The same formula, when applied to a particle whose original ordinate was x+k, gives this result, About the particle whose original ordinate was x + k, the elastic pressure of the air upon a unit of surface is The mass of air included between these two particles, taking a tube whose section is 1, is Dk; and the pressure urging it forward is II-II' or H.D.NX+H.D.N ka dx 1.2 + &c. Hence, remarking that in Article 12 all our pressures are estimated by weights, we have for the motion of the included air*, When a body whose weight is W falls freely under the action of gravity, it is in fact a mass W (estimated in conformity with the rules of Article 12) whose motion is affected by a pressure W (estimated in conformity with the rules of the same article). In this instance, as we know, the increase of velocity downwards produced in the unit of time is g. Hence we have, in this case, Increase of velocity in the direction of the Pressure force, produced in the unit of time Mass xg. Therefore as, by the understood laws of motion, velocity produced is as pressure directly and as mass inversely, we shall have in every case ď2 (x+X) _ _! {H'.D.Ndk+H.D.Nd dt Dk dx2 dX 12 {+&c.}. dx = 0. Taking dt But x is independent of t, so that the rest on the supposition that k is made indefinitely Or, putting for the thermometer-factor in Article 17, provided that the pressures and masses are estimated as in Article 12. Now, if X be the variable ordinate in the direction of motion, increase of ordinate or the limit of increase of velocity increase of time dX di' is the velocity; and the limit of (which for such a force as gravity is the same as increase of velocity produced in the unit of time) is We shall often have occasion to use this equation. 22. The equation is independent of Local Gravity. The factor g is purely an atmospheric element. For it will be remarked in Article 12 that, in applying measures to a given state of air, D is independent of the gravity at the place of experiment, but P is inversely as the gravity; and therefore, as P=H.D, H is inversely as the gravity. Hence, wherever the experiments are made, gH is invariable. We shall put for it the symbol a'. Now at Paris where the weights of air, &c. were determined, the length of the seconds pendulum =39-12877 inches (Encyclopædia Metropolitana, 'Figure of Earth,' section 8), whence g=32·18212 feet. In Article 8 we have found H=260876 feet. Using the English foot as the unit of length and the mean solar second as the unit of time, a =√gH=916·2722. Our partial differential equation now is In the observed phænomena of sound we have very strong reason for believing that n does not depend on the nature of the motion of the particles of air; but we have no means of knowing how it may depend on the temperature of the air. In any case it may be combined with 0 to form one factor. We shall in calculation consider n constant and equal to 1.2. 23. General Solution of the Equation. The general solution of the equation above (see the Author's Elementary Treatise on Partial Differential Equations, Article 35, making a = 0), is X= (noa. t − x) + ¥ (n@a. t + x), where the forms of the functions and are absolutely undetermined by the theory of the solution, and are to be determined so as to answer to the physical conditions which are to be satisfied. Thus the solution admits of infinite variety. If we suppose X = mx (noa. t − x) + m × (noa.t+x), or = 2mnoa. t, we have simply a uniform current through the tube, with equal velocity for all the particles. If X=-mx (noa. t − x) + m × (noa.t+x), or = 2mx, so that the original ordinate x is changed into x+X or x+2m, we have the air in a quiescent state, with the original intervals of its particles multiplied by 1+2m, denoting a uniformly increased or diminished. density throughout the tube, and implying that the ends of the tube are stopped. With second or higher powers, we should have movements produced by variable densities. But, for our Theory of Sound, we shall most frequently treat each of the functions in a general form. D 24. One term of the solution indicates a Wave travelling Forwards; the velocity is independent of the character of the wave. - So far as depends on the function (noa.t-x), whatever may be the form of 4, the following property holds. Suppose t increased by t; and consider the state (at that increased time) of a particle whose original ordinate was +noa.t. In the function, for t substitute t+t, and for a substitute x+noa. t. Then the value of X becomes (noa.t+noa. t' - x - noa. t'), or (noa.t-x); which is exactly the same value as that for the particle x at the time t. That is, if we consider the motion of a point whose quiescent ordinate was xor x+noa. t, we find that, at the end of the time t+t', its displacement is exactly the sanie as was the displacement of a point whose quiescent ordinate was x, at the end of the time t only. That is, if we increase the time, we may find certain particles in the same state of disturbance as the first particles at the first time, but we must go to a larger value of x in order to find these disturbed particles. This is exactly the characteristic of a wave. And since it appears that, upon increasing the time by t', we must go to a value of x increased by nea. t', it follows that the velocity of the wave is nea, or n0 × 916·2722 feet per second. It is important to observe that this result is entirely independent of the character of the wave. It may be |