14. On the Law of Error in the Case of Correlated Variations. If we have a great number, N, of independent magnitudes, each liable to variation according to any law of its own, but remaining always finite, the chance that their sum, each divided by ✔N, shall lie between x and x + dx is proportional to hd, where h is a constant. The proof of that proposition is originally due to Poisson. The first application of it was to errors of observation, each of the 'magnitudes' aforesaid being such an error, and the N observations being supposed independent. Modern writers, among others Mr. F. Galton and Mr. F. Y. Edgeworth, have substituted for the single magnitude given by each independent observation a group of mutually dependent or 'correlated' magnitudes, and for the single square forming the index in e- a quadratic function, i.e., sums of square and products of the correlated magnitudes. If, for instance, they be denoted by x and y, the expression corresponding to e-hdr will be (ax+by+cydady. The coefficient b expresses the fact of 'correlation' between a and y. The object of the following paper is (1) to extend the purely mathematical investigation hitherto applied to the case of single magnitudes to the case of groups of magnitudes, the members of each group, although 'correlated' with one another, being still supposed independent of any other group-in this I only follow the lines of the known proof; and (2) to show how in certain cases the method may be extended to groups which do not possess this property of mutual independence. PART I.-ARTICLES 1-15. 1. Let fa(a,. an), or fa, be a continuous function of the variables a1 an. Let f(b... bn), or f, be the same or a different function of the variables b1... bn, and so on to fl¶1 In), there being N functions. Denote by Pa the integral 2. Assume the b's to be independent of the a's, so that the variables a are not contained explicitly or implicitly in f. . . or fq, the variables b are not contained in fa, or in fe.. fa, and so on. Then where the coefficients are numerical and of the order of magnitude subject to the condition that the linear functions lie respectively between limits 81+ds1, &c. ... 8n for a1 4. To do that, substitute 8, an by the linear equations, and then perform the integration according to b c, &c. That reduces the integral to 8n)ds,... dsn, where p denotes some function. Now let and ... S du, fdu, . . . ♬ (u, . . . u,) cos (0 ̧μ‚ + 0 ̧« ̧ + · · · ) = p cos r, Then if 01 = 02 = == =0=0, p=1. If any 00, p<1. From this it follows that unless every 60, each of the quantities X,X less than unity. 7. Now let a1 an be any set of correlated variables, and let fa(a, . . . a«) da... dan be the chance that they shall simultaneously lie between the limits a1 a1 + da,, &c. Let fo(b,... b) have the corresponding meaning for another set of correlated variables independent of the a's, and so on for fe, &c. Then, if we form the linear functions the chance that they shall lie between the limits r1 and r,+dr1 &c. is, by Proposition I., 8. In this expansion substitute for Xa &c. their values, and expand the exponential factor. If any differs from zero, each of the factors X is, by Prop. II., less than unity, and therefore (since we are dealing not with the product of integrals, but with the integral of the product Xa... Xa) we may neglect in that expansion all powers of 0 above the second. The expansion is then reduced to the form And performing the integrations according to 1 On, we obtain as result the expression in which the coefficients are known quantities when the functions fa, fb, &c., and the X's, u's, &c., are known. It thus appears that the chance of the linear functions is a quadratic function of (r11), (r1-r1), &c. The exponential form always occurs in the result, whatever be the forms of the functions fa, fu, &c. Only the coefficients of the quadratic function are determined by the forms of these functions. PART II. 9. Up to this point we have assumed that any one of our sets of correlated variables, e.g. a,. an, or, as we may call it, any one 'association,' is independent of the variables in any other. It is now proposed to treat the several associations is a1 an, b1... bn, &c., as representing the state of the same material system, defined by n variables, at successive points of time. That is, if a an be the variables defining the system, fda, ... dan is the chance that when t=0 they shall lie within the limits ... ~1 = a1 and x1 = a1 + da1, &c. Similarly fodb db, is the chance that when t=r they shall lie within the limit a, b, and a1 = b1 + db1, &c., and so on. are not independent of a affects the chance that when t=7, a1 shall ап, Then generally the variables b1 . . . br because given that when t=0, x,a,, &c., that b1, &c. We cannot therefore at once apply our investigation of Part I. to the series of functions fa, fv, &c. 10. It is now shown that in cases where the condition of complete independence is not satisfied, another condition, called the modified condition of independence, may be satisfied. That is, namely, that although the b's are not independent of the a's for such values of the time 7, yet it may be the case that when = or >T, df where T is a definite time, the b's are independent of the a's, or vanishes. It is shown that if that modified condition be satisfied for every pair of associations, or states of the system, separated by an interval of time not less than T, we may legitimately apply the method of Part I. to the whole series of associations, N2 in number, at intervals of time at or exactly as if they were mutually independent. If, that is, we form n linear functions of the type T N' r1 + dr1, and find the chance that they shall respectively lie within the limits &c., by the method of Part I. we obtain correct results so far as the exponential form is concerned. The result is of the form e-Edr1... drn, with R a quadratic function of (r-r1), (r2 −r2), &c. ... 11. We may now make our variables x, . .n, the time differentials of another set §1... En, that is, de de, &c. X2 = dt da, is now the chance that when t=0 12. Then we may make dt infinitely small and N infinitely great, while Nôt = T and our linear functions become 19 or §-X1, 2-X2, &c., if X1, X2, &c., now denote the initial values of §1, §2, &c. And our proposition now assumes the form that the chance of within assigned limits varies as where the index is a quadratic function of (§,- X1), (§, −X2), &c. ... lying We thus get the exponential form, or law of error,' in all cases for which the modified condition of independence is satisfied. In order that it may be satisfied the time variation of έ,... En must depend not only on the values of §,... § for the time being, but also, in some appreciable degree, on external fortuitous causes. And the greater the comparative importance of these fortuitous causes, the shorter is the time T which, as the interval between two successive states of the system, makes the variables in one of those states independent of those in the other. Department II. METEOROLOGY. 1. Probable Projection Lightning Flashes. The classification of lightning flashes is already beset with difficulties. The object of this paper is to suggest the possibility of Projection Lightning Flashes, the existence of which would increase the difficulties of classification. Sheet lightning reflected on clouds is made up of numerous images of the lightning flash superimposed one upon the other. If there is a cloud with a sufficiently small opening in it between the lightning discharge and a reflecting surface of clouds, the latter being in an adequate position, on the clouds that otherwise would have been merely illuminated with sheet lightning will appear the optical projection of a lightning flash. If the surface of clouds upon which the image falls is level, the image will be a perfect reproduction of the lightning flash, save that it is inverted, and to some extent dulled in brilliancy. If, however, the surface of the clouds is irregular, such as those of the cumulus type, the image of the flash will take the shape of the irregular surface. In this way a zig-zagged flash with long angles could be formed very like the lightning flashes depicted by artists. If there happen to be more openings than one in the clouds between the flash and receiving surface, there will be a corresponding reduplication of the flashes, which may perhaps explain some of the multiple effects observed. If projection flashes occur in nature it becomes a question whether the photographic plate can register them. 2. Report on Solar Radiation.-See Reports, p. 81. 3. Report on Earth Tremors.-See Reports, p. 184 4. Reports on Earthquakes in Japan.-See Reports, pp. 81, 113. 5. On some Experiments made with Lord Kelvin's Portable Electrometer. By ARTHUR SCHUSTER, F.R.S. Experiments made during a recent trip to Switzerland have demonstrated the great convenience of Lord Kelvin's portable instrument for the observation of atmospheric electricity. The electrometer can easily be carried in a knapsack to almost any place the mountaineer can reach, and the observations are readily performed in a short space of time. The object of the experiments was to learn, if possible, whether the electric force at great heights is sensibly different from that at the sea-level. But owing to the difficulty of comparing together observations made in valleys with those made on mountain peaks, the measurements made during the past year were entirely of a tentative character, more for the purpose of testing the working of the instrument than of obtaining any definite results. The numbers given are only relative, and are referred to an arbitrary unit. Numerous experiments made in fine weather in the centre of a small meadow in front of the hotel at Pontresina gave for the electric force, on the average, about 30, while the number obtained on September 13, in Ipswich, in a place rather more sheltered by surrounding houses, was between 80 and 100.1 On the so-called 'Diavolezza Pass' the force was as high as 147; on the Isola Pers, a rocky island, projecting well out of the surrounding glaciers, the number found was 116. The instrument was taken on August 19 to the top of the Glüscheint (12,000 feet), which consists of a ridge of rocks just wide enough for one man to stand upon. The electric force reached the number 816, while on the snowfield a few hundred feet below the top, the force was only 170. The top of the Schafberg gave 357. On the Morteratsch glacier the numbers were, as a rule, very low, except on one occasion, when a strong downward wind was blowing; even then the force was only 42. Half a mile below the foot of the glacier, the numbers were very irregular, and seemed to depend a good deal on the direction of the wind. On one occasion, when that direction was frequently changing, a negative electrification was observed whenever the current of air was away from the glacier. This may have been due to the negative electrification caused by the glacier stream in a way similar to that in which a similar electrification is produced by a waterfall. The diurnal variation was observed on one day by taking measurements every half-hour during the morning and every hour during the afternoon till seven o'clock. A maximum of 36 was observed at 11 A.M. The numbers then fell In Manchester, during the fine weather in the last fortnight of September, the numbers went up to 250. 1895. SS |