According to this view, it is necessary to assume that, in all unsaturated compounds, two, or some even number of affinities are disengaged; and also that all elements which combine with an even number of monad atoms cannot combine with an **odd number**, and vice versa, - in other words, that the number of units of affinity active in the case of any given element must be always either an even or an **odd number**, and that it cannot be at one time an even and at another an **odd number**.

Again, when tungsten hexachloride is converted into vapour it is decomposed into chlorine and a pentachloride, having a normal vapour density, but as in the majority of its compounds tungsten acts as a hexad, we apparently must regard its pentachloride as a compound in which an **odd number** of free affinities are disengaged.

The problem of finding the sum of r terms is aided by graphic representation, which shows that the terms may be taken in pairs, working from the outside to the middle; the two cases of an **odd number** of terms and an even number of terms may be treated separately at first, and then combined by the ordinary method, viz.

By similar methods, a circular plate may be made to exhibit nodal lines dividing the surface by diametral lines into four or a greater, but always even, number of sectors, an **odd number** being incompatible with the general law of stationary waves that the parts of a body adjoining a nodal line on either side must always vibrate oppositely to each other.

The formula shows that except for numbers of the form (3n 2 n) the number of partitions without repetitions into an **odd number** of parts is equal to the number of partitions without repetitions into an even number of parts, whereas for the excepted numbers these numbers differ by unity.