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where $R_{47}^{wrg}$ is the 47/44 ratio of the working reference gas (wrg) and is determined as:  \[R_{47}^{wrg} = 2 \cdot R_{13}^{wrg} \cdot R_{18}^{wrg} + 2 \cdot R_{18}^{wrg} \cdot R_{17}^{wrg} + R_{13}^{wrg} \cdot {(R_{17}^{wrg})^2}\]   Note that implicit in this treatment is an assumption that the mass spectrometer working reference gas has a stochastic distribution of isotopes. It is self evident that this is incorrect since the working reference gas has been equilibrated with water at the laboratory temperature. However, since the $\Delta$ values are <1‰ we can make this assumption and carry out a later linear transformation of the data to take account of the actual reference gas $R^{47}$ $R_{47}$  value without introducing any significant errors. The ratios R13, R17 $R_{13}$, $R_{17}$  and R18 $R_{18}$  are determined from the δ13CVPDB, δ17OVSMOW $\delta^{13}C_{VPDB}$, $\delta^{17}O_{VSMOW}$  and δ18OVSMOW $\delta^{18}O_{VSMOW}  values of the working reference gas: \[{R_{13}} = \left( {\frac{{\delta _{wrg - VPDB}^{13}}}{{1000}} + 1} \right) \times R_{13}^{VPDB}\]  (4)  and similar equations for R17 $R_{17}$  and R18. $R_{18}.  The reference gas ratios are R13VPDB $R_{13}^{VPDB}$  = 0.0112372 (reference), R17VSMOW $R_{17}^{VSMOW}$  = 0.0004023261 (reference) and R18VSMOW $R_{18}^{VSMOW}$  = 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.