Examples of Groups 



13 



Consider now Glaisher's functions. 



„ (3 — 8 x — a 



S7l i W = r jv 



a — o x- — p 



„ a — (3 X—S 



a — 8 X — (3 



dn 2 w = 



p" 



cd 2 w = 



-y , 



a — y .r — /? 

 a — y x- — 8 



c-6dshv ■ 



a — 8 x — y 



a — (3 x — y 



.yrze; = 



j3 — y X — a 

 (3 — 8 x — a 



We have — 



si?i 2 6sii 2 w 



tan' 6 en 2 w 

 sec*Qdri 2 w 

 sin 2 6cd 2 w 

 csc 2 dds 2 w = 

 cos 2 6sc 2 w = 



/?— 8 



together with twelve equations gotten by taking reciprocals of both 

 members of each of these. 



S?l"W 



The set — sc 2 w 

 11c 2 w 



cn'w 

 ns 2 w 



is 



cos 2 4> 



—cof 2 4> 



csc 2 <$> 



and so of a sort frequently considered 

 these changes are indicated to the left 



sin 2 4> 

 — tan?<f> 

 sec 2 4> 



The substitutions producing 

 Call any one of these forms 

 f\{w, k)—f\{zv)=f\. Then any other 

 substitution than that of the group 

 in a, (3, 8 will change any f\ into a 

 new form/2 and the six forms fo will 

 be related precisely as are the six 

 forms yi; for to operate upon 1 — f\ 

 or i-r-/i with any substitutions is 

 merely to operate upon theyi giving 

 I — -fi and 1-5-/2. In particular, con- 

 sider the three substitutions other 

 than unity which leave k 2 unaltered, 

 viz.: • (a8)(/3y), (a/3)(yS), and 

 (ay)(/3S). These belong to the 

 rectangle whose corners are a, (3, y, 8 and may be symbolized by it. 



243 



