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



show the results of an attempt to bridge over the change of 

 curvature at 16 per cent, by a single equation : any four 

 points could be represented with absolute exactness by an 

 equation such as is here used, and yet we cannot get an 

 equation to represent seven of the points in this part of the 

 figure — three on either side of 16 per cent. — without an error 

 twenty-one times greater than is legitimate, and even six or 

 five points cannot be thus represented without this error being 

 five times greater than it should be. This would seem to be 

 quite conclusive that there is a change of curvature here which 

 cannot be smoothed over by any simple equation of this sort. 



I have also examined mathematically the portion from 16 

 to 32 per cent, taken separately, to ascertain the result of 

 representing it by two curves instead of one, and find that 

 such a representation, and also representations in which any 

 six or eight of the nine points are taken, give practically the 

 same total error as a representation of all the nine by one 

 equation — namely, from '021° to '025° ; so that any splitting 

 up of this portion of the figure is unjustifiable. 



Thus it will be seen that the mathematical examination 

 leads to precisely the same conclusion as does the graphic ; and 

 for two different methods to lead to the same conclusion as to 

 the existence and position of the breaks must be a strong 

 argument in favour of the reality of these breaks, even if it 

 can be urged that neither of the forms of curves used in the 

 two methods is a really suitable one. 



It may be noticed that the errors given by the one method 

 are sometimes larger and sometimes smaller than those given 

 by the other, though their general tenour is always the same. 

 This must inevitably be the case ; for the bent-lath curve is 

 not a parabola, and, in addition to this, there are different 

 sources of inaccuracy in the two methods. In the graphic 

 method we have various errors introduced by imperfect plot- 

 ting, reading, and drawing ; while in the mathematical method 

 the curve deduced necessarily makes the sums of the positive 

 and negative errors exactly equal, which, as I have stated 

 above, does not necessarily give us the truest representation. 



It is perhaps not altogether unnecessary to correct an 

 erroneous opinion sometimes held, that a change of curvature 

 in a series of freezing-point determinations implies that a 

 different substance Or hydrate crystallizes from the solution. 

 This is, of course, not the case : the solvent, water in the 

 present instance, is the crystallizing substance throughout ; 

 and the temperature at which ice can be separated from the 

 liquid is as much a continuous property of the solution as is 

 any other property : the breaks in the freezing-point curves 

 are precisely similar to those in the density &c. curves. 



