overlapping periods is applied to the mean of the 4, producing a probable value of the fifth between the years 1720 and 1820. This probable value is inserted in parenthesis in the table and all 5 values added up for an average. As a rule, groups are carried back only far enough to make assumed values of this kind a minimum in number.
There is one other problem in this immediate connection, namely, that of "gross rings." By gross rings I mean certain regions in a section where the average size of the rings becomes 2 to 5 times as great as normal. This is a problem by itself, both as to cause and as to method of treatment. Some study of its prevalence in different trees has been made, and it is usually safe to say that where an epoch is shown to have gross rings in one tree, the chances are at least even that the same years will have gross rings in the next tree. Since gross rings may not come oftener than once in several hundred years and last only 10 to 15 years, it is evident that we are dealing with something more than mere accident. The phenomenon probably has a climatic character. Yet, gross rings are not universal at any one time, and while one epoch may show gross rings in half the trees of a group it does not show it in the other half, judging by the groups examined. It is considered best to allow the ring values to enter the curves just as they are found, for while the gross rings disturb very greatly the size of a series of 10 to 20 rings, they do not seriously disturb the relation in size between a ring and its immediate neighbors. They therefore, as a rule, do not render the rings unidentifiable. It is likely, therefore, that they should be included in the means, and if some better way of handling them is discovered later it will not be difficult to apply it.
In general the smoothing of a curve means removing some of the minor variations, so that the larger variations may be perceived. In the early part of the work the use of overlapping means was adopted. At the very start, overlapping means of a considerable number, such as 11 or 9, were used. This was quickly changed to overlapping means of 3. These overlapping means were done by the calculating machine (Brunswiga). On this machine three were added and then contin- uously the one next in sequence was added, while one at the other end of the three was dropped. However, this was changed to Harm's formula, because his formula is normally easier to apply and it gives a little more individuality to each observation. The method of applying Hann's formula consisted in adding to a table two columns consisting of, respectively, first and second intermediate values. This can be done rapidly and without taking too much space. To express the differences between overlapping means and Hann's formula graphically, we only need to say that if we take successive groups of three in any curve, forming a triangle, the center of gravity of the triangle is the value