12 - REPORT—1874. 
of partitions miy be reduced to that of the expansion of algebraical fractions. In this 
way Cayley has regarded the question in his memoir in the Philosophical Transactions, 
1856, p. 127, where, besides considering the general decomposition into partial frac- 
tions of the expression to be expanded, he has given the values of P(1, 2)2, P(1, 2,3) 
2,..++, P(A, 2, 3, 4, 5)x, and in the Philosophical Transactions, 1858, p. 52, also of 
PCI, 2, 3, 4, 5, 6)2, PC, 2, 3,....7)x denoting the number of ways of partitioning 
x into the elements 1, 2, 3, 4....7. The subject is also considered by Sylvester 
(Quarterly Journal of Mathematics, vol. i. pp. 81 and 141, 1857), who likewise has 
treated it (though very differently) as an expansion problem. It may, however, be 
regarded in another light as follows :— 
It is known that if we form the literal derivations of a power of a letter, say a‘, 
according to Arbogast’s rule, viz. 
a, 
a*b, 
ae, ab’, 
ad, a’be, ab? 
ae, a*bd, a’c’, abe, b', 
then each term corresponds to a partition; thus if a=1,d=2,....a* corresponds to 
1,1,1,1, a to 1,1, 1, 2, the third line to 1,1, 1,3, and 1, 1, 2, 2, and so on, viz. 
we have the partitions into four parts of the numbers 4, 5, 6,...., the m+ 1th line 
(the nth derivation) giving the partitions of n+-4 into four parts. But by a known 
theorem the number of partitions of x+r into 7 parts is equal to the number of 
partitions of n into the elements 1, 2, 3,....7. Thus the number of terms in the 
ath derivation of a‘ is equal to P(1, 2, 3, 4), and generally the wth derivation of 
a” contains P(1, 2, 3,....%)# terms. From these considerations the value of 
PC, 2, 3,....n)2 can be found in the way which will now be briefly explained. 
Consider a?, and let 2” denote the number of terms in the wth derivation; then, 
writing down @? and its first two derivations, 
a’, 
ab, 
ac, b?, 
we see that 2?=1-+-2°, whence 2” +2_ 971; oy writing wv, for 2” and E for 1+, 
(i? =1)u,=1, 
the solution of which, by the ordinary rules for the treatment of linear equations of 
differences with constant coefficients, is 
mat Aes Be 
“where a and f aré the square roots of unity. The complementary function can also 
he written in the far more convenient form A+B(1,—1) per 2., adopting Cayley’s 
notation, in which (A,, A,..°. A,_1) per adenotes A,a,+A,a,_1... +Ag_14%-a4p 
a, being a quantity which =1 when a==0 (mod. a), but which =0 in every other 
case, and the coeflicients A,, A,....A,, being such that for every factor b of a 
(including unity but excluding a itself), A,+A,....-+A(e—1),=0, A, AAs 41+ ++ 
+A(e—1)041=0,- ++» Ag_y-. »- +A,g_1=0, where be=a, and for the case of b=1, 
A,+A,....+A,_;=0. Determining, then, the constants from the conditions 
2° 2'=1, we ind 
2*=P(1, 2)r=7{2e+3+(1,—1) per 2,} 
Now consider a’, and let 3° denote the number of terms in its th derivation ; 
then, by writing down the first three derivations of a’, we see that 3°=1°+-2'+3°; 
so that 8°+—3"=142"+1 and the equation of differences is 
(B° —1)u, =F {2¢+9+(1,—1) per2,44}- 
