Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/132

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100
MAGNETIC MEASUREMENTS.
[454.

If we suppose that the probable errors of these equations are equal, as they will be if they depend on the determination of only, and if there is no uncertainty about , then, by multiplying each equation by and adding the results, we obtain one equation, and by multiplying each equation by and adding we obtain another, according to the general rule in the theory of the combination of fallible measures when the probable error of each equation is supposed the same.

Let us write

for &c.,

and use similar expressions for the sums of other groups of symbols, then the two resultant equations may be written

,

,

whence

,

and


The value of derived from these equations ought to be less than half the square of the length of the magnet . If it is not we may suspect some error in the observations. This method of observation and reduction was given by Gauss in the 'First Report of the Magnetic Association.'

When the observer can make only two series of experiments at distances and , the value of derived from these experiments is

, \quad .

If and are the actual errors of the observed deflexions and , the actual error of the calculated result will be

.

If we suppose the errors and to be independent, and that the probable value of either is , then the probable value of the error in the calculated value of will be , where

.