EQUILIBRIUM OF AN ISOTROPIC ELASTIC ROD OF CIRCULAR SECTION. 965 



{(I) (XII) is the only one of the twelve integrals involving displacements only. 



Hence 11 is the only one of the four functions which can be determined in this 



way when it is the displacements that are given at one or both of the ends. 



Jn the general case when the given displacements at an end are arbitrary, there 



seems to be no simple way of determining the exact conditions satisfied at that end by 



the functions, other than 11, which define the permanent modes. But in the case, the 



only one of practical importance, when the given end displacements are iiull, the end 



conditions to which F, G, H are subject can be found approximately. 



We will show that, at a fixed end. 



the two lowest terras of F + F_2 + F,-, = 01 



and the lowest term of —-(F + F^,) = 0. 



az " J 



Let 



(1) 



(2) 



u. 



be the displacements of a transitory mode of the zaj-flexural type, so that u^, u, 

 contain cos co, sin «, cos w respectively as factors. 



Obviously, when the displacements at an end vanish, the displacements due to such 

 modes, taken along with those due to the mode F, and with those of similar type in 

 the exact particular solution defined approximately in Art. 38 (9), (12), must vanish 

 independently. 



We prove in the first place that if the constants A, A', A^ are such that the three 

 equations 



A + 2A^Ua =0, 



2A^V^ =0, 



A'.c/a + 2A^W^ = 0, 



hold, then will 



A = 0, A' = 0, A^ = 0. . . . . 

 Consider the semi-infinite cylinder, z = to z= ao , and the displacements 



u^= - A'z/a + A + 2e « A^gU^ , 



(3) 



(4) 



^e'aApVp, ' 



ti, = A'.r/a 



_pz 



+ 2.6 aA,3W^. 



(5) 



By (3), the displacements of this Iree deformation vanish at 2 = ; and the stresses 

 at a section where z is very great are very small. This being true for the real and 

 imaginary parts of (5) separately, the potential energy of each of these parts for the 

 semi-infinite cylinder vanishes. Hence (5) can only represent a rigid body displace- 

 ment, which must be null, since the end 2 = is fixed. Thus (4) follows. 



Now the condition that the displacements vanish at any given section leads 

 (assuming for brevity that the series of transitory modes converges there), by equating 



