magnetism. Dynamics, in the abstract, regards matter in general, without relation to species; chemism generates therefrom mineralogical or so-called chemical species, which, theoretically, may be supposed to be formed from a single elemental substance, or materia prima, by the chemical process. Dynamics and chemistry build up our inorganic world, giving rise to geogeny and, as applied to other worlds, to theoretical astronomy. "Proceeding now to the organic kingdom, its physiographical study leads us first to organography, and then to descriptive and systematic botany and zoology, two great sub-divisions of natural history. Coming next to consider the physiological aspect of organic nature, we note, besides the dynamical and chemical activities manifested in the mineral, other and higher ones, which characterize the organic kingdom. On this higher plane of existence are found portions of matter which have become individualized, exhibit irritability, the power of growth by assimilation, and of reproduction, and moreover, establish relations with the external world by the development of organs, all of which characters are foreign to the mineral kingdom. These new activities are often designated as vital, but since this word is generally made to include at the same time other manifestations which are simply dynamical or chemical, I have elsewhere proposed for the activities characteristic of the organism the term biotics (biotikos, pertaining to life)." "The philosophy of matter in the abstract is dynamical, that of mineral species is both dynamical and chemical, while that of organized forms is at once dynamical, chemical and biotical. The study of the biotical activities of matter leads to organogeny and morphology, while the relations of organisms to one another, and to the inorganic kingdom give us physiological botany and zoology. We thus arrive at a comprehensive and simple scheme for the classification of the natural sciences, which is set forth in the subjoined table." On the Law of Facility of Error in the sum of n Independent Quantities, each Accurate to the Nearest Degree. By CHARLES CARPMAEL, M.A. (Read May 26, 1882.) Let a1, az, aza, be the n quantities true to the nearest unit, and let the absolute values be a1 + x1, a2 + X, Az + X3,..... A3 ann, so that x1, x2, x3............, are each between the limits - 1 and + 1, and all values between these limits are equally likely. The chance of an error in a between 2, and 2, + da, in a, between 2 and x + dx2, &c., occurring simultaneously is x X2 X2 there will be a portion of (ii) in which r chosen variables and these r only are greater than unity. Let us call this part R. Sec. III., 1882. 2 Subtract unity from each of the n variables in turn, we obtain n integrals each equal to Of these n integrals r, and r only, contain the part R; namely, those obtained by subtracting unity from one of the r chosen variables. Similarly if we subtract unity from two of the variables, and take every possible combination of the n variables two at a time, we obtain - n. n 1 integrals each equal to contain the part R; namely, those obtained by subtracting unity from any two of the r chosen variables. n So ifs be any number not greater than t, if we subtract unity from s of the variables, and take every possible combination of the n variables s at a time, we shall obtain integals each equal to $ n-8 or (11) times, i.e., it will not contain it at all. Now this is true for any r particular variables, and for all values of r from r 1 to rt. The series (iii) then, contains those portions, and those only, of (ii) which are due to values of the variables all less than unity, it is therefore equal to the series to be continued as long as the part raised to power n is positive. n 2 This then is the chance that the error lies between and y, and differentiating with respect to y we find that the chance that the error lies between y and y+dy is The result in this form would be of very little use except for small values of n. But Laplace has obtained approximate values for the above series when y is small, and his * This formula may also be proved by induction, for if it be true in the case of any number n of independent variables, we shall have for n + 1 variables the chance which is the same formula as in the case of n variables with n + 1 written for n. Now when n= = 1 the formula (iv) reduces 4 + y and is therefore obviously true. Hence it is also true when n = 2 and therefore when n = 3 and so on. |