EQUILIBRIUM OF AN ISOTROPIC ELASTIC ROD OF CIRCULAR SECTION. 953 



39. Solution f 07' a distribution of body force — the 'part arising from the transitory 

 source solutions. 



We return now to Art. 37 in order to consider the part of the solution for any distri- 

 bution of force which arises from the transitory- terms in the solutions for a single force. 



We shall suppose not only that the applied force vanishes outside the interval 

 %<z<2;2, but also for the most part that within that interval each component of the 

 force is a continuous function of z possessing a certain (perhaps unlimited) number 

 of continuous z-derivatives. The results obtained can be extended at once to the 

 case where the interval z-^ to z^ can be divided into any finite number of intervals 

 fulfilling this condition. No restriction is placed upon the character of the applied 

 force as depending on x and y, except the condition of integrability. 



For 2;>22 or 2<2i, that is, outside the region of applied force, the matter is simple. 

 The solution obtained by such integrations as those of (6), (7), (9) of Art. 37 is obviously 

 compounded of permanent and transitory free modes, for they are obtained from such by 

 integrations the variables and limits of which do not depend on the current coordinates. 



Within the region of applied force, however, this is no longer true, for the limits of 

 integration in 37 (8) depend on z, and the character of the solution is completely altered. 



We wish to consider this part of the solution with reference more especially to 

 two questions, 



(i) the question of the expansibility of the solution in terms of ascending 



order in a ; and 

 (ii) the question of the behaviour of the solution at the critical sections 2; = % 

 and z = Z2, where the applied forces or some of their z-derivatives are dis- 

 continuous functions of z. 



Throughout we are considering separately the m-component of the solution, that is 

 to say, that part of the general solution the displacements of which involve cos ww or 

 sin mw as factors. Of course any transformations efiected on w,„ do not aff"ect the 

 convergence of the w-series ^'u,,^. 



As in Art. 37 so in the following transformations we contemplate, to fix ideas, the 

 displacement u^ due to a distribution of internal P (radial) force. The case of any 

 other displacement or force, or of surface traction, may be treated in the same way. 



Consider the second line of 37 (8), say, 



Repeated integration by parts gives, if F{z) be put for P{x', y', z), 



I 



P. " ■'-fl'W-|.iI'(^)+- •■•+( -')■-'!; ^W 





(2) 



