On the Directions which are most altered by a Homogeneous Strain

Extract

The cosine of the angle through which a unit vector ρ is turned by the homogeneous strain ϕ is

This is to be a maximum, with the sole condition

Differentiating, &c, as usual we have

Operate by S.ρ and we have

Hence the required vector, its positions after the strain, and after a subsequent application of the conjugate strain, lie in one plane; and the tangent of the angle between ρ and its first distorted position is half of the tangent of the angle between it and its doubly distorted position.

When the strain is pure, the required values of ρ are easily found. Let the chief unit vectors of ϕ be α, β, γ, and its scalars g₁, g₂, g₃.