OF AETS AND SCIENCES. 79 



III. 



GRAPHICAL INTEGRATION. 



Bt Edatakd C. Piceebing. 



Presented, Oct. 13, 1874. 



When determining the relation between two physical quantities, we 

 sometimes are able to measure only the relative rates at which they 

 alter, instead of the alteration* themselves. Or, to speak mathemati- 

 cally, if i/=f(x), instead of measuring various corresponding values 



of X and y, we can obtain only the values of x and J^ =f' (x). Of 



course, if the form off (x) is known, the ordinary methods of integra- 

 tion give/(a;) and y. But in general this is not given, and the usual 

 methods of approximation are liable to introduce large errors, since by 

 the summation the error adds, and the deviation continually becomes 

 greater and greater. The problem is perhaps better understood by 

 some familiar examples. Thus, given the velocity of the wind at certain 

 times, to determine its total distance travelled per hour; given the 

 velocity of a river, at various points of its cross-section, to find its 

 total discharge ; given the strength of an electric current, to find the 

 total quantity transmitted. The case which actually suggested this 

 problem was in calibrating a thermometer tube, having given the 

 length of a mercury column at various points in the tube, to determine 

 the correction to be applied for unequal diameters of the tube at vari- 

 ous points. Here the various lengths of the column give the values of 



-j^, and the distance of its centre from one end gives the corresponding 



values of x. Construct a curve with these two quantities as co- 

 ordinates, and the area included between this curve and the axis 

 of X serves to measure the true values of y. To determine this area, 

 draw a number of equidistant ordinates, and read from the curve the 

 length of each. Then compute by Simpson's formula, A = §a 

 [(yo+ 4 1/i-i-y-i) + (2^2+ 45^3 +2/4) + &c.], the area included between 



