80 



PROCEEDINGS OF THE AMERICAN ACADEMY 



ench second ordinate, the curve, and the axes. It gives the ordinate 

 of points of the required curve, y =f (x), the abscissa being of course 

 that of the limitinsr ordinate. 



To test these principles by an actual example, the following method 

 was emplo}'ed. A sraooih curve was drawn by a pencil on a sheet of 

 I)aper divided into squares, and the co-ordinates of six points on it noted 

 as follows: a:- = 0,7, c: = .84; x= 2.3, 2:= 1.14 ; a;=4.4, 2 = 1.65; 

 a: = 5.8, 2=2.05; x=7.6, 2=2.09; a; =9.6, 2=3.54. It was 

 then assumed that by some measurement these observations had been 

 obtained, and that while x represented one of the variables 2 gave the 



relative rate of change, or -~. These points were then laid off on a 



fresh piece of paper, and a smooth curve drawn through them. Of 

 course this should agree with the oiiginal curve, were there no errors ; 

 and the deviation serves to show the amount of error to be expected. 

 To obtain two independent results, a third curve was constructed, like 

 the second, on another piece of pajier. The values of z for r = 1, 2, 3, 

 &c., v/ere then determined on curves two and three, with results given 

 in Table I., columns two and three. Applying Simpson's formula gives 

 the numbers in columns four and five, which it will be seen agree very 

 nearly, the difference being but little more than the errors of observation. 

 Of course, if necessary, still closer results could be obtained, by residual 

 curves and other methods ; but in general the accidental errors present 

 in the original observations render this refinement unnecessary. 



TABLE I. 



As another example, suppose the following measurements made in 

 calibrating a thermometer tube: a? = 5°, 2=10°.0; a; =28°, 2 = 

 10°.4 ; X = 54°, 2 = 10°.7 ; x = 83°, z = 10°.9, in which 2 gives the 

 length of the mercury column, and x the position of its middle point. 



