S. C. BROOKS 65 



curve, the more individuals are succumbing per minute. The ordi- 

 nates of the mortahty curve are then at any time proportional to the 

 steepness of the time curve. The slope or steepness is best found by 

 plotting the time curve on suitable coordinate paper, drawing straight 

 lines tangent to it at several points, and counting the number of 

 squares passed over vertically by such lines for a given number passed 

 over horizontally. This ratio is the trigonometric tangent of the slope 

 of the time curve. If we draw ordinates proportional to these ratios 

 at corresponding points on the x axis, and connect the tops of these 

 ordinates by a smooth curve, we shall obtain the rate or mortality 

 curve. Since the process by which we have gotten the rate curve is a 

 graphic method of differentiation, we may consider the curve to be 

 the differential of the time or integral curve .^ 



Under certain conditions the mortality curve may be identical with 

 the variation or frequency curve of individual resistance. In the 

 following paragraphs the relationship between the time curve and 

 the variation curve is considered, starting with the simplest imagin- 

 able conditions, then varying the frequency curve alone, the course of 

 the fundamental reaction alone, and finally both together. 



If the rate of hemolysis is uniform, its time curve would be a 

 sloping straight line (the integral curve, a, Fig. 2), while since its 

 tangent or slope is the same at every point, the differential or mortality 

 curve would be a straight line parallel to the axis of the abscissae 

 {h, Fig. 2); this condition could be expressed by the differential 

 equation 



dn 



— = k 

 dt 



where n is the number of cells, t the time, and ^ is a constant. 



If we now assume the cells to be divided into classes differing from 

 each other by one unit of resistance {r) (defining as a unit of resistance 

 the power to resist one unit of fundamental reaction; e.g., the forma- 

 tion of one mol of toxic substance) , and further assume that the fun- 

 damental reaction is proceeding at a uniform rate, the differential 

 equation may be replaced by the equation of a frequency curve, or 



"This relationship has been suggested by Davey (/. Exp. Zool., 1917, xxii, 

 573), in connection with curves representing the per cent of death occurring 

 among flour beetles {Tribolium confusum) at various times after X -radiation. 



