218 FURTHER APPLICATIONS OF BESSEI/S FUNCTIONS OF [380 Similarly, if we write ka = a'+ iy', where at = X/JJL, y' = y/fi, Dn (x1 + iy'} Dn W + iy D»' (<0 ' and in virtue of (10) 2)n" (aT) = - ^£ Dn' (aO + sinh2 /3 Dn (a/\ Tl> where cosh /3 = n/so'. Thus D«tf+W) _ A/ 60 f , v / COsh £ * l+W- ~ + smh Dn' D Accordingly with use of (36) . ., , . , „ Q n Dnf = - smh £ n + » 02t + iw' I --- - + smh2 ^ /r? ^ n , / , • /v = - --- - /r? - T Dn(oc +iy'} ^( J \ n Dn D Equation (32) asserts the equality of the expressions on the two sides of (38) with (37) If we neglect the imaginary terms in (38)3 (37), we fall back on (34). The imaginary terms themselves give a second equation. In forming this we notice that the terms in y' vanish in comparison with that in y. For in the coefficient of y' the first part, viz. — n~l cosh (3, vanishes when n is made infinite, while the second and third parts compensate one another in virtue of (33). Accordingly (32) gives with regard to (34) ~pksmhp~ p sinh/3 ' '. in which cosh/3 = ,u ............................... (40) In (39) iy is the imaginary increment of ha, of which the principal real part is n. In the time-factor eikvt, the exponent ., rrj_ ihaVt inVt L i (39)) ikVt = - = - il + — — -k pa fj^a ( n ) In one complete period r, nVt/fM undergoes the increment 2?r. The exponential factor giving the decrement in one period is thus e~*«wn} ................................. (41) or with regard to the smalmess of (39) 27T//0- gr-anfl-tonhfl •!•""" '10 • ....... • ................ (.-H^1) np sinhp This is the factor by which the amplitude is reduced after each complete period..... (30)