324 UNIVEBSITY OF VIRGINIA PUBLICATIONS 



Let X — ^ (x" + x') = .Tmand y = |(rc" — x') = h, in (14). Then 



^(X") - <p{x') = -2 ^{Xrrd ^Pih), 



xl^ix") - Hx') = + 2^(a;J ypih). (17) 



7. The even function tp{x), which can never be greater than 1, is con- 

 tinuous and since ^(0) = 1 the function is positive in the neighborhood of 

 zero and cannot be equal to zero save for some finite or infinite value of x. 

 Also i/'(0) = 0. If \l/{x) remains constant and equal to in the neighbor- 

 hood of equations (16) and (17) show that i/{x) is and ip{x) is 1 for all 

 values of x which is not possible since then e''' would be constant and equal 

 to 1 which requires a; = 0. Hence }p{x) must be different from for some 

 value of x in the positive neighborhood of 0, say at x = h, where \p(x) has 

 either a positive or negative value numerically less than 1, we assume a 

 positive value. Equations (17) show that as x' increases from 0, the inter- 

 val x" — x' = h remaining constant, the function (p continues to decrease 

 from 1 and i/- to increase from as long as (p{x^) remains positive. If 

 <p{x) remains positive for all values of x the function xp must continually 

 increase and since \p can never be greater than 1 the increasing variable \l/(x) 

 must have a superior boundary equal to or less than 1 for some finite or 

 infinite value of x. It is not possible for \p{x) to increase continually to a 

 superior boundary s when a; = + <», for then (p must attain an inferior 

 boundary 



c = l/ 1 - s^ < 0, 



and (17) gives 



= c-c= - 2s\P (h), 



which is not possible since neither s nor \l/(h) is zero. Therefore ^p cannot 

 continue to increase for all values of x but must attain, since it is continu- 

 ous, a superior boundary for some finite value of x, say at x = J tt. 

 Equations (17) give 



H^tt) - -/-(Jx - 2h) - 2^{\ir - h) 4,{h), 

 ^(Jx + 2h) - rPi^w) = 2^(i7r + h) ^Pih). 



The first of these shows that ^o(|7r — h) is greater than zero. The second 

 that <p{^ir + h) cannot be greater than zero. Hence the common Hmit 

 ipi^ir), for small values of h, must be zero. Therefore ^(|x) = and 



