Examples of Groups n 



and if, moreover, we define <£ and k by 



(^— yKa— 8) 2 /?— 8 x— a 



k -(a-yXfl-S)' ^~^=8 *-0' 



it follows that 



w, or 





, v x — k 2 sm 2 <f> 

 l /_ (a— y)(/3— 8) rx dx 



i V (x~a)(x—(3)(x—y)(x—8) 



%x dx 



= say, MV- 



Vs* 



If now we perform upon sin <£ the 24 possible changes in the ar- 

 rangement, substitutions we call them, of the letters a, (3, y, 8 en- 

 tering into its expression, it will pass over always into some constant 

 multiple of one of the Glaisher functions. 



The substitution group on the letters a, (3, y, 8 is 



0'24 ==: 



f I, (a/3), (ay), (aS), (/?y), ((38), (y8), 

 (/3yS), (/?8y), (ay8), (aSy), (a/38), (aS/3), (a(3y), (ay/3) 

 (ap-)(yS), (ay) (08), (a8) (/8y) 

 (apyS), (aSy/3), (ay/88), (a8/?y), (a/?8y), (ayS/3) 



Each substitution in the second line is the square of one next 

 to it in that line. Each in the third line is the square of each of 

 two in the fourth. Each in the fourth is the cube of one next to 

 it in that line. 



These 24 substitutions produce in k 2 only six changes, as 

 follows : 



k 2 = say, sin 2 6= • — — = is left unchanged by 



a- — y (3 — o 



I, (a/?)(yS), (ay) ((38), (a8) (/3y) ; 

 is changed to 



^ =l -k 2 =COS 2 d= a -^-y^by (ay), ((38)(a(3y8), (a8y(3) ; 



24I 



