352 REPORTS ON THE STATE OF SCIENCE, ETC. 
These equations, it will be noted, are quite independent of stress-strain relations, 
and hold equally in a plastic and in an elastic solid. 
If we start from a point O, where the P-stress is Py, and proceed along the corre- 
sponding line of principal stress to any point A, we have, integrating (3), 
A ~ 
P=P—[*P-Qdain 2 
And, similarly, if we proceed from O along a line of Q-stress to a point B, we have 
B 
Q=%-["P-AQdsln 2 sO) 
Now there are two ways in which we may conveniently compute the integrals 
on the right-hand sides of (5) and (6). Call ¢ as before the angle which ds, makes 
with the positive direction of the axis of x. 
Then 1/p, = oe 
(7984. A.) 
Fic. 3 
Now consider a point A (fig. 3) through which passes the isoclinic of parameter ¢, 
and let CD be a near isoclinic of parameter ¢ + dd, which meets the line of P-stress 
through A at C and the line of Q-stress through A at D. Then 
1/p. = dp/AD, 
and ds, = AC. .*. ds,/pp =do x (AC/AD). 
Let y be the angle through which the line of P-stress has to be rotated (counter- 
clockwise) in order to bring it upon the isoclinic. [ 
Then from fig. 3, AD/AC = — tan y, and thus 
[pee 1 +(2 (P—Q) cot ddp . . ‘ sevtg('7) 
0 
Equation (6) will then take the symmetrical form 
Q=Q +(° (Q—P) cot ydp, . : Oe fea] 
0 
where yf is the angle through which the line of Q-stress has to be rotated to bring it 
upon the isoclinic. : 
