The mathematical theory of birotation is given by Müller as follows.[121] In proportion as the unstable modification _A_ is transformed into the stable modification _B_, the rotation will vary. Let ρ = the specific rotatory power of _B_ and _a_ρ = that of _A_, both in the anhydrous state. Let now _p_ grams of the substance be dissolved in _V_ cubic centimeters of solvent and observed in a tube _l_ decimeters in length. The time from making the solution is represented by θ. The angle of rotation α is read at the time θ. Let _x_ = the mass of _A_, and _y_ = that of _B_, and the equation is derived. _a_ρ_xl_ ρ_yl_ α = --------- + ------: _V_ _V_ But _x_ + _y_ = _p_ ρ_l_ whence α = [(_a_ - 1)_x_ + _p_] ----. _V_ If now there be introduced into the calculation the final angle of rotation αₙ, which can be determined with great exactness; we have _p_ρ_l_ (_a_ - 1)_x_ αₙ = ------- and consequently α = αₙ[1 + ------------], _V_ _p_ (_a_ - 1)_x_ α whence ------------- = --- - 1. _p_ αₙ This equation gives the quantity _x_ of the unstable matter which is transformed into the stable modification in the time θ. It must be admitted that the quantity _dx_ which is changed during the infinitely small time _d_θ is proportional to the mass _x_ which still exists at the moment θ, whence _dx_ = -Cʹ_xd_θ where Cʹ represents a constant positive factor. From this is derived the equation _dx_ ----- = -Cʹ_d_θ. _x_ Integrating and calling _x_ the quantity of matter changed to the stable form at the moment θ, corresponding to a rotation α₀, we have 1 _x_₀ Cʹ = ------- log. nap. ----, and taking into consideration θ - θ₀ _x_ the equation given above, and substituting common for superior logarithms we get 1 α₀ - αₙ C = --------- log. ---------. θ - θ₀ α - αₙ Experience has shown that such a constant C really exists, and its value can be easily calculated from the data of Parcus and Tollens.[122] The mean value of C from these data is 0.0301 for arabinose; 0.0201 for xylose; 0.0393 for rhamnose; 0.0202 for fucose; 0.00927 for galactose; 0.00405 for lactose; 0.00524 for maltose, and for dextrose, 0.00348 at 11° to 13° and 0.00398 from 13° to 15°. The constant C as is well known, increases as the temperature is raised. The constant C, at a given temperature, measures the progress of the phenomenon of the change from the unstable to the stable state. It will be noticed that among the sugars possessing multirotation properties the pentoses possess a much higher speed of transformation than the others.