Examples of Groups 15 



It will be noticed that we get the substitutions in any/ from 

 those in any other by performing upon the symbols in the substitu- 

 tions of the first / the same substitution that carries the forms 

 connected by the substitutions of the first / into those connected 

 by the substitutions of the second / It is not difficult to see 

 why this is. Consider for example the four forms to the left 



and their connections. 

 j,7«v (aS)(0y) c d'w Plainly we can pass from 



cdP-w to dc 2 w by the route 

 (aS)(/?y).(a/3).(aS)(/3y). 

 But (a8)(/3y) changes 8 

 • dc'w an d y to a and (3 respect- 

 ively; the second inter- 



iaSH/3/) 



changes them as transformed; the third transforms them back again 

 to 8 and y; so that it is the same as if 8 and y had been interchanged.* 

 The substitutions of order 4 of which (a/?yS) is the type are not 

 expressed in our scheme. They are all products of a substitution 

 of order 2, within a set, by one of the substitutions that connects 

 like-placed functions in the different sets. Thus — 



(a/3y8)=:( / 88).(a/3)ry8) = (a/3)(y8).(ay). 



The five other pairs of equations similar to this can be obtained 

 by merely permuting the letters a, (3, y, 8. 



Similarly can we find the effect of any substitution not occurring 

 in a set upon a form that does there occur. 



E.g. (ay) does not occur in the/i set. 



But (ay) = (/3S).(ay)(/38) 



and (/38) does occur in the set. 



So cn*w(ay)==nc*zv. (ay) (f38)=—tanWcn 2 w. 



Again, because (a/3y) = (aS/3). (ay)(/?8), 



we have c?i 2 w{aPy)=ns' l w. (ay) (08) =sin 2 6cd 2 w. 



* It is usual to consider substitutions performed in the order in which they 

 are written, i.e., from left to right. 



245 



