169] THE GOLDFISH AS A TEST ANIMAL— POWERS 49 



consideration the continuous action of the poison on the rate of reduction 

 of metabolism. But at any given time the speed of reduction of the rate of 

 metaboUsm due to the continuous action of the protoplasmic poison is in 

 proportion to the amount of poison acting at the given time and the actual 

 rate of metabolism, i.e., (M-z). Thus the speed of the inhibition of the rate 

 of metaboUsm by the continuous action of the protoplasmic poison can be 

 compared to the velocity of a simple mass action. And since the fish is in a 

 solution the volume of which is large as compared to the volume of the fish 

 and since the poison is continuously diffusing into the protoplasm to the point 

 of equiUbrium, the amount of protoplasmic poison can be considered as con- 

 stant throughout the entire process. The equation representing the speed 

 of reduction of rate of mxCtabolism by the continuous action of the protoplasmic 



dz , , 



poison then becomes — =KiX(jM-z). Ki = the efficiencv of the protoplasmic 



dt ' 



poison in reducing the rate of metabolism and X = the amount of poison em- 

 ployed and is a constant for any definite amount of poison employed and 

 appears in the equation only for the sake of comparing the action of different 

 amounts of protoplasmic poison employed in any two experiments. 



Since both of the processes indicated by the above equations are acting 

 simultaneously to reduce the rate of metabolism, the actual rate of reduction 

 is the sum of the two left members. If we now denote this actual rate of 



reduction by — ^, then --^ = KiX(M-z)-|-K2z(M-z). 



Integrating we have t-j-C (constant of integration) = 

 1 , K1X+K2Z 



MK.-fKiX K2(M-z) 



1 K X 



But when t = 0, z = 0. Then C= loge — ^' 



MK2+K1X K2M 



By substituting the value of C and transforming we have: 



1 , K1X+K2Z 1 - KiX 



logp losfp 



iVIKs-fKiX ^'K2(M-z) MK2-I-K1X ^"KaM 



1 , ,M , K2MZ 1 , 



- loge ( — h r— 7^— • - )• 



MKa-fKiX M-z Ki(M-z) X 



^ 1 ^. MKo+KiX 



Or -= \ = 



t , , M , K2MZ 1 , 



loge ( h • ) 



M-Z Ki(M-z) X 

 The reciprocal of the survival time of the goldfish - is represented as ordi- 

 nate in the graphs of the velocitv of fatality curves, thus - = Y. 



t 



