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Performing the experiment on the first isotope (isotope 1) it is possible to determine the magnetic field B_J as you know the spins I_1 and J_1 and hence F_1 and measure a_1. Now you can do the exact same measurement on the second isotope (isotope 2): you get as a result a_2, you know J_2 as we suggest that the electron cloud is not changed when a nucleon is added (is already an approximation), B_J you can then take from the previous experiment. Since you know I_2, the only unknown left is the nucleaur maggnetic moment in the second isotope ยต_2. Its value can now be extracted.
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