218 MESSRS. A. V. HARCOUET AND W. ESSON ON THE LAWS OF CONNEXION 

 and if at the commencement of the reaction the substances had been present in infi- 



nitely large quantities, 



y= 



1 



nx 



(8) 



The curve (6), which expresses the reaction of two substances originally present in 

 equivalent quantities, is a rectangular hyperbola, and when the original quantities are 

 infinite, the residue varies inversely as the time. This result has been already referred 

 to at pages 202 & 203, where experimental evidence of the relation has been adduced. 

 That evidence is, however, somewhat impaired by the fact that the numbers which 

 express the percentage of chemical change during the first four or five minutes fail to 

 satisfy this relation; but it may be shown that this failure is probably due to the 

 gradual formation of one of the substances which take part in the reaction. 



Let us suppose that at the commencement of the reaction there are present a equiva- 

 lents of a substance A, which during the course of the reaction is gradually changed 

 into an equivalent quantity of a substance B, and that B reacts with a substance C of 

 which a equivalents are originally present ; also let u be the number of equivalents of A 

 which remain after an interval x, and v the number of equivalents of B which remain 

 after the same interval ; then, since the velocity of diminution of u is proportional to its 

 quantity, and the velocity of diminution of v proportional to the product of its quantity 

 into the quantity of c, and the velocity of increase of v equal to the velocity of dimi- 

 nution of u, we have the following equations, 



£ = -(3- (9) 



| = -«(»+.)+(3t. (10) 



The solution of (9) is 



u=zae-^; (11) 



so that if the residue of u could be measured separately from that of v, the rate of 

 change of u into v could be determined, but in the actual experiments u and v are 

 determined together, and the relation between the total residue y[=u-\-v) and the 

 duration of the reaction x is consequently very complex. 

 Adding (9) and (10), we have, 



J+«^=0' (12) 



substituting for dx from (9), and for v its value y—u,vfe obtain the equation 

 the solution of which is 



^-4-— ---0 as) 



|e''"|c— logM+^tt-j^^MJ +...|y=l . (14) 



If we replace for u its value ae~^% we obtain an equation connecting the residue y mth 



