382 



MAJOR P. A. MACMAHON ON COMBINATORIAL ANALYSIS. 



lively, and in the successive columns the elementary functions a>^ Vt , 

 respectively, and the number A enumerates the lattices possessing this property. 



We may give this case a purely literal form by writing 100 = a, 010 = 6, 001 = c, 

 and then we have a lattice of s columns and t rows, such that the products of 

 letters in the successive rows are a Pl b q 'c ri , a p '& ?1 c r ', . . . a 1 *b q 'c r ' respectively, and in the 

 successive columns a*'& Ml c l>1 , a*'b*c**, . . . a^tfc" 1 respectively. 



Art. 14. Stated in this form the problem appears to have a close relationship to 

 the problem of the Latin Square. It is in fact a new generalization of that problem ; 

 for put s = t = 3 and 



Pi = 1l = r l = P-2 = ?2 = 8 = PS = ?3 = r Z = l 

 ^1 = /*! = v \ = X 2 = M2 = "2 = X 3 = P-S = "3 = 1 



so that the operation is DJ,, and the function a* u . One lattice is then 



or in literal form 



which is a Latin Square. Hence the numbers of Latin Squares of order 3 is 



in general of order n 



D s 

 *r 



B 



m ... 



a very simple solution of the problem. If reference be made to the solution arrived 

 at (loc. cit.) by considerations relating to a single system of quantities, it will be 

 noticed that the peculiar difficulties intrinsically present in that solution disappear 

 at once when n systems of quantities are brought in as auxiliaries. The Latin 

 Square appears at the outset of this investigation, and in a perfectly natural 

 manner. 



Art. 15. Now put 



Pi=Pz = 

 9i = ft = 



r i = r z = 



. j 



v + it. + 



A \ 11 + >) 



so that 5 = < = 



We have then lattices enumerated by 



DA + ,1 H K .A + M + y 

 Autt Ul,,^ . 



