n = C/(83 + 1.8 w). (13) Hence, using Equation ll, V5 26000 Cw 563 C ¢ eae t+ (14) From Equation 14 we find that woe 26000/V_¢ - 563). Hence, if both V.,, be solved for w. n. (15) and C are specified, where V_C > 563, Equation 15 may This may then be substituted? in Equation 13 to obtain As an example, suppose our budget allows $200 per field sample for Pu analysis and we require V_, the variance of x, to be no more than 100. Then V_C = 20,000 and Equation 15 gives w= 1.4. Using this value for w in Equation (13) we find n = 2.3. Hence, for C = $200 and V_ = 100 we should analyze two aliquots, each weighing 2 g. Equation 12P indicates this would cost $173 per sample. We note from Equation 14 that V_ will attain its minimum value of 563/C when w is infinitely large. V Poan be made as close to its minimum value as desired by choosing w sufficiently large. Figure 4 is a plot of V_ Vv computed using Equation 14 for C = $100 and $200. The curves for d¥op off rapidly for w between 1 and 10 g then decrease more gradually tbr w greater than about 10. For each value of w, the corresponding n may be computed using Equation 13. It is clear from Figures 3 and 4 that there is little to be gained by using aliquots larger than about, say, 20 g. However, further studies should be conducted under a variety of contamination levels, sources, and environmental factors to determine the generality of our results. It is likely, for example, that the regression relationship (log s versus log w, Equation 2) is site specific, and of course, Equation 2 is based on only five data points. In Appendix B we derive general analogs of Equations 7, 8, 14, and 15. These may be used tp estimate w and n for variability and cost functions of the form s = aw and C = n(o + Bw), i.e., for any values of a, b, a, and § that may be applicable in a given situation. Optimum Allocation of Numbers of Field Samples and Aliquots The above sections do not address directly how to determine the optimum number of field samples in relation to the number of aliquots per field sample. An approach to this problem is given by Cochran (1977, pp. 280-283). 418