Composition of Dilute Sulphuric Acid. 305 



the results of observation appears in this case to be freer 

 from exceptional, and possibly accidental, causes of disturbance 

 than in others. 



I have therefore tried to find an equation which will express 

 a considerable portion of this curve to within the limits of the 

 error of experiment. 



If a curve, determined from isolated points, is discontinuous, 

 it is nevertheless often possible — though perhaps difficult — to 

 express it by a moderately simple continuous equation. To 

 do this, however, terms would have to be introduced which 

 express the rapid change of curvature in the neighbourhood 

 of the point of discontinuity. 



My equation, however, consists of three parts, which are 

 not suited to turning awkward corners. 



The lower part of the curve is nearly a straight line. The 

 differences between the ordinates of the upper part of the 

 curve and those of this line are very accurately in geometrical 

 progression. 



If this were all, the equation would be of the form 



y = a-\-bx — cd x . 



As a matter of fact, for the middle region this curve lies 

 a little below that given by experiment. The difference 

 gradually increases, reaches a maximum, and dies out ; but 

 the change is very gradual, extends over a considerable range, 

 and does not in the least suggest discontinuity. 



To make the two curves, given by observation and calcula- 

 tion, fli, I have introduced a term which makes the form of 

 the equation 



y=a + bx—cd x + 



n x + n' 



A curve of this kind has in all seven arbitrary constants, 

 for if we change the origin to x Q y it may be written 



, 7 , 7 ink 



ij = a +bx—c ,d x + --- 

 J n x + krn * 



where 



k = 71*0. 



In the following Table I give the values of the three 

 terms, a + hv, ccl x , and wi/(w + »"*), so that the fact that no 

 term introduces violent and sudden changes may be obvious. 



I have carried out the calculations to one place further than 

 the results of the observations are given, as Mr. Pickering 

 uses that place in the readings from his smoothed curve, 

 given in Table III. of his paper (Trans. Chem. Soc, March 

 1890, p. 145). 



