (Fy, Fs, -.- Fn), 
i.e. the aggregate of all expressions 
m 
IGFs 
1 
where G,, Go, . . . Gm are arbitrary polynomials. 
The difficulty we feel may be illustrated by taking 
the case of four homogeneous variables and con- 
sidering those polynomials which, equated to zero, 
represent all surfaces of given “deficiency” p; do 
these all belong to a finite modulus of the 
Kronecker type? To speak, as Prof. Glenn does, 
of polynomials as “formed according to any defi- 
nite laws” is so very indefinite as to make us fear 
some tacit and illegitimate assumption in the 
proof. Very likely our difficulty is owing to our 
stupidity, but there it is; of course there is no 
great trouble in showing that the theorem does 
apply in a large number of important cases. 
The present work is in many ways similar to 
that of Messrs. Grace and Young; it is rather 
more analytical in character, though there are a 
fair number of articles on geometrical applications 
and interpretations. We sometimes wish for 
another Clebsch to appear and give geometrical 
embodiment to these immaterial formule. The 
trouble is that the simplest analytical concomi- 
tants do not, as a rule, correspond to the simplest 
geometrical derivatives. For instance, a binary 
quintic has four linear covariants; suppose we 
represent the quintic by five points on a conic, 
each of the linear covariants must give a point 
on the conic derivable by projective construction 
from the original five; but it is not by any means 
