METHODS EMPLOYED IN CALIBRATION OF MERCURIAL THERMOMETERS. 163 



If these rules are applied to the numbers in Table V. the correction 

 at the principal points are found to be 



118, 153, 106, 113, 134, 110, 110, 60, and 10, 



as given by the three Gay-Lussac curves, and 



121, 156, 109, 117, 138, 113, 112, 61, and 10 



by the Thiesen curve. 



As these nowhere differ by more than 0004, and by that only near a point 

 where the Thiesen curve is certainly in error (116°), and as they nowhere 

 differ fi-om the results given by Bessel's curve by more than O'OOS, except 

 at 136, where the cause of the difference has been explained, it follows 

 that both methods attain the second standard of accuracy. The three 

 Gay-Lussac curves would involve altogether sixty thread-measurements ; 

 the Thiesen and two Gay-Lassac curves would involve 104, so that on the 

 whole it would seem better to adopt the former plan, and if when the 

 calculation is concluded there seems any reason to doubt the result, to 

 make another Gay-Lussac correction-curve, and include it in obtaining 

 the final results. 



The general result of the investigation may, therefore, be summed up 

 as follows — that labour is saved, and equal accuracy is obtained, by the 

 repetition of the simplest method of correction (Gay-Lussac's) instead 

 of the employment of more elaborate and, theoretically, more perfect 

 schemes. 



Part II. 

 Details of Calculations, with Examples of each Method. 



(21) The following system of symbols will be adopted. 



The upper and initial points of a thread will be indicated by the 

 letters u and i. 



An uncorrected thread-length is therefore v, — i, indicated by t. 



The corrections at any point x to the first, second, &c., degrees of 

 approximation are indicated by ^jj (x), (p2 (a;), &c. Since the calculations 

 often give the differences between these quantities, it is convenient to 

 define 



f (^^) = 92 (^) - <P\ («) 



and so on. 



A corrected thread- length is indicated by T, which is defined by the 

 equation 



'J: =, u + 9 (u) - (i + 9 (i)) 



In principal point methods it is convenient to follow Thiesen (Zoc. 

 citS) in the use of the symbol S, such that 



B(x + l) = (t,(x + l) -9 (x), 



where the symbols on the right refer to the x + V^ and x^^ principal 

 points respectively. 



In cases where the true positions of the thread-ends are not coincident 

 with their theoretical positions, the latter are often indicated by u and i,. 

 the actual positions by ?t + Am, and-i +Ai. 



When in a step-by-step method the correction is referred to the mean 

 point between the positions of the upper end of the thread and that of 

 the lower end, when next shifted, that mean point may be indicated by u. 



M 2 



