INSTRUMENTS AND TECHNIQUE 43 



were obtained differing by exactly one year, such as 17.1, 18.1, and 

 19.1 years. This, of course, arose from retaining annual points in 

 the plot and in the cycle one was apt to select some multiple of unity, 

 that is, simply a whole number or very close to it, instead of an actual 

 fractional value. 



Smoothing — Accordingly, some form of smoothing is now always 

 used, and the Bloxam formula, which I have sometimes called Hann's 

 formula, is generally accepted. But there are several variations of 

 this process. 



Numerical Harm — The first is the simple application of the Hann 

 or Bloxam formula, in which three successive (overlapping) values are 

 merged into a substitute for the middle one by averaging the three, 

 with double weight given to the second. It is this double weight 

 applied to the original whose substitute is desired that differentiates 

 this formula from a running mean of three. The place of this emphasis 

 will be referred to below. This process may be done on a set of tabu- 

 lated values by two successive sets of intermediates, as explained in a 

 previous volume. 



Geometric Hann — This is the same process, done graphically on a 

 curve already plotted, by taking each three successive points as the 

 corners of a triangle. Consider that the first and third points form 

 the base. From the center of the base, one-third of the way to the 

 middle point will be the running mean of three, while one-half of the 

 way from the base to the middle point will be the weighted mean or 

 the "Hanned" value. This forms in practice a very easy way of 

 smoothing a curve and has been very largely used in a slightly abbre- 

 viated form which I have called the graphic Hann. 



Graphic Hann — The plotting paper used in the cyclograph needs 

 to be 4 inches wide by some 45 inches long, fairly opaque, and with 

 parts of the curve cut out so that light may pass through. All this is 

 best done on rough brown paper cut in strips of the proper size. The 

 present process, therefore, is to plot the tabular averages directly on a 

 long strip of coordinate paper, using a rather large vertical scale, so 

 that variations will generally be an inch or two high. This strip is 

 placed upon the heavy strip of brown paper with carbon paper between 

 and a blunted needle or pointer is passed slowly along the plotted 

 curve, touching the points which by eye estimation and occasional 

 measure should constitute the geometric Hann. Century dates at 

 the same time are touched, so that the curve thus transferred becomes 

 a satisfactory working smoothed plot of the standardized group 

 average. This is called the graphic Hann and can be done quickly 

 and accurately. This process of smoothing has a perfectly definite 

 ideal to look to in case of doubt and I believe is almost entirely free 



