Unlike in the wavelength domain, the change of refraction index would not affect the frequency of the transmitted light. I concluded that the proper choice of integration domain in which to perform tristimulus integration should be a matter of experiment. This conclusion was incorrect. As often happens when one poses a paradox, closer examination provides a resolution. The tristimulus values should not change if the integrals are performed properly. I describe the resolution in a tutorial note recently posted for publication in

*Color Research and Application*[2].

It is reasonable and respectful of physical principles that a tristimulus value [say, M = the integral over λ of E(λ) m(λ)] should not depend on the domain of integration used to obtain it. In [2] it is shown that the domain invariance from wavelength λ to frequency ν = c/λ is assured if the spectral power distribution of the test light in frequency is represented as E*(ν) =|dλ/dν| E(λ(ν)).

Now imagine a transformation from frequency to two alternative wavelength domains λ

_{1}= c

_{1}/ν and λ

_{2}= c

_{2}/ν , where c

_{1}and c

_{2}represent speeds of light in different refractive indices (possibly functions of ν ). Here, M doesn’t change either; rather, the spectral power distribution E*(ν) is multiplied by the Jacobian |dλ

_{1}/dν |

^{-1}or | dλ

_{2}/dν |

^{-1}, respectively. (Things get a bit more complicated when λ

_{1}or λ

_{2}is a nonmonotonic function of ν , but the principle is the same.) Of course, changing from λ

_{1}to λ

_{2}is equivalent to changing from λ

_{1}to ν and then from ν to λ

_{2}. There is no predicted variation of M, by construction.

So my “gentle paradox” evaporates by construction. Obeying the rules of mathematics leads back to where one started: tristimulus values don’t depend on the refractive index of the transmitted medium.

Where does that leave the role of experiment? If one were to measure the camera version of tristimulus values in different refractive media (say, with M being a measured camera value for a light E through a camera sensitivity function m), we would expect no change in those values. But experiment can trump theory. If M were observed to change, then a new paradox would surface. From “paradox lost” we would see “paradox regained.”

**References:**

**[**1]. HQ Fuller, RM. Fuller, and RG Fuller, Physics Including Human Applications. Harper and Row, 1978. Revised electronic version copyright 2009 by RG Fuller. http://physics.doane.edu/hpp/Resources/Fuller3/pdf/F3Chapter_19.pdf exercise 2, p. 436.

[2] MH Brill, Ways to define a tristimulus value. Color Res. Appl. 2016, Early View, DOI 10.1002/col.22079.

Michael H. Brill

*Datacolor*

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