Examples of Groups 9 



It is interesting to note the trigonometric meaning of the trans- 

 formations (xr) and ( yr). Connected with every number 9 there 

 is a number u, such that 



si?i 9 = tan iu.* 



We call 9 the Gudermanian of u, writing 



9=gud ti=$in~ x ian iu. 



Expressed in terms of u and like-placed on the hexagon with their 

 equals in terms of 9, the six functions are 



ran Mr . sec t'uj 



s,n tu : - i, \ C6C /u 



cos iu ■• i cot tu 



It thus appears that (yr) changes each function of iu into its 

 complementary function. 

 Similarly, if we write 



co-9=gud v 



the six functions become 



sec '* tan/'i/ 



esc /is '■ -. ■ <S/n_f# 



cot IU * 'cos fv 



*i is, as usual, V — 1. 



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