934 



DR JOHN DOUGALL ON AN ANALYTICAL THEORY OF THE 



J / 2 ox , 3 2 , 1 o / , 3\ 2 1 1 C^^F 



l\ d^F 





d'¥ 



where F is a function of z only such that ^^^ = 0. 



dz 



(ii) Flexure in the plane zy. Sg, S^, Sg, Siq. 

 (iii) Extension. Sg, Sg. 



M^= - (TX 



az 



23=E 



M^=H, 



(iv) Torsion. Sg, Sjg- 





where 





'Wj, = j;IT, 



where — -„ = 0. 



dz^ 



27. The coefficients of the transitory modes in the solution for a single force 

 determined by Betti's Theorem, assuming the expansibility of the solution in permanent 

 and transitory modes of the form already found. 



Let a force (P, O, Z) act at {p, «', z'). 



We assume the existence of a solution in terms of permanent and transitory modes. 

 We assume also that only such of the transitory modes occur as vanish at infinity at 

 either side of the source, and we are to determine their coefficients. 



On the side 2>2', let the transitory mode 



n^' ' = ' , ' e aj IX . Tow (1) 



occur with the coefficient M. 



We take 



(A', B', C) = (Ai , B, , Ci)Po + (A2 , B2 , 02)^0 + (A3 , B3, C3)Z„ , . . . (2) 

 as at (8) and (9) of Art. 3. 



In the investigation we make use of the transitory mode of similar type to (1), hut 

 with — /3 instead of /3 ; say this is 



(3) 



>A 



A", B",\ ^, Bp cos 



;<// eaj '— ^ • TOO), 



where A", B", C" are proportional to A', B', C. 

 We write the displacements of (1) in the form 



Up =e a Up'(/3) cos moi, 

 uj = e a Ua,'(p) sin tooj, 

 u^ = e a \Jz'{p) COS moi, 



(4) 



