Let us consider a system of two identical prisms, of apical angles A, and of
refractive index n_{p}, placed in a position where they both disperse a ray of
light passing through them, as in the above figure. Let the angle between them be M.
The system of two prisms, amounts to an imaginary prism A''BB' with apical angle A'',
and refractive index n_{2p}. We have for the minimum deviation angle for one
prism:

ε_{p,min}=2*arcsin{n_{p}/2}-60° (1)

This comes from the minimum deviation formula for one prism:

n_{p}=sin{(ε_{p,min}+A)/2}/sin(A/2) (A=60°) (2)

For an equivalent imaginary prism, the same formula applies, but with changed ε and A. In that case:

n_{2p}=sin{(ε_{2p,min}+A'')/2}/sin(A''/2) (3)

Now in order to calculate with (3) we need ε_{2p,min} and A''. It
can be shown using geometry that M=ε_{p,min} (4)

Now, A''+240°+C2=360° =>A''=120°-C2, C2+C1+120°=360° =>C2=240°-C1 =>A''=C1-120° (5)

Next,
C1+180°+ε_{p,min}=360°=>C1+180°+ε_{p,min}=360°=>C1=180°-ε_{p,min}
(6)

(5)(6)=>A''=180°-ε_{p,min}-120°=>A''=60°-ε_{p,min}
(7)

We also have: ε_{2p,min}=2*ε_{p,min} (8)

(3)(7)(8)=>n_{2p}=sin{(2ε_{p,min}+60°-ε
_{p,min})/2}/sin{(60°-ε_{p,min})/2}=>
n_{2p}=sin{(ε_{p,min}+60°)/2}/sin{(60°-ε_{p,min})/2}
and using equation (2) =>
n_{2p}=(n_{p}/2)/sin{(60°-ε_{p,min})/2} =>
n_{2p}=(n_{p}/2)/sin{30°-ε_{p,min}/2} (9)

Using (1) we get:

(9)(1)
=>n_{2p}=(n_{p}/2)/sin{30°-arcsin(n_{p}/2)+30°}=>
n_{2p}=(n_{p}/2)/{sin60°cos[arcsin(n_{p}/2)]-cos(60°)*(n_{p}/2)},
which becomes
n_{2p}=(n_{p}/2)/{sqrt(3/2)*sqrt(1-(n_{p}/2)^{2})-n_{p}/4},
which after simplifications becomes:

n_{2p}={sqrt(3/2)*sqrt((2/n_{p})^{2}-1)-1/2}^{-1}
(10)

The function's graph is shown on the following figure:

It has a singularity exactly at n_{p}=sqrt(3), since then,
n_{2p}->∞. Note that the system doesn't make sense for
n_{p}>2 so the domain of the function is precisely the set
{1<n_{p}<sqrt(3)} U {sqrt(3)<n_{p}<2}.

Since at n_{p}=sqrt(3) the prisms are in such positions of minimum deviation
that M=60°, the system resonates at that point. Below sqrt(3) the quantity
n_{2p} is positive, above, negative, to reflect the fact that there is no
equivalent prism, since the apical angle is essentially negative. The frequency of
resonance can be found using interpolation, or the program on Measuring Wavelengths section. It is N=60°, M=60°,
λ=5615.97363281 Angstroms.

System resonance is shown on the figure, above. When it happens, the ghost image for λ=5614.97363281 Angstroms coincides with the real image for the same wavelength.