p, q = Principal Stresses. m, n = Principal Stress Directions. $,,, q~n = Angles between directions m, n and the positive direction of the x axis. q~ = General notation for either @m or ~b~.
Stressfield studies can be carried out either by means of two polarizing prisms, two Nicols for instance, or an interferometer or extensometers. Comparatively few laboratories are equipped with interferometer and suitable extensometers, however, and those active in research along such lines under such conditions are, and always will be, faced with the problem of obtaining complete information from crossed Nicol observations alone, a problem which dates back to 1815, when Brewster made the discovery of forced double, refraction.
Maxwell 1 was the first to show 35 years later that the information from crossed Nicol observations is all that is required, fundamentally, for a complete stressfield analysis. Namely, in combining (p -q), $ and boundary values from such observations with the differential equations obtained by formulating the equilibrium requirements of an element, see appendix I, and then integrating from free boundaries to points inside in accordance with the derived equations, the stresses p, q are obtained separately for all points of the structure. Consequently, as @ is also known for all points, the stressfield is fully determined in such a case, as is well known. l Maxwell, "On the Equilibrium of Elastic Solids," Trans. Roy. Soc. Edin. 185o , and Collected Papers. 457 * Eq. (II) and ( ) hold for @., < ~r/2. For ยข~ > 7r/2 use eq. (IIa) and (I2a), see appendix II.