OF ARTS AND SCIENCES. 



81 



The problem is to determine the correction to be applied to the ob- 

 served temperatures, assuming the 0° and 100° points to be correct. 

 Constructing a curve with the co-ordinates given above, we deduce the 

 points given in columns one or two of Table II. Now, calling m the 



volume of the mercury drop, we have z:m=^dx: dv, or -r- = ^, Hence, 



we must use for ordinates in our summation the reciprocal of z as given 

 in column three. Treating these as before, we obtain by the formula 

 column four, and dividing by the total sum 283.8 gives in column five 

 the true temperature, and subtracting the observed readings from these 

 gives the correction in column six. 



TABLE II. 



To determine how rapidly the errors diminish, increasing the number 

 of ordinates, the area included between the axis and the curve y = 



— sin X was computed for 2, 4, 6, 12, and 18 divisions ; the errors in 



these cases were .030047, .001454, .000276, .000019, .000003, so that 

 a high degree of accuracy is readily obtained. M. Chevilliet has re- 

 cently shown (^Comptes Rendus, Ixxviii. p. 1841) that the error in 



h* d^ u 

 Simpson's formula depends on j^ -j4^, while the method of summing 



by trapeziums gave r^ -f-. In an example he finds that the area of 



the curve x log x, between a;= 10 and x=. 20, is given correctly by 

 Simpson's formula, taking ten intervals, within .000005, while by the 

 method of trapezoids the error is .001809. Evidently, then, it is easy 

 to obtain by the first of these formulas as great an accuracy in the result 

 as is needed in almost any physical research. 



VOL. X. (n. S. IT.) 



