220 MESSRS. A. V. HAECOUET AND W. ESSON ON THE LAWS OF CONNEXION 



a„ a^ being the quantities of the substances originally introduced into the system, and 

 a„ Ka the fractions of them which disappear in a unit of time. 



If, however, the substances are not independent, but are such that one of them is 

 gradually formed from the other, we have a different system of equations to represent 

 the reaction. 



Let u, V be the residues of the substances after an interval x, y{-=.u-\-v) being the 

 total residue actually measured at that time. Let the initial values of u and vhQU=a, 

 t>=0 ; let au be the rate of diminution of u due to its reaction with one of the other 

 elements of the system, and /3w its rate of diminution due to its reaction with another 

 of the elements of the system, by means of which v is formed, and let yv be the rate of 

 diminution of v, then 



'^"=-(«+/3K (18) 



dx 



dv 

 dx 



J=/3r.-y., (19) 



whence 



M=ae-'"+^)', (20) 



'>^=^^y{e-^'-e-''''") (21) 



3/=„T^^{P^-^'+(«-yK'"^''"}- (22) 



There are several particular cases of these equations which require to be considered 

 separately. 



(1) j3=0. Fraction of V formed=0. 



In this case the system of equations reduce to . 



o— a* 



u=.ae , 



i)=0, 



y=iae~". 



The experimental case corresponding to this is that recorded on page 209. 



(2) y>a. Fraction of v decomposed in a unit of time, greater than the fraction of 

 u decomposed in a unit of time. 



In this case the last equation of the system is of the form 



The experimental case corresponding to this is that recorded on pages 210, 215, n=l. 



(3) y=a. The fraction of v decomposed in a unit of time equal to the fraction of 

 u decomposed in a unit of time. 



In this case the last equation of the system reduces to the form 



y=.ae~". 

 The experimental case corresponding to this is recorded on pages 210, 215, n=3. 



