1910.| The Chemical Dynamics of Serum Reactions. 497 
take smaller and smaller values, this asymptote approached closer and closer 
tox—0=0. At the same time, it was clear that, in general, these inclined 
asymptotes cut the axis of y at some point other than the origin; and, on the 
other hand, this point of intersection reached the origin when z became zero 
that is, there was no lysis in the absence of both amboceptor and complement. 
I chose, then, as the final equation for this asymptote, 
x 
——y+cz = 0. 
a 
Hence the equation of any of these curves, in which z was constant, became 
y(—y+e2] =i), (3) 
where d was in general some function of z, to be determined. 
The experimental curves showed that the lytic value for any one of these 
curves could not be attained if the complement fell below a minimal amount, 
depending on the constant value of the lysis for the particular curve in 
question. Seeing that the phenomenon of “diversion” begins here, the 
corresponding point on any hyperbola may appropriately be called its 
“point of diversion.” Ata point of diversion the tangent to the hyperbola 
is parallel to the axis of y; hence to find these points we have only to find 
x and y from equation (3), and from the equation got from putting dz/dy = 0 
derived from it when 2, and therefore d, are constant. This latter equation is 
~——2y + ez == 10) (4) 
Whence, by means of (3), we get at once 
P= xf, and "2 = 22,/d— cz? (5) 
the co-ordinates of the “ point of diversion” in terms of z. 
The co-ordinates of the “point of diversion” of any of these hyperbolas 
satisfy equation (4) for z= constant. The envelope of this line, for varia- 
tions of 2, is 
UP = Gy. (6) 
If we assume that all “ points of diversion” lie on this parabolic cylinder—a 
view which, at least, seems favoured by the appearance of the curves—we 
are able to determine d as a function of z. For, writing equation (4) in 
the form 
“w—2yz+c2 = 0, 
by equation (6) it becomes 
y?—2yce+e2=0, or y= eX, : (7) 
whence @ =o 
