288 
DR. G. B. JEFFERY ON PLANE STRESS AND 
rupture occurs, the crack will begin on the straight edge according to the greatest 
tension theory, and on the edge of the hole if the greatest stress-difference theory 
holds. 
It will be noted that the stresses produced will become large if the hole is near to 
the straight edge. The formulae are so simple that it is hardly worth tabulating 
their numerical values, but a single example will serve as an illustration. If the 
shortest distance from the hole to the straight edge is one-tenth of the radius of the 
hole, the maximum tension in the straight edge is 19'5 times the pressure in the 
hole. 
§ 7. A Semi-infinite Plate Containing an Unstressed Circular Hole and 
Under a Uniform Tension Parallel to its Straight Edge. 
Let the circular boundary be defined by a = a u so that if r is its radius and d the 
distance of its centre from the straight edge, 
r — a cosech a l5 d = a coth a x , dfr = cosh a x . 
At a distance from the hole the stress-function may be taken as y = ^T y* where 
T is the tension, so that, if a > 0, 
hx o = |-«T sinli 2 a/(cosh a —cos ft) 
= JaT sinh a (1 + 2 2 e~ na cos nft) ...... (65) 
We have to add to this a stress-function which gives no stress at infinity and no 
stress over a = 0, and is such that the complete stress-function gives no stress over 
a — a x . 
We may omit the term in K in (28), since in this case the region considered lies 
entirely on one side of a = 0, and clearly the required stress-function is even in ft. 
It may readily be seen that the condition that act and aft shall vanish over a = 0 is 
satisfied by (28) if 0„(O) = 0 and <p' n ( 0) = 0 for n= 1 , and hence from (26) and (27) 
A„ + B n = 0 and (n+l) C„+ (n+l) D„ = 0. We may therefore take for our complete 
stress-function 
hx - «T 
\ sinh a -I 1+2 2 e ” a cos nft\ + B 0 a (cosh a —cos ft) + A x (cosh 2a—l) cos ft 
+ 2 
L n = 1 
A n [cosh (n + 1 ) a — cosh (n — 1 ) a] 
+ E„ [(n— l) sinh (n+ 1) a — [n+ 1) sinh (n— l) a ] 
cos nft 
] 
( 66 ) 
At infinity a = 0, ft = 0 the first series diverges, but may of course be replaced by 
the alternative form in (65). If the second series converges it is clear that at infinity 
X = Xo- 
