The Journal of Geology

doi: 10.1086/626391pmid: N/A

In part I we compare vector ($$\bar{a}$$) and arithmetic ($$\bar{x}$$) means of circular-type variates by showing how each changes with addition of one measurement (an) to (n - 1) measurements. Differentiating, $$\sigma x/\sigma a_n = 1/n$$ (no dispersion term appears here); $$\sigma a/\sigma a_n = (R_1 cos a_n + 1)/(R^2_1 + 2R_1 cos a_n + 1)$$, where R1 the vector resultant of the first (n - 1) measurements, is a measure of dispersion. As $$a_n \rightarrow 0$$ and as $$R_1 \rightarrow (n - 1)$$ (i.e., dispersion of the $$[n - 1] measurements \rightarrow 0$$), then $$\sigma a/\sigma a_n \rightarrow \sigma \bar{x}/\sigma a_n$$. Thus we obtain requirements for approximate equivalence. In part II, using trigonometric symmetry to simplify computations, a short method for computing vector mean and vector resultant is presented. We then examine ten sets of earth-science data; to nine of these we attempt to fit linear normal and circular-type normal density functions, testing goodness of fit with x2. The former (i.e., linear) fits satisfactorily in five out of nine comparisons. Circular-type frequencies are computed in four of nine examples; two of the four fit satisfactorily. In part III, we define the dispersion measure ß' as the angle enclosing a fictitious uniform distribution having the same vector strength ($$\bar{a} \equiv R/n$$) as the observed data. After showing a high linear correlation between β' and s (standard deviation) of our earth-science data, we derive the basis for this relationship in terms of equivalent uniform distributions. Further, the corollary relation between $$\bar{a}$$ and s is shown to be $$\bar{a} = (sin \sqrt{3} s) / \sqrt{3} s$$. Concluding remarks deal chiefly with this study's shortcomings.

doi: 10.1086/626392pmid: N/A

A mechanism for recrystallization and differentiation due to shearing and pressure is proposed. The stratiform ultrabasic rocks have been chosen as a specific example, since they commonly lie in the zone of high pressure and strong shearing stress along the flanks of folds. The associated serpentine commonly occupies the low-pressure area along the crest of the fold. Under this hypothesis, such stratiform ultrabasic rock masses are interpreted as being a residual accumulation forming a dense Fe- and Mg-rich rock in a region undergoing a volume decrease because of the depletion of the initial $$SiO_2$$ and other more acid elements. The stratiform ultrabasic rock is considered an extreme end-member in a system involving shearing stress and high pressure. The presence of such a residual mineral assemblage in a depleted area is demanded by the common existence of pegmatites and ore bodies which are the extreme end-members characteristic of low-pressure zones. A spectrum of conditions, compositions, and rock types must lie between these two general extremes. A low-pressure area in the structure will favor reactions with an increase in volume and will therefore be a scene of enrichment. The formation of a mineral assemblage characteristic of low-pressure areas contemporaneous with the formation of an assemblage characteristic of high-pressure sheared areas suggests a correlation between metamorphic facies and their structural environment. A series of reactions is presented and an explanation offered to account for this association of mineralogy and structure. These mineralogical reactions result from and conform to the structure; consequently, even if they result in a volume increase or decrease, they are unable to cause a distortion of the structure. The influence of shearing stress and pressure in determining the new mineral assemblage is through the control of the composition. The greater density of the new suite of minerals developed in a high-pressure and shearing zone is a direct result of the new composition rather than of an accommodation to increasing pressure with no bulk compositional change. The new mineral assemblage contains whichever minerals can grow from the remaining material under the existing stress, temperature, and pressure conditions. A thermodynamic evaluation of some of the mineral reactions in the system $$MgO-SiO_2-H_2O$$ is used to demonstrate this model. Heat-of-formation data are presented for serpentine (-39.53-0.36 kcal/mole) and talc (-35.53-0.36 kcal/mole), and petrological interpretations are drawn from the reactions which produce them. A comparison is made between an open-system and a closed-system reaction, both of which produce the same mineral assemblage. The open-system reaction is shown to take place at lower temperatures.