. A treatise on the mathematical theory of elasticity . ™ This result is the same as would be found by adding the expressions for e^, e/3/3, e^yin (.36). * This method is due to Lord Kelvin. (Sir W. Thomeon, Math, and Phys. Papers, Vol. 1,p. 25. The date of the investigation is 1843.) 20, 21] CURVILINEAR COORDINATES 55 .f I A T ^^P^^^^^ ^^ 2^v we form the line integral of the tangential componentot the displacement along the edge of the element in the face y + dy. The contributions ofthe tour portions of the edge can be written down by the help of Fig. 3 as follows •— contribution oi RP = aa ,
. A treatise on the mathematical theory of elasticity . ™ This result is the same as would be found by adding the expressions for e^, e/3/3, e^yin (.36). * This method is due to Lord Kelvin. (Sir W. Thomeon, Math, and Phys. Papers, Vol. 1,p. 25. The date of the investigation is 1843.) 20, 21] CURVILINEAR COORDINATES 55 .f I A T ^^P^^^^^ ^^ 2^v we form the line integral of the tangential componentot the displacement along the edge of the element in the face y + dy. The contributions ofthe tour portions of the edge can be written down by the help of Fig. 3 as follows •— contribution oi RP = aa ,, „ R(^= da „ QR= dff t>T>i .. ^/3 dd U^-r-+dl 3 / dei\. Fig. 3. On adding these contributions, we obtain ^(1)}- This must be the same as issydad^ jk^h^, and we have thus an expression for xsy whichis given in the third of equations (38); the other equations of this set can be obtained inthe same way. The formulse* are --?MK?)-!©)) 2cr^=A3Ai \dy\hj da\hs)r 2tsv -k,h, Jg^ 3/3 ( ?, .(38) * The formulsB (38), as also (36) and (37), are due to Lam^. The method here used to obtain(38), and used also in a slightly more analytical form by CesSro, Introduzione alia teoria matematicadella Elasticita (Turin, 1894), p. 193, is familiar in Eleotrodynamios. Cf. ,PW2. Soc. vol. 178 (1888), p. 150, or J. J. Thomson, Recent Researches in Electricity and Magnetism,Oxford, 1893, p. 367. The underlying physical notion is, of course, identical with the relationof circulation to vortex strength brought to light in Lord Kelvins memoir On Vortex Motion,Edinburgh, Soy. Soc. Trans., vol. 25 (1869). 56 CUBVILINEAR COORDINATES [CH. I 22. Cyl
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