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



A three-curve drawing, in which a change of curvature comes 

 at the same point, would of course show an error intermediate 

 between the errors of the two-curve and four-curve drawings ; 

 whereas, with a three-curve drawing in which the breaks do 

 not come at this point, as in the case where the three curves 

 taken are of equal length, we get a far greater error, namely, 

 36 times the experimental error (column IV.) ; and a drawing 

 of as many as even five curves of equal length (column VI.) 

 gives an error eight times greater than it should be. 



As far as a two-curve drawing, therefore, the figure may be 

 simplified either with no appreciable increase, or with an actual 

 diminution in the apparent error ; but further simplification 

 is impossible, for an attempt to draw it as one curve gives a 

 result of which the total error is many thousand times greater 

 than it ought to be (column IJ.). The two-curve drawing is, 

 therefore, the simplest, and the only legitimate representation 

 of the experimental values. 



The next point examined was whether the two curves in 

 which the figure must be drawn might be made to meet at any 

 point other than 16 per cent. Columns vm. to xm. give the 

 results obtained, and show that shifting the meeting-point to 

 even the next experimental point on either side of 16 percent, 

 increases the total error to from 270 to 20 fold in the various 

 cases, and that the farther it is shifted the more is the error 

 increased. 



In all the drawings here mentioned, the four forces applied 

 to the ends of the lath were such that the direction of the curva- 

 ture was the same throughout {vide supra, p. 141); and it was 

 evident that the use of a wavy curve would in some cases 

 produce better concordance. I therefore examined such 

 cases by the application of such curves, and give the results 

 in columns vn., xiv., xv., and xvi. ; these results, though 

 considerably better than those given by the other curves, still 

 exhibit errors far too great to permit of the drawings being 

 considered acceptable. 



For the mathematical examination, the method which I 

 have employed is the more usual one described by Mr. Lup- 

 ton (Phil. Mag. xxxi. p. 418), but not applied in the way in 

 which he applied it, for that method of application neces- 

 sitates the employment of several experimental points beyond 

 those which are actually under investigation. I have also 

 introduced one more term into the equations than he did. 



If there be n experimental points, and the values at p l} p 2y 

 &c. percentages are i/i, y 2 , &c, then the values for the con- 

 stants of a parabolic equation of the form y = a -f bx + ex 2 -f dx z 

 may be found from 



