Context for the question: Nye (1985, p.25) states the following.

The concept of the magnitude of a property in a given direction is one that needs careful definition, because of the lack of parallelism between the vectors involved.

In general, if $p_i=S_{ij}q_j$, the magnitude $S$ of the property $[S_{ij}]$ in a certain direction is obtained by applying $\mathbf{q}$ in that direction and measuring $p_{\parallel}/q$, where $p_{\parallel}$ is the component of $\mathbf{p}$ parallel to $\mathbf{q}$.

Nye states that this definition is appropriate because it is $p_{\parallel}$ that is measured in simple experiments.

It is not until Chapter XI of Nye that we learn that the simple experiment to which he referred on p.25 was actually two experimental setups given the sample's geometry: 1) to measure the flux of $p_{\parallel}$ across a flat plate, which is suited to measure the magnitude $S$ of the property $[S_{ij}]$ in a certain direction; 2) to measure $q_{\parallel}$ down a long rod, which is suited to measure the magnitude $R$ of the property $[R_{ij}]$ in a certain direction. $[R_{ij}]$ is the reciprocal of $[S_{ij}]$, and the relationship between $\mathbf{q}$ and $\mathbf{p}$ is $q_i=R_{ij}p_j$.

Back on pp.24-25, Nye states:

In discussing second-rank tensor properties of crystals one often uses phrases like 'the conductivity in the direction [100]' or 'the susceptibility in the direction [112]' or whatever direction it may be.

Indeed, I've seen works concerning fields other than crystal physics that defined the magnitude of $[S_{ij}]$ in the direction of the flux vector when the measurement is made on a sample whose geometry is that of a long rod. For example, see Carslaw and Jaeger (1959), p.46, where they define the "thermal conductivity in the direction of the (heat) flux vector"; and Scheidegger (1957) pp.63-66, where he defines the viscous fluid permeability through porous media in the direction of (fluid flow) flux vector.

Per these descriptions, they take on definition that the magnitude $S$ of $[S_{ij}]$ in the direction of flux as $|p_i|=S(|q_j| \cos \alpha)$, where $\alpha$ is the angle between the vectors $p_i$ and $q_i$, and $|q_j| \cos \alpha$ is the component of the potential gradient vector $q_j$ that is co-linear with the flux vector $p_i$.

So it seems to me that, without attributing the definition (or convention?) stated by Nye (1985) on p.25, perhaps he is the one that established the definition (or convention). And this is my question, who (and when) established the definition of the magnitude a 2nd-rank tensor property in a given direction?

Edit: Giving this more thought, perhaps Nye's material-tensor's-magnitude definition stems from the mathematical definition of a second rank tensor and the associate transformation laws?

$$S=S_{ij}l_i l_j$$ $$S'_{ij}=a_{ik}a_{jl}S_{kl}$$

Nye, J.F. (1985). Physical Properties of Crystals: Their Representation by Tensors and Matrices (1st ed. 1957). Clarendon Press, Oxford.

Carslaw, H.S., and J.C. Jaeger (1959). Conduction of Heat in Solids, 2nd ed., (1st ed. 1946), Oxford University Press, London,

Scheidegger, A.E. (1957). The Physics of Flow Through Porous Media, University of Toronto Press.


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