It has long been recognized that the least-squares estimation method of fitting the best straight line to data points having normally distributed errors yields identical results for the slope and intercept of the line as does the method of maximum likelihood estimation. We show that, contrary to previous understanding, these two methods also give identical results for the standard errors in slope and intercept, provided that the least-squares estimation expressions are evaluated at the least-squares-adjusted points rather than at the observed points as has been done traditionally. This unification of standard errors holds when both x and y observations are subject to correlated errors that vary from point to point. All known correct regression solutions in the literature, including various special cases, can be derived from the original York equations. We present a compact set of equations for the slope, intercept, and newly unified standard errors.

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In a recent publication, O'Nions et al. [1979] used geochemical constraints to solve transport equations describing the evolution of the outer 50 km of the earth. The paper includes a very helpful compilation of the available data useful in such a solution and provides a valuable overview of the problem. However, it can be shown that the paper contains serious computational errors which put in question a number of the quantitative conclusions. The analysis is based on transport equations of the form dni•/dt = a(t)niM-13(t)ni• (1) dniM/dt = 13(t)ni • -a(t)ni M The symbols n• and n• represent the total number of atoms of element i in the 50-km layer and in that part of the mantle with which it interacts, respectively. The symbols a and/3 are first-order transport coefficients. For potassium, for the O'Nions model II, the transport coefficients are stated to be time-dependent quantities of the form a(t) = 2.6 exp (-2.0t) /3(0 = 20.0 exp (-0.65t) (2) where the numerical quantities are in units or reciprocal gigayears (Ga-•). The computational difficulties of the O'Nions solution are immediately evident from a consideration of these equations.It is not explicitly stated in the reference cited whether the authors considered a crustal layer of constant dimensions with composition evolving from an initial value equal to that of the mantle, or whether the crustal layer was assumed to evolve from zero initial size. Because they criticize earlier models for starting with a finite crustal size, there is a strong implication that their model starts with zero crustal size. (Incidentally, so do the models of Russell [ 1972] and Russell and Birnie [1973], which were included in the criticism.) Therefore, this present comment will proceed on the basis of assumed zero initial layer thickness. The other asumption was examined and shown to lead to substantially similar conclusions. At t = 0 (corresponding to the origin of the earth 4560 Ma ago) n/c • is zero and n•/w is equal to n/c ø, the total number of potassium atoms in the differentiating system. Thus, the initial rate of transport to the outer layer was, according to the equations, dn•/dt = 2.6 rtK ø atoms per Ga This quantity is not zero, and as will be shown below, is about 14 times as great as the corresponding value at any later transport maximum. This clearly violates an initial constraint of O'Nions et al. [1979] that requires the rate of crustal development to peak between 3500 and 2500 Ga ago. Geochronologicai studies suggest that the net rate of crustal growth should be close to zero, initially, as suggested by the illustrations of O'Nions et al. [1979]. Apparently, the illustrations were arbitrarily drawn to zero initial rate. It seems likely that the difficulty can be attributed to the use of a finite number of 50-Ma stages to represent the continuous evolution, but it is not immediately evident what consequences this would have on the various numerical conclusions. To determine this, equations (1) and (2) were solved with careful attention to the computational pro...