Applying simultaneouse linear equations to physics and engineering, the measured values may involve some errors inevitably and sometimes the number of equations may be larger than that of unknowns. The most probable values are generally found from the normal equations derived by the least squares method. While, absurd solutions which are not to be in the problems of physics or engineering are sometimes obtained from the normal equations.
As a way to overcome those difficulties we developed an algorithm of constrained simultaneous equations for processing the data under the conditions that satisfy the pertinent physical meanings of solutions. The method is based upon an idea that the optimum solutions of unknowns to be solved are obtained by taking the minimum value of a penalty function. The penalty function is the sum of squared solutions which violated the restrictions. One correction term is added in each equation. When all values of correction terms are put zero, the solutions obtained from these equations are used as the initial values. The penalty function is calculated slightly changing the value of any one of corrections. Same calculations are carried out against each correction. Next, the values of penalty function which are smaller than that of former step are chosen. Among the chosen values, the minimum value which are the sum of squared corrections are taken as a value of new step. Thus the solutions obtained by iterative calculations approach to the optimum solutions.
In this paper, the algorithm of constrsined simultaneous equations is described indetail. Last, two examples are presented which are the applications of this method. The one is related to simultaneous linear equations which is ill condition. The other is concerned in particular problem which deals with simultaneous linear equations composed of measured values for obtaining spectral transmittances of an optical filter by means of retarding potential method. The later case, the simultaneous linear equations include some unavoidable errors.

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Experimental studies have been made of the valt-ampere characteristics of Ne-Hg and He-Hg lowpressure arc discharges in tubes 320 mm in length and 30 mm in diameter, operated at 50 Hz AC currents between 0.1 and 5 amperes. The positive volt-ampere characteristics with strong radiation of He and Ne spectral lines have been obtained.
Measurements have been made of the axial and radial distributions of radiation intensities of He, Ne, Hg and Hg ion spectral lines emitted from the positive column of positive characteristics. Within experimental accuracy the positive column was found to be axially homogeneous, and it was confirmed that Hg and Hg ion depletion occurred along the center of the tube.
Luminous efficacies of the positive column with thin phosphor coatings were about 12-19 lumens per watt for Hg-20 Torr Ne mixtures, 5-12 lumens per watt for Hg-40 Torr Ne mixtures and 5-10 lumens per watt for Hg-6 Torr He mixtures.

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