Basis for the equations:

I. Sum of masses of radium G (atomic weight, 206)
   actinium D (atomic weight, 207)
   thorium D (atomic weight, 208)
   ordinary lead (atomic weight, 207.2)

equals mass of lead of atomic weight, A, found in the mineral.

II. Sum of numbers of atoms of radium G,
     actinium D,
     thorium D,
     ordinary lead

equals the number of atoms of lead of atomic weight, A, found
in the mineral.

III. Number of atoms of parent substance disintegrating in each of
the above cases,
results ultimately in the same number of atoms
of the end product which is an isotope of lead.

HST.400026.035

Mean value of L from the 3 equations is

$$
L = \frac{1}{3} \left[ \frac{I_{208} \cdot \text{Pb} - \text{ThD}}{0.4955} + \frac{I_{207} \cdot \text{Pb} - \text{AcD}}{0.2020} + \frac{I_{206} \cdot \text{Pb} - \text{RaG}}{0.2775} \right]
$$

= ordinary lead by basic equations

$$
= 207.2 \left[ \frac{\text{Pb}}{\text{A}} - \frac{\text{AcD}}{\text{C}} - \frac{\text{RaG}}{\text{G}} - \frac{\text{ThD}}{\text{D}} \right]
$$

A, C, G, D are atomic weights respectively of the
lead in the numerator.

This equation in exponential form can be used also to
solve for the accurate value of t, the age; or be used
as a check.

HST.400026.039

Number of actinium D atoms, c, is

$$
c = (r \cdot \lambda_u \cdot u \cdot I) \cdot \frac{e^{\lambda_{acu} \cdot t} - 1}{\lambda_{acu}}
$$

True mass of AcD, at. wt. 206.983,

$$
\text{AcD} = \left( r \cdot \lambda_u \cdot \frac{U \cdot I}{W_u} \right) \cdot 206.983 \cdot \left( \frac{e^{\lambda_{acu} \cdot t}}{\lambda_{acu}} - 1 \right).
$$

"False" mass of AcD, assuming at. wt. 207.2, is

$$
(AcD)' = \left( r \cdot \lambda_u \cdot \frac{U \cdot I}{W_u} \right) \cdot 207.2 \cdot \left( \frac{e^{\lambda_{acu} \cdot t}}{\lambda_{acu}} - 1 \right).
$$

Fractional error is

$$
\frac{(AcD)' - AcD}{AcD} = \frac{207.2 - 206.983}{206.983} = 0.00105
$$

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TABLE I.

<table><thead><tr><th>Mineral</th><th>Locality</th><th>Pb</th><th>U</th><th>Th</th><th>A</th><th>Analyzed by</th></tr></thead><tbody><tr><td>*1. Uraninite</td><td>Flat Rock Mine, N. Car.</td><td>3.86</td><td>76.96</td><td>2.44</td><td>206.40</td><td>Richards-Lambert (18)</td></tr><tr><td>2. Thorite</td><td>Brevig, Norway</td><td>0.35</td><td>0.45</td><td>30.10</td><td>207.90</td><td>Hönigschmid (7)<br>Fajans (19)</td></tr><tr><td>3. Bröggerite</td><td>Raade, Norway</td><td>8.64</td><td>61.67</td><td>6.30</td><td>206.12</td><td>Gleditsch (20)<br>Richards-Wadsworth (21)</td></tr><tr><td>4. Cleveite</td><td>Aust-Agder, Norway</td><td>10.61</td><td>66.73</td><td>0.82</td><td>206.17</td><td>Gleditsch (4)</td></tr><tr><td>5. Cleveite</td><td>Saetersdalen, Norway</td><td>7.83</td><td>53.30</td><td>4.61</td><td>206.08</td><td>Gleditsch (4)<br>Richards-Wadsworth (21)</td></tr><tr><td>6. Bröggerite</td><td>Karlhus, Raade, Norway</td><td>8.018</td><td>61.158</td><td>4.377</td><td>206.19</td><td>Fenner-Piggot (25)<br>Aston (14)</td></tr><tr><td>7. Uraninite</td><td>Keystone, S. Dak.</td><td>15.24</td><td>66.90</td><td>1.89</td><td>206.071</td><td>Davis, C. W. (22)<br>Richards-Hall (23)</td></tr><tr><td>8. Uraninite</td><td>Sinyaya Pala, Carelia</td><td>18.10</td><td>61.14</td><td>0.13</td><td>206.06</td><td>Nenadkevič (24)</td></tr><tr><td>9. Thorianite, I</td><td>Ceylon</td><td>2.34</td><td>11.8</td><td>68.9</td><td>207.21</td><td>Hönigschmid (7)</td></tr><tr><td>10. " II</td><td>"</td><td>3.11</td><td>20.2</td><td>62.7</td><td>206.91</td><td>"</td></tr><tr><td>11. " III</td><td>"</td><td>3.50</td><td>26.8</td><td>57.0</td><td>206.84</td><td>"</td></tr></tbody></table>

* Unfortunately, the atomic weight A was not made on the same specimen as was used for obtaining the other data for this mineral.

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