# Equation Definitions for Land Surveyors

**equation, angle—**A condition equation which expresses the relationship between the sum of the measured angles of a closed figure and the theoretical value of that sum, the unknowns being the corrections to the observed directions or angles, depending on which are used in the adjustment. Sometimes termed “triangle equation,” it is used to make the sum of the three observed angles of a triangle, with corrections applied, equal to 180° plus the spherical excess of the triangle.

**equation**, **azimuth—**A condition equation which expresses the relationship between the fixed azimuths of two lines which are connected by triangulation or traverse. When a survey (triangulation or traverse) connects two lines whose azimuths are fixed, by direct observation or by previous surveys, an azimuth equation is used to make the azimuth of either line as computed through the adjusted survey from the other line agree with its azimuth as previously fixed.

**equation, condition—**An equation which expresses exactly certain relationships that must exist among related quantities, which are not independent of one another, exist a priori, and are separate from relationships demanded by observation. For example: In measuring the angles of a triangle, no condition exists until all three angles are measured. The condition equation will then express the condition that the three measured angles plus certain corrections must equal 180° plus the spherical excess of the triangle. The various condition equations set up in surveying are defined under terms which are descriptive of the conditions—angle equation, side equation, length equation, latitude equation, longitude equation, and azimuth equation.

**equation, correlate—**An equation derived from an observation or condition equation, using undetermined multipliers, and expressing the condition that the sum of the squares of the residuals (or corrections) resulting foul the application of these multipliers to the observation or condition equations shall be a minimum. In the least-squares adjustment of triangulation, correlate equations are formed directly from the observation or condition equations, there being as many correlate equations as there are corrections to be determined, but only as many undetermined multipliers (correlates or correlatives) as there are observation or condition equations. From these correlate equations, the normal equations are formed, equal in number to the undetermined multipliers which constitute the unknowns in the normal equations. The solution of the normal equations determines values for the multipliers (which, when substituted into the correlate equations, give values for the corrections which will satisfy the observation or condition equations, make the observations and their functions consistent among themselves, and at the same time make the adjusted values the most probable that can be derived from the given observations).

**equation, error—**The probability equation which expresses the laws of the occurrence of accidental errors. The error equation is the basis of the method of least squares, used in the adjustment of observations for determining the most probable value of a result from those observations.

**equation, latitude—**A condition equation which expresses the relationship between the fixed latitudes of two points which are connected by triangulation or traverse. When a survey (triangulation or traverse) connects two points whose latitudes have been fixed by direct observation or by previous surveys, a latitude equation is used to make the latitude of either point, as computed through the survey from the other point, agree with its latitude as previously fixed.

**equation, length—**A condition equation which expresses the relationship between the fixed lengths of two lines which are connected by triangulation. When a section of triangulation connects two lines whose lengths are fixed by direct measurement or by previous triangulation, a length equation is used to make the length of either line, as computed through the adjusted triangulation from the other line, agree with its length as previously fixed.

**equation, longitude—**A condition equation which expresses the relationship between the fixed longitudes of two points which are connected by triangulation or traverse. When a survey (triangulation or traverse) connects two points whose longitudes have been fixed by direct observation or by previous surveys, a longitude equation is used to make the longitude of either point, as computed through the survey from the other point, agree with its longitude as previously fixed.

**equation, normal—**An equation derived from observation or condition equations or from correlate equations, expressing the condition that the sum of the squares of the residuals (or corrections) resulting from the substitution in the observation or condition equations of factors obtained from the normal equations, either directly or through the correlate equations, shall be a minimum. In a least-squares adjustment, corrections are desired to observed values which are connected by a series of observation or condition equations, the number of such equations being smaller than the number of observed values on which they depend. The basic equations are transformed into normal equations, either directly or through the medium of correlate equations, which express the conditions previously listed and which contain the same number of unknowns as there are equations. Factors obtained from the solution of normal equations, either directly or through the correlate equations, are applied to the observation or condition equations to obtain the desired corrections.

**equation, observation—**An equation containing an observed or measured value, its residual errors, and unknown parameters. One equation is written for each observation.

**equation, personal—**The time interval between the sensory perception of a phenomenon and the motor reaction thereto. Personal equation may be either positive or negative, as an observer may anticipate the occurrence of an event, or wait until he actually sees it occur before making a record. It is a personal error, for which the term personal equation is reserved.

**equation, side—**A condition equation which expresses the relationship between the various sides in a triangulation figure as they can be derived by computation from one another. A side equation is used to make the computed length of a triangle side the same for all routes through the triangulation from which it can be derived.

**equation of time**– See *time, equation of*

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Source: NSPS “Definitions of Surveying and Related Terms“, used with permission.

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