938 



DR JOHN DOUGALL ON AN ANALYTICAL THEORY OF THE 



The solutions for an internal force given in this and the preceding Article must of 

 course be equivalent to those of Arts. 18 and 19. From the point of view of physical 

 interpretation there can be no question as to which of the forms is to be preferred. 

 For such analytical manipulations, however, as are sometimes required in applications 

 to problems of continuous distribution of force, the original forms are more useful, 

 as the functions of /S involved in these can be expressed as the residues at the poles 

 of D,„, of functions of /3 which have no other poles. {Cf. Art. 34.) 



It may be observed that the forms now given are rigorously proved. The method 

 of Art. 27 assumes indeed the expansibility of the solutions in terms of the permanent 

 and transitory modes, but this expansibility was proved in the method given earlier. 

 The comparison of the two forms for the various cases leads to a variety of relations 

 among the coefficients A,^_,„„ B^^,„, . . . , A^, Bj, . . . , J(/3), J'(/S), G(/3), and G'(/5). 

 Some of these relations depend on I)„i being zero, others not, and their independent 

 proof is by no means immediate. I have verified a few of them by direct algebraical 

 work, with the help of the relation 



G^;8J,;/3-GJy8J,„/3 = l//3. 



Among the relations mentioned the following are the simplest. They are 

 independent of D,„=0, and will be found to be the necessary and sufficient conditions 

 that the displacements calculated from the original forms should possess the reciprocal 

 properties proved in Art. 24. 



mAe 



vAd 



(Arts. 3 and 16.; 



e^m^ >'.ti>/(, m+ ^e, m = U, 

 A0, m + vA^^ „i - ?«A^, m - Bfl, TO = 0, 

 inMe^ m - vi^^, m + C(|,, m + vCg, ni " mC^^ ,„ = 0. 



(5) 



D„ = 



30. The zeroes of the function Dto(/3). 



If, as in Art. 3, we write J and J' for brevity instead of J^/^ and J„/|S, we liave 

 (v - 2)/32J + 2/33,J', (2^2 _ 2/32)J - 2j8J', %np^' - 2?wJ 



- 7H/32J, - 2n?/3J' + 27kJ, {(r- - 2«i2).T + 2^'J' 



{m? - 2/32) J + v^J', - 2/? J', - ?kJ 



= J3{4^6 - 12m2^4 + (i2?«2 _ 2v - 4)m2/32 + im^{l - m'^)}\ 

 + J2J'{8/5''> + (4./ - 8)m2/33 + 4,'Hi2(i _ „j2)^} 



+ JJ'2{4y8'5 - (8^2 + 2v)/i-' + 4?h2(to2 _ 1)^2} 



+ J'3{8/35 + 4v(m2-l/3'3}. 



Put 



Sq = — _ — - and a = /3 - 80 



(1) 



(2) 



■ The earlier terms of the asymptotic expansions of J and J' are, when the real part 

 of ^ is positive, 



,J,;/j = / "^ { ros a - \ f m2 + AVi-1 sin « - A (^,„,2 _ 1Y,,,2 + 15\ , cos a I . 



\ nfi [ 2 \ 4 / b \ 4 / \ 4 / ) 



(3) 



