1 α density d, by 1 additional atmosphere; d(-a)=d+a= additional compressibility of air at density d, by a additional atmospheres. It seems reasonable to suppose that the ratio of compressibility between the liquid and air of the same density may be some function of the compressibility of air under the same pressure. This function may perhaps be represented by the ratio of homologous representatives of equal weights of air and liquid under the ordinary atmospheric pressure; for example, by the ratio of the diameter of a sphere of air or liquid at density d, to the diameter of the same sphere at density 1, which is the ratio of 1. If such 1 = X compressibility of liquid by 1 addid (d+1) is the case, The assumption of the specific gravity of air as the unit of the ratio, is undoubtedly, to some extent, arbitrary. But as soon as the pressure of the air is removed, in any degree, from the surface of a liquid, a portion of the liquid assumes the form of vapor, and it would be impossible to approximate so nearly to the average density of the mixed liquid and vapor, at any other density of the air, as we can at the one we have assumed. It is true that even at the ordinary pressure of the atmosphere, liquids are constantly evaporating; but as that pressure is appointed by the Creator, as the one of natural equilibrium, it seems reasonable to presume that we shall not materially err, in taking it as the starting point of our hypothesis. To test that hypothesis, let us examine 66 d=642.7 Mercury, 66 which has d: = 11023.7 Water, which has d=812.5 compressibility of air under pressure of a atmospheres, by the pressure of a addi tional atmospheres. According to the above formula, the compressibility by a single additional atmosphere, would be, for In the following Table, the theoretical are compared with dif ferent experimental results. The experimental results are undoubtedly all subject to correction for friction, imperfections of apparatus, and difficulty of minute observation; and the theoretical results should likewise be modified by considerations of temperature, vapor, imperfect elasticity of the air, and perhaps other unknown influences. Our data are so few and imperfect, that they cannot be regarded as deciding the truth or falsity of the theory. They may, however, be sufficient to call attention to an almost untrodden field of investigation, awakening an interest which will be followed by more numerous and accurate experiments, and lead to the modification of the formula here suggested, in the discovery of a new one, which will more accurately represent the law of fluid compressibility. It is already known that the law of MARIOTTE is only approximately true. When the compressibility of various different gases and vapors has been determined with the greatest possible precision, we may, perhaps, theoretically determine, with equal precision, the compressibility of the condensed vapors in the liquid form. The first experiments which proved with any tolerable degree of accuracy the compressibility of water were those of PERKINS. The results of his early trials gave a degree of compressibility more than twice as great as his final estimate; but the ratios of compression between the successive experiments correspond very nearly, in many instances, with the theoretical ratios. I can hardly believe that this correspondence is entirely accidental. In the following table, the first column gives the number of atmospheres (a) to which a column of water, 190 inches in length, was subjected; second column, the compressibility, in decimals of an inch, according to PERKINS'S table, with the ratios between the results of the successive experiments; third column, the theoretical compressibility in decimals of an inch, with the successive ratios. The numbers in the third column are obtained by the formula 190 X × 1.* α ds (d+a) THE ELEMENTS OF QUATERNIONS. By W. P. G. Bartlett, Cambridge, Mass. THE following brief essay is intended to present, in a direct method, the fundamental principles and notation of the Quaternion Analysis; so as to enable one who has mastered it to proceed at once, without difficulty, to the geometrical applications and further developments given in the VIIth of HAMILTON'S "Lectures on Quaternions" or elsewhere. With a few trifling exceptions, HAMILTON'S notation is strictly retained. Nothing will be given, at present, upon the differentials of quaternions, which are introduced in Lecture VII., § xcvii., of HAMILTON. The author intends hereafter to take up, as a continuation of these papers, some special subject, such as Spherical Trigonometry, or Analytic Geometry, and discuss it in the language of Quaternions. 1. Equal lines are such as have the same length and the same direction in space. 2. One line is the negative of another when it has the same length, but the opposite direction. 3. The sum of two lines is the diagonal of a parallelogram, of which these lines are two adjacent sides; that diagonal being taken, whose direction lies between the directions of the two given lines. Hence the sum of two or more lines is the same, in whatever order they may be placed or added together; and when any set of lines in space form a closed circuit, their sum is equal to zero, and therefore each line is the negative of the sum of all the others. 4. In operations on lines in space, it is convenient to substitute for some of the given lines other lines, equal, and therefore parallel, to the given ones, but passing through a point in space com mon to themselves and the other given lines; so that all the lines may be coöriginal. In what follows this is supposed to be the case. 5. The abstract operation of so changing the length and direction of one line as to make it coincide in length and direction with another line, is considered as the quotient of the second line divided by the first, and is called a quaternion. The plane in which, by 4, both these lines are situated, is called the plane of the quaternion. Conversely the product of multiplying the first line by the quaternion is the second line. In writing a product the multiplier always precedes the multiplicand. 6. Lines are generally denoted by the letters a, B, 7, &c.; quaternions by p, q, r, &c. The angular distance in the plane of a and from the positive direction of a to that of is denoted by &. β 7. The quotients ẞa, and 87, are equal, when α, B, 7, and β 8 = and the ratio of the length of to that α y 8 being coplanar, Hence if there are given any two quaternions, q='a', and py'÷d', there can always be found three other lines, a, B, and y, such that B÷ a=ß' ÷ a' =q, or ẞ=qa; and y÷ay'÷d'=p, or 7 = pa; that is, some line, a, can always be found such that it may be multiplied by both the given quaternions. The three lines a, ẞ, and 7 might have been so determined that ßaß'÷ α' and yẞy'÷d' = p. or, = 8. The sum of two quaternions is defined by the equations p = B÷α, q=r÷a, p+q=B÷a+r÷a= (ß + r) ÷ α ; ß÷a, pa=B, qa=r, (p + q) a=ß+r=pa+qa. By §§ 3 and 5, p+q=(B+r) ÷ a must be a quaternion. Hence |