952 



BR JOHN DOUGALL ON AN ANALYTICAL THEORY OF THE 



Force parallel to the axis. 



(/2F_i 



+ H^' - f) .:^' + -^y-£i-' - -^: + Fi j 





M„ = G, 



-1 





rf2^ 



d22 



(/Z 



I 



r 



M, = - .-c^ 1 - ^-—J + H(, , 



dF_ 



lb: 



dz 



(12) 



where 



a=-E/^-E/-^ + Efo, 



(13) 



^- = ^^^^"2(.^)' 



F,= - 



■Ea* 



Dri2[ { l^(a,-2 + ^2) _ (^^ + 1 _ ^^«2.^. I ^ 



2 



G_,= 



-^^^^-^2^^^)' 



^^^=-;i^^^^"^2 



||^(,.2 + y2)_^,+ | 



^^'/[^^^ 



H„ 



rEa^ 



Dr^^z. 



(U) 



The suffixes attached to the various functional symbols F_2, F^, etc., are intended 

 to indicate the order of the functions with respect to the infinitesimal a, subject to 

 the following understanding as to the order of the applied forces. 



The functions X, Y, Z are supposed to be functions of xja, yja, and z, so that when 

 we put ^, '; for xja, yja, the functions become explicitly independent of a. X, Y, Z 

 are therefore of order zero. 



On the other hand, X^, Y^, Z^ are supposed to be functions of ^, »;, z all multiplied 

 by a, so that they are of order one. 



On this understanding 2aj is of order 2, 2(a!X) of order 3, and so on. 



2 is defined at 38 (2), and D;*, D;^ D;^ D,"' at 38 (7). 



The tables give correct approximations as far as they go for the displacements and 

 such of the stresses as are mentioned. Any stress which does not appear in the tables 

 is of higher order than Siwy which does. Cf. Art. 41 for a fuller account. 



It will be seen that the tables give the displacements as far as terms of order one, 

 and all the stresses as far as terms of order zero. 



If we write /for the sum of F (Art. 26) and F_2 + F_i + F0 + F1, so that / is the 

 approximate a;-displacement of a point on the axis [cf. Art. 43), we have obviously 



dz^ 



rEa* 



d^ 



2{-i<;«^ + J/2)+i(cr + |-T)a%}z]. . (15) 



Similar equations can be written down at once for the sum of the functions G, 

 or H, or II. 



