Multiplying by xa, and transposing, the equation becomes x2 m x + am = o.. 2x mo and m = But from the first equation m = reduced gives x = 2 a. When radical quantities enter the proposed function, it does not appear that the common rule for reducing equations having two equal roots is generally applicable. We shall, therefore, investigate one by which examples of this kind may be resolved. Since the equation fry has two affirmative roots, let p = less root and p +e greater (e being their difference, and represented in the illustration of the lemma by the line P P'). Then p and p + e substituted for r in the equation fr=y give the same result, viz. y. Therefore fp = y = f (p + e) (A). After developing the second member of this equation, and taking away the quantities that are common to each side, all the remaining terms will be divisible by e, and we shall have an equation containing p, e, and constant quantities, which will be true for every value of the function. But when y = m, e = 0; therefore all the terms containing e and its powers will vanish, and we shall have an equation expressing the relation between p and given quantities, from the resolution of which, p or its equal will be known. Let us apply this method to problem 2, where we have given y, find r when y = m. This equation has two affirmative roots (by the lemma). Let p = less root and pe greater. Then these substituted for a in the proposed equation give the same result, viz. y. Therefore, p p3 = y $ po 4 ? e ( p + e)3, or p p3 = p + e - 3 p2 e Bees! Taking away pp from each side, and dividing the remaining terms by e, we have 1 - 3 pee2 0. Now make eo (because when y=m, e vanishes), and the last equation becomes 1 - 3 p2 =0. 3 p2 p = ps = x. From this example, it is evident that in developing the functions of (p + e) only two terms of the series need be taken, because the succeding ones contain e, e3, &c. and, therefore, ultimately disappear. Problem 4.-Given in position two points A, B, and the line CK; it is required to determine the point E in this line, so that (A E + B E) may be the least possible. From the points A and B, let fall the perpendiculars A C, B D, upon C K, and let E be the resto quired point. Pute A Cha BD = b, CD e and CE སྐ 1 Therefore a + x2 + √ (c = x2) + b2 = y, and it is required to find x, when y = m (minimum). Substituting p and (p + e) for x, as in the last example, we have √ u2 + p2 + √ (c − p)2 + b2 — y = √ a2 + (p + e)3 (p + e)2 + b2. Now √ a2 + (p + 2e) = √1 a2 + på + √ (c Emerson, in his Algebra, has given a rule very analogous to this, for resolving problems relating to the maxima and minima; but it appears to have been suggested to him by the differential calculus. At any rate his notion of the symbol e is very different from ours; for he supposes it to represent an infinitely small quantity, and he rejects the terms containing its powers, with out assigning a very satisfactory reason. We conceivebe to be the difference of two affirmative roots of the equation fry, which is always a real finite quantity, except when y arrives at a limit, and then it actually vanishes. We shall conclude this article with a concise demonstration of the well-known theorem, that the fluxion or differential of a function = 0, when it is a maximum or minimum. For developing the second term of the equation marked (A),by Taylor's theorem,* we have ƒ p = ƒ (p + e) = ƒ p + f dfp e + e2 + &c.ado d p2 Taking away fp from each side of the equation, and dividing dfp dfp by e, we have + e+do. But when the function dp d p2 attains a limit e = o, and, therefore, all the terms, except the first, disappear from this equation; and we have in this case df x #.0.5 But p = xv. dx dp = 0, or dƒ x = 0,d od dóndw in nada edo itiw yilgozolidg sesɗT * For a demonstration of Taylor's theorem, sec Calcul. differentiat et intégral de Lacroix. Examination of the Influence of the Time of the Day upon Barometrical Measurements. Extracted from the Researches of M. Delcros.* M. DELCROS undertook to resolve this problem: "Suppose two barometers, separated from each other by a certain space, both horizontal and vertical; at what time of the day ought they to be observed, that the height of the stations resulting from calculation may approach the nearest to accuracy?" To obtain the solution, M. Delcros made choice of two stations, conveniently situated for observing at the same time two barometers, well constructed, at five different periods of the day, each separated from the other by an interval of two hours; namely, at eight o'clock in the morning, ten o'clock, noon, two o'clock, and four o'clock in the evening. One of the stations was at Strasburg, in the cabinet of M. Herrenschneider, Professor of Natural Philosophy in the Academy, and a very accurate observer. The other station was the castle of Lichtemberg, upon an insulated summit of the Vosges, about ten leagues north from Strasburg, and about 264:35 metres above that city, and connected with it by one of the triangles belonging to the great base of Ensisheim. Colonel Henry, who had the superintendence of the geodesical observations executing in the east of France, had resolved to make a set of astronomical observations at Lichtemberg, to determine the amplitude of the celestial arc of the meridian from Geneva to Luisberg, an arc which this point divides into two parts nearly equal. M. Delcros being obliged to make a considerable abode at Lichtemberg, in order to assist his superior, took advantage of the opportunity to make a complete series of barometrical observations in the same place at the five times of the day above-mentioned, These observations were simultaneous with others made at Strasburg by Professor Herrenschneider, They were continued for 22 days, which gives 100 observations to compare, disposing them in five groupes of corresponding observations, which "may be compared with each other. This comparison has been carefully made by M. Delcros, which, in his opinion, may add some rays to the luminous pencil Fin collected by the celebrated philosopher Ramond, to whom the barometrical method is indebted for so many labours-for so many profound investigations-for so many precious memoirs, in which he has united the principles of a simple and luminous philosophy with the charms of style." s These observations have been arranged by the author in two very interesting tables. * Translated from the Bibliotheque Universelle, vii. 236. (April, 1818.) In the first, divided into 14 columns, and which we are prevented from publishing by its great size, we find the dates (days and hours), the heights of the barometer observed, and the temperature of the mercury and the air at Lichtemberg; the same elements for Strasburg; the numbers given by the tables of Oltmans; the corrections for the temperature of the mercury and the air; for the latitude, for the diminution of gravity in the vertical; the difference between the heights of the barometers, derived from the calculation of each of the corresponding observations; and, finally, all the meteorological circumstances that accompanied each observation. This table contains all the elements of the second, the object of which is to show the influence of the time of the day, by the way in which the results are grouped. This table accompanies the present article. Its general title, and that of its several columns, indicate sufficiently its object. We perceive the results of each observation grouped respectively into each of the horary epochs that furnished it: at the end of each observation is given, in metres and centimetres, the quantity by which it differs from the true height of the station as determined geometrically. At the bottom of each of the five columns of these differences is given the mean number of metres round which the results oscillate; and the greater this number is, the less is the time of the day which it represents favourable for accuracy. But by casting the eye over the bottom of the five columns, we shall perceive the results of which we form this very instructive little table. That is to say, that if we choose eight in the morning for the simultaneous observations of two barometers placed as above stated, we have the mean chance of an error of 3 metres in the 264; that is to say th of the whole; at noon the error is only 0.62, or, which is very small; but at two o'clock, p. m. the error is still less, being only 0.59 in the 264 metres, or 452 |