52 Proceedings of the Royal Irish Academy. 



five quaternions are linearly connected; and the number of distinct 

 permutates is reduced in this way to 



„ n.n-l.n-l.n-^.n-4: _„. 



n» jj^ . (28) 



Similarly the conjugates and permutates of a function of the 

 fourth order are connected because ^af{bcde) is a particular case of 

 the function of the fifth order. 



The number of conjugates and permutates formed on the plan of 

 Art. 5 is, in the first instance, II {n + 1). This reduces, for the 

 reasons given, to 



n (n + 1) - Jll^-^i) ^ . (29) . 



^ ^ n5n{n-5) n4n(»-4) ^ ^ 



