practical Instrument of Physical Research. 493 



V. A particular Integral in the Theory of Elasticity. 



Consider an elastically isotropic solid in equilibrium under 

 arbitrary external bodily forces. Let e be the (small vector) 

 displacement at any point. Also let F be the external force 

 per unit volume. Then [Thomson and Tait's ' Natural Philo- 

 sophy/ § 698, equation (7)] the equation of equilibrium is 



Nv 2 e + M V Sve = ^ = F; .... (37) 

 where M, N have been put for Thomson and Tait's m, n 

 respectively, and where ^ is a symbolic self-conjugate linear 

 vector function of a vector, 



v t"=-T 2 v(No>-MUvSa>Uv), 

 we have 



^- , ©=-T- a v{N- , © + (N- 1 -[N + M]- 1 )UvS©Uv}. 

 [Here it is assumed that if i 



Aft) — Bi Sz'ft) = ifrcQ = r, 

 then 



T =^- 1 ft) = A- 1 ft)+(A- 1 -[A + B]- 1 >'Szft), 



which can be easily verified, if not obvious, by operating on 

 the last equation by yjr]. Therefore by equation (37), 



e=^- 1 F = N- 1 v- 2 F-MN- 1 (M + N)- 1 v- 1 Sv~ 1 F. . (38) 



By equation (5) a particular value of v~ 2 F is given by 



47rv- 2 F = jjJwFd9; 



and the integration may be supposed extended over the given 

 region, or, in addition, over any external region where F may 

 be given any convenient values. Again, 



47rv- 1 Sv- 1 F = 47rv- 1 SvV" 2 F = V~ 1 SvJJj^^ = Jj , JSF V .V~ lw ^- 



Since we are seeking only a particular solution any particular 

 value of v _1 w will serve. Putting <r for the pb—p a of equa- 

 tion (4), it is known that 



so that 



V" I t«=-Ucr/2. 



Hence 



8ttN(M + N)6= JjJ{2(M + N)«F + MSFv.U<7^/ ? . (39) 



is a particular solution of equation (37). 



If these symbolic methods be objected to — though they are- 

 just as legitimate as the ordinary symbolic methods adopted 

 for discovering particular solutions — they may be regarded as 



Phil. Mag. 8. 5. Vol. 33. No. 205. June 1892. 2 L 



