COMPLEX STRESS DISTRIBUTIONS IN ENGINEERING MATERIALS, 353 
Now the value of P — Q can be read off at any point from observation of the 
isochromatic lines or by direct measurement with a compensator. 
The values of y are obtained from the isoclinics by subtracting, from the inclination 
to the z-axis of the tangent to the isoclinic, the parameter of the isoclinic itself. 
The isoclinics are usually well-defined brushes, of which the direction, at any 
point, can be observed with considerable accuracy. Accordingly y should be read off 
with an error of only 1° or 2° in most cases. 
It is necessary, of course, to draw the lines of principal stress, since these are the 
paths ofintegration. But, given the isoclinic lines, this usually presents no difficulty ; 
and, as a rule, this is the most accurate and immediate deduction from the optical 
data. 
Once the lines of principal stress have been drawn, P — Q and y should be noted 
for the various points at which a given line of stress meets the successive isoclinics. 
The values of (P — Q) cot y can then be calculated and plotted to ¢, and the area of 
the curve so found obtained by any of the usual methods. 
3. In this way P and Q can be obtained for any point on the line of stress, provided 
Py (or Qo) is known. 
Now, at the boundary, both principal stresses can be obtained if the applied 
traction is entirely known. If the P-stress line makes an angle x with the normal 
n to the boundary, s denoting the direction of the boundary itself, 
ss — nn = (Q— P) cos?x, ! 
and nn, ns being supposed known, 38 is given by this equation. 
The simplest and most important case occurs when we are dealing with a part of 
the boundary entirely free from traction. In this case nm = ns =0, the boundary 
itself is a line of principal stress (say Q-stress) and 
a= Q=P 
as given optically ; P, of course, being here zero. 
We may thus follow any one line of principal stress which starts from a free portion 
of the boundary, and the stresses P, Q are completely known along this line. 
, It may be that this will allow us to reach directly all the region of the plate which 
_ we wish to explore, in which case the problem is solved. 
If, however, this is not the case, we may now take any point already reached as 
; starting-point and proceed from it along the orthogonal line of principal stress. In 
; this way the whole of the plate will ultimately be reached. 
; 4. The formula (7) becomes highly inaccurate if y be small, in which case a slight 
error in y makes a large error in cot y; and also, the isoclinics being then nearly 
parallel to the line of principal stress considered, the intervals along the path of 
integration are too large for the method of quadratures to give accurate results. 
In this case we have to treat equation (5) by a different method. 
Let Ay (fig. 3) be the intercept, measured perpendicular to the path of integration 
between two near isoclinics in the neighbourhood of the point considered, whose 
parameters differ by Ap. Then L/p,= 4/4y approximately. And if we take the 
interval Ap constant throughout, which will usually be the case since the isoclinics 
should be drawn for constant differences of , we have equation (5) leading to 
Ap 
P=P)— A9|. 9 ae, Sout. ammtve med) 
» 
_and the integration can be proceeded with graphically as before. 
This method is specially useful when, as not infrequently happens in cases of 
symmetry, there exists a line of principal stress which is straight, and therefore is 
_ also an isoclinic. Ay is then the intercept between this line and the nearest (curved) 
 isoclinic, measured perpendicular to the (straight) path: of integration, --. 
BB 2 
