936 DR JOHN DOUGALL ON AN ANALYTICAL THEORY OF THE 



If wi == 0, the result requires a slight modification. 



It may be remarked that, if the solutions for a unit element of surface traction be 

 assumed, the value of M found in (11) may be deduced more easily. 



Thus, for a linear distribution of force on the circle z = Zo, p = a, of intensity 

 (PqCOswco, l^osinmo), —^Zq cos mco) per unit of «, the solution is, for z<Zo (Arts. 6, 



+ certain permanent terms. 



Apply Betti's Theorem to this for one system, and to the solution for the force 

 (P, f2, Z) at (jo', w', 2'), assumed to be expressible in terms of the permanent and 

 transitory modes, for the other system, and M will be found without difficulty. 



For the purpose of comparison of the result with (11), note that 



Up':U„':U; = Up" : Uo." : - U/'. 



It is also worth noting that, since Pq, Oq, Z^ in (l), (2) are arbitrary, the values of 

 cf), 0, and \// in (1) cannot depend on their relative values. Hence 



Up"(a) : VJ'{a) : jSU;'(«) = Aj : A2 : AA 



= B,:B,:bA (13) 



= Ci : Cg : Cg j 



These relations are easy to verify by Betti's Theorem and the solutions for surface 

 traction. {Cf. Art. 24.) 



28. Simplified expressions for the transitory modes for an internal force. 



The results of the preceding article can be stated in a neater and clearer form if we 

 consider separately the cases of a P, O, and Z force, and vary the choice of Po, 12o) and 

 Zo to suit the particular case. 



Take Up, U„, U^ proportional to the functions of p which occur as factors in the 

 displacements of a permanent mode. The ratio Up(/>') : Up(a) is perfectly definite, 

 whether the mode belong to the type (4) or the type (5) of Art. 27. 



The following are the transitory modes of the m, ^ type which occur, with their 

 coefficients. The complete transitory part of each solution is obtained by taking 

 2'2^ of these. 



Ic* force at (//, «', z'). 

 Take Po=l, ^0 = 0, Zo = 0. 



for 2>2' ; for z <^z', interchange z and z'. 

 Q. force at {p, of z'). 

 TakePo=0, fio=l, Zo = 0. 



r<^. 0^\ ^ _ ^ IUp)(A., ,„ , B, ,„ ,\/''-f ( - sin) _ ^ .3. 



W; ^i^dDjciiiVMy c;,„„ / COS "*^" ->J'"„'- • ^'' 



for 2> z' ; for z<z' , interchange z and ;/. 



