Results of Experiments. 421 



When the terms of an equation of the second degree are 



dv 

 differentiated, and the differential — is replaced by a new 



ordinate z suppose, an equation of the first degree represent- 

 ing a straight line is obtained. Hence the hydrates of sul- 

 phuric acid are apparently made to depend upon the flexibility 

 of the steel lath used. 



(iii.) The empirical method — the form of the equation is 

 guessed at, often by the aid of the plotted curve ; the con- 

 stants are determined from the experimental results, and the 

 calculated are compared with the observed numbers. 



In the great majority of cases in Physics and Chemistry 

 the value of the variant for any value of the variable may be 

 expressed with sufficient accuracy by one or more terms of 

 the equation 



Ux = </>(#)= a + bx + ex' 2 + dx 3 + 

 where a, b, c, d are constants which may be zero or negative, 

 and must be determined by experiment. 



The assumption that (f>(x) can be expanded in ascending 

 powers of x is by no means universally true. In some cases 

 a derived function becomes a constant and the series becomes 



finite ; in other cases some of the differential coefficients 



_i 

 become infinite and the series fails, e. g. e * when # = 0. In 

 some practical cases, as in that of the tension of aqueous 

 vapour, the series required is unmanageable and another 

 equation must be chosen. 



Four values of u are determined by widely separated expe- 

 riments, in one of which, if possible, x is made equal to to 

 determine a ; the constants are then solved for by the theory 

 of determinants, or by the method of cross multiplication due 

 to Lagrange, which gives : — 



_ u Y - a(xl^ 3 - x l x l) + u 2 ~ a(x 3 A ~ 3; *A) + u z - a{x\^ 2 ~ ^ x x\) 



b = 



7/., 



«i(«J«J — x\x\) + x 2 {x\x\ — x\x\) + x^x^xl — x\x%) 

 — a (xlx B — x 2 xi) +u 2 — a(x x xt — x\x^] 4- w 3 — a ( x\x 2 — x x x^) 





x\ (x\xs — x 2 x\) + x\ (x^l — x\x s ) + x\ {x\x 2 — x x xl) 



u x 



— a (x 2 x\ — x\x^) + u 2 — a [x\x z — x^ci) + u B — a [x x x\ — x*x 2 ) 



d= 



x\{x 2 x\—x\x^) + x^x^—x^f) + x'Kx^l— x\x 2 ) 



Were all the results absolutely accurate, the same values 

 for the constants would be obtained whatever experiments 

 were used; but owing to errors of observation the values 

 found differ, and the most probable ones can be obtained by 

 the following method. 



