EQUILIBRIUM OF AN ISOTROPIC ELASTIC ROD OF CIRCULAR SECTION. 903 



(2) 



where N, T, L are given functions, which we shall suppose to vanish at all points out- 

 side a certain finite portion of the cylinder lying between two cross-sections z = Zy and 

 z = 22) where z^z^. 



The functions N, T, L are expressed in forms suitable for analytical treatment by 

 means of Fourier's series and integral. 



Take the function N, for example. 



We have 



N((0, z) = :7- I N"(a) , Z)(h,i + — / N(a)', z) COS (o) - w)<lw + 



zttJo it Jo 



= - ^ / N((j)', ?j) cos m{(i> - (o')ch}', 



IT m JO 



to use a well-known and useful notation. "* 

 Also 



N(<,)', z)^l[ (Ik rN{m', z) COS k{z - z')dz. 



By combination of (2) and (3), 



N(w, z) = -- ^ I '/o) I ifK I N((o', z') COS k(z - z') cos 7n(o} - <ji)ilz'. 



Tt'- m Jl) Jo Jz, 



The expansion (4) suggests that we may be able to deduce solutions for the three 

 general problems of (1) from solutions of the three simplified problems in which 

 N, T, L in (I) are replaced by cos ^(z — 2') cos m(w — w'). 



5. Normal traction. Simplest case. Any distribution over a finite area. Unit 

 element. 



For the simplified problem of normal traction, assume 



A, B,\t • / '\ cos / '\ /i\ 



,) „,iKpcos k(z - z) . m(oi-u>). . . . .• . (1) 



'" ' ^ ' sin ^ ' ' 



(3) 



(4) 



iff J^\ C 

 These values of <p, 0, \f^ give at /o = a, from 2 (8), 



pp^ = I A(2 - VK%2J - 2iK%3j') + B( _ 2K^a\V') + C( - 2m){3 - iKaJ') } cos k{z - z) cos m(o> - 0/), 



.a^ 



0(0 — = {AjHK^ay 



+ E2m(.J - tKaJ'} + C(K-a-^J" + iKaJ' - vi^J)} cos k(z - z) sin m{w - oj'), 



r (2) 



„ffl 



i^— = { A(2K2rt2j + „^va.l' + m2J) + B( - 2?V«J') + C( - mJ) 



JU.K 



I sin k[z - z) cos »«(oj - w'), 



where the argument of the functions J, J', and J'' is iica. 



It will be convenient to take advantage of the notations introduced ip Art. 3 by 

 writing /3 instead of ixa in certain of the terms of (2). Thus (2) become 



pp— = (Aftj + B&i + Ccj) cos k{z - z) cos ?k(o) - w'), 



(Afflj + Bi2 + CC2) cos kI'z - z) sin 77?(o) - 0/), 



pz — = (Artg + B&3 + Ccg) sin k(z - z') cos to(w - w). 



/XK 



(3) 



* Anothernotation, also well known, is sometimes convenient, viz. we define €„, as equal to 1 for m>0, but 

 equal to | for m = 0. Thus 2''"m= 2 em"m = 5^0 + "1 + '"2 + ■ ■ ■ ■ nd co . 



TRANS. ROY. SOC. EDIN., VOL. XLIX. PART IV. (NO. 17). 124 



