734 MR DUNCAN M. Y. SOMMERVILLE ON 
Hence if 10 is excluded, we must have 4 and 6. 
For 4 is necessary in order to give squares, and if we exclude 6 we also © 
exclude 5, and therefore 3. We are then left with only 1, 2, 4, which do 
not involve hexagons. 
(d) 6 must be accompanied by 4 or 5. 
Excluding 4 and 5, the hexagon must be surrounded by triangles, and 
squares can never be introduced without producing 4. 
1 must be accompanied by either 4, or 5 and 6. 
Exclude 5 and 6; then T, must be surrounded by either triangles, or tri- 
angles and squares. Since hexagons are excluded at this stage, either of 
these introduces 4. 
— 
fa 
y 
— 
Exclude 4 and 6; then T, must be surrounded by hexagons. Now squares — 
can only be introduced after the concavities have been filled up. If we fill - 
them with hexagons fresh gaps are produced, and if we fill them with tri- 
angles there are always two triangles together, and the addition of squares is 
impossible without giving 4. Hence we cannot exclude both 4 and 6. 
Exclude 4 and 5; then 6 is also excluded by (d), and T, can only be 
surrounded by triangles. 
Also 1 can only be continued by 4, 5, or 6; hence if we exclude 4, we 
must have both 5 and 6. 
Rejecting according to these rules, we are left with forty-seven combinations, each of 
which gives a composite type. The combinations may be represented by the following 
notation. Let C,(a,,....,a,) stand for any one combination of 7 or more of the a’s, 
then the forty-seven combinations are 
5+C,(4, 6, 10)+C,(1, 2, 3) 
4+C,(6, 10)+C,(1, 2) 
10+ C,(2, 5). 
§ 15. Class C. II. Triangles, Squares, and Dodecagons (1, 2, 4, 7, 11). 
(a) 2 must be accompanied by 4 (fig. 11). 
(Gi) as Z ; oS (ie): 
After rejection there are left eleven combinations, all of which give com- 
posite types except 1, 7, 11. There are therefore the following ten composite 
types in this group: 
Ast eC) On Ter. wl palilnemeaasy, lee 
III. Squares, Hexagons, and Dodecagons (2, 3, 12). 
Excluding triangles, 12 can only be continued in one way, hence there are 
no composite types in this group. | 
§ 16. Class D. ‘Triangles, Squares, Hexagons, and Dodecagons (1, 2, 3, 4, 5, 6, 7, 
10, 11, 12). 
