of the Fifth Degree. 359 



VI. Dependent interchanges are subject to the relations 



y/ ab\ /bc\ _ y(^'^\ h^\ — Y(^^\ (^^\ 



VII. We can operate with the first and second of any three 

 interchanges as if the third did not exist, and in operating with 

 the second and third we may neglect the first. 



14. Using the affix I to denote either of the equivalent sub- 

 stitutions 



(fOOOC"")' ("*)(•■) CO' «="•■ 



we find, Y being suppressed for a moment, 

 and 



15. Writing out the equations 



Y=;0(a, A, 0, «?) = 04, 



Yl=(f>{b, d, a, c) = <^j, 



Y/^=^(c,fl,«?,^) = ^3, 



YP = (f){d, c, b, a) = 02, 

 and remembering that 



/''"+'' = f, 

 we see that, if ^ be a symmetric function of 0,, 02> 'P^ ^^^ ^v 



e=6i=ei^=ep. 



16. Let X, fjb and v be any affixes of substitution. Then 

 eithef of the equations 



Z/A=Zv and Z/iaA=ZvX 

 is a consequence of the other, whatever Z may be. 



17. By the aid of this proposition and of art. 13 we obtain, 

 from d = ei, 



and, from d = 6P, 



