62 CLIMATIC CYCLES AND TREE-GROWTH. 
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 imme- 
diately smooths the curve, and therefore it is evident that the pre- 
liminary 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 periodo- 
graph, 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 con- 
vention 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, espe- 
cially 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 else- 
where. 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 represent- 
ing 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 
1 The three-score of curves which are now specially prepared for examination with the periodo- 
graph carry the mean values without smoothing. 
