with equation (5) of the preceding paragraph, we find Yer = +2 sin C cos C e-t/Tjdt et (e-f)

±(sin2-cos2 ) e-T Sdt efTe.

.

(8)

$5. Using equations (3), (4), (5) of § 2, as well as (3) of § 1, the preceding equation (8) may be written, after some transformation,

Yi = { (sin3 © —cos2) e='/TS dt e'/TM (sin3ht—cos3ht) 74 sin ✪ cos ✪ e-/T Sdt et sin ht cos ht}{-2}. (1)

The integrals entering into this expression are easily calculated. We find finally

Let the liquid be traversed by a ray of light whose direction is parallel to the axis of rotation. The double refraction produced, referred to unit length, is, according to F. E. Neumann's theory, proportional to the value of the corresponding component yr of the actual deformation. Thus the observable optical effect will in the first place depend on the kinematical conditions of the experiment, such as the velocity of rotation, the nature of the function q(r), &c. In the second place, it will vary with the duration, for the liquid in question, of the time of relaxation T. It will finally depend on a purely optical coefficient; but it would be difficult, if not impossible, to advance any definite conclusions as to the nature and exact value of this coefficient. Even if it were supposed that its value does not differ greatly for different liquids (certain very special cases being excepted), the only conclusion which one could arrive at is the one already pointed out by Maxwell, viz.: the double refraction observed in using different liquids under identical kinematical conditions, depends above all upon the time of relaxation T of the liquid. Now the viscosity of a liquid does not depend solely on the value of T; it is equally dependent on the value of an entirely different constant, the momentary rigidity of the substance. To sum up, there is nothing to justify the assumption that accidental double refraction in a liquid depends solely on its viscosity.

term

6. Equation (2) of the preceding paragraph contains the

already considered by Kundt in the Memoir referred to above. In order to calculate it, Kundt had recourse to the formule given by Stokes for the case of steady motion, of which the optical experiment is a realization.

Sir G. G. Stokes evidently started in his analysis from the classical equations of motion of a viscous fluid. In the particular case with which we are concerned, it is seen at once that the generalized equations which we have developed in our previous communication are incapable of assigning to the term (1) a value other than that found by Stokes; this results from our supposition that the motion of the liquid tends to become more and more steady. This conclusion is easily verified. Let w=0, X=0, w=0 in the first equation of motion, § 10 of the previous communication. We then have

da ду

(2 a)

. . (3)

By the equations

;

lastly, the last term of the second member of equation (2 a) has the value

±μ(2+1dq_ 2) sin - hT cos ✪
-3)si
dr.2 r dr

1+h2T?

(5)

In virtue of these results, equation (2 a) teaches us that the motion of the fluid, while approaching the limiting case of steady motion, must more and more accurately satisfy the equations

which are precisely those given by Stokes. On integrating d taking into account equations (1), § 1, we find from

$7. Let N stand for the number of revolutions of the cylinder per unit of time. Let R be an optical coefficient, A the double refraction referred to unit length. Let further

Using the relations (8) and (9) of the preceding paragraph, we find from equation (2), § 5,

According to this equation, the quantity A/N should not be constant, as was at one time supposed: it should decrease as N increases.

§ 8. The numerical results given by Umlauf in the paper quoted above are probably among the most exact hitherto published on accidental double refraction in liquids. But the accuracy of the results is not the same for the various substances studied. On close examination it is easy to see that the most reliable data are those obtained with gum tragacantha fact which is also confirmed by Umlauf (p. 311 of the cited volume of the Annulen). The following are the results of the corresponding series of experiments; we have supplemented them by the numbers contained in the last column of the table. In order to calculate this column, we have assumed T=0·0014 sec. for this series; as regards the coefficient B, we have calculated its value by means of formula (1), § 7, making use of the data supplied by Umlauf regarding the dimensions of his apparatus.

The way in which the values of A/N and ▲(1+BN2T2)/N vary according to these experiments is in perfect agreement with our predictions. It is to be noted that the temperature of the liquid was never strictly constant in these experiments; it varied from 13°2 C. to 15°.7 C.

It is a matter of doubt as to whether the results obtained

by Umlauf for a number of other liquids can satisfactorily be used in a calculation of such refinement. Nevertheless, here is a series of experiments relating to collodion (p. 309 of the paper quoted):

The following is another series relating to the same substance (p. 314):

These two series were obtained with two different forms of apparatus; hence the coefficient B did not have the same value in each case. The temperature of the liquid varied within about two degrees, the mean being 16° C. In calculating the figures in the last column we have taken T=0.00219 sec. A comparison of the two series has already been made by Umlauf himself. The mean of the results entered in the last column is 152.5.10-5 for the first table, and 110. 10-5 for the second; the ratio of these two means is therefore 1.387. If we calculate the ratio of the values which we should have to give to the coefficient A according to (1), § 7, in these two series of experiments, we find 1:561.

The results of the experiments carried out by G. de Metz lend themselves still less readily to this kind of calculation. The following is the only series of experiments which we have found it possible to utilize. It refers to castor-oil at 25° C. (de Metz, l. c. p. 504). T has been put =0·0013 sec.

A(1+BN2T2)/N.

69.10-4

It would certainly be rash to pretend that the values of the time of relaxation T which we have found for certain liquids closely approach the true values. Nevertheless the order of magnitude which we have been led to assign to them does not appear to be impossible.

XLVII. Simultaneous Volumetric and Electric Graduation of the Steam-Tube with a Phosphorus Ionizer.-IV. By C. BARUS*.

1.

FTER graduating the colour-tube in terms of the volume-influx per second of air saturated with phosphorus emanation, it is next necessary to investigate corresponding data for the degree of ionization of air as related to the colour-effect. Indeed both graduations may be made simultaneously by passing a known volume of saturated air per second through a suitably constructed tubular condenser, and observing coincident values of the electrical leakage of the latter and the colour of the enclosed steam-jet. Clearly the colours of the tube will each correspond to a definite electrical current passing through the condenser. Moreover, while the volumetric equivalent of a given colour is dependent on the degree of initial saturation of the phosphorus emanation conveyed by the current of air, the electric equivalent should be independent of it. The final graduation cannot at once be carried out, however; for in the case of the colour-tube constructed and used as below the nuclei are injected into the air-current maintained by the steam-jet. An arbitrary element is thus introduced, and the results will only be comparable when all observations refer to a tube the action of which has been left quite undisturbed. True, there seems to be no objection to putting a vaneanemometer into the influx-pipe (enlarged) of the colourtube, in which case the arbitrary factor would be specified; and other methods of eliminating the factor will be indicated, but the data following refer to the earlier methods of experiment.

2. In fig. 1, CC' is the colour-tube, with the jet j, the thermometer T, and the influx-pipe C', bent for convenience. KL is the tubular condenser, consisting (as shown in detail in fig. 2) of a brass tube d, 6 centim. in internal diameter and effectively 50 centims. long, surrounding a steel rod a, 318 centim. in diameter, coaxially. Rod and tube are separated symmetrically throughout by the short hard-rubber

* Communicated by the Author.-See previous communications, Phil. Mag. [5] xxxviii. pp. 19-35 (1894); [6] i. pp. 572-578 (1901); ii. pp. 391-403 (1901).