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MATHEMATICS: H. BLUMBERG 
A combination of electron bombardment with high pressure should 
give larger displacements for Hnes which are strengthened by strong 
electrical conditions. The following evidence bears this out: (a) The 
lines which are strong in the arc relatively to the furnace have large 
pressure displacements, {h) Enhanced lines as a class are found to be 
displaced by pressure more than arc hnes. {c) For a given pressure 
the enhanced lines are displaced more in the spark than in the arc. 
The details of this investigation, with corroborative evidence from 
other spectra, will be pubhshed in the Astro physical Journal. 
ON THE FACTORIZATION OF VARIOUS TYPES OF 
EXPRESSIONS 
By Henry Blumberg 
DEPARTMENT OF MATHEMATICS, UNIVERSITY OF NEBRASKA 
Presented to the Academy. April 30, 1915 
Not a few mathematicians have dealt with the problem of setting up 
criteria by means of which the irreducibility of certain expressions 
in certain domains may be seen at a glance from the character of the 
expressions. Gauss, Kronecker, Schoenemann, Eisenstein, Dedekind, 
Floquet, Koenigsberger, Netto, Perron, M. Bauer, and Dumas have 
written on the subject.^ The work of these authors may be said to 
center around the Schoenemann-Eisenstein theorem, which, however, 
is an exceedingly special case of various theorems obtained (for example, 
of theorem IV when interpreted for situation 1). With the exception 
of Floquet and Koenigsberger, the authors mentioned deal exclusively 
with the case where the expressions are polynomials, and chiefly with 
the case of polynomials whose coefloLcients belong to the domain of 
rational numbers. Floquet and Koenigsberger have extended the 
investigation to the case of hnear homogeneous differential expressions. 
A variety of methods have been employed. Thus the theory of 
algebraic fields, the theory of algebraic functions, the character of the 
solutions of Hnear homogeneous differential equations, and even geo- 
metric representation (Dumas, loc. cit.) have been used. Elementary 
methods, requiring no such means, have succeeded in yielding only 
the less general results. 
One of our objects is to show that elementary and comparatively 
short considerations may be made to yield results more general than 
any hitherto obtained. In fact, for a complete comprehension of the 
proofs, little more is required in the way of specific knowledge than an 
understanding of the definition of the various expressions considered 
