On account of this difficulty, the atomic weights published by Dalton, and the more accurate ones of Berzelius, were not always identical with the values now accepted, but were often simple multiples or **submultiples** of these.

In addition to these wave-lengths there are other groups centred round the wave-lengths which are **submultiples** of the principal one - the overlapping spectra of the second and higher orders.

We may therefore put y=a sin -T (x - e) +b sin (x - f)+c sin 6 (x - g) +&c. (21) where the terms may be infinite in number, but always have wavelengths **submultiples** of the original or fundamental wave-length A.

It follows from this that any periodic disturbance in air can be resolved into a definite series of simple harmonic disturbances of wave-lengths equal to the original wave-length and its successive **submultiples**, and each of these would separately give the sensation of a pure tone.

3.3 (iii.) Multiples, **Submultiples** and Quotients

**Submultiples**.-The relation of a unit to its successive multiples as shown in a multiple-table is expressed by saying that it is a submultiple of the multiples, the successive **submultiples** being one-half, one-third, one-fourth,..

- Instead of regarding the 153 in 27.153 as meaning o h, we may regard the different figures in the expression as denoting numbers in the successive orders of **submultiples** of i on a denary scale.

The following tables are given as illustrations of the arrangement adopted elsewhere in this article; the entries in any column denote multiples or **submultiples** of the unit stated at the head of the column, and the entries in any row give the expression of one unit in terms of the other units.