552 COLLECTED STUDIES IN IMMUNITY. 



bases are neutralized by weak acids (ammonia and boric acid). This 

 similarity constitutes the basis for their mathematical work, which 

 leads them to conclude that toxin and antitoxin are simple substances 

 whose reaction is reversible. This reaction finds its expression in 

 the curve just mentioned. Let us examine their conclusions and 

 see whether they are justified. 



The two graphic methods referred to are equally correct. Never- 

 theless it cannot be denied that the one employed by Ehrlich, the 

 so-called "poison spectrum," has certain advantages, for it brings 

 out more clearly any deviations from the regular curve. Speaking 

 mathematically we say that the "poison spectrum" is the graphic 

 representation of the differential quotients of Arrhenius and Madsen's 

 curve. In this sense, the ordinates of the spectrum represent the 

 direction of the neutralization curve, i.e., the trigonometric tangent 

 of the angle which the tangent forms at every point with the 

 axis of the abscissas. Hence, if the course of the neutralization 

 curve is that of a straight line, the direction therefore being the same 

 at all points, we must represent the poison spectrum as a rectangle. 

 If, as is often the case, the addition of a small amount of antitoxin 

 causes no decrease in toxicity (prototoxoids), so that the neutraliza- 

 tion curve in this part of its course lies parallel to the axis of the 

 abscissas, we must represent the poison spectrum as having a gap 

 at this point, for the angle between tangent and axis of abscissas 

 is 0. This brief statement should make it clear that in the poison 

 spectrum, by representing the direction of the separate parts of the 

 curve as ordinates, deviations from the regular curve-like course 

 will be more clearly shown. It may be well to study these conditions 

 by means of a diphtheria poison investigated by Madsen. 1 See 

 Figs. 1 and 2. 



These figures show that the deviations from the hyperbolic curve 

 demanded by Arrhenius and Madsen's views are much more clearly 

 shown in the representation employed by Ehrlich. Entirely aside 

 from the question whether the sharply defined zones of the poison 

 spectrum actually exist, or whether a gradual transition must be inter- 

 polated, it is certain that the changes should always occur in the same 

 way; for they merely represent the differential quotients of the 

 neutralization curve, and should therefore, if this curve were hyper- 

 bolic, show a successive decrease. The manifestly very irregular 



1 The sole object in employing this poison is to illustrate the two methods 

 of graphic representation. 



