results are most easily explained by periodic variations concurring in a node at the point of reversal and having a master-period of eleven years. The relation has not continued undisturbed. Constructed in 1904 the figure for 1905 was a successful forecast, that for 1906 was not; and there are reasons why it should not be. The most interesting from the point of view of periodicity is that the natural period of the subject under investigation is eleven years, but the period of the "cause" is nearly but not exactly eleven years, in which case there are "beats" between the cause and the effect, one of the maxima of which was in process of demonstration during the years in question.

The periodogram

On the analogy of the analysis of light in the spectrum, arranged according to the frequency of oscillation of the waves of which it is composed, Sir Arthur Schuster has developed a method of dealing with the periodic oscillations, in the curves of observations of a quantity, to which he gave the name of periodogram. This method is in a sense the opposite of the method of residuation in that it proceeds by adding ordinates in such a manner as to retain one component and annihilate all the others. It depends on the fact that if in the equation of p. 275 m is equal ton, the resultant curve is 2a, sin t,

2π

n

in other words the amplitude of the resultant is doubled. If we take half the sum the amplitude will be the amplitude of the vibration in the original curve. But the vibration of every other period which is not a harmonic component of n will be reduced and if the process be repeated by the addition of ordinates for a great number of successive periods n, the vibrations of periods not harmonic sub-multiples will be annihilated.

Hence by dividing the original curve into equal intervals and adding the ordinates of all the intervals and taking the mean, we get a curve which represents the amplitude of the component oscillation of that period in the original. Final values for this oscillation being obtained its first harmonic is determined and that gives us the amplitude of the vibration of that period in the original curve. This process is repeated for the whole series of possible components with a series of yearly values components of two years, three years, four years, etc., and for monthly values components of two months, three months, four months, and so on.

The process is laborious. Sir Arthur Schuster1 applied it to the periodicity of sunspots and found components with periodic times of 4.81, 8.38, and 11.125 years, the latter 11-year period being noticeable in the years 1825-1900 but not in the years 1750-1825. More doubtful periods of 5·625, 3·78, 2·69 and 4.38 years were also indicated. Sir William Beveridge2 applied the method to the periodicity of adjusted wheat prices and found components 3'415, 4415, 5·100, 5.671, 5·960, 8.050, 12.840, 19.900, 35.500, 54.000 and

1 'On Sunspot Periodicities,' by Arthur Schuster, Proc. Roy. Soc. A, vol. LXXVII, 1906, p. 141. 2 'Weather and Harvest Cycles,' Economic Journal, vol. xxx1, 1921, pp. 429-52; 'Wheat prices and rainfall in Western Europe,' Journ. Roy. Stat. Soc., vol. LXXXV, 1922, pp. 412-78.

68.000 years, and it has been applied by Captain D. Brunt1 to the periodicity of Greenwich temperatures which show components of one year with the second and third harmonics, 5 years, 594.5 days for part of the interval only, and a doubtful period of 26.21 months.

CURVE-PARALLELS

As an illustration of the method of curve-parallels we give a reproduction (fig. 112) of part of a diagram from W. J. Humphreys' Physics of the Air which draws "parallels" between pyrheliometric values of solar radiation, 1883-1913 (P), sunspot numbers 1750-1913 (S), a hybrid curve (P + S),

Fig. 112. Relation of pyrheliometric values and mean temperature-departures to sunspot numbers and violent volcanic eruptions (W. J. Humphreys).

and temperature-departures as evaluated by Köppen (1750-1913). The most striking similarity is curiously enough between the hybrid curve and temperature. Below the curves are symbols representing notable eruptions of volcanoes the dust from which may have affected the temperature-curve shown next above them.

1 'A periodogram analysis of the Greenwich temperature records,' Q. J. Roy. Meteor. Soc., vol. XLV, 1919, pp. 323-38.

Curve-parallels for pressure are given by W. J. S. Lockyer extending over forty-five years (fig. 113) from which a see-saw of pressure in complementary parts of the world was inferred1.

One of the most assiduous exponents of this method of representation is H. H. Hildebrandsson in five papers on 'Quelques Recherches sur les Centres d'Action de l'Atmosphère.' We reproduce in fig. 114 two pairs of parallels, the first for the winter rainfall of the Faröe Islands and the summer rainfall of Newfoundland and Berlin, the second for the winter rainfall of British Columbia and the subsequent autumn rainfall of the mid-Atlantic2.

Fig. 113. The variations of pressure from year to year at Bombay and Cordoba, 1860-1905.

(W. J. S. Lockyer.)

It is necessary to remark that the comparison of curve-parallels is sometimes suggestive of relationships that cannot be rigorously defended. When there is a succession of notable ups and downs in two curves one under the other the succession of prominences often bears a superficial aspect of concurrence which on closer examination is somewhat impaired by differences of a year or more in the maxima while others give no sign of lag.

1 'Barometric Variations of Long Duration over Large Areas,' Proc. Roy. Soc. A, vol. LXXVIII, 1906, p. 43.

2 K. Svenska Vetensk. Akad. Handlingar, vol. xxix, no. 3, vol. XXXII, no. 4, vol. XLV, no. 2, vol. XLV, no. 11, vol. LI, no. 8, 1897, 1899, 1909, 1910, 1914.

We revert to the methods which statisticians have developed to guard against a false impression of that kind. They deal with the numbers themselves, thus denying any opportunity for the deception, willing or unwilling, of the eye. The most instructive process is the construction of a Cartesian dotdiagram of which the co-ordinates are the corresponding departures of the several observations from their mean value. Thus in the case of the pressure at St John's and Berlin we should form a set of departures or deviations of the several observations from their mean values and set them out on a diagram. The dot-diagram may be drawn for simultaneous values of the departures of

Fig. 114. Curves of variation of rainfall from year to year: (1) for winter at Thorshavn (Faröe Islands), for the preceding summer at St John's (Newfoundland), and for the succeeding summer at Berlin, and (2) for the winter in British Columbia and the succeeding autumn at the Azores (Hildebrandsson).

the two elements or the value of the departure of one of the elements may be plotted against that of the second element after a definite period of time; in either case the time does not appear in the diagram the formation of which is the geometrical equivalent of the algebraical process of elimination of a common co-ordinate from two equations, although in the geometry we have no knowledge of the algebraic equation which would represent the curves.

The mere arrangement of the dots in the diagram will show whether there is any prima facie reason to consider the two sets of events to be connected. If all the points lie on a straight line through the origin it is evident that there is direct proportionality between the two. If the straight line passes from

the left lower quadrant to the right upper one the corresponding relation is evidently direct proportion, but if the line connects the other pair of quadrants the relation is inverse in the sense that an increase in the one element corresponds with a proportionate decrease in the other.

In so far as the points do not lie definitely on a straight line but show some "scatter" over the area the relationship is correspondingly ill-defined and therefore uncertain.

The dot-diagram leads us back again to the consideration of arithmetical manipulation by statistical methods based upon the theory of large numbers of observations with which the chapter began.

REGRESSION EQUATION AND CORRELATION

In so far as the scatter of the dots in a dot-diagram is restricted and suggests little deviation from proportionality between the two quantities we are induced to look upon a linear equation as being the best expression of the relation between them. And the next step is to determine the straight line which expresses the relationship in the best manner. For that purpose we assume an equation to a straight line passing through the origin of co-ordinates. y = bx and we seek the best value of b among all the various possibilities. A line can be drawn by eye, but if we desire to adhere to arithmetic in order to avoid bias, on the part of the investigator, or guessing, which is by some regarded as even more heinous than bias, we may note that if the equation y = bx were a true representation of the conditions, y - bx would be zero, and we may therefore seek arithmetically to choose b so that when all the values are taken into account the difference of y - bx from zero shall be numerically as small as possible. Since we are not here concerned with the sign of the difference we make the differences least if we choose b so that the sum of the squares of the differences is as small as possible. Thus Σ (y- bx)2 is to be a minimum. By the ordinary rule of differentiating, for the minimum dx value of any quantity X depending upon b, must be taken as zero. Differentiating therefore, we get for the best value of b

If we had approached the question from the other aspect of the relation of x b'y, we should clearly have arrived at the result

=