20 MATHEMATICAL BIOPHYSICS OF THE CENTRAL NERVOUS SYSTEM 



a(x) = a(x) ~ XI , (16) 



where 



I=f [*(x') -h]dx',k = pN. (17) 



Now, if we knew the limits of the region over which a > h , we could 

 substitute expression (16) for a into the integral in expression (17), 

 integrate, solve for / , and finally place this value in relation (16) to 

 obtain a . Not knowing these limits, we proceed as follows. Since o- 



and a differ only by a constant, the limits of the region can be defined 

 by the equation 



«{x)=[i, (18) 



for a suitable ii . Leaving /u for the moment undetermined, we carry 

 through the steps as outlined except that the range of the integration 



is to be defined by o- > n . We first obtain 



fa(x')dx' - hM(fx) 



/</*) = (19) 



1 + XM(ti) 



where M([i) is the measure (length, area, or volume, according to the 



dimensionality) of the region o- > ii . But since a > ii and <r > h define 

 the same region, it follows from (16) that 



it-XI(ii)=h. (20) 



Hence if we solve this equation for /a , then we find, by equations (16) 

 and (20), that 



(x) =7(x) +h- p. (21) 



si 



It is evident that the procedure here outlined is applicable, with obvi- 

 ous modifications, in case AT is a function of x' but is independent of x . 

 From the fact that a and cTdiff er only by a constant, certain prop- 

 erties of the solution are at once apparent. If o- anywhere exceeds h , 

 then o- must somewhere exceed h . For if a nowhere exceeded h , then 



I = , a = a , and we have a contradiction in the fact that o- itself 

 somewhere exceeds h . Further, o- is a decreasing function of X . For 

 if an increase in X led to an increase in a, then by relation (17) / 

 would increase and by relation (16) a would decrease, which is a con- 

 tradiction. To suppose that a decreases as h increases leads likewise 

 to a contradiction, so that <r is an increasing function of h . If o- is 

 everywhere increased by an additive constant, a also is increased by 

 an additive constant, but the increase of o- is less than that of a . If a 



