1(50 MAJOR P. A. MACMAHON: MEMOIR ON THE 



and observing that this last result may be written 



(aa;) = (40;) (a, a-4;) + (31;)(a-l, a-3 ;) + (22 ;)(a-2, a-2;), 

 it is natural to suspect the law 



(aa ;) = 2 (at ;) (a-t, a-s ;), 

 where 



s + t = constant, 

 and it is easy to establish it. 



For consider the graph 







The four lowest numbers at the nodes are 



(i) The last four of the second row, 



(ii) The last of the first row and the last three of the second, 



(iii) The last two of both rows. 



Taking case (ii), the nodes marked x, the numbers may be 



234 



431 or 421 or 321, 



and these arrangements are enumerated by (31;), we see, by subtracting each number 

 from the number 5. Hence, in particular, 



(88 ;) = (40 ;) (84 ;) + (31 ;) (75 ;) + (22 ;) (66 ;), 

 and, in general, 



(;) = (at;) (a I, as ;) where s + t constant. 



19. Representation of (aa ;) as a Sum of Squares. 

 Putting s + t = a, we find 



(aa;) = (aO;) 2 +(a-l, 1 ;) 2 +(a-2, 2 ;) 2 + ..., 



the last term being 



(|a,-|a;) 2 or {(+!), i( a - 1)} 2 , 



ascending as a is even or uneven. 

 Hence the identity 



