from overlapping means, but the point midway between the vertex and the middle of the base is the point from Hann's formula. In the present work Hann's formula has been used frequently, and in order to shorten description of processes the word "Hann" has been used as a verb.
In the analysis of curves already performed by the periodograph, the curves have sometimes been smoothed by Hann's formula before plotting and photographing. But a trifling error in the focus immediately smooths the curve, and therefore it is evident that the preliminary smoothing of a curve before plotting need not be done.
Such preliminary smoothing helps the eye to judge variations in the curve. The effect of out-of-focus position in a photograph may be called optical smoothing. It is evident that optical smoothing may be done in two directions, vertically and horizontally. In plotting a curve it is evident that the desired smoothing must be in a horizontal direction, but in the differential photographs made with the periodograph, the directions of optical smoothing may have a very important bearing on the judgment of the significance of the photograph. Of course, in the differential pattern, long interference fringes are sought and these are emphasized by optical smoothing parallel to them. Some illustrations of this will be given under the subject of the periodograph. Perhaps no feature of this subject of tree-growth and climatic and solar variation has received more adverse comment than the matter of smoothing curves. The author is entirely open to conviction as to the advantage and disadvantage of such process, but it seems well to remember that our views as to this are likely to be a matter of convention rather than of actual thought in relation to the subject in hand. For instance, a monthly mean is a smoothed result. The rainfall, instead of being taken as it came, mostly in a few days, especially in the summer, is treated as if it were the same for every day in the month. Yearly means are smoothed values. The ordinary method of plotting yearly means is a smoothed representation of those quantities. The unsmoothed representation consists of what one may call a columnar plot. Examples of plots of that type may be found in connection with some rainfall records published by the United States Weather Bureau and in a representation of the London rainfall for more than 100 years published by the British Rainfall Association, and elsewhere. In this kind of plot the rain for a year is not represented by a dot, but by a block column which extends from the base-line up to the required amount and it has a width equal to the interval of one year according to the scale of the plot. Now, the ordinary way of representing rainfall places a dot at the middle of the top of this column, and these dots are connected together by straight lines. It is immediately
- The three-score of curves which are now specially prepared for examination with the periodograph carry the mean values without smoothing.