334 



TRANSACTIONS OP SECTION A. 



7, 8, 9, &c, years. Thus the value of a 2 -ft 2 ought to be (say) 1,500 in the earlier half 

 and 700 in the later half : or if one of them falls short of this, there must be com- 

 pensation in the other. No values approach this combination. In seven years the 

 high value 802 in the first group is at once discounted by the low values 54 and 193 in 

 the second and third. The best combination is, perhaps, that at 19 years, suggesting 

 a nutational effect depending on the moon's nodes. 



But what difference will it make if the true period differs somewhat from an exact 

 number of years ? 



With one qualification, the case will be no better. It might at first seem that 

 if the phase of the fluctuation in the first half did not accord with the phase in the 

 second, we could improve matters by slightly altering the period so as to make the 

 phases accord. But we have, as a matter of fact, tacitly assumed that the phases 

 were already in accordance. If they are not, then the case for periodicity is so far 

 weakened. Indeed, the table presents the case on the most favourable assumptions, 

 not only for periodicities of the exact values indicated but for others of neighbouring 

 values at the same time. 



But with this qualification. If the true period is 19J years let us say, then by 

 taking five periods together on the assumption of 19 years, we reduce the amplitude. 

 The phases of the two extreme periods are a whole year out from that of the mean, 

 i.e., about 18° ; and those of the two intermediates are 9° out. Hence the maximum 

 will be 1 + 2 cos 9° + 2 cos 1 8° = 5 — T 2 instead of 5, i. e. , is reduced in the ratio 1—0 -024 

 to 1 ; and its square is reduced in the ratio 1—0-048 to 1. This is not very serious ; 

 but the reduction is much greater near seven years. If the true period were 7£ years 

 the reduction of amplitude is 



(1 + 2 cos 24° + 2 cos 48°)/5=0-83 



and of the square is thus 0-69. Hence instead of a 2 +6 2 exceeding the average 

 five times (say) we need only look for 5 x 0-69= 3 \ times. 



But a second glance at the table shows that no supposition we can make will 

 grant even this lower indication. For illustration consider the case presented by 

 the groups for 15 years, the second of which (512) is 3*6 times the mean value (140) 

 of a 2 +b 2 for the second half ; but the 314 for the first group dilutes it considerably. 

 If we deem the matter worthy of further examination we must take into account the 

 phases. The following table shows the phases for the whole of the groups, and we see 

 that a very slight adjustment would make 211° and 238° agree. But a slight adjust- 

 ment will not suit our hypothesis that the period is sensibly different from 15 years, 

 and that the amplitude has accordingly been reduced by taking five periods together. 

 The change of phase in five periods must be that due to about 1\ years, i.e., about 

 one-sixth of a whole period or 60°. We can make this in the direction of the observed 

 change, though it does not agree in amount. There is thus room for compromise ; the 

 further we depart from 15 years the more we can allow for the reduction due to five- 

 year grouping, but against this must beset an increasing difference of phase between 

 the two groups. 



Table II. — Phases for the same Groups as in Talle I. 



