TRANSACTIONS OF SECTION A. 439 
larity, such as is connected in molecular theory (¢9. gas theory) by the 
existence at each point of a definite temperature, is adequate for this purpose. 
On the other hand, if the Réntgen rays consist of absolutely independent pulses, 
devoid of all regularity in their succession, thus comparable to the traffic 
along a street or to the fire of a skirmishing company of soldiers, they would 
undergo no regular refraction. 
DEPARTMENT OF MATHEMATICS. 
The following Papers were read :— 
1. A Fragment of Elementary Mathematics. By Professor F. Morury. 
The text-books of elementary analytical geometry are introductory to Projective 
Geometry. It is easy to work out an analytic introduction to Inversive Geometry, 
and such an introduction is, I have found, of much interest to students at an early 
stage, because it connects very directly with the class of facts handled in Euclid or 
his substitutes, or in such books as Casey’s ‘ Sequel to Euclid,’ and also because the 
algebra is transparent, the geometric operations being in evidence. And such an 
introduction is of value to students later on in the geometric side of the theory of 
functions of a complex number, and in vector analysis, 
We work in a plane with reflexions in lines, two reflexions being a rotation. 
The operation of rotating may be called a turn (or ort, Heaviside), and denoted 
by ¢. A turn is thus a complex number of absolute value or length unity, but 
there is no necessity to speak of complex numbers. By itself, ¢ may be thought of 
as a point on a base circle of unit radius. The centre of this circle is the base 
point or origin ; a fixed line through the base point is the base line or axis of reals. 
The reflexion of a point « in the base line may be denoted by x or y as is 
convenient ; this reflexion is the conjugate of «. 
The map-equation of a line is 
a 
t= P| 
Se 
and forms of the conjugate equation of a line are 
vja+azja=l, 
a+t?x=pt. 
It is important here to note that between a point « and its reflexion 2, in this 
line we have the relation 
vlat+e2,/a=1. 
A similar remark applies to circles—namely, that the conjugate equation 
VL-ALX-aX= p* 
is a special case of the relation x z,—a x, —a x= p*, which connects inverse points. 
After adequate explanation of these preliminaries, we have next (1) the metrical 
theory of the n-lines, including in particular convenient proofs of the known 
metrical facts of the triangle; and (2) the theory of rational curves, such curves 
as, for example, the “imagon type: 
v=a polynomial in ¢; 
o=¥ 
the hyperbola type: 
Bias 
t -¢’ 
