EQUILIBRIUM OF AN ISOTROPIC ELASTIC ROD OF CIRCULAR SECTION. 935 

 and those of (3) in the form 



Mp" = eo Up"(p) cos vio), 



. (5) 



Up" = eo Up"(p) cos mo), 



uj' = ea \JJ'(p) sill ?HOJ, 

 U^" = e a U/'(p) COS ?)20J, 



For an application of Betti's Theorem, take as first system the solution for the 

 force (P, O, Z) at [p, w', z'), and as second system the mode (3), and apply the theorem 

 to the part of the solid enclosed between the planes z^z^ and 2 = z.2, where 2i<z'<2;2. 



At z = Zi the work difl:erence vanishes, for it is the same as at z= — 00; at 2 = ^2 the 

 work difference reduces to that of (1) . M and (3), or of (4) . M and (5). 



Hence 



M ■ work difference of (4) and (5) = Viip\p\ to', 2') + QuJ'(p', w, z) + Zuz"{p', w', z'), . . (6) 

 which allows M to be calculated. 



The direct calculation of the work difference of (1) and (3) by integration over a 

 cross-section would be very complicated, but by an artifice we can express it as an 

 integral over the contour of a section, easy to evaluate. 



Apply Betti's Theorem between 2 = and 2 = 2 to (5) and —^ (4). 

 Thus 



fj{u;p^..^..-^pfu;-..-..yB 





■ (7) 



(3z 



If we write for a moment zp = e'^Z^' cos wiw, then in the first integral of (7) 



Wp ^-p = Up {p) COS vjiwi — Zp - -Zp j cos ?HW, 



so that the contribution of this term to the difference of the two integrals over the 

 sections 2 = and 2 = 2 is 



- - \ \ Up"(p) cos miaZp COS mmd^, 

 i.e. --{[u;'^'(lS. 



Hence the first line of (7) = - — . work difference of (5) and (4). ...... (8) 



As for the second line of (7), we have by 3 (10) 



From (7), (8), and (9), 



Work difference of (5) and (4) = Trp.dD/dl3{F^Vp"{a) + n^,VJ'{a) + ^Z^VJ'{a)}. . . (10) 

 From (6) and (10), 



■^^ _ 'Pup"{p', w, z ) + CluJ'{p', oi', z) + Zv z\p', 0)', 2' ) /^^^ 



7r/xcZD„,/(;/3{PoUp"(a) + OoU<«"(«) + /3ZoW'(«)}' " ' " 

 a simple expression for the required coefficient. 



TRANS. ROY. SOC. EDIN., VOL. XLIX. PART IV. (NO. 17). 128 



