According to the foregoing examples, then, it is the angle of rotation above all that one needs to determine with the utmost accuracy, and the more so the less the angle is. The error in the measurement of tube length influences the result but slightly, that in the determination of specific gravity rather more, the effects of both being greater the higher the specific rotation. As regards the error in determining percentage composition and concentration of solutions, this of course exerts a greater effect the more dilute the solutions. In practice, however, the amount is not so great as shown in the foregoing calculations, because the amount of error in determining concentration and percentage composition is there assumed to be the same for all solutions, whereas it diminishes with increased dilution.

In general, indeed, the errors are seldom so large as they are assumed to be in the foregoing examples, and the smaller they can. be made the less of course will the result obtained differ from the true value. Besides, errors made by any single observer may be partly positive and partly negative, and so become eliminated from the result. In any case it will be seen that careful working is requisite to get even the first place of decimals correct, and that in all researches of this kind it is necessary to know the degree of accuracy with which the several measurements have been made before any judgment can be formed on the value of the final result.

VI.

PRACTICAL APPLICATIONS OF ROTATORY POWER.

A. Determination of Cane-Sugar.

OPTICAL SACCHARIMETRY.

§ 76. The determination of the percentage of sugar in aqueous solutions is based upon the following propositions established by Biot:

1. The amount of deviation of the plane of polarization is proportional to the length of the liquid column.

2. The deviation is proportional to the concentration—that is, to the number of grammes of sugar in the unit-volume (100 or 1,000 cubic centimetres) of solution.

Accordingly, by determining once for all the angle of rotation. given by a single saccharine solution of known concentration in a tube of a certain length, we are able by simple proportion to calculate the number of grammes of sugar in 100 cubic centimetres of any solution of unknown strength from its observed angle of rotation.

As before explained, §§ 23 and 37, Biot's second proposition is not strictly correct, inasmuch as the specific rotation of cane-sugar decreases somewhat with increase of concentration, so that a solution of double percentage does not give exactly double the angle of rotation, but rather less. The errors from this source are, however, small, and apart from very accurate experiments may be entirely neglected. In the following account of the saccharimeters, the proposition is at first taken as strictly true; corrections for varying rotation being always discussed separately..

It is evident that any of the forms of polariscope already described