Approximate calculations by means of infinite series

The reader will, perhaps, have been impressed with the frequency with which experimental results are referred to a series formula of the type  

In the absence of any knowledge as to the proper setting of the Umv ant - Digitized by Microsoft  

When the progress of any physical change is represented by the above formula, the approximation is closer to reality the greater the number of terms included in the calculation. This is best shown by an example. The specific gravity s of an aqueous solution of hydrogen chloride is an unknown function of the amount of gas p per cent, dissolved in the water. (Unit water at 4° = 10,000.) 

The first two columns of the following table represent corresponding values of p and s, determined by Mendel6eff. It is desired to find a mathematical formula to represent these results with a fair degree of approximation, in order that we may be able to calculate p if we know s, or, to determine s if we know p. Let us suppress all but the first two terms of the above series, 

Formula (2), therefore, might serve all that is required in, say, 

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