157 
On dividing this expression by 2?+p,2+p,, we obtain by 
the ordinary algebraic rules a remainder of the form MV. x12 + No, 
where M,,_, and NV, are functions of p, and P2 Whose weights are 
2s—1 and 2s respectively, and which may accordingly be written 
in the forms 
Mie Dass + P2bos—s TF veeeee + piby, 
Nos = Cog + PaCng 10+... 1 pe", 
where the b,c are of an order in p, indicated by their suffixes. 
On writing down (by Professor Sylvester's Dialytic method) 
the result of eliminating p, between these equations, it is at once 
apparent that this resultant is of the order s(2s—1). Thus the 
determination of a quadratic factor of an expression of degree 2s 
is reduced to the solution of an equation of order s(2s—1). But 
this number is one degree more odd than the original number 2s; 
that is to say, if the number 2s is 2’ multiplied by an odd num- 
ber, then s(2s—1) is 2** multiplied by an odd number. Hence 
_ by a repetition of this process we shall ultimately arrive at an 
equation of odd order, which, as is well known, must have a real 
root. By then retracing our steps the existence of a quadratic 
factor of the original expression is demonstrated. 
(3) On the Space-Theory of Matter. By W. K. Cur- 
FORD, B.A., Trinity College. 
[ Abstract. ] 
RIEMANN has shewn that as there are different kinds of lines 
and surfaces, so there are different kinds of space of three di- 
mensions; and that we can only find out by experience to 
which of these kinds the space in which we live belongs. In 
particular, the axioms of plane geometry are true within the 
limits of experiment on the surface of a sheet of paper, and 
yet we know that the sheet is really covered with a number 
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