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:

å_{p,min}=2*arcsin{n_{p}/2}-60° (1)

This comes from the minimum deviation formula for one prism:

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

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

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

Now in order to calculate with (3) we need å_{2p,min} and
A''. It can be shown using geometry that M=å_{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°+å_{p,min}=360°=>C1+180°+å_{p,min}=360°=>C1=180°-å_{p,min}
(6)

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

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

(3)(7)(8)=>n_{2p}=sin{(2å_{p,min}+60°-å
_{p,min})/2}/sin{(60°-å_{p,min})/2}=> n_{2p}=sin{(å_{p,min}+60°)/2}/sin{(60°-å_{p,min})/2}
and using equation (2) => n_{2p}=(n_{p}/2)/sin{(60°-å_{p,min})/2}
=> n_{2p}=(n_{p}/2)/sin{30°-å_{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°, ë=5615.97363281 Angstroms.

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