DR. G. B. JEFFERY ON PLANE STRESS AND 
■7 7 9 
Lj i Li 
We have therefore from (16) 
2 (\ + n) 
and from (12), (13) and (14), (la) 
) M 7 OY 7 oQ 
> jfxU = . 
A + /U cot cp 
= 7T- A -A + h 
A + /x rp ca 
( 18 ) 
(19) 
( 20 ) 
It is readily seen that these equations determine u and v apart possibly from rigid 
body displacements, for, although owing to the double integration an arbitrary 
function of a and an arbitrary function of (3 will appear in AQ, these will be 
determined by (17), except for functions of a or /3, which make its left-hand side 
vanish identically. The only possible arbitrary terms in hQ are therefore given by 
AQ = aA (cosh a + cos /8) + B(cosh a —cos f3) + Ca sinh a + D« sin (3, or 
Q = Ar 2 +aB + Qz+Da? 
where r is the distance from the origin. These give rise to terms in u. v corresponding 
to motions of pure translation and rigid body rotation about the origin. 
§ 3. The Stress-Function. 
Turning now to the consideration of the possible forms for the stress-function in 
these co-ordinates, we note that the differential equation (8) can readily be solved by 
the ordinary method, and that its general solution is 
A^ = 6 (j>i (ot + l/3) + e a (p 2 (ot + 1/3) + 6 a 0 3 (a— l(3)-\-6 a <p i (a— (/3). 
If we seek a solution of the type Ax = f(ot) cos n/3 or f (ot) sin n{3, (8) shows that 
the differential equation for/(a) is 
the solution of which is 
f (ot) = A„ cosh (n+ l) a + B„ cosh (n— l) a + C n sinh (n + l) a +1),, sinh (n—l) ot , 
unless n = 0 or 1. In the latter case we have 
f (a) = A] cosh 2a + B! + C, sinh 2a Tl^a, 
and when n = 0 
/(«) = A„ cosh ot + B 0 a cosh a + C 0 sinh a + D 0 a sinh a. 
