50 ILLINOIS BIOLOGICAL MONOGRAPHS [170 



Any mathematical expression of the velocity of fatality curve must comply 

 both with actual experimental data and with theoretical demands. 1. The 

 mathematical expression should show that the velocity of fatality increases 

 very slowly with increase in concentration when very low concentrations of the 

 poison are used. 2. It should show that the velocity of fatality increases very 

 rapidly with an increase in concentration when higher concentrations of poison 

 are used. 3. Finally \vith still higher concentrations of the poison it should 

 show a less rapid increase in velocity of fatality with increase in concentration. 

 4. At very high concentrations of poison it should show that the velocity of 

 fatality curve approaches a straight Une. See discussion page 43. 



1 K2M+K1X 



When X varies in the equation Y 



t , , M , KoMz 1 , 

 loge ( \- • — ) 



M-z Ki(M-z) X 



and all other factors on the ]*ight hand side of the equation remain constant, it 

 is found that when X is very small so that KiX is very sma.ll as com.pared to K2M 



the numerator (K2M+K1X) approaches a constant. Thus the value of — or Y is 



t 



controlled by the reciprocal of the logarithm of a number v/hich is conti oiled by 



the reciprocal of X, i.e., loge ( 1 — • • - )• When X is neither very 



^ ' ' o ^,j_^ Ki(M-z) X ^ 



large nor very small as compared to the other factors neither the numerator 



nor the denominator approaches a constant and Y is controlled by the increase 



of X in the numerator and the reciprocal of the logarithm of a number which is 



increased by the reciprocal of X. Finally, when X is very large the denomina- 



, , M , K2MZ 1 , , M , 



tor loge ( 1 • - ) approaches a constant, becomes very large 



M-z Ki(M-z) X M-z 



as compared to -^ '- Thus Y is controlled by the increase of X in 



Ki(M-z) X 



the numerator (K2M+K1X) since KiX is large as compared to K2M. From 



these three conditions it is seen that the value of Y at first increases very 



slowly, being controlled by the reciprocal of the logarithm of the reciprocal 



of a number. After this it increases more rapidly, being increased by the same 



factor as the first and in addition is increased by a multiple of the number 



itself, i.e., numerator (K2M+K1X). Finally, in the third case Y increases 



only by some multiple of the number since X is very large as compared to 



K2MZ_ - . K2MZ 1 , T^l •. • U 



the expression • — approaches zero. Ihus it is seen when 



Ki(M-z) Ki(M-z) X ^ 



— or Y is plotted as ordinate and X as abscissa a curve, the velocity of fatality 



curve, will be formed which at first has a very gradual rise followed by a rapid 

 rise which again is followed by a less rapid rise depending on the value of KiX 

 which finally approaches a straight line at very high values of X. This curve 



