EQUILIBRIUM OF AN ISOTROPIC ELASTIC ROD OF CIRCULAR SECTION. 945 



between the part of a solution which arises from the permanent source terras and 

 the part which arises from the transitory terms becomes still more marked. 



We suppose that the applied force per unit volume has the form F(x/a, yja, z), 

 the form of the function F being such that F(^, >;, z) is explicitly independent of a. 

 We also suppose that F vanishes unless z lies between a certain pair of finite values z^ 

 and Z2, where z^<Cz2- Seeing that the forms of the source solutions depend on whether 

 z is greater or less than z', the ^'-integration must, if Zi<iz<jz.2, be taken in two steps, 

 one integration being from z^ to z, and the other from z to z^. 



The cases when 2<%, 2>32 do not need much consideration. Obviously the part 

 of the solution arising from the transitory source terms is of infinite order in a, while 

 that arising from the permanent terms is of finite order. 



Suppose then that z-^^z^z^. 



When we multiply a function (a factor in a displacement or stress) of the form 

 {z — z'y^f[xla, yja ; x' ja, y' jo) by ¥{x'/a, y'/a, z') and integrate over the cross-section z', 

 the result is of order a^, and of the form, say, a^{z — zYfi{z'). Then the integration 

 of this as to 2' does not alter the order. 



But when we multiply e~ » '/'(•s/a, y/a ; x'/a, y'/a) by F{x'/a, y'/a, z'), and so 



obtain by integration over the cross-section a^e~ « f^i^'), then integration as to z 

 from % to z gives 



a^j^e-~V-f,(z')dz' (1) 



lfy^(0') is finite, this can be shown to be of order a^. 



For let l^=p + iq, where jp and q are real. Then, by the First Theorem of 

 the Mean, 



the real part of a'^ j e « f^{z')dz' 



= ■< a mean of a^ cos?^^ — —Ai'O \ ' I e~~«~ dz 



-{ " Ifi-- !■ ■ • ■ (^) 



which is of order a^. Similarly with the corresponding integral from 2 to z^. 



To extend this result from a single term of the /3-series to the whole series, we 

 may proceed as follows. The displacement, or stress, at 2 is partly due to the force 

 at points 2', where 2'<2, and partly to the force at 2', where 2'>2. Consider the 

 former part. For this, the displacement or stress at an internal point 2 is the limit 

 for e = of the displacement or stress at 2 -h e ; for the displacements and stresses are 

 continuous within the body even when the applied force is discontinuous. (This does 

 not hold for any higher derivatives of the displacements than the first.) 



Now 



2.6 -^ A¥)dz' 



,-, _) 3(Z+e-z' ) 



= 2j h " f2(^')dz', 

 8 J', 



