NATURE 



[Jnly 26, 1877 



takes the two hourly observations of the barometer for 

 hve successive days at Madras, and shows that — 

 yhs ge}ie}-al mean difference = 0'030 inches, 

 'Vhs periodic ,, ,, = 0'0I4 ,, 



Here, he says, a true period existing, the periodic mean 

 difference becomes much less than the general mean 

 difference. 



1 shall now venture to show that this is no criterion of 

 periodicity. If we represent variations of any quantity 

 lor a given time by a curved line, and if we have several 

 such lines of exactly the same form placed one over the 

 other, a straight line passing through the curves, with as 

 much space between the straight and curved lines above 

 as below, will represent the general mean. In a simple 

 curve of two branches the general mean difl'erence will be 

 nearly one-fourth of the amplitude of the oscillation ; while, 

 as all the oscillations agree with each other, and therefore 

 ■with the mean oscillation, the periodic mean difference will 

 be zero. If, however, we displace the individual curves 

 so that as many shall be above as below their mean, 

 both the general and the periodic mean differences will 

 increase, and the difference between these quantities will 

 diminish, till the individual curves are so separated from 

 the mean that none of them is cut by it, when the two 

 mean differences will be equal : between this case and 

 that of general coincidence the two mean differences will 

 have values which will differ more or less from each 

 other, according as the individual curves are nearer to, 

 or more remote from, the mean ; and the ratio of the one 

 mean diflerence to the other will tend to a constant value 

 as the number of cycles increases, a ratio which will 

 depend for its value on the mode of distribution of the 

 individual curves and of the irregular deviations from the 

 mean. 



Gen. Strachey's illustration is from a case approaching 

 coincidence ; hundreds of cases, however, may be found 

 of the other class, especially when, as in this instance, 

 only a few periods are in question. Thus, taking two 

 hourly observations of the barometer at Simla during six 

 days in the beginning of January, 1S45, I find — ■ 

 Tlie general mean difference = o'o634 inches, 

 Tlie/tvmZ/f ,, ,, = 00615 >> 

 and if the last day of the six be omitted so as to have an 

 odd number of days, I find — 



The general mean difference = 0'o656 inches. 

 IXk periodic ,, ,, = 00634 ,, 



Gen. Strachey's conclusion from the Madras rainfall 

 observations is in fact that because the periodic mean 

 difference was only one-tenth less than the general mean 

 difference, there was no evidence of periodicity whatever ; 

 here we have a large and regular semi-diurnal period (the 

 whole mean range being o'cyo inch) where the periodic 

 is not one-thirtieth less than the general uisSin difference. 

 I may add that when the true sun-spot period of ten 

 and a half years is employed, for the Madras rainfall 

 observations, I find — 



The ^I'mva/ mean difference = 1 2'4 inches. 

 The periodic ,, ,, = io'2 ,, 

 quantities which differ by five times as much as those 

 found for the true periodic variation of the barometer at 

 Simla. 



I have taken the variation chosen by Gen. Strachey to 

 illustrate this question, but the fact that the difl'erence of 

 ihc i^eneral and periodic mean difference is no criterion 

 of periodicity might have been shown equally well with 

 cases more resembling that of the rainfall, where the 

 irregular variations are large compared with those following 

 a known period ; I cannot here, however, enter into 

 details and notice only the objections offered by me to 

 Gen. Strachey's paper when it was read before the Royal 

 Society. 



It would be easy to show that the Madras rainfall 

 observations, laieu alone, give results which are remark- 

 able in several respects. Thus, ist, They show a mean 



oscillation larger for about ten (nine to eleven) years than 

 for any other duration. 2nd. When the mean variations 

 for a period of ten and a-half years are represented by a 

 function of sines they give the yearly mean rainfall (y) 

 in the period, ^ = 6 2 sin (^ + 310°), showing the large 

 range of i2'4 inches. 3rd. This representative equation 

 gives the epochs of maximum rainfall in the years of 

 maximum sun-spots, or as nearly so as would be given 

 by the mean sun-spot areas represented by a similar 

 expression.' 



On the other hand, the irregularity in the amount of 

 rainfall from year to year is so great that the probable 

 error of the periodic means is too consid'^rable to give 

 any great weight to this result alone.^ When observa- 

 tions during a sufficiently large number of cycles have 

 been obtained, so as to make the probable error of the 

 means small compared with the range of the periodic 

 variation, then there will be a general acceptation of 

 Gen. Strachey's remark : " It is hardly conceivable that 

 there should be a coincidence with the sun-spot period, 

 such as is supposed to be found at Madras, based on any 

 physical cause which should not in some way be discern- 

 ible in the rainfall at Bombay and Calcutta" (Nature, 

 vol. xvi. p. 172). fie has then taken Jii'e cycles of 

 eleven years' rainfall at Bombay, and pour cycles at 

 Calcutta, and testing them by his criterion he obtains 

 results quite similar to that for Madras. 



I have only the periodic means for the five eleven-yearly 

 cycles at Bombay now before me, but seeking from these 

 the representative equation of sines as for Madras, and 

 repeating the latter for comparison, 1 find — 



Bombay y = 6'I sin {$ + 316°). 



Madras y = 6-2 sin ($ + 310°). 



Both equations give almost exactly the same range of the 

 oscillation and nearly the same epochs of maximum and 

 minimum as the sun-spots.^ This result, which was 

 wholly unexpected by me, is all the more remarkable that 

 the two places are on the opposite coasts of India, and 

 have their rains from different quarters. Calcutta, with 

 a sufficiently large number of cycles, might also have 

 agreed with Bombay and Madras, which is not the case, 

 however, with four cycles only. In each case the criterion 

 would show that no periodicity exists. 



1 cannot, then, agree with Gen. Strachey as to his test 

 of periodicity nor to the conclusions he has deduced from 

 it. I will not enter here into the consideration of the 

 weights which may be given to results founded on the 

 known principles of the calculus of probabilities, nor 

 into the question whether the rainfall, not at one or two 

 stations only, but over a country or the whole globe, may 

 not show some relation to the sun-spot period as Mr. 

 Meldrum believes, and as I think quite possible, judging 

 from other results of solar actions. This relation, how- 

 ever, it appears to me has still to be proved, though the 

 observations considered by Gen. Strachey are, on the 

 whole, so much in its favour as to encourage further 

 investigation. JOHN Allan Broun 



Lyndhurst, New Forest, July 18 



In the paper read by Gen. Strachey before the Royal 

 Society, May 24 (sec Nature, vol. xvi. p. 171), "On 

 the alleged Correspondence of the Rainfall at Madras 

 with the Sun-spot Period, and on the True Criterion of 

 Periodicity in a Series of Variable Quantities," certain 

 conclusions are arrived at which render it desirable to 

 test the value of the criterion of periodicity employed. 

 This is the more necessary when it is considered not 

 merely that the principle, if a sound one, must be of 



' The years given by the equalion, the series commencing with i8i3'5, 

 are 1817 8, 1828 8, 1838-8, 1849-8, 1859-8, T870 8. The condition that an 

 oscillation should agree in its epochs of maximum and minimum with (hose 

 ot a known phenomenon (a very weighty ote when the chances are to be 

 coi sidered) has bten neglected by uen. Strachey altogether. 



^ T his refers to the periodic means deduced from the observed quantities : 

 the above equation for Madras gives the observed meaus with a probable 

 error of le-s than 3 inches. 



3 The first cycle at Bombay begins with the same year, (1824-5) as the 

 second cycle for Madras. 



