22 PROCEEDINGS OF THE AMERICAN ACADEMY. 



the corresponding operation for water, and in a manner which has 

 the advantage of preserving the irregularities which are shown by 

 experiment actually to exist, but which would have been effaced if 

 the attempt were made to give smooth second differences, as was done 

 for water. In performing this smoothing, the fact was used that the 

 change of volume of nearly all the liquids can be represented by 

 a curve of nearly the same shape. Ethyl chloride is an exception, 

 and a special computation had to be applied to it. As a first approxi- 

 mation the change of volume of the other eleven liquids was found to 

 be reproducible by a formula of the type; 



^ ^. fp- 500Y, ^fp- 500Y. fp - 500Y, . fp - 500Y2 



In order to apply this formula to any one liquid it is necessary 

 simply to multiply all four constants by the same factor. The meaning 

 of this is that to this degree of approximation the chief difference of 

 the liquids with regard to changes of volume is in respect to the abso- 

 lute, not the relative magnitudes of the change. The constants of 

 the above formula were computed, therefore, so as to apply to the 

 average of the eleven liquids. This was done by finding the average 

 change of volume for the eleven liquids at 1500, 3000, 7000, and 

 12000 kgm. and determining the four constants of the formula so that 

 the curve should pass through these four points. It will be noticed 

 that in this formula the zero of pressure is at 500 kgm., so that it 

 applies directly to the changes of volume as found above. The 

 formula was now applied to the eleven liquids in succession by multi- 

 plying the four constants by the appropriate multiplier for each 

 liquid. The multiplier was so determined that the formula should 

 give the observed change of volume at 7000 kgm. The changes of 

 volume were now calculated with this formula at intervals of 500 kgm. 

 up to 5000 kgm., and beyond 5000 kgm. at intervals of 1000 kgm., and 

 compared with the observed values. The differences between the 

 observed and the computed values were plotted on a large scale and 

 a smooth curve drawn through the points. The values obtained 

 from these smooth curves and the formula were combined to give 

 the final values for the volume at 40°. The advantage of the method 

 is that the final results lie on perfectly smooth curves, and that the 

 curves show the various slight irregularities which correspond to the 

 experimental facts but which would be smoothed out if the second 

 differences were made uniform. The values for ethyl chloride were 



