(v), from A to A+B, π D. sin (v-4)=2. versin (v – A). 2π B B In the second form, Newton has expressly remarked that the wave may be solitary. The functions just exhibited possess this property, that the value of X, beginning from 0 when v = A, becomes a real value, which increases, and again decreases, till it is again 0 when v=A+B; that is, the particle returns to its original place. But in some cases (as on the explosion of gunpowder) it is desirable to have a form of function which will shew that the particle, after undergoing the wave-disturbance, is left in a place more advanced than the original. Such functions as the following satisfy that condition, retaining also the other conditions of freedom from numerical discontinuity:— the value of the integral commencing from v A, and the function expressed by the integral being used for the value of X from v = A to v = A + B, after which the value of X is to be constant, and is to be that given by the definite integral from v = A to v = A + B. The precise form and extent of application of the functions must bo detormined from consideration of the initial circumstances. (See the Partial Differential Equations, Article 53, &c.) Suppose that the impulse which generates the wave is given to the particle where 20. For that particle, v, or nea.t-x, is noa. t. Therefore, knowing the displacement of that particle for a sufficient number of values of t or of nea.t, we can express it as a function (algebraical or merely numerical) of noa.t, so that X. (noa. t). Then, at every other point, X= p (noa. t-x), with the same form of p. The interpretation of this, subject to the conditions at the beginning of this Article, will best be given by examinations referring to two considera tions, = First, what is the state of all the particles at a certain time T? At that time, v=noa.т-x, or x noa.T-v; where v A, xnoa. T-4; where » -A+B, x=noa. T-A-B. Thus the conditions (beginning with the third) are these: For the particles where x is less than noa.T-A-B, there is no displacement. For the particles where x is greater than noa.T-A-B and less than noa. T-A, there is displacement. 1 For the particles where x is greater than noa. T-4, there is no displacement. Second, what is the movement of the one particlo whose original ordinate is ? For that particle, v=ndu.t- &, ort="+; when v A+ when v = A, t= πιθα ηθα Thus the conditions (beginning with the first) become These two exhibitions give a complete account in an intelligible form of the meaning of the discontinuous function. 32. Formation of the Equation when small quantities are not neglected, and approximate Solution. In Article 21, the investigation was completed by neglecting the powers of dX above the first. In the present Article, we shall treat of the solution when superior powers are taken into account. For the reasons mentioned in Article 26, we must confine ourselves to a single wave. The accurate solution is by no means easy. We would refer our readers to a paper by Mr Earnshaw, in the Philosophical Transactions, 1860, where the solution is exhibited by an elimination between two functions which cannot be effected in a general form. The process which we shall use here is the more cumbrous one of successive substitution. The equation obtained in Article 21, retaining all powers, but omitting for convenience the symbols n'ớ”, and remembering that gH= a', is To transform this into an equation in which the independent variables are u = at +x and v=at-x, we shall use the process in the Partial Differential Equations, Article 35. This makes We shall now proceed with the successive steps of solution. First step. Neglect all the terms on the right hand. 0; X (see Partial Differential Equations, Then ďX dv.du = Article 30)(v) +↓ (u). As we propose to consider only a wave travelling in the direction of x increasing, Article 24 of this treatise, we shall neglect the second function, and adopt X = 4 (v). Second step. Substitute the value (v) for X in the first term on the right hand, and we have, since |