970 



DK JOHN DOUGALL ON AN ANALYTICAL THEORY OF THE 



The most general solution of these equations is 



"-2 = ^-2 - V^-2 > 





2' 



(17) 



where U_2, V_2, W_2, 0_2 are independent of ^, »; and are therefore functions of z only. 



We have just seen that the terms in «'* in (6), (7), (8) and those in a~^ in (9), 

 (10), (11) disappear. Consider now the terms succeeding these next in order. 

 If the forces can come in at this point, the equations for w_i, v_i, w_i will be 



8i(m_i , w_i) + X$— w_2 - X^ = 0, 



oz 



oz 



Y. = 0, 



[ . 



■ (18) 



8^{W-t) + 1^^ (^_2 + -qV.^) - Z. = 0. 



These equations for u_-^, t;_i, w_-^ are of the same form as (12) and (13) for 

 w_2, ''-'-2, M^-2- The extra terms here are known from (17) and are to be counted in 

 along with the given applied forces. 



Now it can be seen in a moment that these extra terms fulfil of themselves the 

 four conditions (14) and (15); for \^ds, \rids, over the boundary, vanish. Hence the 

 given applied forces would also need of themselves to fulfil the four conditions, and 

 we must conclude as before that they cannot yet be brought in, or thatp>— 3. 



The equations for u_-^, v_-^, w_i are therefore (18) with X, Y, Z and X„ Y^, Z, 

 left out, that is, 



r)i(M_i , w_ J = 0, 



D2(M-i , W-i) = 0' 



8>_,,.;_i) + X|-^2 = o, ^ 



dW 



and 



DaO--i) = 0, 8,{w_,) + ^{i^^-^ + y,^^] = 0. . 



(19) 



(20) 



A particular solution of (19) and (20) is 



dz 

 dW_„ 



dz dz 



