Changes of Curvature by Means of a Flexible Lath. 441 



of various drawings, whether we consider any one of the three 

 errors separately, or the combined results of all of them. 

 By taking account of the extent to which e l is less than the 

 experimental error, and of the extent to which e 2 and e 3 are 

 less than unity, it would obviously be possible to estimate the 

 improbability of a drawing which follows the experiments too 

 closely, as well as that of one which does not follow them 

 closely enough, such as is here alone considered. 



Examination of Case I. 

 The first series of results examined consisted of the freezing- 

 points of solutions of propyl alcohol in water from to 32 per 

 cent, in strength. They are given in the first part of Table I.* 

 The average experimental error is *0215°. The figure which I 

 consider they form is represented by A B, fig. 1 (p. 445). Such 

 a drawing represents an average apparent error of '0275° in 

 the experimental points (column in. Table II.), and, as the % 

 and e B errors are less than unity, the total error of the draw- 

 ing is *0275 also, or 1*28 times the ascertained experimental 

 error, an agreement as close as could be expected when deal- 

 ing with a comparatively small number of points f. When 

 drawn as four curves of equal length, we get a change of 

 curvature at the same place as in the two-curve drawing, 

 and, consequently, a slight diminution in the total error — 

 to 1*07 times the experimental error (column v.), — but so 

 slight that it would afford us no justification for accepting 

 such a drawing in preference to the much simpler two-curve 

 one, this latter being also in harmony with the experimental 

 error f. 



* It was subsequently ascertained that the propyl alcohol here used was 

 not quite pure, but this will not affect the results for the present purposes. 

 Series of determinations have been made with a purer specimen, and the 

 nature of the figures and the position of the breaks were found to be the 

 same, though the magnitude of the depression was slightly different. 



t Any drawing must, in the case of the present results, pass through 

 the zero point, the freezing-point of pure water, for all the other points 

 are determined by comparison with this. Also, any two consecutive 

 curves must always be drawn so as to meet, since the property must be a 

 continuous function of the strength. It has not been thought necessary 

 to give the readings of the curves here, or the values given by the equa- 

 tions in Table III., but only the differences between them and the 

 experimental points. 



% Since writing the above I have done a good deal more work on the 

 subject, and consider that a closer agreement between the apparent error 

 of a drawing and the experimental error than is found in this and in some 

 of the other cases here considered may be expected. But to obtain such 

 an agreement the experimental error must be determined by some means 

 more certain than that of the comparison of duplicates. The results of 

 an examination of about ten series of results are now appearing in the 

 Ber. d. deutsch. ehem. Gesel. 



