

12 



is changed to 

 1 2 



is changed to 



Elle'ry Williams Davis 



J^ y f=f h y ( a ^)> C**), ( a ^ 8 )> ( aS ^): 



-^ 20 = -~j~ 8 h Y ( aS ). (ftO, («£8y), (ayS/3); 



is changed to 

 is changed to 



T7= COti. 



a— y /?— S 



.by (a8j8), (/My), (a/3y), (ayS) ; 



a— f3 y— 8 



a ^~8 hy (a/3S) ' (/8y8) ' ra ^' (a8y) - 



The substitutions from G?i that leave unchanged some one letter, 

 say 8, form a subgroup of order 6, producing all the changes above 

 recorded in kr or sin 2 6. The group leaving out 8 is 



jl, (a/?), (ay), (/?y), (a/3y), (ay/3) } 



and obviously the s ime as 



|ii OnO> (**>. O0> C*y*)i (**y)\> 



sm<6 a > 



cos ' e 



-Jon 



sec' 9 csc'0 



i 



those which produce the same changes in U 2 



242 



To the left I have dia- 

 grammed the changes 

 produced by the group in 

 a, ft, y. The arrow indi- 

 ,tg cates that the change pro- 

 duced by the substitution 

 is as the arrow flies, and 

 not in the reverse way. 

 In place of the substitu- 

 tions written can be put, 

 in this particular diagram, 



