TRANSACTIONS OF SECTION A. 311 
Sciences, M.DCC.LXXIXx. Its status and importance were made apparent by 
Kronecker’s theory of Divisor Systems. 
The central problem (of which no satisfactory solution has yet been given) 
is to find one or several ways of expressing the complete conditions that a 
polynomial F must satisfy in order that it may be capable of taking the form 
Be XW, tn 5, tong 
where F,, F,,..., Fx are given polynomials (in general non-homogeneous) in 
m variables z,, 2,,..., 2%”, and X,, X,,..., Xx are any polynomials in the 
same variables, not given. The whole system of polynomials F is called a 
module (of polynomials), and is regarded as a single entity or class, and is 
symbolised by a letter M. Each F is called a member of M, and the set of 
polynomials F,, F,,. .. , Fx is called a basis of M. 
A module M is considered from its two conjugate aspects, viz. (i) its content 
as represented by its members, and (ii) its content as represented by its modular 
“equations, i.e., the linear equations which are identically satisfied by the 
coefficients of each and every member F of M. (ii) has received little attention 
hitherto. Hither (i) or (ii) gives a complete representation of the module; but 
the difficulty is to obtain (ii) from (i) and vice versa. In combination (i) and (ii) 
give a very complete view of a module and afford together a simple answer to 
most general questions. Thus the members of the G.C.M. of any number of 
modules consist of all the members of all the modules; the modular equations 
of the L.C.M. of any number of modules consist of all the modular equations 
of all the modules; the members of the product MM’ of two modules M and 
M’ can be obtained at once from the members of M and M’; and the modular 
equations of a residual M/M’ can be obtained at once from the modular 
equations of M and the members of M’. 
The most important types of modules were considered ; these are the unmixed 
module, prime module (corresponding to a prime number in arithmetic), primary 
module (corresponding to a power of a prime in arithmetic), simple module (that 
is, a module containing one point only), module of the principal class, perfect 
module, and closed module. The method of obtaining the modular equations 
from a basis of the module was discussed, and also the resolution of a module 
into primary modules. 
6. A Theory of Double Points. By F. 8. Macaunay, M.A. 
BRISBANE. 
FRIDAY, AUGUST 22. 
Professor E. W. Brown, F.R.S., Vice-President of the Section, delivered the 
following Address :— 
To one who has spent many years over the solution of a problem which is 
somewhat isolated from the more general questions of his subject, it is a satisfac- 
tion to have this opportunity for presenting the problem as a whole instead of in 
the piecemeal fashion which is necessary when there are many separate features 
to be worked out. In doing so, I shall try to avoid the more technical details of 
my subject as well as the temptation to enter into closely reasoned arguments, 
confining myself mainly to the results which have been obtained and to the 
conclusions which may be drawn from them. 
In setting forth the present status of the problem, another side of it gives one 
a sense of pleasure. When a comparison between the work of the lunar theorist 
and that of the observer has to be made, it is necessary to take into consideration 
the facts and results obtained by astronomers for purposes not directly connected 
with the moon : the motions of the earth and planets, the position of the observer, 
the accuracy of star catalogues, the errors of the instruments used for the 
measurement of the places of celestial objects, the personality of the observers— 
