CONFERENCE ON MICHELSON-MORLEY EXPERIMENT 399
the arms of the frame are longer in my apparatus than in his. I should like to mention again that my experiments have been carried out under a great variety of conditions. My assistant moved around the apparatus to see if his position affected the distribution of temperature or the stability or level of the instrument. The light was placed in different positions, both inside and outside the house. At Mount Wilson, the instrument has been mounted in two different buildings, differently oriented. The effect has persisted throughout. After considering all the possible sources of error, there always remained a positive effect.
Professor E. R. Hedrick: Mathematically speaking there cannot be any question as to the correctness of the computations which Professor Lorentz has presented to us. The result for the second-order terms seems beyond question. It is conceivable, however, that there is introduced an error when the path of the beam of light is changed by the motion of the apparatus into a new one. The instrument might not be always in the ideal position assumed in the calculations.
I should like to call your attention to a second point. We start from a certain number of assumptions. Now our aim in mathematics is always to reduce the necessary number of assumptions to a minimum. We make use in this special case of the two principles of Huyghens and Fermat. Can we trust them to terms of the third order? We do not know. Might not a combination of third-order effects eventually affect the magnitude of the second-order effect? Anyhow, if we could reduce the number of physical principles involved in our calculations to a single one, it would be very desirable. That is what Righi and also I have attempted to do.1
Lorentz : I should like to defend my theory. Hedrick says we should try to reduce the number of our assumptions. Now the two
1 It should be stated clearly that the operations of differentiation and integration, freely used in these discussions, cannot be trusted to the extent that is often assumed. The derivative of an approximation to a true formula is not necessarily an approximation to the derivative of the true formula. It is true also that the integrals to successive approximations to a true formula are not necessarily successive approximations to the integral of the true formula, unless the successive approximations are uniform. These conditions cannot be said to hold in such fine approximations as those of the Michelson experiment. Therefore it has seemed to us, and it still seems to us, to be necessary to proceed by direct calculations from definitely stated assumptions, rather than through an intermediate proof (e.g., Fermat’s principle) that is thus questionable.
principles of Huyghens and Fermat are not independent. The second may be deduced from the first. It is easy to prove that this is true. There is then no question of having two assumptions.
Hedrick: Is this really generally true?
Lorentz: Yes; the relation between Huyghens7 principle and Fermat’s principle is absolutely general. I might repeat more precisely some of the features of the reasoning I gave yesterday.
Suppose P (Fig. 5) to be a luminous point. (The difficulties might of course begin here, if we were obliged to state exactly what we mean by this.) Suppose, further, that rot w = o, which is Fresnel’s idea. Making use of Fresnel’s coefficient and entrainment, we find the influence of a motion of the apparatus on effects of the first order to be the same for each of the paths h and l2.
There is still one point to be considered which I did not mention before. If we take into account effects of the second order, the path of the rays will be changed by the motion of the apparatus, so that we should have to use in one moment I and in the next V. Still I think that for the effects under consideration it does not make any difference which one we take. [Hedrick remarks: “Yes, that is all right.”] It can easily be seen that the difference between I and V produces only an effect of the fourth order. We are thus justified in using the path existing without motion of the ether.
Of course the value of the light-path I must be exact to the second order. For those cases in which we are concerned with the propagation in ether only, this value follows from the expression for v (velocity of light in the moving system) :
i = ? [1--C0S û+% (cos2 #+§ sin2 #)
V C c c
[See expression (3) in Report II.] But the question arises, and this is what I wanted to add, what will be the form of the equation when we deal with light passing through the moving glass plates? In this case w2/c2 would be replaced by k2 uf/c2, where 1 — k = (n2 — i)/n2 is Fresnel’s coefficient. Now this value for k might not be quite rigorous in this connection. The expression wkdt due to the entrainment by matter might be doubted if terms of the second order are to be considered. This indeed might necessitate a change of magnitude for these second-order effects. It is to be remarked, however, that the dis