THEORY OF THE PARTITIONS OF NI'MIU-ks. 87 



Art. 13. The next step is to establish the fundamental relations 



ft ft n \ L (l'< i.a. 3. ) 



- 



GF ft m n) = L & OT) n ) 

 (l)(2). 



We have to take account of the circumstance that the part magnitude is now 

 restricted not to exceed I. Take again the case, previously considered, of two rows and 

 three columns. I recall the five distinct parts of the summation 



(i) p > q 2S r 2: s 2: t i u giving 2x B+JD+3C+iB+&A+6tt 



( (11) p S: S > q > r > t > M 



(iii) j)>9>0>r>t>u 

 I (iv) ^a/a>r2=M 

 (v) p =s s > 5 >: > r> M 

 For the condition (i) we put 



s = w+A + B+C+D, 



from which it is clear that 



w+A+B+C+D+E 



cannot exceed I in magnitude ; hence the sum 



B + 2D + 3C + B + 5A + ttu 



is the generating function of partitions on a line into I parts not exceeding 6 in 

 magnitude, and is therefore 



Similarly in each of the cases (ii), (iii), (iv), belonging to Lj ( oo, 3, 2), we put 

 p _ M + A+B+C+D + E + l, and it is clear that u + A + B+C+D + E cannot exceed 

 ^-1 in magnitude ; the corresponding portion of the generating function is therefore 



Finally, since in (v) we put 



p = W+A + B+C+D+E+2, 



