1912] THROUGH A STRATIFIED MEDIUM, ETC. 83 As an example, the case where kl = n2/x- may be referred to. Here fkdx — n logx—e, and (67) gives s(pt — nlogas + e) ...................... (70) as an approximate solution. We shall see presently that a slight change makes it accurate. Ke verting to (64), we recognize that the first and second terms are real, while the third is imaginary. The satisfaction of the equation requires therefore that a*v = G, .................................. (71) and that #» = Cy* - - ?? 5 ............................ (72) ?? dor while (63) becomes £ = r, e ~ ic*h~*dx ................................. (73) Let us examine in what cases 77 may take the form Da?. From (72), 7c2=a4D-4ar-4r-r(r-l)tfr2 ....................... (74) If r = 0, kz is constant. If r = I, Jc- = 0'iD~isD~i, already considered in (68). The only other case in which k* is a simple power of x occurs when r = ^, making k* = (C*D-+ + $}w-2 = n*x-*(s&y) ................... (75) Here 77 = Dx^ , C2 t]~n- dx = 02/D2 .logos — e, and the realized form of (73) is (f> = D^cos{pt-^/(n2~&logx + e}, ............... (76) which is the exact form of the solution obtained by approximate methods in (70). For a discussion of (76) reference may be made to Theory of Sound, second edition, § 148 6. The relation between a and 77 in (71) is the expression of the energy condition, as appears readily if we consider the application to waves along a stretched string. From (53), with restoration of eipt, d(f) 'int. -i\adx . dd> ipt = • = * If the common phase factors be omitted, the parts of 'd<j>/dt and which are in the same phase are as py and arj, and thus the mean work transmitted at any place is as aif. Since there is no accumulation of energy between two places, atf must be constant. When the changes are gradual enough, a may be identified with k, and then i) oc & , as represented in (67). If we regard 77 as a given function of x, a follows when C has been chosen, and also 7c3 from (72). In the case of perpendicular propagation &a cannot be negative, but this is the only restriction. When ?? is constant, 7c2 is constant ; 5—2 / )