296 
Proceedings of the Royal Society of Edinburgh. [Sess. 
and differentiate by the aid of equations (22) and (23), and remember that, 
for the weight alone, 00 and rO are each equal to zero at (9 = ±a, four 
equations are obtained to determine the four unknowns A, B, C, and D ; 
these give finally 
X = pgr 3 \s(3 sin 2 a sin 0 - sin 3 #) 4- c(3 cos 2 a cos 6 - cos 3 0)]/l 2 , . (24) 
and 
fr = - pgr[s(cos 2 a - sin 2 0) sin 0 - c(cos 2 (9 - sin 2 a) cos 0 + 2 cos (/3 - 0)]/ 2 , 1 
00= p^r[s(3 sin 2 a - sin 2 0)sin 0 + c(3 cos 2 a - cos 2 <9) cos 0 - 2 cos(/3 - 0)~\/2 , j (25) 
rO = - pgr[s(sm 2 a - sin 2 0) cos 0 + c(cos 2 0 - cos 2 a) sin 0]/2 , ' 
where 
s = sin /3/sin 2 a, and c — cos /3/cos 2 a . 
If rectangular co-ordinates are used it can be shown, by a process 
analogous to the above, that the stresses due to the weight are 
xx— - pg ( sin f3 cot 2 a y + cos /3.x )/ 2 , j 
yy = ~pg(cos/3 tai\ 2 ax + sin f3.y)/2 , ( (26) 
xy = - pg (sin f3.x + o,os f3.y)/2 . 
Equation (19) admits of easy transformation to rectangular co-ordinates 
and gives for pressure Pr on 0 = « 
= -pf 
Q q r 3 ? ,S 
i • — (y 3 + x 2 y)- 1 (x 3 + xy 2 ) 
l_sin a v cos a x ' 
co-> 3 a siu 3 a 
/ 24 • 
(27) 
The stresses due to the weight and pressure Pr on 0 = a are 
^ PI 
XX = - — 
4 1 
3 sin 2 a - 1 x 
■ o y + 
L sin d a cos a_ 
- [sin /3 cot 2 a y + cos /3 . x] * 
A 
II 
l 
y 3 cos 2 a - 1 
[_sin a cos 3 a 
- OjL [cos (3 tan 2 a x + sin f3.y], 
<§) 
II 
[.jl + jl] 
l_Slll a cos a_ 
- [sin /3.x + cos f3. y] 
(9) The principal stresses are, by a well-known theorem, given by 
_xx + yy J(xx - yy) 2 4- 4xv/ 2 
(28) 
and the direction of the principal stresses with reference to the fixed axes 
is given by 
tan 2</> = 2xy/(xd: - yy ) , 
and from equations (28) the values (xx-\-yy)/2, (xx — yy), and 2 xy can be 
written out. 
For example, if a = /3 = lS° and p = 2'30 and the pressure Pr is due to 
* For an equivalent solution by Professor Levy, see Comptes Rendus , vol. cxxvii. pp. 10-15, 
Paris, 1908. 
