84 MAJOR P. A. MACMAHON: MEMOIR ON THE 



Thus the numerator finally determined is necessarily the lattice 



function L( oo ; 2, 2) found by the specified rules. 



Art. 9. Next take a case which is not quite so simple 



p q r 

 s t u, 



where p 2: q ^ r, s^t^u,p^s,q^t,r^u; the associated lattice arrangements 

 and the Greek letter permutations are 



654 643 653 652 642 



321 521 421 431 531 



a* * v? x" x * 



yielding 



We have five non-overlapping systems 



(i) p>q^r>s>t>iu, 



(ii) p^s > qZzr^tZzu, <*fi\ aa 



(iii) p>q>s> r^t^u, 



(iv) >2:<j'is>:> r>:ti, aa^8/8 1 aft, 



(v) j>S*>gl>r>tt, a^|a)8|ai8, 



wherein the positions occupied by the symbol > are to be compared with the 

 positions of the dividing lines in the corresponding Greek letter permutations. It is 

 clear that the summations derived from the systems (i), (ii), (iii), (iv), (v) give powers 

 of x in the numerator of the generating function exactly corresponding to those 

 which enter into the lattice function by the rules given. Hence 



_ 

 (!)().. .(8) (2)(3)-(l)(2)...(6)' 



