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Day of the month.

brevity, as it is in every case 29 inches. The computation was extended to four places of decimals, though it has been thought unnecessary to give more than three.

TABLE I.

Atmospheric pressure for every day in the year—from the periodic function.

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7791 7255 6854 6696 6636 6822
7740 7365 6732 6780 6588 6806
+48 +16

Means

Means from record..

Means from corrected record

6999 7354 7835 7558 7118 .7557 7057.7250 7968 7506 7068 7624

+57 -110 +122 -84
-58 +104 -133 +47 +50 -67
7727 7351 6732 6818 6594 6822 6978.7280 7954 7516 .7094.7624

It here appears that the extremes of the annual fluctuation are the maximum 29.7902 on the day of the autumnal equinox, with the minimum 29.6612 on May 10, while a subordinate wave occurs in the beginning of the decline from the September maximum, reaching a minimum value of 29.7066 in the middle of November, with a subsequent rise to 29.7877 on the 8th of January.

The most marked deviation between the monthly means of the computed normal values and the means which were the basis of computation, aside from those which are due to maxima or minima already exposed, is found in the months of March and April; and this is due to a deviation from the regular progress occurring in the end of March, which will be noticed hereafter in its proper place.

It is well known that the accidental barometric oscillations have a greater range in the winter than in the summer months, and for the purpose of obtaining a quantitative determination of the normal degree

Day of the month.

of this atmospheric disturbance in every part of the year, as well as a more accurate account of the annual fluctuation of pressure, the daily means of the barometer for the twenty-five years were next arranged under each day of the year in two columns, one column containing those means which exceed the normal value for the particular day in question, derived from Table I, the other column containing the means for like-named days in the remaining years, all falling below the computed normal value. Each column having been added, the difference between its sum and the product of the normal value by the number of means in that column was found. Then if d and d' represent these differences, N the normal value, and n the number of means in both columns. d+d' taken together, (twenty-five, except when an omission occurred,) n is the mean of the departures of the daily means from the normal value for d-d' 1 that day of the year, and N+ equal to th of the sum of the two

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columns, is the mean pressure for the same day. The mean pressures for every day of the year, thus deduced, are as follows:

TABLE II.

Almospheric pressure for every day in the year-directly from observations.

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20.37920. 826 20. 504 20. 459 20. 396 21. 603 22. 528 23. 889 23. 248 21. 160 23. 560 21. 132 S

.7280 .6718 .6835 .6600 .6799 .6970 .7267 .7963 .7499 .7053 .7600

As might be expected in so short a series as twenty-five years, the values embraced in this table are far from escaping the effect of acciS. Mis. 31-30

December.

dental variations, but we may attempt to eliminate that effect by combining them in groups. The following are the mean values of atmospheric pressure in ten-thousandths of an inch, obtained directly from Table II, for thirty-six equal periods of 10.15 days each, beginning with January 0 and embracing the entire year. (The integral part, 29 inches,

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In this series few oscillations appear beyond the two principal ones already noted. Beginning with the yearly minimum on May 10th, which falls in the thirteenth period, the rise to the September maximum, in the twenty-seventh, appears to be broken by a depression culminating in the twentieth period (about July 17), and in like manner the fall thence to the November minimum is checked by a subordinate maximum in the end of October, the thirtieth period. The only remaining noteworthy departure from the normal value is more marked than either of the preceding, and consists of a depression in the ninth period—the ten days following the vernal equinox-which appears to amount to a true equinoctial storm. Thus each of the two equinoxes is a definite barometric epoch, though they are marked by opposite characteristics.

A tendency is apparent in one or both of the periods adjacent to a maximum or minimum to depart from the normal values in an opposite direction, so that the gradual approach to the critical value, on the part of the more distant periods, appears to culminate ten or fifteen days from the true epoch, and a double maximum or minimum is the result. Examples may be noticed in the fourteenth period, immediately following the yearly minimum in May, while a double maximum occurs in September; also, in the case of minor oscillations, the minima in March and July are each followed by a single period of high barometer, and the January maximum is both preceded and followed by single periods slightly lower than those adjacent to them. The phenomenca appears suspicious, as though due to accidental variations, yet some ground for

a contrary opinion may be derived by examining the deviation of each period from the normal pressure.

As will presently be shown, the probable error of one of the values in Table III is a variable quantity, having for its mean value 0.0095, while at its maximum it is about 0.0122 and at its minimum 0.0058. If now we compute a value for the middle day of each period from the periodic function, in the same way as Table I, but to four places (which may be regarded as representing accurately enough for the present purpose the normal value of pressure for the period, aside from the irregularities now under discussion), these normal values, subtracted from the successive values of Tables III, will leave the following residuals:

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It here appears that the periods following the March and May minima and the January maximum deviate from the normal value in the opposite direction from the adjacent critical value, and to an amount exceeding the estimated probable error for the time of year, while the periods preceding the January and September maxima appear to be normal, the double maximum in these cases being produced by a relatively high value of the period in which the anticipatory subordinate maximum occurs, rather than by a relatively low value in the period separating it from the true maximum. The effect of the disturbance in March-April appears to be confined to the ninth and tenth periods, and it produces that divergence of the computed and observed means for the two months, previously noticed in connection with Table I.

The division of the year into thirty-six equal periods, in Table III, gives the opportunity for a recomputation of Bessell's periodic function. The following are the coefficients of ten terms, based upon the thirty-six periodic means. The first five or six terms may be regarded as replacing by more accurate values the formula previously derived, and from which Table I was computed; the concluding terms are, perhaps, chiefly

interesting as showing the impracticability of obtaining actual convergence from a series of observations no longer than the present:

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Mr. E. L. De Forest, in the Analyst for July, 1877, refers to a previous paper of his own in which he has pointed out a simple test of the accuracy of the adjustment of a series of values subject to accidental variation. The test quoted is in substance as follows: If the original (observed) values be subtracted from the adjusted values, term by term, the result is the series of residual errors. If the terms of this series be pointed off into groups, by inserting a point of division at every change of sign, the most probable number of terms occurring in groups of three or more is N±.533 √N; where N denotes the whole number of terms of the series, and the expression following the sign is the probable error. If the number of terms occurring in such groups falls short of N by more than the probable error, the inference is that the inequalities of the series have not been smoothed out enough; but if it exceed N by more than the probable error, the series has been smoothed too much. From the nature of the periodic function, the adjustment effected by it would be expected to err (if at all) in the latter of these two directions. In fact, when a series of residuals is formed by subtracting the terms of Table II successively from the corresponding terms of Table I, the number of terms or signs occurring in groups of three or more exceeds half the whole number of terms by 79, while the probable error, .533 √366, is only 10.3. Though a closer approximation to a perfect adjustment might probably be derived by the use of the more exact coefficients just obtained, it seems preferable, for the purposes of comparison, to use a different process of adjustment, and hence the method of successive means has been employed. The arithmetical means between each term of the series of Table II and the next succeeding term form the first order of means, the second is derived in the same way from the first, and so on until the tenth order of means is obtained, which constitutes Table IV.

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