100 Mr. E. H. Hayes on Objections to Mr. Pickering^ 



differential equation of the second order, the equation of the 

 curve (which can be expressed in terms of elliptic integrals) 

 contains five constants. It is therefore of a generality equal 

 to that of the conic 



ax 2 + hxy + by* + gx +fy = 1, 

 or of the curve 



y = a + bx + cx 2 + dx d + ex 4: (1) 



In the bent lath therefore we possess a means of drawing a 

 curve through the experimental points, which is of a more 

 general nature than those usually dealt with by arithmetical 

 methods, and in which there can be no abrupt change of 

 curvature, however slight, still less change of direction. 

 To the former change corresponds an abrupt change of 

 direction in the first differential curve, and a breaking up of 

 the second differential curve into two curves which do not 

 meet ; to the latter a breaking up of the first differential 

 curve into two curves which do not meet*. 



Difficulties of accurate drawing excepted, there appears to 

 be no a priori reason why the bent-lath curve should represent 

 the actual facts less faithfully than an equation such as (1) 

 above, the arithmetical treatment of which would be terribly 

 laborious. It is also tolerably obvious that a single equation 

 of this nature would be incapable of representing within the 

 limits of experimental error the results of a single series of 

 Mr. Pickering's experiments (e. g. one of the direct first 

 differentials of the densities): so that more than one equation, 

 probably three at the very least if the number of constants in 

 each was five or thereabouts, would have to be employed ; 

 and their precise number, as well as their respective ranges, 

 would be a mere matter of individual taste. 



If it were possible to divide the experimental results into 

 groups in all sorts of ways, calculate the constants for the 

 equation representing each group, and determine in every 

 case the difference between the observed and calculated 

 values, curves fitting the experimental points as closely, if 

 not more so, than Mr. Pickering's might no doubt be ob- 

 tained, and most valuable evidence deduced from them as to 

 the presence or absence of breaks t at particular points. 



* In the former case -— „ in the latter — is discontinuous. 

 dx~ dx 



t The word hreak is used to denote a point at which, in an ideal 

 curve which gives an absolutely correct representation of the facts, the 

 values of one of the differential coefficients of the ordinate with respect to 

 the abscissa either become discontinuous or exhibit a rate of change 

 enormously great as compared with that in the neighbourhood of the 

 point. 



