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By William Elwood Byerly

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Xx cos 2 ct. 6) 2 2 X~, = (-Xx + Yy)sinexcosex + Xy{cos ex-sin a). After obvious transformations these formulae become Xx' == I I Y y' , = X x +Yll 2 X +Y 1J Xy' = - X 2 +X - X - Y lI 2 X -Y X 2 Xx-Y y • SIn 2ex 2 1I cos 2ex + Xv SIn 2~, cos 20':: - + Xy cos 2rx. 7) CHAP. 8) + 2iXy)e2ia.. The first of these formulae has been known for a long time and it was proved above for the more general case [cf. 2)J. The second,very important and convenient formula was stated by]. H. Michell [3J and it was found independently by G.

By a known formula the cosine of the angle between two vectors (o~v 1)1 is given by 1t') ( cos :2 = EI/Z + 0111' a~1) and (a~2' (1)2' ~2 + a~2) + (1)1 + 01)1)C'lj2 + ~~1(~2 + O~2) vo~~ + ("1]1 + 0"1]1)2 + o~i . vo~~ + O"1]~ + (~2 + O~2)2' a~1a~2 CHAP. AL YSIS OF STRAIN But when E yZ is small, ~ cos ( <:1/z) = - <:1IZ' neglecting infinitely small higher order terms. 15) Thus the quantity 2e llz represents the decrease of the angle between two vectors having originally the (positive) directions of the axes Oy and Oz.

Ox oy 8z v Remember now that this equation must hold true for any region V in the body. This can only be so, however, if the function under the integral signs IS zero at each point of the body. 1 ) 8Z z -ox + "+ + Z == 0. 8y OZ These equations, to which reference will often be made, will be called equilibrium equations. 1) is based on the follo\ving reasoning. If F(x, Y z) is a function continuous in a given region and I fff F(x, y, z) dV = 0 V for any part V, contained in that region, then F(x, y, z) ::= 0 in the entire region.

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