972 



DR JOHN DOUGALL ON AN ANALYTICAL THEORY OF THE 



If we write X", Y", Z" and X/', Y/', Z/' for the applied forces which arise from 

 the part of the solution already obtained, then, takinii; account of (24), 



7" 9 / t^^'U_2 d^Y_.d^W_-,\ 



From these 



(28) 



(29) 



2X" = 0, 2Y" = 0, 2;z" = ttE 



?2W_i 



f/22 



>:(|Y"-,X") = !|:^^-. 



(30) 



Thus, just as at (23), we must still defer the introduction of the applied forces, 

 and we must have 



d'W_,_ dm_,_ 

 dz^ ' dz^ 



I'he equations for Ui, Vi, w-^ are therefore (using ^^ + rj^=l in 4'i^i)> 



(31) 



(32) 



and 



«3W-v(«^-'^')=». 





-.2 



c?z 



dz 



(33) 



The solution of these is 



-1 = -^V^ + ^-{r,^ - ef^' - <^rf^' + V: + ^^ , 1(34) 



If we write X'", Y'", Z'" and XJ", YJ", ZJ" for the applied forces arising from the 

 part of the solution already obtained, and take account of (31), then 



^ =<^ ■*■">& 11+'' a?' 



- x;" = x^- w^ , 



-IT- /// ■> 9 

 02 



■ (35) 



