T1IKORY OF THE PARTITIONS OF NUMBERS. 



369 



X 







(1)(2)(3) 



GF(Ac,e*;4) 



x 



X(c) 

 X 



x , X M (b+l), X 31 (c)(c+l) 







b 



(!).. .(5) 



(a+1), (b)(b+l), 

 x , X(b+l), 

 ,' x 

 , , 







xX 41 (d-l)(d) 



x , X M (c+l) , X 4S (d) 



, x , X 43 



GF(a,b,c,d,e;!>) 



5+1), .r 3 (d-2)(d-l)(d)(d+l), x>-3)(e-2)(e-l)(e) 



X 31 (c)(c+l) , xX 41 (d-l)(d)(d+l) , .r'X 51 (e-2)(e-l)(e) 



Xsa(c+l), , X(d)(d+l) , xX ( 



x X43(d+l) X5a(e) 



x X M 



and tlie law is evident. 



Art. 32. In general, supposing the lattice to l)e in the plane of xy, that of the 

 paper and the axis of z perpendicular to the plane of the paper, if we project the 

 partition on to the plane of yz, we obtaiu a partition at the nodes of a lattice of I 

 rows in which the part magnitude in the s lh columns is limited by the number a r 



The general formula for GF(Z; a,, a a , ..., a,) is remarkable from the fact that 



GV(l ; a,, .... a.) = GF(Z ; 6,, 6,, .... 6.), 

 where (a t , a a , ..., a,), (b lt l> 3 , ..., b m ) are any two conjugate line partitions. 



,' 



Adumbration of the Three-dimensional Theory. 



Art. 33. 1 conclude this Part by pointing out a path 

 of future investigation into the Theory of Partitions 

 in space of three dimensions. 



I consider a complete or incomplete lattice in 

 three dimensions, the lines of the lattice being in 

 the direction of three rectangular axes of x, y, z -4 

 respectively. Just as an incomplete lattice in two dimensions is defined by a one- 



VOL. ccxi. A. 3 B 



