128 THE ROTARY PROPERTIES OF 



Sources of Error. 



A careful investigation of all possible sources of error, both 

 in determining the strength of the solutions and in measuring 

 their rotary power, led to the conclusion that by far the most 

 important source of error lay in the observer himself. The 

 values found at different times for a single setting of the 

 polarimeter, each the mean of thirty or forty readings and 

 having a probable error of less than 0.01, would frequently 

 differ among themselves by 0.05 or more, especially when 

 the intensity of the sodium flame varied or the observer's eyes 

 became fatigued. The error thus introduced into the value 

 of the specific rotation was from 2 to 5 per cent, accord- 

 ing to the size of the angle measured, while the errors 

 introduced by the methods of determining the percentage of 

 proteid matter hi solution, which ranked next in magnitude, 

 never exceeded 0.3 or 0.4 per cent. 



To find the probable error of the measured angle of rotation, 

 the probable error of the initial setting with an empty tube 

 was calculated from a large number of readings covering an 

 interval of some days, and the probable error of the mean 

 rotation reading was calculated in each case from the results 

 of a number of independent series of readings. The square 

 root of the sum of the squares of these gives the probable 

 error in the value of the total rotation, and the probable error 

 of the specific rotation is easily calculated from this. In 

 calculating the probable error of a number of measurements 

 of the same angle the abbreviated formula* 



0.8453 2v 



was used, where r is the probable error of the mean, v the 

 deviation of each observation from the mean, 2 the usual sign 

 of summation, and n the number of observations. 



* See Merriam's Text of Least Squares, 2d ed., p. 93. 



