■ 



36 Dr. C. E. Weatherburn on Two Fundamental 



to formula (11). It' instead of p we take a boundary point t, 

 the first member must be replaced by 27rD(£), and we may 

 write the equation 



D(*) = JD(*) .V(st)d8-$®(t8) .¥(s)ds. . (11') 



If, then, any displacement D(p) possesses zero surface 

 traction, it satisfies the relation 



D(t) = $l)(s).y(st)ds, 



which is identical in form with (56'). But the only regular 

 deformation corresponding to zero surface traction is a 

 translation or rotation of the body as a whole *. Now there 

 are three independent translations, i, j, k, and three in- 

 dependent rotations, i X p, j x /?, kxp, where p is the position 

 vector of the particle p referred to the cm. of the body, and 

 the unit vectors i, j, k are taken along the principal axes of 

 the body. These, then, are the six independent solutions 

 of the homogeneous equations (560. If follows then that 

 the associated homogeneous equation 



v(0 = j>(te) .Y(s)ds .... (57) 



also admits six linearly independent solutions 



«,(«), (i=l, 2, . . . , 6) ; 



and in order that (56) may possess a solution it is necessary 

 and sufficient that f(s) be orthogonal to each of the 

 functions a^s) ; that is, it must satisfy the relations 



Jf (*) .«,(*>fo = 0, (i = l, 2, . . . . , 6) . . (58) 



If these conditions are satisfied (56) admits a solution v(s) 

 which, when substituted in (26), gives a displacement for 

 the outer region whose value at the boundary is —f(t). 



Even when the conditions (58) are not satisfied the 

 problem may be solved for the boundary value —f(t) as 

 near as a displacement of the surface as a whole. For by 

 § 12 there exists a solution to the modified problem corre- 

 sponding to the boundary value —f^t), where 



fifo=f(o-,y.*(»)-* + i(*o<fr 



= f(*)-[A 1 i + A 2 j + A 3 k + B 1 ix / 3 + B 2 jx/)-f-B3kx/>] 



= f(0-a-o)X> (59) 



* Cf.j e. g., Marcclong'o, loc. cit. p. 196. 



