962 



DR JOHN DOUGALL ON AN ANALYTICAL THEOEY OF THE 



' IjzjjdS^O, (5) jjyzJ:dS = 0, . 



j j a.r.!/a + I lcr(//2 - ,,.2) - i- ra^ | zJ) + Eyu, dS = 0, 



'[jzzdS = 0, 



/ I {a:fzx + aryzij + Ew,)^7S = 0, 



\jjiy^-xzj))dS = 0, 



I j ix{yu^-.ru,^)dS = 0. .......... 



tZS = 0, 



• (6) 



■ (7) 



• (8) 



• (9) 



• (10) 



. (11) 



• (12) 



Exact evaluation of these twelve integrals in terms of the force applied in the body 

 and at the cylindrical surface, and of functions defining free permanent modes. 



It follows from the results (1) to (12) for the transitory modes, and from the 

 argument used towards the end of Art. 33, that the values of the integrals over a 

 cross-section in (l) to (12), as calculated for the particular solution which is derived 

 by integration from the source solutions, can be found by evaluating them for the 

 permanent terms of the source solutions and then integrating for the given distribution 

 of force. Then, by Art. 43, the further addition of their values as calculated from 

 the permanent modes of Art. 26 gives the exact values of the twelve integrals for any 

 deformation of a finite cylinder. 



The results in terms of the functions of z partially defined in Art. 26, and of those 

 explicitly defined in Art, 38, are as follows. We write (1) for the integral on the left 

 of (1), and so on. 



(I)=-l.Ea4|^(F + F_,), 

 (II)=_^^Ea*^'(F + F_2 + F_,), 



(III) = - ~^E«*;|(F + F_2 + r_i + F„), 



(IV) = + l7rEa4(F + F_2 + F_i + F„ + F,), 



(V)=-l.Ea^|3(G + G_,), 



(VI) = - l,rEa4|!(G Ar G,., + G..i), 

 4 az- 



(VII) = - l-7rEa*-'^(G + G_., + G_, + G„), 

 4 r/z 



(Vni)= + 1 ,rEa4(G + G_, + 6_i + Go + G^), 



(IX) = 7rK«2|(H + Ho), 

 (X) = 7rEfl2(H + H^-l-H,), 



