MEASURE AND PROPERTIES OP SHEAR. 23 



tion of the minor axis is 1 — 1 / «. The sum of the two is a — a -1 = 2 s. 

 While 2s measures shear and is not unfitly called the amount of shear, 

 s might e |ually well have been regarded as the measure of shear ; indeed, 

 this would have been more convenient, because it would have accorded 

 with the received nomenclature of stresses. 



Many of the properties of shear can be inferred in the simplest manner 

 from its definition. Since it involves neither change of volume nor of 

 the area of the strain ellipse, it can consist only in re-arrangement of 

 matter, each fiber perpendicular to the plane of shear, retaining its 

 original thickness, length and direction, though shifted to a new position. 

 Since the major axis of the shear ellipse exceeds unity and the minor 

 axis falls short of unity, there must be four intermediate radii of unit 

 length, and the symmetry of the conditions shows that these four radii 

 form two diameters. Thus there are two diameters winch have the same 

 length after strain as before strain. These diameters are the traces on the 

 x y plane of planes passing through >> :. ami these planes undergo no dis- 

 tortion through strain. In them the circular sections of the strain 

 ellipsoid evidently lie. All planes parallel to these are also, by the 

 properties of homogeneous strain, planes of no distortion. Any two 

 planes of no distortion must stand at the same perpendicular distance 

 apart after strain as before, for were it otherwise the volume of the ellip- 

 soid would lie changed. 



Thus a shear can consist only in the sliding of planes of no distortion 

 upon one another and in changes of the angles between the two systems 

 of undistorted planes. 



The behavior during the straining process of the planes of no distor- 

 tion is of great geological importance ; but as this behavior depends to 

 son^e extent upon rotation, it appears appropriate to. defer its discussion 

 until some of the simpler compound strains have been explained. 



COMPOUND STRAINS. 



How treated. — For the immediate purposes of this paper it is needful to 

 examine compound strains of several varieties. It seems desirable also 

 to examine the simpler combinations in somewhat more detail than is 

 absolutely essential to the results which will be deduced from them in 

 the subsequent sections in order to give assurance that the geological 

 deductions are not vitiated by the omission of important properties of 

 strain. It is to be hoped also that the treatment here submitted may 

 facilitate the solution of geological problems not touched upon in the 

 present investigation. 



Pure Deformation. — Any pure deformation is resoluble into two shears 

 at right angles to one another, one axis being common to the two ele- 



V— Bum.. Geol. Soc. Am., Vol. 4, 18112. 



