(b) If **nx** >o, (1+x).

Thus we arrive at the differential coefficient of f(x) as the limit of the ratio of f (x+8) - f (x) to 0 when 0 is made indefinitely small; and this gives an interpretation of **nx** n-1 as the derived function of xn (ï¿½ 45)ï¿½ This conception of a limit enables us to deal with algebraical expressions which assume such forms as -° o for particular values of the variable (ï¿½ 39 (iii.)).

} **Nx** ..., it is clear that, if x,.

=n, then we have in the development of the product a term x n, and hence that in the term **Nx** of the product the coefficient N is equal to the number of partitions of n with the parts I, 2, 3, ..., without repetitions; or say that the product is the generating function (G.

Observing that any factor 1/I-x l is=l+x l +x 2l +..., we see that in the term **Nx** the coefficient is equal to the number of partitions of n, with the parts I, 2,, ..,withh repetitions.

+**Nxnzk**+.., we see that in the term **Nx** n z k of the development the coefficient N is equal to the number of partitions of n into k parts, with the parts I, 2, 3, 4,, without repetitions.

We see that in the term **Nx** z of the development the coefficient Nis equal to the number of partitions of n into k parts, with the parts I, 2, 3, 4, ..., with repetitions.

Let somehow or other retardations be introduced so that the optical length of the successive parts increases by the same quantity **nX**, n being some number and X the wave-length.