384 



MAJOR P. A. MACMAHON ON COMBINATORIAL ANALYSIS. 



Again, suppose the row products to be a?b, a 2 6 8 , a 2 6 3 , and the column products 

 a s , a~b, a?b, b* 



DD?2 30 a 



03 



and the lattices are 



= D 31 D' K (fo 3 )(To'oT) 2 (oT) 

 = D* 2 (To 2 )(T6oi) 2 (oi 2 ) 



= 2D 22 (TO) 2 (OI) 2 =2, 



Art. 17. In general, we may state that the lettered lattices, which are such that 

 the row products are in order 



ifliffi &*t rtP-7>-/ r - p. 



./ t/ /!/* Lv I/ ly Ar 



and the column products in order 



ctf*' l b ttt c Vl k i cfrb^^c 1 *' &** ci^"Zy ill c 1 '* yfc^* 



one letter being in each compartment, are enumerated by 



a very interesting development of the Latin Square problem. 



We have found above the nature of the lattices enumerated by the number 



any number of systems of quantities being involved, and the mere fact of the 

 existence of the lattices indicates a law of symmetry which may be stated as 

 follows : 



If 



then 



Art. 18. The next case that comes forward for examination is that connected with 

 the homogeneous product sums h^ v . . . We require the theorem 



and also 



