have taken the principle of its formation from the paper of M. Decroizelle, in the Annales de Chimie. London, Aug. 28, 1829.. I am, Sir, Respectfully yours, W. G. COLCHESTER. The apparatus consists of a glass jar about one inch in diameter, containing about five cubic inches, and graduated into inches and tenths; a dropping tube about seven or eight inches long, divided into thirty equal parts; a porcelain mortar and pestle; a weight of 100 grains, and a bottle of sulphuric acid, so diluted that the quantity contained in twentytwo divisions of the dropping tube will just saturate fifty grains of crystallized sub-carbonate of soda. To determine the point of saturation litmus paper may be used, or, what is much more convenient, infusion of cabbage. METHOD OF USE. The sample to be examined having been pounded sufficiently to pass through a coarse sieve, rub up some of it in the porcelain mortar until it be reduced to a very fine powder; from this weigh 100 grains and return it into the mortar; add thereto boiling water, a small quantity at a time, and continue to rub it as long as any grittiness appears under the pestle ; suffer it to stand a short time, and pour off the liquid into a pint or half-pint vessel with a lip; add more boiling water to what remains, and again use the pestle, repeating this to ensure the perfect solution of all the soluble part of the sample, until about half a pint of boiling water has been employed; transfer the whole into the same vessel, stir it well together, and allow it to stand for the insoluble part to subside; when this is effected, measure off the clear liquor by pouring it into the graduated jar and set it by for use; measure also the remainder, first shaking it up, and having noted the total quantity, this remainder may be thrown away. Take of the clear solution just one half of the whole amount of the two quantities, and add thereto about a table-spoonful of the infusion of cabbage; then, having filled the dropping tube to the upper division with the test acid, drop so much into the sample, constantly stirring the mixture, as will just change its green colour to crimson; the quantity of acid used, as indicated by the divisions on the tube, will show the per centage of alkali in the sample, if it be barilla, kelp, or manufactured soda; but, if the sample be pot or pearl ashes, augment the proportion of test acid used, by adding to the number of divisions indicated by the dropping tube, one half such number, and the total will be the per centage of alkali in such sample. Should it be desired to ascertain the quantity of carbonic acid contained in the sample, we need only note the point at which the solution becomes blue in the foregoing process, and deduct the divisions then indicated by the test tube from the subsequent total amount; every ten of the remainder will then indicate seven per cent. of carbonic acid, whether of barilla or of pot-ash. The apparatus is made and sold by Mr. Bate, Philosophicalinstrument Maker, 21, Poultry. Memoir on the Mean Results of Observations; read before the Academy of Sciences, April 20, 1829. By M. Poisson. THIS Memoir is the continuation of that which I inserted in the Additions à la Connaissance des Temps for the year 1827. My object is to add some new developements to that part of it which treats on the probability of arithmetical means between the results of a great number of observations. When there is no reason for believing some more exact than the rest, the mean of them should be taken for the unknown value sought; and one is naturally led to think that this mean result approaches the nearer to the truth as the number of the observations is more considerable. But La Grange is the first person who subjected this question to mathematical analysis*, and who investigated the probability that the arithmetical mean between any number of observations does not differ from the * Vol. v. of the old Memoirs of the Academy of Turin. 92 M. Poisson's Memoir on the 1 : 918aibro mez ɔdi ing07612 le 695 ust 976d 100 yem true value by a quantity greater than some assigned limit! To solve it, he supposes known the law of probability of errors in the observations, or the values which they may give for the thing which it is sought to determine; an hypothesis which prevents the formulæ deduced from it being applicable and of any use in practice. It is to Laplace we are indebted for havi ing rendered the probability of the mean result independent of this law in the cases where the observations are numerous Tso that from the sole numerical data of the observations the pro bability may be calculated of a determinate limit of error to be apprehended in taking this result for the value of the und known quantity. I hope the details on this subject, into which I have entered in my Memoir, will be well calculated to dissi pate the doubts which might still remain as to the degree of approximation of this probability*. To form a precise and general idea of the limit to which the mean result of observations approaches indefinitely in propor tion as the number of them increases, we must suppose the construction of a curve, the ordinates of which are proportional to the probabilities of the values of the unknown quantity, which last is expressed by the corresponding abscisste. If the law of probability change from one observation to another, this curve will change also; and another will be constructed, the ordinates of which will be means, for each abscissa, between those of all the particular curves. This being so, the limit in! question, in every case, is the abscissa which corresponds to the centre of gravity of the area of the mean curve. This limit to which the mean result of the observations converges, is not necessarily one of the values of the unknown quantity which have the most probability, and are given most frequently! by isolated observations. It may even happen that the proba bility is altogether none, and that it cannot be given by any single observation; which, in fact, will be the case if the ordia nates of all the curves of probability are null for the same abscissa, and symmetrical on each side of it. In the general case where the curve of probability varies from one observation to another, it may also happen that the areas of all the curves * Supplement to Théorie Analytique des Probabilités, p. 7. may not have their centres of gravity on the same ordinate; the abscissa which corresponds to the centre of gravity of the mean area will then vary with the number of observations; and if this number be divided into several parts, which still consist of considerable numbers, the mean results of these partial series will not be the same, although the error to be apprehended from each of them is very small, and all possess a very great probability. The calculation of mean life is one of the most ingenious applications which has been made of these principles. A great number a million, for example-of children are considered as born at the same epoch, and the future duration of the life of each infant is assimilated to an eventual gain, of which the probability is unknown. The sums of all the possible durations of life, from zero to the greatest age which men can attain, mul tiplied by their respective probabilities, and relative to this infant, will then form his chance or his hope of life consequently, mean life will be the sum of these quantities for all the infants divided by the number of them; now it is easy to see that this quotient is nothing else than the abscissa of the centre of gravity of the mean area, which was mentioned above. Thus, by taking the mean time that an equal number of individuals, born in the same country as the children under consideration, have lived, and at a period as near as possible to that of their birth, we shall obtain an approximate value of mean life; and from these observed durations of human life may be calculated the probability that this value does not differ from the truth, such as it has been defined, by more than a given time. The probability of living to an assigned age is, doubtless, not the same for a million of infants born at the same period. But it may be admitted that the mean of its unknown values varies but slowly by the extinction of maladies and the improvement of society; experience alone can teach us if this mean law of probability, and consequently the mean duration of life, remains stationary, or sensibly varies in long intervals of time. b.15 52-bindi It is also by the same principles that the mean advantage, and the probability of it, which may be expected from a very great number of speculations, is calculated, from the known losses and gains of another very considerable number of similar operations, that is to say, of which the mean law of probability is supposed the same. In other questions depending upon the same theory, where the subject is the greatness of a phenomenon or the measure of any thing which is to be determined by a series of observations, it is supposed implicitly that, among all the values of which this thing is susceptible by its nature, there exists one from which it is equally probable that there will be an equa difference in excess, or defect in each observation; it is supposed, moreover, that this value of the unknown quantity is the same for all the observations, and it is this value, thus defined, that it is sought to discover. That amounts to saying, that the curves of probability relative to all the observations are symmetrical on each side from one of their points, and that this point corresponds to the same abscissa for all these different curves. In this hypothesis the centres of gravity of their areas and that of the area of the mean curve will be situated on a common ordinate, the abscissa of which will represent the true value of the unknown quantity. By multiplying the observations, the quantity by which we shall indefinitely approximate will be constant and independent of their number; and although their laws of probability may be different, their mean result will give a value nearer and nearer to the unknown quantity; and at the same time, from the whole of the observations collectively, the probability of its degree of approximation may be calculated. But however small may be the error to be apprehended in taking the mean result for the value of the unknown, and however probable may be the limit of this error, it must not be lost sight of that the value of anything drawn from observations is always subordinate to the hypothesis already stated. If any unknown cause render the errors of the instruments or the variable circumstances which influence the phenomena, preponderant one way or the other, or if the thing to be determined varies progressively during the continuance of the observations, this hypothesis will not hold good, and the observations should be rejected as improper for |