930 



DR JOHN DOUGALL ON AN ANALYTICAL THEORY OF THE 



Betti's Theorem therefore gives 



- XuJ'Xx', y', z') - YuJ'-ix, y', z') - Zu^'\x', y , z) 

 + work difference of the system (2) and S,, !- = 0. 



- work difference of the system (2)' and S,. 



The last two lines on the left here are equal, so that we get 



- Xw/'(a;', y\ z') - YuJ'-\x', y', z') - Zu^Kx', y', z') 



_^ 2 f XW(1, r) + YW(2, r) + ZW(3, r) + {y'Z - 2'Y)W(4, r) + (z'X - cc'Y)W(5, r) ) 



( +(a;'Y-y'X)W(6,r) + a,W(7,r) + a8W(8,?-)+ ... +a^^W{U,r)) 



(4) 



= 0. (5) 



Now the object of inserting the rigid body rotations involving the coefficient r 

 in Si and Sg (Art. 21) was to secure that W(l, 5) and W(2, 4) should vanish ; that 

 the others of the form W(r, s) vanish when r, s = l, 2, . . .,6 can be seen at a glance. 

 Also of W(7, r), . . . W(12, r) when r=l, 2, , . ., 6 the only one that does not 

 vanish is W(r4-6, r), and its value is — |^. 



Hence 



a,,+6 = - XM/»(a'', y', z',) - Yu/\x, y', z) - Zu}'\x', //', z), .... (6) 



r=l, 2, . . ., 6. 



It may be noticed that we might have left the coefficients of S^, . . ., Sg in (2) 

 undetermined, and found them by the present method, taking r — 7, . . ., 12. 



It will be seen that there is a very definite correspondence between each of the 

 rigid body displacements and one of the other six permanent modes, S7 corresponding 

 to Si, Sg to Sg, and generally S,.+6 to S^. Modes which correspond in this way may 

 be called conjugate. 



23. Table of the twelve permanent free modes with their coefficients in the solution 

 for a force (X, Y, Z) at [x' , y\ z'). 



We may put down in full the permanent terms of the solution S for the force 

 (X, Y, Z) at (x', y' , z'). The following belong to the region z^z' ; for z<z' , change the 

 signs of the expressions for the displacements throughout. 



u,j = 2(TZxy, 



M, = - z^x + ^■x{x'^ + ?/^) - ( o- + — - T ja%, 



'u^ = 2(TZxy, 



1 / 3 \ 



X- 



rEa* 



rEa^ 



u^= - <tx, 

 n,, = - o-//, 

 a. = 2, 



Z( - J-) 



