24 ON A PHYSICAL INTERPEETATION OF [352 We conclude that " F1(Q), ............ (9; o which may be considered as expressing F in terms of/. If in (6), (9) we take F(R) = cos R, we find* f """ Jo Differentiating (9), we get " * Jo (R sin 6) sin 0 d8 = R~l sin R. e get F(R) = f "f(R sin 0) sin 6 d6 + R \ * f (R sin 0) (1 - cos2 6) d0. . . .(10) Now f ^cos2 6 f (R sin 6)d6={ cos 0 . d/(£ sin 6} Jo' J = -/(o) + f "/(-& sin 0) sin Accordingly JF (E) = /(O) + R I * f (RamB^dff ................ (11) Jo That f(r) in (1) may be arbitrary from 0 to TT is now evident. By (3) and (6) 2 f^77 f(r) = - dcfr {b0 + b,. cos (r cos <j£>) + b2 cos (2r cos c/>) + . . . j 771 J o where &„ = JP (£) d£ 6» = - f'cos^^)^ .......... (13) " Further, with use of (11) * * ............ (14) ............ (15) by which the coefficients in (12) are completely expressed when / is given between 0 and TT. The physical interpretation of Schlomilch's theorem in respect of two-dimensional aerial vibrations is as follows : — Within the cylinder r = TT it is possible by suitable movements at the boundary to maintain a symmetrical motion which shall be strictly periodic in period 27r, and which at times t — 0, t = 2-7T, &c. (when there is no velocity), shall give a condensation which * Enc. Brit. Art. "Wave Theory," 1888; Scientific Papers, Vol. m. p. 98. becomes a function only of R, the radius vector in space. If 6 be the angle between R and one direction of f, £ = R cos <9, and we obtain, as the mean