XI.The application of least squares

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Abstract
The principle of least squares is applied to the general problem of finding the most probable values of observed quantities and of parameters that are related by equations of various sorts. The entire derivation is based on the statement of the principle that the weighted sum of the squares of the residuals is to be minimized with respect to the most probable values of the observed quantities and of any parameters that enter relations between these quantities. These relations will vary in form and number from one type of problem to another. The general solution takes the form of symmetrical equations, and the term “general normal equations” is suggested for them. For a specific type of problem these reduce to a form characteristic of that type.

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