206 REPORTS ON THE STATE OF SCIENCE.—1914. 
lines of principal stress make an angle a with the axes, and if R is the 
principal stress-difference, it is well known that 
P=OQ=R cos 2a 
2S=R sin 2a. 
Thus a determination of R and a at every point Yeads to the value 
of 8 at all points. 
On the other hand, considering the equilibrium of a small rectangle 
dz, dy and neglecting body-forces, we have the well-known body stress 
equations for generalised plane strain, 
sy eae ie wis 
Se Sp ae ayaa. 
Now, at a point of the boundary, all the stresses will be known. 
For the normal stress across an element of the boundary where the 
outwards normal makes an angle with the axis of wis 
P cos? 6+Q sin? 64258 cos 6 sin 6 
z= ae + ae cos 20+8 sin 20. 
S and P—Q being known from optical data, and the normal stress across 
the boundary being also known from the boundary conditions, the above 
equation determines P+-Q and hence (P—Q being known) P and Q. 
Consider now a point A of the plate. Draw a line through A parallel 
to the axis of « to meet the nearest boundary at a point Ao (@p, y). 
Then, integrating the equation 
éP , 38 
bu OY 
along the line Ay A, we find 
P— Pa -\5 da, 
oy 
where P, is the value of P at Ay. 
Similarly, if a line through A parallel to the axis of y meets the nearest 
boundary at a point Bo (x, yo) when the value of Q is Qo, 
Yo 
Now, if we know the value of S at all points, the values of the partial 
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differential co-efficients dy’? dy, can be obtained approximately by 
taking differences. P and Q can then be found as above by the ordinary 
process of graphical integration, Pj, Q) being known, as explained. This 
method can be used with any set of experimental data, provided only that 
these are accurate enough to allow of differences being taken to calculate 
3S 
Se’ By In any case, before actually applying the method, the curves for 
S when either « = constant or y = constant should be ‘smoothed’ so 
