314 REPORTS ON THE STATE OF SCIENCE, ETC. 
We can further simplify by taking the axisof y parallel to the force resultant 
on the inner boundary. We have then X,=0, and therefore a,=0, so that only the 
dislocation due to translation parallel to a need be considered (fig. 12, right-hand). 
The stress function for this dislocation has been given by Timpe (Joc. cit., p. 31). 
It can also be deduced from Volterra’s general results. In this case 
. 22 Vind 
E,=K sin 0| ©? —* 4+ (a2+5%)r log x 
= K sin of SF + (a+ br log r | 
leading to the following stresses, 7, 0 being polar co-ordinates 
i mas » g (b?—7*) (7? —a*), 
i} 
70, =a eae (b?—7°) (7? —-a*), 
0, = K oe 8 fab? - B74 +72 (a? + B®), 
a" 
We find also 
y, = 2K f3a3— avy? + (a+ b*) (@ log 7—y8—2)}, 
whence 
Cy(y,) = —4aK(a? + b*)y, 
and therefore 
cy(*) = 4 4eK (024 0), 
by 
so that equation (18) gives 
Qu,/(1—o,) = —40K (a? +02), 
or 
K = — 2 (1 +m) 
4 (a? +b)’ 
and, using (22 
CK =F (n—)Y¥, : 
' 4 (a? +b?) 
The corrective terms to the stresses, which we may call for shortness Arr, Ar, 00, 
are then 
Me: a rts (1 —m)Y, (b° ne +3) (7? — a”) 
Arr =ar7, = tn (a?+ 08) 3 sin 6 
Ard =a,70, = + (1-10) Yo Cet er a a (24) 
dm (a* + b”) rs 
4 232 — 3744 92 (a2 2 
A6e = a,60, =-— (a=) Yo é B—3ri+ (a +) | sin 0 
4m (a? + b”) a J 
Note carefully that these corrective terms depend only upon the total force 
resultants applied to the hole and not in any way upon the manner in which these 
force resuitants are distributed ; and this remark holds good, not merely for the 
circular ring, but for the general case of a plate of any shape. 
11. We have now to consider where these corrective stresses are greatest. 
It is clear from equations (24) that for each stress the maximum occurs for the 
same value of 7 on each radius vector. These maxima will themselves reach their 
greatest values; (1) in the case of Arr and A06 when 6= 47/2, that is along the 
a “-s 
diameter parallel to the resultant applied force ; (2) in the case of Ar@ when 8=0 or 
a, that is along the diameter perpendicular to the resultant applied force. 
Consider now the maximum corrective stress Ab0. It will occur when ¢ = a@7b?/73 — 37 
+ (a? + b*)/7 has its greatest numerical value. Now do/dr = —3a*b?/73—3—(@? + 8) 7" 
