•052 Dr. S. R. Milner on the 



Since z is an indefinitely small quantity, the value of this 

 integral will not be altered by altering its lower limit from 

 i to 0, and since 



we have 



,. .2, 1-e X l " 1, / 7^5 



!L m JJ„ WTiy =i V--(t) d (*-^) 



Introducing this into (19) we obtain 



Cx=o = c x + 2 \ / 5. § g r F V^ • . . (20) 



V 7T r=0 



(20) is thus the solution for the case in which F(t) consists 

 of sudden finite variations separated by intervals during 

 which the function remains constant. But it is clear that 

 the result will hold however small the variations and the 

 intervals ; it will consequently hold in the limit when F(t) 

 undergoes a continuous variation. Thus let F(t) be a con- 

 tinuous function, the value of which is zero before the instant 

 t = 0. Expressing it in the form 



F(O=fV(0)d0, 



we may replace in (20) 



t r by 0, S r F by F'(0)tf0, and i by C . 

 We thus obtain 



e tm ^« l +ay/^* > /i^e.F / (e)d$, . (2i) 



and from this expression numerical values may at once be 

 obtained when the form of the function F(t) is known. 

 This form of expression may be considered to include the 

 case in which the function undergoes sudden finite changes 

 of value as well as a continuous variation ; for if a sudden 

 variation occur at the instant 0, F'(0) is infinite, and 

 F'(6)d6 may be put equal to BF the finite change in the 

 value of the function ; the corresponding part of the integral 

 then simply gives rise to a term 



2\/-8F s/t^O 



V 7T 



in the expression for the concentration, which is in agree- 

 ment with the corresponding term obtained from (20). 



