30 PROFESSOR HORACE LAMB ON THE PROPAGATION OF 



If we take, as the typical solution of (113) and (116), 



^ = A-J (^r), X = Rr"J (&r) ..... (119), 



where a, ft have the same meanings and are subject to the same convention as in 

 Art. 3, we have, from (118), 



q = (- f Ac- + fiSRT*) J, (frr) "I 



12 . 

 w = (_ a Ae- + f 3 Be-*) J (frr) ' 



and tlience for the stresses in the plane z = 



- (2f * - tf) f B} J, (frr) ] 



f ( / 



- F A - 2^/3B} J (&r) 



J 



The formula- differ from (108 i and (109) only in the substitution of igR for B. The 

 notation of (119) is adopted as the basis of the subsequent calculations. 

 If we are to assume, in place of (1 19), 



...... (122), 



the corresponding forms of (120) and (121) would be obtained by affixing accents 

 to A and B, and changing the signs of a and ft where they occur explicitly. 



1 1. As in Art. 4, we begin by applying the preceding formulae to the solution of a 

 known problem, viz., where a given periodic force acts at a point in an unlimited solid. 



Let us suppose, in the first place, that an extraneous force of amount Z . J (CT) e'P', 

 per unit area, acts parallel to z on an infinitely thin stratum coincident with the 

 plane z 0. The formulae (119) will then apply for : > 0, and (122) for z < 0. The 

 normal stress will be discontinuous, viz. : 



[^],= + o-[p4=-o=-Z.J (^) ...... (123), 



whilst p :a . is continuous. Hence 



(2f* - **) (A - A') - -zeft (B -f B') = - Z | 



P > - (124). 



2a (A + A') - (2f- - F) (B - B') = J 

 Also, the continuity of q and CT requires 



--(B-B') = o| 

 We infer 



