1911] CHLADNI'S FIGURES FOR A SQUARE PLATE 49 in which q is Young's modulus, and 2h the thickness of the plate (§ 214). Also for the kinetic energy per unit area we have T=phw~, .....................................(2) p being the volume-density. From the symmetries of the case w must be an odd function of x and an odd function of y, and it must also be symmetrical between x and y. Thus we may take w = q^xy + q2xy (a? 4- y2) + qsxy (& + 2/4) + q^y3 +..........(3) In the actual calculation only the two first terms will be employed. Expressions (1) and (2) are to be integrated over the square; but it will suffice to include only the first quadrant, so that if we take the side of the square as equal to 2, the limits for x and y are 0 and 1. We find ...........................w drw d ^e2 d Thus, if we set ^ = 57-, -—: V', ...........................(6) 6(1 + /ju) • / we have V^^tf + 2giga + $ql + _" ......................(7) In like manner, if rp _ *P'1 rnt /ON 2 —g"^ ' .................................W When we neglect q2 and suppose that ql varies as cosj9i5, bhese expressions give ™^ 7,>.......................(10) if we introduce a as the length of the side of the square. This is the value found in Theory of Sound, § 228, equivalent to Ritz's first approximation. In proceeding to a second approximation we may omit the factors already accounted for in (10). Expressions (7), (9) are of the standard form if we take A =1, = 2, n — _ 6 18 R. VI.n Theory of Sound, "Vol. II. Appendix 1 I had supposed that this treatise abounded in applications of the method i question, see §§ 88, 89, 90, 91, 182, 209, 210, 265 ; but perhaps the mo; explicit formulation of it is in a more recent paper J, where it takes almos exactly the shape employed by Kitz. From the title it will be seen th* I hardly expected the method to be so successful as Ritz made it in the cas of higher modes of vibration.