350 MAJOR P. A. MxcMAIION: MEMOIR ON THE 



Art 7 Continuing, for the present, to regard the part magnitude as unrestricted, 



we now proceed to the equations satisfied by the inner lattice funct 

 Guided by the relation 



IL(oo;a 1 ,a 2) ...,a,) ) 



we find 

 (2a)IL( oo; a, a) = (a) IL( a> ; a, a-1) ; 



(3a)IL(oo;a,a,a) = (a)IL(oo; a, a, a-1); 



(a+2b) IL( oo ; a, b, 6) = (a+2) IL (oo ; a-1. b, 6)+(b) IL( ; a, 6, 6-1) 



-(a+2) (b) IL ( oo ; a- 1,6,6-1); 



(a+b+c)IL( oo; c , 6, c) = (a+2) IL( oo; a-l, b, c) + (b+l) IL( <x ; a, 6-1, c) 



-f(c)IL(oo; a, 6, c-1) 



-(a+2) (b+1) IL ( oo ; a- 1, b-l, c)-(a+2) (c) IL ( ; a-1, 6, c-l) 



+ (a+2) (b+1) (c) IL( QO ; a-1, 6-1, c-l), 

 and, in general, for n unequal rows, if we write symbolically, 

 q. !L(oo ; a lf Oj, ....a,) = (a,+n s) IL( oo ; a 1; a a , ... 



; a,, a a , .... 



To these may be added 





k* el- 



. , y 



which can be readily generalized. 



Art. 8. I proceed at once to find an expression for the inner lattice function. 



It appears to be right to seek an expression of the function which shall show at 

 once that the sum of the coefficients therein is that which it is otherwise known to 

 be. Thus the result 



shows at once that the sum of the coefficients is a 6 + 1. 



