78 MAJOR P. A. MAOMAHON : MEMOIR ON THE 



Clarendon type can be transformed into identities involving Roman type by sirnply 

 cha ,Hng the Type in the bracketed factors ; this ensues because the fractions (. M t 

 rSndon Kernes equal to (.)+(/) in Roman in the lumt when X equal 



* Art** The graphical representation by a three-dimensional lattice shows that the 

 generating function GF (/, m, ) is unaltered by any permutation of the letters I, m, n 

 The subjoined notation is designed to show clearly the six alternative expresses of 

 ,1,, .^...rating function, arising from this circumstance, which it a principal 

 if this paper to establish. 

 Tims, write 



TM i a+B)a+8+i)...(i+m+-i). 



s-l 







, M| 



IVTT 



(8) (8 + 1).. .(1 + 8-1) 



It is to be shown that 



GF(?,w,n) = |LM|, |LM| 2 



= | ML , |ML| 2 

 = |MN|, |MN|, 

 = |NM|, |NM| 2 

 = |NL|, |NL| 2 

 = |LN| 1 ]LN| 2 



|LM| n , 

 |ML| n) 



|NL 

 LN 



Art. :l. Every known particular case agrees with these formula, but only two 

 general results have been established prior to this paper. One is the well-known 

 case of partitions on a line, viz. : 



GP 1 n) = 



