356 

 whence 



or 



MAJOR P. A. MxcMAHON: MEMOIR ON THE 



IL ( oo ; , 6, c) = 



+* 3 (b) (b+1) (c)+z 3 (c-1) (c) (a+2)-* 4 (c-1) (c) (b+1), 



(c) -=- a-, 1, 1 



x 1 * 3 , <r, 1 



a sa 



satisfactory representation of the inner lattice function. 



Art 14 Passing now to the consideration of the inner lattice function of 

 order 4, viz., IL ( oo ; a, b, c, d), and guided by the above results, I put 



Al = (a+3), A 2 = (a+2) (a+3), A 3 = (a+1) (a+2) (a+3), 



63 = 



C 2 = (o) 



D,= (d) , D 2 = (d 

 and I consider the twenty-four products 



A 3 DA, 



B 3 A 2 Ci, B 3 A 2 U 1; L) 3 A 2 Ui, 

 BgCgAi, -D3.L) 2 Ai, _U 3 \-/ 2 A.i, 

 C,A,Bi, D 3 A 2 B 1( C 3 A 2 D 1 , C 3 D 2 B 15 



C^A,, DsBA, CaDA, CaBA, 



which, to suffices 3, 2, 1 in descending order, involve every permutation of the letters 

 ABCD, three at a time. 



Art, 15. I shall show that each of these products is a solution of the functional 



equation 



y^TL( oo ; a, b, c, d) = (l-q,) (l-q a ) (l-$ 3 ) (l-q,) IL ( oo ; a, 6, c, d) ; 



for, looking at the definition of the symbol q, it is clear that 



9,A,BA = (a)A 3 BA, gA 3 BA = (b)A a BA, gaAaBjd = (c)A 3 B 2 C 1( ? 4 A 3 BA = (d) A 3 BA, 



= (a)(b)A 3 BA, giMa^ACi = (a)(b)(c)A 3 B 2 d, &c. 



Hence 



-?4) A 8 B 2 C 1 



establishing that A 3 B 3 Ci is a solution. 



-(b)} (l-(c)} (l-(d)} A 8 BA = 



