6 Ell cry Williams Dai'is 



give the group generated by (xy) (xy) and (xx) corresponding 

 respectively to c and s; the group, namely: 



1 1, (xy)(xy), (xyxy),(xx),(xx)(yy), (xy)(xy), (xyxy), (yy)\. 



There is, of course, a one-to-one correspondence between the 

 operations of these various groups. We can best, indeed, regard 

 the various groups as merely several out of many ways of writ- 

 ing the same group. It is the multiplication table, not the names 

 we give the multipliers, that determines the character of the 

 group. 



The correspondence is exhibited below: 



(xx) (xy) (xy) (xyxy) (xyxy) (xy) (xy) (yy) (xx)(yy) 



The six trigonometric func- 

 cos tions are connected by a group 



somewhat similar to the one we 

 have been studying. In the 

 diagram to the left any pair of 

 - (Ct* expressions joined by full lines 

 are reciprocal; any pair of ex- 

 pressions joined by dotted lines 

 have unity for their sum. It 

 follows that, if k 2 be the value 

 of any one of the expressions, 

 the six are 



-tan' 



<er<0 



csc'tf 



k 2 , i—k 2 , i:k 2 , k 2 —i:k\ i:i—/b\ k 2 :V—i. 



If r denote, as above, reciprocal -of, while m denotes I — , the 

 six may be written 



k 2 , mk 2 , rk 2 , mrk 2 , nnkr, mrmk 2 , 



giving the group |i, m, r, mr, rm, mrm\, 



236 



