Complement and Imaginary Geometry. 605 



shears of finite amount are reducible to two lying in planes 

 at right angles to one another having one axis in common, and 

 neither produces any relative tangential motion or slide in 

 the plane of the other. In a pure shear of ratio a.-e u the 

 planes of maximum tangential strain stand at an angle to 

 the major axis of JG(w)= cot -1 a. For this case they are 

 circular sections of the shear ellipsoid, and in them lie the 

 isocyclic diameters. If a shear o£ ratio 7, at right angles to 

 the plane of a, 1/a, is superposed on this strain, then the 

 sheets of particles subject to maximum tangential strain by 

 the first shear are deflected into a new position, and now 

 make an angle a) with the axis a. It has been shown and is 

 easily seen that 



tan ft>= — =6. 



If the 7-shear had been applied first there would have been 

 maximum slide at cot -1 7, and this by the a-shear would be 

 reduced to cot -1 (y . a). Thus each of the four sets of planes 

 of maximum tangential strain makes with the least axis an 

 angle 7r/2 — oo. 



In finite extensions as well as in infinitesimal ones a ten- 

 sile load Q is divisible into thirds, of which one produces 

 dilatation and each of the others a shear. From two of 

 Cauchy's stress quadrics it may be shown that the load on 

 any central plane of the shear ellipsoid is the same or Q/3. 

 On planes whose directrices are the principal axes this load 

 is normal. There are two other planes on which it is wholly 

 tangential, and these make with the greatest axis an angle 

 whose cotangent is the ratio of shear. In other words, they 

 are planes of maximum slide as well as of maximum tan- 

 gential load. Maximum tangential stress, on the other hand, 

 occurs at 45° to the axis of stress for any homogeneous 

 strain, and on planes at this angle the slide falls short of a 

 maximum. 



Many substances, like mild steel, if ruptured by tension, 

 part along surfaces which make angles with the axis of stress 

 not very different from 45°, and most homogeneous sub- 

 stances ruptured by pressure break at similar angles to the 

 axis of stress. The fragments into which rupture divides a 

 specimen must show at least a little elastic recovery, which 

 in the case of tensile loads would increase the apparent angle 

 of planes of parting to the axis of stress, and in the case of 

 rupture by pressure would decrease these angles. Hence, if 

 rupture actually occurred on planes at 45° to the axis, the 

 specimens broken by tension would show surfaces inclined 



