294 
Proceedings of the Royal Society of Edinburgh. [Sess. 
These stresses represent a moment M per unit length of 0 axis, and satisfy 
the conditions 00 = r0 = 0 at 0 — ± a. 
Fig 2. 
(5) For a force F * 
direction of y put 
giving stresses 
per unit length of 0 axis directed along the positive 
X = F rO cos 0/( 2a - sin 2a) , .... (9) 
rr = - 2Fr 1 sin 0/(2a - sin 2a) , 
66 = r6 = 0 everywhere. 
( 10 ) 
For a force F per unit length of z axis directed along the positive 
direction of the axis of x put 
X = - F r6 sin 0/( 2a + sin 2a) , 
giving stresses 
rr = - 2Fr -1 cos 0/( 2a + sin 2a) , ^ 
66 = 6r = 0 everywhere. { 
( 11 ) 
(12) 
(6) Uniform Pressure . 
For uniform pressure P on 0 = a and uniform tension P on 0 = — a put 
X= - Pr 2 (sin 20-2 cos 2a0)/2(sin 2a- 2 a cos 2a) 
jj - Pr 2 (sin 20 - 2 cos 2a0)/2m , say, 
• (13) 
giving stresses 
rr = P(sin 20 + 2 cos 2a0)/m , , 
00 = - P(sin 20 - 2 cos 2a0)/m , [> . 
■ (11) 
r6 = P(cos 26 - cos 2 a) jm . ' 
For uniform pressure on both sides of the wedge put 
x =-ivy 2 , 
giving 
n 1 — 00 = - P , and r6 — 0 every where. 
The stress function for an unlimited solid with a plane upper boundary, 
when one half of this is loaded with a pressure P per unit of area, is 
X = - Pr 2 (sin 26 + 26 + 7r)/47r . 
* Force at an angle has been treated by Professor J. H. Michell, Proc. Lond. Math. Soc., 
vol. xxxiv. See also Love’s Elasticity , p. 208. 
