186 



EEPOET 1882. 



The entire continuous curve in fig. 2, Plate II., made up partly of tlie 

 initial point curve, partly of the first mean curve, partly of the two com- 

 bined, is called the first approximation curve. 



(30) If a second approximation is required, the theoretical procedure 

 is to insert in Table XXI. the corrected values of the initial and upper 

 points as given by the first mean curve. Hence, to deduce a corrected, 

 Table XIX., and to proceed as before. 



A considerable saving of labour can, however, be effected by dealing 

 with the corrections in the quantities in Table XXI. alone, instead of 

 with the quantities themselves. 



Thus the corrections for the upper ends of the thread taken from the 

 first approximation curve are entered in Table XXVI. 



In Tables XXVII. and XXVIII. the same operations are performed 

 upon these as are performed in Tables XXIII. and XXIV. npou those in 

 Table XIX. Hence a series of numbers are obtained by which the correc- 

 tions in Table XXIV. would have been altered had the measurements 

 been made with the scale corrected to the first approximation instead of 

 with the uncorrected scale. 



By adding the numbers in Tables XXIV. and XXVIII. the quantities 

 entered in Table XXIX. are obtained, which constitute the corrections or 

 the second approximation for the upper points. 



Table XXIX. 



The numbers in Table XXIX. are plotted down in fig. 3, Plate II., 

 and the mean of the ordinates of the curves is taken as in the first 

 approximation, and these when added to those of the first mean curve 

 form the second mean curve. 



It is, however, necessary below 116° to combine these curves with 

 the corrected initial point curve. It has been shown that if f(ir) be the 

 coi'rection for the r* initial point, <p(i,.) = h,. = h^ — i,. plus a constant 

 which, as differences only are required, may for the present be neglected. 

 Now, by taking the mean of the initial and upper point curves at that 

 part of the scale where they overlap, ^(iV) is changed into 



'/'(v) = f(i) + '^r, 



where -cr,. is the difference at that point hetween the first mean curve and the 

 first initial point curve. 



