932 



DR JOHN DOUGALL ON AN ANALYTICAL THEORY OF THE 



after (1); the one deformation being due to a unit force at P, the other to a unit 

 force at Q. 



Apply Betti's Theorem to these two deformations over a part of the cyhnder 

 bounded by two sections enclosing P and Q between them. Owing to the symmetry, 

 the work differences at the two ends neutralise each other, as in Art. 20, and the 

 result is simply that the displacement at Q, in the direction of the force there, due to 

 the force at P is equal to the displacement at P, in the direction of the force there, due 

 to the force at Q. 



Now Art. 23 gives the displacements at {x, y, z) due to a force at {x', y', z'), where 

 2>2' ; the displacements due to a force at {x',y',z') due to a force at {x,y,z) are 

 obtained from these by interchanging accented and non -accented letters, and then 

 changing all the signs. 



Hence, in the table, the coefficient of X in u^ ought to change sign merely when 

 we interchange accents and non-accents ; similarly with the coefficients of Y in Uy, and 

 of Z in u^. Also the coefficient of X in Uy ought with the same interchange to become 

 the coefficient of Y in u^ with sign changed ; similarly with coefficient of X in u^, 

 coefficient of Z in u^, and with coefficient of Y in v^, coefficient of Z in Uy. 



These properties are easily verified by a glance over the table. 



It may be remarked that these properties are almost, but not quite, sufficient to 

 determine the coefficients of the rigid body displacements after those of the six principal 

 modes have been determined from elementary statical considerations. 



25. A more compact form of the table of permanent terms in the solution for a 

 force (X, Y, Z) at (x', y', z'). 



It is obvious from symmetry that we can deduce the solution for the force (X, Y, Z) 

 at {x', y , z) from the solution for the same force at {x' , y' , 0) by writing z — z' instead 

 of z in the displacements. 



This allows the results of Art. 23 to be put in a more compact form ; namely, in 

 these results we put z' = 0, and then write z — z' instead of z. We thus find, for z'>z', 



^^= y(2 - ^y + {^ - ^')H-'^'' - y') - ra-"], 



u^=2a-{z-z')xy, 



u^= - (z- z')^x + — x{x^ + y^) ~ {'^ '^ IT ~ ''' )^^-^'' 



u,: = 2<r(z - z')xy, 



u, = 1(. - 2')3 +{z- z'){cr(!/^ - r/:2) _ ra^}^ 



u,= -{z- zfy + -^ ?/(x2 + 2/2) - (^o- + — - T V^//, 



X 



•Ea* 



rEa-i 



M, = 



- o-./-, 



U,, = 



- o-.'/, 



w, = ; 



'' ~ ''''' _ 



z - 



27rEa2, 



