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the same might be said of all frequencies between nine years and nine years and 

 one month : for the extreme difference of phase would not exceed 40°. But in 

 course of time when the series of ninety years' observations become 900 years, the 

 differences of phase will approach or exceed a complete cycle, and we inu.st 

 accordingly narrow the intervals between frequencies chosen for examination. 



The length of the series of observations is thus an important factor in our 

 procedure, for which Professor Schuster has indicated a beautiful analogy. Our- 

 illustrations hitherto have been provided by the science of sound, but we may 

 also gather them from that of optics. Testing a series of rainfall observations 

 for a periodicity is like examining a source of light for a definite bright line. 

 The process of computation indicated by Fourier gives us what corresponds to 

 the measured brilliance of the bright line ; and the complete process of analysis 

 corresponds to the determination of the complete spectrum of the source of 

 light, which may consist of bright lines superimposed on a continuous spectrum. 

 And the length of the series of observations corresponds simply to the resolving 

 power of the optical apparatus. The only point in which the analogy breaks 

 down is unfortunately that of ease and simplicity. In the optical analogy, an 

 optica] instrument performs for us with completeness and despatch the analysis, 

 which in its counterpart must be performed by ourselves with much numerical 

 labour. 



Let us consider how we should most conveniently proceed to the complete 

 delineation of a spectrum. We should ultimately need an apparatus of the 

 greatest possible resolving power, but it might not be advisable to begin with it : 

 on the contrary a small instrument which enabled us to glance through the 

 whole spectrum might save much time. Suppose, for instance, that there was 

 a bright line in the yellow; our small instrument might suffice to show us that 

 it was due either to sodium or helium, but no more : the decision between these 

 alternatives must be reserved for the larger instrument. On the other hand, if 

 no line is seen in the yellow at all, we have ruled out both possibilities at once, 

 and so economised labour. Hence it is natural to use first an instrument of low 

 resolving power and afterwards one of higher. 



Now in the work for which this serves as an analogy this procedure is actually 

 imposed upon us by the march of events. It has been pointed out that the resolv- 

 ing power of the optical apparatus corresponds exactly to the length of our series 

 of observations. Hence our resolving power is continually increasing. Quite 

 naturally we begin with a short series of observations, which shows us our lines 

 blurred and confused : to define and resolve them we have but one resource — ■ 

 ' wait and see ' ; wait and accumulate more observations, to lengthen the series. 

 But the lengthening must be in geometrical progression : we must double our 

 series to increase the resolving-power in a definite ratio; and double it again. 

 We begin to get a glimpse of the important part to be played by the observer in 

 the future, and of his increase in numbers. 



Let us glance at a few illustrations of the use of this method. Professor 

 Schuster has applied it, for instance, to the observations of sunspots. Now it 

 may fairly be said that the genera] law of sunspots was thought to be known : 

 the variation in a cycle of about 11£ years has long been considered to repre- 

 sent the facts : it catches the eye at once in a diagram, and though there are 

 also obvious anomalies, they had not been deemed worthy of any particular 

 attention (with one exception presently to be mentioned), until Professor Schuster 

 undertook his analysis. To his surprise, when he calculated the periodogram of 

 sunspots, he found two entirely new facts : 



Firstly, that there were other distinct periodicities, notably of about four, 

 eight, and fourteen years ; 



Secondly, that the eleven-year cycle had not been continuously in action, but 

 that during the eighteenth century it had been much less marked than the eight- 

 year and fourteen-year cycles. 



A further most interesting fact seems to emerge, viz., that several of the 

 periodicities are harmonics of a major period of some thirty-three years or more, 

 and it seems just possible that a connection may ultimately be established with 

 the Leonid meteor-swarm, which revolves in this period. But it would take us 

 too far from our main point to follow these most interesting corollaries : the 

 point well worthy of our special attention is this, that we have here an undoubted 



