362 Densities of Solutions of Soda and Potash. 



given in column A of Table II. are obtained*. In this case 

 the total error is 1*6 times the known experimental error, 

 but this comparatively small excess above what it should 

 be is, as will be seen, due entirely to the magnitude of the 

 e 2 error, and might be caused by even one experimental 

 point showing an accidentally large error. Very little weight 

 ought therefore to be attached to it, and the drawing cannot 

 be regarded otherwise than as agreeing very closely with 

 the known experimental error ; indeed, the mean apparent 

 error of the points (^) agrees almost absolutely with the 

 experimental error— -00000627 against -00000631. 



If the breaks shown by this drawing have no real existence, 

 and if it is merely by some strange chance that the drawing 

 has been made in precisely the right number of sections to 

 produce this remarkable concordance between the apparent 

 error of the points and the known experimental error, we 

 ought to get a similar concordance with any other drawing of 

 a similar degree of complexity independent of the particular 

 positions at which the breaks are made to come. But this is 

 not so. The results given in columns B show that if the 

 breaks are made to come at points intermediate between those 

 in the previous drawing, the apparent error is increased from 

 1*6 to Q'Q times the experimental error, every one of the factors 

 which make up the total error being in excess of what it 

 should be. Therefore the breaks shown by the A drawing must 

 be in reality singular points in the figure. 



In the B drawing, it must be noticed, it is necessar}'- for the 

 sake of fair comparison to omit the half sections which are left 

 at each extremity of the figure, and to deal only with the re- 

 maining eight sections which are comparable in length, and in 

 the number of experimental points on them, with the nine sec- 

 tions in the A drawing. It will also be noticed that the sums 

 of the + and — errors are not very equally balanced ; but in 

 this case, as in some others which have been investigated (see, 

 for instance, Ber. xxv. p. 1593), a drawing which gave better 



* The situations of the experimental points are indicated sufficiently here- 

 by the approximate values in the columns headed p. e x is the mean apparent 

 error of the points, e 2 is the error due to the existence of apparent errors 

 of improbable magnitude, e 3 that due to runs of errors with the same 

 sign; E, the total error, =e l Xe 2 Xe 3 , and the relative error is the 

 ratio of the total error to the experimental error, e. The relative error, of 

 course, in an acceptable drawing should be nearly unity, as also should 

 the e 2 and e 3 errors, while e l and E should be nearly equal to the experi- 

 mental error. For further details see Phil. Mag. xxxiii. p. 437. 



