91G 



DR JOHN DOUGALL ON AN ANALYTICAL THEORY OF THE 



The sources {(p. 6, \|/) = Df-7'"^ give certain stresses at p = c(. To get solutions for 

 these sources we add to the sources solutions free from singularity within the cylinder 

 and giving at the surface tliese stresses reversed, or, as we may say, we balance the 

 sources at p = a. And the method of solution will be to balapce separately the 

 components into which the sources are analysed by the expansion (3). 



16. Solution for the auxiliary sources 



{ct>, e, ^) = 



4\ ( , ,., . ,,, . 1 



^ cos k(2 - Z )'),jKp (T,„tKp - -- 



) t —'Ik . COS m((D - w), 

 m\p/ ) K- 



(1) 



'>p ; 



that is to say, the "^n-components, (m>0), of the sources {(p, 0, \|/)--=D/V \ 



The source 



^ = G,„iKp cos k{z - z) cos m(oD - o)') . . . . . . (2) 



can be balanced by 



(Y')=-(-'*c;,!r">>'>«»'(-^') .">("-"') p) 



In the notation explained in Arts. 3 and 5 the equations for the coefficients in (3) are 



and their solution is 



or, say, 



A^, TO = (ajAj + a2A2 + a3A3)/D„i ,' 

 B^, m = (ttil!] + ttgBg + ajlSg)/!)™ , 



It follows that the strain defined, for p^p , by 



4>= - ~\ {Gmixp - A,|,_ m-^miKp)-Jm'iKp' COS /((s - 2')— COS ?h((u - w') L 



(4) 



(5) 

 (6) 



4f 

 ^1 



P',/,, wUn^Kp'^miKp COS k(2 - z')— COS ??^(oJ - o)') > 



l/^ = - - i - C^, m-TmiKp-'iwiKp' COS k(2 - z') — sill ?H.(a) - (//) I , 



TT { K- J 



(7) 



gives null traction at jO = a. 



The functions of (7) are uniform, even functions of «:. 



The function Gc.Jxp contains a non-uniform term — log kJ^^Vjo, but it can be seen in 

 a moment that A^^ ,„ has the non-uniform term — log k, so that 



GmiKp - A^,, m^mixp (and similarly B,(,, m and C^, ,„) 

 are uniform. 



The functions (7), when expanded in ascending powers of «-, contain terms of 

 negative degree, say 



TT I 2VI\ p J ) K- 



_Aj e„|i^cosm(o,-co'), y (8) 



