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\[{R_{13}} = \left( {\frac{{\delta _{wrg - VPDB}^{13}}}{{1000}} + 1} \right) \times R_{13}^{VPDB}\]     and similar equations for $R_{17}$ and $R_{18}. $R_{18}$.  The reference gas ratios are $R_{13}^{VPDB}$ = 0.0112372 (reference), $R_{17}^{VSMOW}$ = 0.0004023261 (reference) and $R_{18}^{VSMOW}$ = 0.0020052 (reference). Similarly the R47* ratio for a sample corresponding to a stochastic distribution of the isotopes is given by:   (5)  with R13, R17 and R18 determined using equation 4 and the measured δ13C, δ17O and δ18O values for the sample.