The goal of the measurement is to determine the ration between a1 of the reference isotope and a2 of the isotope with known spin I2 but unknown mu2.
Isotopes only differ from each other by the amount of neutrons. This means that J1 = J2 and B_J1 = B_J2.
Taking the ratio of a1 with a2 gives then:
a1/a2 = (mu1*I2)/(mu2*I1)
a1 and a2 are measured, mu1, I1 and I2 are known => this gives us mu2