262 
ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. 
[406 
We thus obtain the symbolic parts of the several expressions for [14], [23].... [11111] 
respectively: the sign of each term is + or — according as the number of factors in 
( ) is even or odd; thus in the expression for [11111], the term (14) (13) (12) (11) 
has four factors, and is therefore + , the term (113) (12) (11) has three factors, and is 
therefore —. 
The numerical coefficients are obtained as follows. There is a common factor 
derived from the expression in [ ] on the left-hand side of the equation ; viz. for 
[11111], which contains five equal symbols, this factor is 1.2.3.4.5, =120; for [1112], 
which contains three equal symbols, it is 1.2.3, = 6 ; and so on (for a symbol 
such as [11222] containing two equal symbols, and three equal symbols, the factor 
would be 1. 2.1.2.3, = 12, and so in other similar cases). In any term on the right- 
hand side of the equation, we must for a factor such as (11), which contains two- 
equal symbols, multiply by for a factor such as (111), which contains three equal 
symbols, multiply by and so on. And in the case where a term (as, for example, 
the term (122) (11) or (23) (12) (11), vide supra) occurs more than once, the term is to 
be taken account of each time that it occurs ; or, what is the same thing, since the 
coefficient obtained as above is the same for each occurrence, the coefficient obtained 
as above is to be multiplied by the number of the occurrences of the term. For 
example, taking in order the several terms of the expression for [1112], the common 
factor is = 6, and the several coefficients are 
6 
1 • 
6 ’ 
6, 6. 6 . 6.^x2, 6.*, 6.^-, 6 . |. I x 2, 
and similarly in the expression for [11111] the common factor is 120, and the coefficients 
taken in order are 
120.1, 120.110.*.*.*, &c., 
without there being in this case any coefficient with a factor arising from the plural 
occurrence of the term. 
The foregoing result was established by induction, and I have not attempted a 
general proof. 
I observe by way of a convenient numerical verification, that in each equation the 
sum of the coefficients (taken with their proper signs) is (—) n_1 1.2.. (n — 1) ; if n be 
the number of parts in the [ ] (n = 5 for [11111], =4 for [1112] &c.), and moreover, 
that the sum of these sums each multiplied by the proper polynomial coefficient and 
the whole increased by unity is = 0 ; viz. for 
[14], [23], [113], [122], [1112], [11111], 
the sums of the coefficients are 
— 1, —1, +2, -j-2, —6, +24 respectively, 
and we have 
1 + 5(— 1) + 10(— 1) +10(2) + 15 (2) +10 (— 6) +1 (24), = 75 -75, =0. 
If we have any five distinct things (a, b, c, d, e), then the polynomial coefficients 
5, 10, 10, 15, 10, 1 denote respectively the number of ways in which these can be 
partitioned in the forms 14, 23, 113, 122, 1112, 11111 respectively, and the last-mentioned 
theorem is thus a theorem in the Partition of Numbers.