THEORY OF THE PARTITION OF NUMBERS. 
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hence :—- 
“ The number of partitions of all numbers which have exactly q parts, a highest 
part equal to p, and s different parts is 
“ The number of partitions of all numbers which have exactly q parts, a hig’hest 
part less than p, and s different parts, or which have less than q parts, a highest part 
equal to p , and s different parts is 
“ The number of partitions of all numbers which have less than q parts, a highest 
part less than p, and s different parts is 
Ex. gr. If p = q = 3, s = 2 the partitions enumerated by these three theorems 
are 
(31*), (3*1), (32*), (3*2) 
(31), (21*), (32), (2*1) 
( 21 ) 
respectively. 
Art. 14. It is clear that all identical relations between binomial coefficients yield 
results in this theory of partitions. Ex. gr., such relations as 
(';•)-("-7‘M'T‘) 
admit of immediate interpretation. 
Art. 15. A line of route has in general right and left bends. The whole number of 
bends may be even or uneven. To determine the number of lines, with a given 
number of bends, we must separate the two cases. If the bends be 2 k in number, k 
must be right and Tc left and the enumeration gives 
