96 MAJOR P. A. MACMAHON ON THE COMPOSITIONS OF NUMHKHS. 



Thus it is easy to verify the three relations 



0{(ab)(cd)(ef)(gh)} 

 = y {(abcd)(ef)(gh)} 



and the further three 



8{(ab)(cd)(efgh)} 

 + 0x{(ab)(cd)(e,f+g,h)}, 



e*{(abcd)(efgh)} 

 + e s {(abcd)(e,f+g,h)} 

 + s {(a, b + c, d)(efgh)} 

 + N {(, b + c, d)(e,f+g, h)} ; 



<t> c {(abcd)(ef)(gh}} 

 -<f> c {(a,b + c,d)(ef)(gh)}, 



4> c {(aV)(cd)(efgh}} 

 -<f) C {(ab)(cd}(e,f+g, h)}, 



4> c {(abcd)(efgh)} 



-<j> c {(abcd)(e,f+g, h)} 



+ <^ c {(a, b + c, d)(e,f+g, h)}. 



Art. 48. From these relations we may obtain new relations by transforming from 



to < c , or vice versd. 



Thus from relations of type 



we obtain those of type 

 < c (a&c) = <j>c{( 

 and from those of type 



} = <j> c (abc)-<j> c (n + b, c)-<j> c (a, 

 we obtain others of type 



These new expressions for 



6 y (abc...) and <f 



with an obviously analogous law to that we have frequently met with, are of great 

 importance. 



