4 Ellery Williams Davis 



Of course a reflection on a line followed by a reflection on that 

 line, or a turn forward followed by an equal turn backward sim- 

 ply leaves alone ; is equivalent to unity. 



Reflection on a line can be replaced by a hajf-turn about that 

 line as an axis. If now we perform the operations, thus modi- 

 fied, upon a square, whose center is the origen and whose sides 

 are parallel to the coordinate axes, they will each bring the 

 square into coincidence with itself. Thus the group and the 

 square "belong to each other," as also do the group and a set 

 of eight angles all having numerically equivalent sets of trigo- 

 nometric functions. 



The group has various subgroups, viz. — 



1. Those of order 2 generated by any self-reciprocal operation. 



2. The group of order 4 generated by sc or cs. 



3. The groups of order 4 generated geometrically by a half- 

 turn about a line and a half-turn, about a perpendicular thereto. 



I write these all out — 



1. J 1, s\, J i, c\, J 1, srs\, 1 1, esc], J 1, (sc) 2 }. 



2. j I, sc, cs, (sc)' 2 \. 



3. \i, s, csc,(sc)' 2 \, \i, c, scs, (scy\. 



The operations 1, other than unity, are half-turns about an 

 axis and leave unchanged all figures having two-fold symmetry 

 about that axis ; for example, a figure whose sections normal to 

 the axis are parallelograms with centers in the axis. 



The operations 2 leave unchanged all figures having four-fold 

 symmetry about the s-axis through O normal to the .ry-plane. 

 Such would be a regular four-sided pyramid whose axis was 

 the .s-axis. 



The operations 3 leave unchanged figures having two-fold 

 symmetry as to the x, y, and .c-axes ; for example a tetrahedron 

 centered in O, whose three pairs of opposite edges are each pair 

 perpendicular to one of the three axes while equally inclined to 

 one and therefore both of the other axes. 



234 



