August 31, 191 1] 



NATURE 



29: 



a series of finite steps, owing to the finite extent of the 

 observations. A definite illustration will make this clear. 

 We have ninety years of rainfall, and we test the 

 record for a frequency of nine years, which would run 

 through its period ten times ; we must certainly test in- 

 dependently for a frequency of ten years, which would 

 onh run through its period nine times, and thus lose one 

 whole period on the former wave ; and so, also, for a 

 possible frequency of nine years and a half, and of nine 

 years and a quarter. But a frequency of nine years and 

 one day would not be distinguishable from that of nine 

 years, lor the phase would only change i° in the whole 

 available period of observation. Indeed, 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 must accordingly narrow the 

 intervals between frequencies chosen for examina- 

 tion. 



The length of the series of observations is thus an 

 important factor in our procedure, for which Prof. 

 Schuster has indicated a beautiful analogy. Our illustra- 

 tions hitherto have been provided by the science of sound, 

 but we may also gather them from that of optics. Test- 

 ing 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 optical 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 re- 

 solving 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 resolving 

 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 accumu- 

 late 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. Prof. Schuster has applied it, for instance, to 

 the observations of sun-spots. Now it may fairly be said 

 that the general law of sun-spots was thought to be 

 known; the variation in a cycle of about w\ years has 

 long been considered to represent 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 par- 

 ticular attention (with one exception presently to be" men- 

 tioned) until Prof. Schuster undertook his analvsis. To 



NO. 2183, VOL - 87] 



his surprise, when he calculated the periodogram of sun- 

 spots, 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 con- 

 tinuously 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 estab- 

 lished 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 advance in knowledge resulting, 

 not from observations made with regard to any particular 

 theory, but from the simple collection of facts and the 

 arrangement of them in all possible ways, the very method 

 which has been despised and condemned. Let us contrast 

 with this the method hitherto adopted, which has been to 

 hunt for some particular possible cause which will give 

 the eleven-year period. Thus Prof. E. W. Brown sug- 

 gested ' in 1900 that the eleven-year cycle was due to the 

 tidal action of Jupiter, altered periodically by two causes : — 



Period Mas. of force 



By Jupiter's eccentricity .. 11 Sb years ... 033 

 By the motion of Saturn ... 993 ,, ... O'H 



and he suggests his contention by an ingenious and 

 striking diagram, which seems to explain not only the main 

 cycle, but its anomalies. (This Paper is, in fact, the excep- 

 tion above referred to.) But if his contention is correct, 

 the periodogram should show bright lines at 11-86 and 9-93 

 years, which it does not. This is worth noting, since it 

 is sometimes said that there is nothing new in Prof. 

 Schuster's method, which is true enough in one sense, 

 since it is simply the analysis of Fourier. The novelty- 

 consists, firstly, in calling attention to the necessity of 

 applying the analysis in all cases, a necessity which I 

 venture to think was overlooked in this instance by so able 

 a mathematician as Prof. Brown ; and, secondly, in the 

 insistence on the examination of all periods, irrespective 

 of any particular theory or preconception. And in this 

 second character the method seems to me to cut at the 

 root of the canons of procedure which have found favour 

 hitherto. 



As a second instance I present with much more diffidence 

 a few results which seem to emerge from a very laborious 

 analysis of the rainfall at three or four stations, for which 

 Prof. Schuster and myself are jointly responsible. There 

 is some evidence for a cycle of 600 days in the Greenwich 

 rainfall, to which a further cycle in the quarter period 

 (150 days) lends support. On analysing the Padua records 

 it is found that these cycles do not exist ; but it seems 

 quite possible that there are cycles of rather shorter period, 

 viz. 594 days and 1484 days, the relation of four to one 

 being maintained. The separate links in this chain are 

 none of them very strong, but they seem to hang together, 

 and there is certainly a case for further investigation. 

 But would this case have been likely to present itself in 

 any other way than by the examination of the whole 

 periodogram? I find it very difficult to think, even now 

 the periods are suggested, of any theoretical cause ; to 

 let the facts speak for themselves took much time and 

 labour, but I venture to think that we might have waited 

 far longer, and cudgelled our brains much more, before we 

 got the clue by formulating hypotheses of causation. 



A new method is not adopted widely all at once. Prof. 

 Whittaker has, I am glad to say, begun to apply the 

 method to variable star observations, and is already hopeful 

 of having obtained valuable information in the case of 

 the star SS Cygni. Possibly we may hear something 

 from him at this meeting. Meanwhile, I take the oppor- 

 tunity to remark that the history of variable star observa- 

 tion affords us many lessons as to the desirability of 

 simplv accumulating observations and letting them speak 

 for themselves, instead of being guided by a theory on 

 hypothesis. Let me give an instance. One of the fathers 

 1 Monthly Notices R.A.S., lx., p. 600. 



