94 MAJOR P. A. MAcMAHOSrON THE COMPOSITIONS OF NUMBERS. 



Also, clearly, if t = I 



= < c {(a,). ..(a.-i)(A<i 

 Art. 43. Therefore, by induction, we can express any form 



<M } 



as a form 



<M } 



The law is well seen by a particular case, viz., 



ex{(a)(b)(c)(d)}=<t> c (abcd), 

 0*{(ab)(c)(d)}=<i> c {(a)(bcd)}, 



e ti {(a)(b)(cd)}=<t> c {(abc)(d)}, 

 N {(a)(bcd)}=<j> c {(ab)(c)(d)}, 



occur 



0*{(abcd)}=<f, c {(a)(b)(c)(d)}. 



We have, in respect of the four letters, 8 = 2 3 relations ; the letters always 



in the order 



a, b, c, d, 



and to obtain the form <f> c { }, which is equated to a form # N { }, we may make use 

 of the zig-zag conjugate law; e.g., connect with 



(ab) (cd) 

 the composition 22 ; take the zig-zag conjugate of this, viz., 121, and then write 



0x{(ab)(cd)}=<j> c {(a)(bc)(d)}, 

 and 



M()(k) (<*)}=*>{(*)(*)}; 

 and so in every case. 



Art. 44. In the general case of p letters we obtain 2 f ~ l relations corresponding to 

 the 2*"" 1 compositions of p ; the relations are obtainable from zig-zag conjugation of 

 such compositions and, in any relation 



we may interchange the form- symbols 



#}> $C- 



Art. 45. In the above investigation we obtained incidentally certain linear relations 

 between the forms 



