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TRANSACTIONS OF SECTION A. 38 
2. On the present Slate of the Theory of Aggregales. 
By Professor E. W. Hopson, I’.8.S. 
Various points which have been raised in the course of recent contro- 
versies relating to the abstract theory of aggregates were discussed in some 
detail. The desirability was pointed out of a new definition of an ‘ aggre- 
gate,’ of a more restricted character than the one due to G. Cantor, and of 
such a character that no difficulties would arise from the ascription of a 
cardinal number to each such aggregate, and also of an ordinal type in case 
the aggregate is an ordered one. 
3. Generalisation of the Icosahedral Group. 
By Professor G. A. Minuer, Ph.D. 
The icosahedral group may be completely defined by the fact that its 
two generating operators (t,, t,) satisfy one of the following three sets of 
conditions :— ‘ 
t?=1,5=(it,)9=1, ?=t%=(44)°=1, t= 1,5 = (t,t,)?=1. 
Several years ago the author considered the groups which result when these 
conditions are replaced by somewhat more general ones, as follows :— 
t?=t,", (é,t.)? =1; t°=1,, (t,t.)° =1; ¢°=2,', (t,t2)? =1, 
He found that each of these three sets of generators leads to only a small 
number of distinct groups. That is, only a small number of groups can be 
generated by two operators which satisfy one of these sets of conditions. 
These results were published in the Transactions of the American Mathe- 
matical Society. In the present Paper he considers the still more general 
sets of conditions :— 
t,2= 4,5, (tyts)®=(toty)®3 G2 = 68, (ht)°=(fh)?s G2=65, (tt)? = (4t)*. 
Among the most important theorems which he established are the following. 
There is an infinite number of groups, each of which may be generated ky 
two operators satisfying one of these conditions. Each of the possible 
groups generated by (¢,, tf.) contains either the icosahedral group or the 
group of Order 120, which is insoluble and does not contain a sub-group 
of Order 60, and it must have one of these groups for its commutator sub- 
group. 
4. A New Proof of Weierstrass’s Theorem. By Professor G. A. Butss. 
The theorem of Weierstrass with which the author dealt is one con- 
cerning the factorisation of power series. Any convergent series in p+1 
variables, F > (v,, 2’, --+ 5 2) y), in which the lowest term in y alone is of 
degree n, can be expressed as a product : 
(y® + ayy™-2 4+ 266 +Qn-1Y + An) B (Hy + + y Up Y)s 
Where a,,a,... a, are convergent series in 2, . . .,%,, which vanish 
with these arguments, while @ is a convergent series in all p+1 variables with 
a constant term different from zero. The theorem is-important because 
it enables one to separate from the function F a polynomial whose roots are 
the only values of y which can satisfy the equation F=0 in the neighbour- 
hood of the origin. The original proof by Weierstrass, and most of the 
later ones, depend upon function theory. Goursat, however, recently 
called attention to the essentially elementary character of the theorem, and 
gave a more direct demonstration. In the present Paper a proof is given 
which seems still simpler than that of Goursat, and formule are set down 
by means of which the coefficients in the series @,, @, . « .) Up, ® may be readily 
computed. 
