ALTER: RAINFALL AND SUX-SPOT PERIODS. 23 



RIGID FOLLOWING OF THE SUN-SPOT PHASES. 



It is evident that the sun-spot period between the minima named 

 above had values of 145 and 141 months, respectively. Let us 

 examine the two maxima occurring between these dates. One oc- 

 curred in 1894, February, and the other in 1906, May, with an in- 

 terval of 147 months. This must have been the average value of 

 the sun-spot period between these dates. It is longer than the 

 period obtained from either pair of minima named above, yet it 

 occurs as part of each of them and contains no part that is not in 

 one or the other of them. We are forced, therefore, to the con- 

 clusion that if continuous (8a I — 



The length of the sun-spot period is continuously varying and a 

 value of the period obtained between successive maxima or suc- 

 cessive minima is merely an average of all values passed through in 

 this interval. 



If we had a curve with time plotted along the axis of abscissae 

 and the corresponding values of the sun-spot period as ordinates, 

 the average value of the sun-spot period between two maxima or 

 two minima occurring at t^ and to would be given by — 



ti — t2 = average value = 



i: 



h-t2 



If we plotted abscissae and ordinates on the same scale, these 

 average values would form squares bounded by ordinates through 

 the dates which limit them. The area between the axis of abscissae 

 and the unknown curve, described above, representing the actual 

 value of the period at all times, would in the interval between two 

 maxima or two minima have to equal the corresponding known 

 square. Since these squares overlap, we know the value of a series 

 of overlapping definite integrals of the unknown curve. From these 

 data it is possible, assuming the simplest curve to be the true one, 

 by the aid of a planimeter, to construct the curve without knowl- 

 edge of its mathematical form. In doing this it is easier to choose 

 some convenient period as the axis of abscissae and to measure de- 

 partures from this period. Changing the axis in this way merely 

 changes all the integrals by a known constant amount and changes 

 the known squares into knowTi rectangles. It is also practical to 

 magnify the scale of ordinates very much over the scale of abscissae. 

 Locating the curve consists first in measuring the area of each of 

 the rectangles; then penciling in what appears to be the curve, 

 measuring the definite integrals of the approximate curve with the 

 planimeter; erasing for a new approximation, and repeating many 



