1904-5.] Networks of the Plane in Absolute Geometry. 393 
Class A. 1. T 6 . 2. S 4 . 3. H 3 . 
Class B. 4. T 3 S 2 . 5. T 2 H 2 . 6. T 4 H. 7. TD 2 . 8. S0 2 . 9* 
Class C. 10. TS 2 H. 11. T 2 SD. 12. SHD. 
Out of these all the semi-regular networks must he built up. I 
distinguish types of networks according to the kinds of angles of 
which they are composed. If there is only one kind of angle the 
type is called simple , otherwise it is composite. The types are 
classified into Groups according to the kinds of polygons which are 
involved, and the groups into Classes according to the number of 
kinds of polygons. There are four classes. Class A. consists of 
the regular networks. 
The simple types are first considered. There are four unique 
types, T 4 H, TD 2 , S0 2 , and SHD. T 3 S 2 admits of an infinite 
number of varieties of the simple type. In T 2 H 2 two distinct 
varieties can be recognised, an infinite number of varieties being 
obtained as mixtures of the two. With TS 2 H there are three 
distinct varieties with an infinite number of mixtures. T 2 SD 
does not admit of a simple type, nor, of course, does Class D. 
The composite types in general admit of infinite variation. In 
any group a composite type corresponds to a possible combination 
of the kinds of angles contained in the group. Thus in the group 
of triangles and squares there are the three angles 1, 2, 4, and 
the composite types 1, 4 ; 2, 4 ; 1, 2, 4 ; the combination 1, 2 
being impossible. The method of investigating these is chiefly 
experimental, and consists in testing the various combinations. 
It is easily seen, however, that certain combinations are impos- 
sible. Tor example, H 3 must be accompanied by T 2 H 2 in order 
that the gap of 120° may be filled up. The following are the 
numbers of composite types in the various groups : 
B. I. (T, S) 3 ■ II. (T, H) 8. C. I. (T, S, H) 47 ; II. (T, S, D) 10. 
D. (T, S,H,D) 169 -Ki- 
ll. The Elliptic Plane. ^Here we get a relation of the form 
A>0, and by giving positive integral values to A an infinite 
number of kinds of angles are found. Only a few of these, 
however, can be developed. Tor example, if there are at a point 
* No. 9 is 2 pentagons and 1 decagon, but this is not a developable angle, 
t I have not exhausted all the composite types in this class. There 
cannot be more than 222. 
