104 
mi L. X. (1. riLOX ox ax ArPllOXIMATE SOLUTIOX Foil FEXDIXG A 
denoting that x — ^ k substituted for x in P, 
X — X, we liave 
Similarly for Q, &c. ; or A\'riting 
P' = 
S' = 
1' P {x') dx 
* x~cd 
U' = 
rx+it' 
Q {x') dx' 
A.-a' 
A^' = 
{x) dx' 
J .. -a' 
P', Q', S', U ^5 V' referring to the stresses and disjdacements due to the nnilorm 
layer. 
We can obtain in this way, at once, as many different forms for P , Q , S^, U , A 
as we had for P, Q, S, U, V. 'fhe series for the latter integrate at once, for they 
are com])osed of terms of the form const. X r" cos or r" sin nc/j, or yr" cos iifjj or 
ijr" sin nrj), where r = ^x^ y", tan (f) = xjij. We have then 
f r" sin mb dx = —- ^ cos n-{- 1 </> 
J a + i 
>’"cos n(f)dx = ;; sin n + 1 (f). 
11 + 1 
Tlie only Ciise wliere this fails is wheii u = ~ f, and m this case it is easy to sir. 
iW 
fliaf 1 ' ^ dx = (j), 
sill (f) 
dx = loir r. 
Terms of the form (/> and log r also occur. They can be infegrated as follows 
^(f) dx = X(f) — y log r, 
jlog r dx = X log X — 
If we a}»})ly these formnhe, and if we call 1)^ and (fig. ii.) the points (— a', + h) 
and (+ a\ -j- b), i.e., the extremities of the layer of stress, and if )'i, denote the 
distances of any point from Dj^, J)^ respectively, and if (f)i, (f)^ he the angles M'hich 
Cl, iv make with the vertical, we find, if we start with the expressions for U, It, Q, S 
in the form (77), (7D), (Hif), (84), 
