736 MR DUNCAN M. Y. SOMMERVILLE ON 
these and found composite types corresponding to 176 of them. Of the remainder it is 
probable that a considerable proportion do not give types. Thus it seems probable that 
the only types which involve 11 without 4 are 1,10, 11, 12; 1, 11, 12; 7, 10,11; and 
7 HOt. We 
The combinations are all included in the following lists. A. contains the 222 left 
after rejection, B. those which I have not verified. 
Ae Ab Creo 8 a7 el) O(6.010): 
4,5 Oslo aO rt Iboae Sy6)) 
A 11 PCC, 2, 1)-0,(6, 1012); 
A 10) 1 CA, 2) 6): 
516) LOM 125 CMI, 887) 
10, 1s -W2e eC. 2, Saar): 
56,10; 12.2.0, (ln 2a) 
10ST C.(2) Oy, le alla 7a 10 
Br 3, 25, 6 7 tl, 1 eC). 
2k 6, 11 HC.(7, 12)4 6,6): 
2, 495, 10; 11,12 €,(7). 4, 11, 12. 
3. 45. 6. Wile 1 4, 5, 6, 11. 
? ? 
BG HlO. dd ISAC. (pearcua): 
10, 11, 12+0,(1, 2, 5,7) [except 1, 10, 11,12 and 7, 10, 11, 12] 
5, 6, 10, 12. 
§ 17. Many of these composite types can be obtained from the simple types by 
filling up the hexagons and dodecagons. Thus, as we have seen, the type 1, 11, 12 can 
be obtained from the simple type 12 by filling up some of the hexagons with triangles. 
From the same simple type can be obtained nine other composite types involving the 
nodes 1, 4, 10, 11, 12. In the same way, having obtained one example of one type, it 
is generally possible to obtain a number of other types from it by some simple substitu- 
tions or displacements. A classification of the composite types might thus be attempted, 
based upon their structure. In this way types which are widely separated in the present 
classification would be brought together, and vice versa. It is to be noted, however, 
that the general variety of a type may fall on the lines of no simple network, so that 
a classification such as that suggested would be difficult to apply in the general case. 
Il. THe Exurpric PLaNe. 
§ 18. We proceed to investigate the semi-regular networks upon the elliptic plane, 
or, what is the same thing, upon the sphere, or in general upon a closed surface of con- 
stant positive curvature. 
We shall first find what species of nodes are possible. Since the angle of a regular 
polygon here depends upon the area of the figure, it is obvious that the number of 
species of nodes is infinite. Whatever holds on the EHuclidean plane regarding the 
number of polygons which can meet at a point will hold @ fortiori for the sphere. 
Hence we have only two cases to consider: viz., at a point there may be 1° two, 2° three 
