METHODS OF PERIODIC ANALYSIS. 87 
bers from 1755 to 1911. The existence of a rhythm in any specified 
period is indicated by a beaded or corrugated effect. A line across the 
corrugations gives in fact the rhythmic vibrations of the cycle. Ona 
moment’s examination this periodogram shows much of the informa- 
tion which has been under discussion for many years. The 11-year 
period is the most pronounced, but it is not so superior to all others as 
would be expected. It may be of any duration from 11.0 to 11.8 years, 
but 11.4 is a good average. There is obviously a period somewhere 
between 9.5 and 10.5 years and one between 8.0 and 8.8, but it is less 
conspicuous. Faint indications of periods are found near 14 years. 
The double of 8.4 is seen between 16 and 17 years. The double of the 
10-year period shows near the 20 and at 22 the double of the 11 begins. 
The preliminary part of producing this periodogram is the construc- 
tion of the “differential pattern” shown in plate 9,8. This pattern is 
the optical counterpart of a set of columns of numbers arranged for 
addition, as when one summates a series of annual measures on a 
10-year period, for example. The series is arranged in order with the 
first 10 years in the first line, the second 10 in the second line, and so on. 
In the case of the pattern the lines are made indefinitely long, so that 
the optical addition may be done in other directions than merely 
straight downward, for by making the additions on a slant a different 
period comes under test. 
In order to produce this pattern the sunspot curve was cut out in 
white paper and pasted in multiple on a black background. The left 
end of each of the upper lines is the date 1755. Each successive line 
is moved 10 years to the left, so that passing from above vertically 
downward each line represents a date 10 years later than its pre- 
decessor. This continues from 1755 to 1911, and the lower 10 lines 
show the latter date at their right ends. It is not necessary that any 
of the lines should be full length, as we use only a part of each. By 
passing the eye downward from the top, a period near 10 years will 
show'itself at once by a succession of crests in vertical alinement. If 
the crests form a line at some angle to the vertical, then the period they 
indicate is not exactly 10 years. It is more if the slant is to the right 
and less if to the left. The horizontal lines are spaced the equivalent 
of 5 years. Hence, if we measure the angle made between a vertical line 
and a line joining two crests in successive horizontal lines, we may 
easily calculate by simple formulas the period indicated. 
Since the photometric*values of all the curves in the diagram are 
proportional to the plotted ordinates, the photographic summation 
of the whole pattern in a vertical direction is almost an exact analogue 
of a numerical summation. This summation is simply done by a 
positive cylindrical lens with vertical axis. This brings down on the 
plate a series of vertical lines or stripes. If, now, we cut across these 
lines with a horizontal slit, the light coming through this slit from one 
end to the other will be the summation of the diagram in the vertical. 
