that line. When it is desired to make longer tables, the pages are pasted together side by side or end to end, and then given a zigzag fold, so that two pages are open at once. In the case of the sequoias, with their 2,000 to 3,000 rings, no attempt has been made to paste the pages together, but enough loose sheets are used to cover the entire series at the rate of 20 years to a page. This gives sufficient vertical space to include all the necessary trees in a group and to use subgroups which may be summarized and averaged by themselves. An attempt has been made to check the addition of these numbers throughout.
AVERAGING.
In simple averaging the sums are placed in ink on the table and divided by the number of trees, using the slide rule for the process. There are several questions in connection with this subject. The first is whether straight averages of trees of widely different size give the best report of the evidence of the trees. It is evident that in taking averages of trees of mixed sizes the larger trees will carry more weight and their variations will be more pronounced in the result. But it is often the case that the smaller trees are the ones which show the greatest relative variations in the rings. This can be so much the case that the omission of a ring becomes a gross exaggeration. It is possible to use the relative values by taking the logarithm of each ring measure, averaging the logarithms, and then coming back to the number. This could be called a geometrical averaging, since it would be the equivalent of multiplying all the values together and then extracting the root equal to the number of values. In this way the small trees of the series would receive more importance. However, this plan is so long that it has not been used in practice.
One of the most common and puzzling problems is the proper method of handling the decrease in the number of trees in a group as the center is approached. A group of 5 may be selected, for example, and perhaps a century from the average center of the trees some one tree whose rings differ from the average may come to its end. It means that for 100 years near the center only 4 trees supply the data and at the point where the 5 change to 4 there is a discontinuity in the curve. In actual practice this lacking tree has usually been supplied by an extrapolation from its subsequent curve. That is, the variations assumed in the non-existent part of the tree follow precisely the variations in the remaining trees, altered to the average size of the missing tree by means of a constant factor, determined by overlapping periods. Thus, if 5 trees carried easily back to 1820, but only 4 of them extended to 1720, and it was desired to carry the full group to 1720, the period from 1820 to 1840 would be taken both for the 4 and for the 1 alone and the ratio between them determined. Now, averages for the 4 are carried back to 1720, and then the factor found in
