1 led to the construction of the optical saccharimeter, which has been extensively adopted and employed in the analysis of other substances besides sugar. Notwithstanding that Biot, in 1860, in a comprehensive paper containing a résumé of all his previous researches in connection with the subject, again called attention to the facts of the case, the belief is still prevalent that all optically active substances behave essentially like sugar, and that the optical analysis of a substance is complete when the deviation caused by any solution of it has been observed, and the specific rotation calculated therefrom by the formulæ the resulting value being regarded as constant.2 In this manner the specific rotations of a very large number of substances have been calculated, and still appear in chemical and physical text-books without reference to the concentration or nature of the solvent media employed. 4 A few years ago, Oudemans, jun.,3 contributed fresh proofs that the specific rotation of substances is susceptible of considerable variations, according as different inactive liquids are employed for solution. Since then, Hesse, in 1875, published a large number of determinations of rotatory power, extending over fifty different active solid substances in solutions of different degrees of concentration. Even for small differences (between 1 and 10 grammes of substance in 100 cubic centimetres of solution), nearly all these substances displayed appreciable variations in the amount of their specific rotation, and nearly always a decrease for increased proportion of active substance. Still greater differences, in some instances exceeding 50°, resulted from the use of different solvents. § 24. Observations like those of Hesse have thus shown conclusively that, as a rule, no value can be attached to specific rotations deduced from an isolated observation of an individual solution. Biot,5 however, in his investigations of the rotatory power of tartaric acid 1 Biot: Ann. Chim. Phys. [3], 59, 206. 2 See, by way of illustration, Buff, Kopp, and Zamminer's Lehrbuch d. phys. u. theoret. Chem., p. 387. 3 Oudemans: Pogg. Ann. 148, 337; Liebig's Ann. 166, 65. 4 Hesse: Liebig's Ann. 176, 89, 189. 5 Biot: Mém. de l'Acad. 15, 205 (1838); 16, 254; Ann. Chim. Phys. [3], 10, 385; 28, 215; 36, 257; 59, 219. long ago showed the significance attaching to these variations in value, and the following considerations point to the same conclusion : The specific rotation of any active liquid can be determined directly, and is constant for any given temperature. But, when such a liquid (e.g., oil of turpentine) is mixed, in different proportions, with an indifferent liquid (as alcohol), and from the composition, density, and angle of rotation of the resulting solution the specific rotation is computed, the values so obtained will differ in a greater or less degree from that of the pure substance. Hence it follows that the specific rotatory power of a substance is in some way influenced by the presence of inactive molecules, so that its value is altered, in the majority of cases suffering an increase, and in rarer cases a decrease for increased proportions of the volume of solvent employed.1 If, however, the active substance be a solid, its rotatory power can only be examined in solution, and then different values for [a] will be obtained according to the character of the solvent, none of which is the actual specific rotation of the pure substance, but a value modified by the presence of the inactive liquid, and differing from the real value by a quantity unknown. When a pure homogeneous liquid is employed as solvent, so that only inactive molecules of one kind are allowed to influence those of the active substance, the variations in specific rotation are best shown by the graphic method, the percentages of inactive solvent (q) being taken as abscissæ, and the corresponding values of [a] as ordinates. The increase or decrease of specific rotation will then in many cases appear as a straight line, increasing therefore in direct proportion to 7. Hence it may be expressed by the formula in which the constants A and B must be ascertained from direct experiment. In other cases, it will appear as a curve, usually a 1 This may be easily shown with a polariscopic tube set vertically with the upper end open. Let oil of turpentine be poured in to a height of 1 centimetre, and the deviation observed. Then on adding successive quantities of alcohol a continuous increase of rotatory power will be found to take place. The number of active molecules here remains the same, but their action is distributed over a greater length of column. On the other hand, if nicotine be used and diluted with water, a continuous decrease will be observed in the rotatory power on successive additions. portion of a parabola or hyperbola, when the relation between the specific rotation and q is represented by an expression of the form— In these formulæ, A denotes the specific rotation of the pure substance. The values B (I.) and B and C (II.) represent the increase or decrease of A for 1 per cent. of inactive solvent. If q = 0, the specific rotation is that of the pure substance. On the other hand, if in equation I. or II., q = 100, we get for [a] a value which may be taken as the specific rotation of the active substance when infinitely diluted. Assuming that, when q = 100, the active substance vanishes and the solution consists of the inactive solvent alone, the rotatory power will then necessarily be nil. As Biot2 has pointed out, this may likewise be deduced from the foregoing expresα 100 1.d.p sions by equating them with the formula [a]= which is the specific rotation calculated from the directly observed angle of вя C+ q 1 The three constants A, B, and C of the formula [a] = A + may, according to Biot (Ann. Chim. Phys. [3], 11, 96, § 69), be calculated in the following manner :— Given three separate solutions with 71 72 73 per cent. of active substance, and three specific rotation values [a], [a]2 [a], respectively, then putting A + B = a, BC b, C = c, : the values a and c may be obtained from the equations : [a]2 22-[a] 21} + {[a], [a], } c — { 92 91} α = 0 equations B' q B wherein B' = and C' C Of course p can be substituted for q in the above formulæ. rotation. Introducing in the latter in place of p, since p + q = 100, the value 100-q, we get by equating with I., Putting in this equation q = 100, we get a = 0; that is, the rotatory power vanishes. If q = 0, then a = 1. d. A; that is, the angle of rotation for a length of 1 decimetre of pure substance of density d. And, as in that case = [a], we get α l.d [a] A, the specific rotation of the active substance in a state of purity. In such active liquids as are miscible in all proportions with some other indifferent liquid the variations in the specific rotation up to extreme degrees of dilution can be determined by direct experiment, and the complete curve be drawn from q = 0 to approximately q = 100. In such cases, if the constant A be deduced from observations of a number of solutions, a value will be obtained more closely approximating to the true specific rotation of the pure substance in proportion as the observations extend over a greater length of curve, and the nearer they approach the point where abscissa 90-that is to say, the greater the concentration of the solutions employed. When the active substance is a solid, the true specific rotation cannot be determined directly, and we can only construct a portion of the curve, larger or smaller according to the solubility of the substance, but always commencing at some distance from the origin of co-ordinates. If from these observations the constants A and B of formula I. or II. be calculated, the values obtained will only be strictly available for interpolation within the dilution-limits of the solutions employed. The question then arises to what extent in such cases we are justified in regarding the value obtained for constant A as the specific rotation of the pure substance. The extrapolation, which is here presupposed, is admissible indeed when the variation in the specific rotation takes the form of a straight line-i.e., can be represented by the formula [a] = A + Bq. But if, on the other hand, it takes the shape of a curve, the value of A, as calculated from the formula [a] = A + Bq + C q2 or some other similar expression, will represent the true specific rotation of the pure substance, the more imperfectly the shorter the length of the experimental curve. How far a correct determination is attainable in such cases will depend on the solubility of the active substance itself. If indeed only dilute solutions can be prepared, and if, moreover, the increase or decrease in the values of [a] is not constantly proportional to q, an estimation of the true specific rotation of the pure substance is quite impracticable. § 25. If we replace q in formulæ I. and II. by p-i.e., the proportion of active substance in 100 parts by weight of solution-the constant ▲ will then represent the specific rotation of the active substance in a state of infinite dilution, and that of the pure substance will be obtained when p = 100. But the employment of q (or, adopting Biot's notation e,-i.e., the proportion of inactive substance per unit-weight of solution) as above is preferable. In estimating rotatory power by the formula [a]= av 1.K = α. 100 1. c (see § 21), no determination of the specific gravity of the solutions is required but merely the concentration c, determined by means of a flask of known capacity. Modifying the foregoing equations I. and II. accordingly, we have [a] = A + B c, and [a] =A+B c+C c2, and putting c = 100, we get the specific rotation of a solution containing 100 grammes of active substance in 100 cubic centimetres of solution. But this would only represent the pure substance if it had a density d= 1. If, as is always the case, d has some other value, say d, we must give c the value 100 8. This, however, is a condition which can be but rarely satisfied, and never with certainty, as it presupposes a knowledge of the specific gravity of the active substance in an unknown amorphous condition. Hence, no specific rotation data where only the concentration of the solutions is taken account of, and specific gravity or weight per cent. composition is neglected, can be employed for the determination of the rotatory powers of the pure substances. $26. When an active substance is influenced by inactive molecules of two different kinds, as when some other substance is dissolved along with it, or the active substance is dissolved in a mixture of two different liquids, the case becomes much more complicated. Each of the inactive substances exerts its own influence on the true specific |