226 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF 
58 bW2)9(Fo+P) KG) 
diy" a5 " =( 
— = —h?B(2/h) IF ip 6 
dn 
We are now left with tractions nn, ns of order zero. In the solution balancing these, 
FR” will be of order —1, W”” of order 2,9,” of order 0. We do not think it worth 
while to write down the equations defining F’”, but it should be noticed that they can 
be found explicitly. In fact, although the residual tractions nn , ns with which we are 
now dealing are partly defined by « series, the integrals il * eam dz and i “2 ns dz will be 
found to vanish to the order concerned so far as they come from these series, in virtue 
of the relation S\g,’Z(«z) = 0, which follows from (iii) (7) of last article. 
The functions g,” give terms of order zero in p’, w’. 
Hence, including in p’ terms of orders -—2,-1 
ee eens q/ song: EE, = ie 
an eee uw Be ot Sie | 
we have 
== Ale oye (Fy eee”) 
(= 41-0): APP +P" y= — en v(F,+F) 
(6) 
= 25 (y/ +y") 
w’ =4(1—o)(F,) + F' 4 F" + F") + 2(02 - 1) y(E, + F). 
The value of ¥ to a second approximation is 
d 5 ' 1 1 , 
Be = — ?B(z/h)3(F + F’) - sls ga F182) 94(B, Ege te (7) 
and =. is given by (5). 
The function F, which (§ 33) defines the permanent solution is 
i i (Np'+Sq'+Zw\dsde | Seis 
For the case of a solid or hollow circular plate all the quantities in the right-hand 
members of (6) can actually be calculated, and we thus obtain the solution for normal — 
traction to a second approximation, and for tangential or perpendicular traction to a_ 
third approximation, in a form, moreover, applicable without modification or addition 
even when the given edge stresses are discontinuous. 
We conclude by deducing the equations corresponding to Kirchhoff’s boundary — 
conditions to a second approximation. (They might be found to one order higher in 
the case of vanishing normal traction. ) 
We suppose that the given tractions N, S are of the same order in h, and that Z is 
of an order one higher. Any case may be reduced to combinations of cases satisfying 
this condition. 
