8 Ellery Williams Davis 



Suppose the uncrossed hexagon 

 regular and prolong the dotted sides 

 to form an equilateral triangle. Con- 

 sider now the movements that bring 

 the figure into coincidence with itself. 

 These are: turning the figure over 

 about any altitude, and turning the 

 figure in its own plane through one- 

 / third, two-thirds, or a whole revolu- 



tion about its center. We thus have 

 a group of rotations corresponding uniquely to the groups just con- 

 sidered. To see this, it is only necessary to let say t\ and fa de- 

 note turning over about any two altitudes, when the multiplication 

 table will be reproduced with t\ and fa in place of m and r. Thus the 

 group we are now considering is related to the equilateral triangle 

 precisely as was the one we started out with to the square. 



If the corners of the triangle are marked x, y, r, and if t x 

 causes an interchange of x and y, while t 2 causes an interchange 

 of y and r, then t 1} t. 2 , and their combinations can be expressed 

 in terms x, y, and r as below. — 



i, h, fa, fafa, fah, fafah, 



1 > (xy)> (yf)> (xyr)> (xry) t (xr). 



sm 0_ cos & But if x and y denote rectangu- 



lar coordinates of a point while r 

 \ and 6 are polar coordinates of the 



\ same point, (xjy) changes each 



r\ \ \ trigonometric function of into its 



T JqJ complementary function, each 



function on the diagram to that 

 joined to it by full lines; while 

 (jj/r) changes each into the one 



% v_„ ^ a joined to it by dotted lines. The 



COS^B esc u J ' . , 



group \d, f\ fits this scheme 



quite as well as the one for which it was devised. The hexagon 



formed by the lines is now singly-crossed instead of triply-crossed or 



quite uncrossed, but, as in those, dotted lines alternate with full. 



238 



