446 Mr. S. U. Pickering on the Recognition of 



14*4 per cent. (C D, fig. 1) ; and such a drawing gives an 

 average apparent error, and also a total error, of o, 028, or 

 •85 times the experimental error (column ill. Table Y.). In- 

 creasing the number of curves used to represent the figure 

 produces scarcely any appreciable diminution of the apparent 

 error till as many as five curves are used (columns iv., v., 

 and VI.) ; but an attempt to represent it by one curve only 

 (column II.) gives an average apparent error of O096, and 

 a total error two hundred times greater than the experi- 

 mental error (column n.). The two-curve drawing is, there- 

 fore, the only acceptable representation. 



The curves here mentioned were non-wavy curves, except 

 in the case of the one-curve drawing, and a difficulty arises in 

 the present case in respect to this: for in attempting to extend 

 either portion of the figure beyond the break we get far better 

 concordance by using wavy curves, and, therefore, we ought 

 to use them. But we cannot compare these results with 

 those obtained with a drawing locating the break at 14 # 4 per 

 cent., unless in this drawing also we use wavy curves. But 

 the substitution of a wavy curve for the straight line in the 

 second part of the figure is evidently (to my mind) unjusti- 

 fiable, for it follows the points more closely than the experi- 

 mental error in this region justifies. For the sake of com- 

 parison with the results obtained by parabolic equations with 

 four constants, it is also necessary to use wavy curves. The 

 only thing to be done, therefore, is to use them throughout, 

 bearing in mind that they give errors which are too small : 

 the use of such curves, moreover, masks the precision of the 

 break to a very large extent ; for the first portion of the 

 figure cannot be extended at all beyond 14*4 per cent, by a 

 non-wavy curve without a considerable increase of error, 

 although as far as this point such a curve gives just as good 

 results as a wavy one. 



The results given in columns vn. to xv. show that there is 

 an increase of error on attempting to make the break occur at 

 any point other than about 14 per cent., although the increase 

 is but small in the cases where it is moved to 16 or 18 per cent. 



Table YI. contains the results of the mathematical ex- 

 amination. Both as to the good results obtained by two 

 equations (which are comparable with those in column xi. 

 Table II.) ; the impossibility of using only one equation ; and 

 the increase in the error when the position of change is 

 shifted to either side of 14*4 per cent. — very small at first 

 but increasing with the distance to which it is shifted — the 

 general tenour of the results is the same as that of those 

 obtained by the graphic method — a break at about 14 per 



