(4) 



904 DR JOHN DOUGALL ON AN ANALYTICAL THEORY OF THE 



For the problem of normal traction cos k[z — z') cos m{w — w') 



Artj + B6j + Ccj = — . 

 I'- 

 Aa^ + Big + Ccg = 0, 



Attj + Bfog + 0(^3 = 0, 



which give 



A = ^'a„„/D,„, B = ^1\JB„,, C = ^C„„/D, (5) 



IX fl fl 



For the general problem of normal traction equal to N(«, z) we now, in view of 

 4 (4), naturally try the result of multiplying the values of <p, 6, ^ defined by (1) and 



(5) by N(w', z'), and taking — '^' [^''dw' ( dKi^^dz' of the functions so obtained for the 



TT" ,„, JO Jo hi 



required values of <^, 0, \|/. 



We find, however, at the outset, that the /c- integrations cannot in general be carried 

 out, on account of the form of the functions of (1) and (5) near /c = 0. But this 

 difliculty is easily surmounted, for, as was remarked near the end of Art. 3, the terms 

 of negative degree in k which occur in (p,9,^ of (l), (5) can contribute nothing to 

 surface traction at p = a, and as they have the form H4/K:* + Hg/f ^, all we have to do is 

 to subtract them from the values of (p, 0, and \|/ given by (l) and (5). 



The tentative solution for normal traction N(ft', z) is now 



where A, B, C are defined in (5) and 3 (6), and 



( "" ', " ' ) are the terms of negative deo;ree in k in tlie ascending power expansions of ) 



A V cos [ • ^"'^ 



( '(^''jJ„Ap cos k(2-z')^°'J ■;»(<,) -0/). j 



We shall not stop here to give an a posteriori verification that (6) actually defines 

 a solution of the proposed problem.* We should require to prove that all the 

 differentiations necessary to show that (f>, 0, and -^ in (6) are harmonic, and to derive 

 the values of the tractions at an internal point, can be performed within the signs of 

 summation and integration, and then to prove that the limiting values of these 

 tractions as the internal point is made to approach the surface are fp = N(w, 2), po) = 0, 



p2-0. 



The main point to be insisted on is that in all these manipulations we may deal with 

 an internal point, so that p<ia. 



Ill (G) we may alter the order of the summation and integrations to 



r'd,^-rN(o>,zy/z'z,'rdK\ \ (8) 



Ja Jz, m J» { ] 



Suppose now that N(«, 2) vanishes except over a small area t„ enclosing the point 

 (a, M, z'), and let it be constant and equal to Nq over this area. Let o■^, diminish and 



* (Jf. Art. 50. 



