the plane xy, and therefore requiring that Z always = () when z = 0." Then our simple solution cannot be made to meet the condition; and we must introduce what is at present an 'undetermined function;' and we must determine it so that, when combined with the simple solution, it shall make Z=0 when z = 0. the external sign being derived, in the investigation of Article 39, from the sign of z within the bracket; and since the value of Z for z = 0 becomes it is quickly seen that, for the destruction of this term at all times and with every value of x and y, & must be the same in the new term; the factors of t, x, and y, must be the same; the external factor must be the same but with different sign, and therefore the factor of z, within the bracket for the general value of Z, must be the same but with different sign; and therefore we must add exactly the term added in Article 40. That term represents a wave precisely similar to the first wave, in law and extent of motion of particles, in velocity, and in inclination of its normal to x, y, and z; but differing in this respect, that the inclination of its normal to z is on the opposite side. And therefore, defining the direction of the wave's motion by that normal, the motion of the reflected wave and the motion of the incident wave make equal angles, but on opposite sides, with the normal to the immoveable barrier xy. This is in all respects the character of an Echo of Sound. 42. Various forms permissible in the expression defining a plane wave of air. If, for simplicity of symbols, we suppose the plane of the wave to be parallel to the plane xy, and its motion to be in the direction z, it is easily seen that the provided that K be a function not containing t or z, which satisfies the equation ďK d'K This troublesome equation (see the author's Partial Differential Equations, Articles 41 and 52) scarcely admits of intelligible solution except under specific assumptions. We may make K= 0.xy+D, or K = C(x2-y)+D or K="+", cos (ny + C') + D, or K= C.log(x2 + y) + D, &c., and thus we obtain, as solutions expressing a plane wave of air, F= {C.xy+D}. & (at − z) ; F = {C (x2 — y3) + D} . $ (at − z) ; F = {e"+". cos (ny + C') + D}. & (at − z); F= {C.log (x2 + y2) + D}. $ (at − z); &c.: and any combination of these with similar terms, or with terms depending on y (at + z). On performing the operations of Article 38, it will be seen that these forms imply motion of the particles in the three directions x, y, z, though the motion of the wave is only in z. It does not appear that these forms have any application in nature. 43. Remarks on the Partial Differential Equations which occur in the investigations that next follow. with the six values for m, 1, 2, 3, 4, 5, 6. It does not appear that equations of this class can be approached by one general method of attack. Some of them yield to the following. Assume, for trial, W=Σ (4 ̧.r"); A being in all cases a function of v or at-r, which satis1 A ďA fies the equation de ; and the index n Σ {− (2n + m) . A'„ . 7′′-1 + n (m + n − 1) . A„ . r"~} = 0, (the accent on A, denoting the derived function of A with regard to v); and, writing down successive terms instead of the symbol Σ, {− (2n+m). A′n} . gothand + {− (2n + m − 2). A'„+ n (m + n − 1). A„}. p”—2 + {− (2n+m−4). A',,+(n−1)(m+n−2). An_1}.7"~ + &c. >=0. [If we suppose A a function of u or at +r, the equation obtained is the same, excepting that the signs of the derived functions are changed]. Making each line =0, we find n= m • 2 and we find each following function in terms of the preceding func tion. And the series of functions will terminate in two cases. Either if one of the numbers n, n-1, n − 2, &c. becomes 0; that is, if becomes 0; that is, if m=0, -2, -4, &c. Or if one of the numbers m+n−1, m+ n − 2, &c., becomes 0; m m that is, if -1, or 2-2, or &c., becomes 0; that is, if 2 m = 2, or = 4, &c. The only numbers here which meet our wants are those for m = 2, m = 4, m = 6; and we are still left without solutions for m=1, m = 3, m = 5. The solution for m = 1 has, however, been found, as we shall mention, in the unsatisfactory form of a definite integral; and the solutions, when m=3, m = 5, &c., may be made to depend on that when m=1. We invite the attention of the student of Partial Differential Equations to these equations. For m=1, m = 3, m = 5, we shall obtain infinite series in descending powers of r, which are practically sufficient for waves diverging to great distances. For waves nearer to the centre, we may assume increasing the index by successive units. Treating the series in the same way, we find n = 1 —m, and |