COMPLEX STRESS DISTRIBUTIONS IN ENGINEERING MATERIALS. 309 
It is easily shown that the displacements are obtained in terms of the stress 
function as follows 
a tT gt 
Sx 5y 
a 3 F ds) 
2 V = ae = | 
i by oa “) i) 
where 
Vw == 0 lege = 3 * , * (8) 
ay oe 
——, “hh - . . . . . 9 
by Vv (9) 
and ¢=4A/(A' +4), so that (1—#) = (1+7)-', where 7 is Poisson’s Ratio. 
FE SE oH 
ba?’ 8y?” Bady 
the same holds good of the third differential coefficients. From this, by the 
theorem of § 4, it follows that pe = are acyclic for a reducible circuit and have 
2 y 
a definite cyclic function for an irreducible circuit. By repeated application of the 
theorem the same result holds good for E. 
dE 
ox 
Taking a circuit enclosing one hole and one only, we have, integrating equations 
(6) along this circuit 
We note first of all that all exist and are one-valued, and that 
The cyclic functions for and “ are very readily found from equations (6). 
Y 
cy()= — [¥as= — Total force resultant on the circuit parallel to y= Force 
resultant on boundary of enclosed hole parallel to y= Y,, say. 
Similarly cy(") = --X,, where X,= Force resultant parallel to 7 acting on the 
y 
boundary of enclosed hole. This result is due to Michell. 
In asimilar manner it can be shown that 
Cy(E)=Y,7—X,y—-M, q : : . (10) 
where M, is the couple resultant at the origin of the forces acting on the inner 
boundary of the enclosed hole. The value of Cy(E) at any point is thus the moment 
about that point of the forces applied to the boundary of the hole about which the 
typical circuit is taken. 
We have next to consider the cyclic functions of as and ue First of all we note 
y 
2. 3 
that since v7E is essentially one-valued, so is 5° and therefore also a and 
~ dxdy 5x37 
3. 3, 3 
say And since 77y =0 ee + inh =0, so that, iby being essentially one-valued. 
3. 3 
so is a4 and, similarly, so is i 
Thus the third differential coefficients of W are essentially one-valued, and 
accordingly so will the fourth differential coefficients be. Hence by the theorem of 
$4 the second differential coefficients have cyclic functions (which inyolves being 
acyclic for reducibie circuits and being unambiguously detined for reconcilable 
paths). The first differential coefficients and y itself will therefore have cyclic 
functions for any typical irreducible circuit. 
Applying Cy to equations (8) <nd (9) and remembering that YE is acyclic, we 
have 
Vv *Cy(v) =0 3 : : : ee Gli)) 
5*Cy(p) _ 
ee ia 0 (12) 
