77 91 



therefore, the same sign throughout, and the Integral cannot vanish. Hence 

 for at least one value s ' between and D the second bracket must either van- 

 ish or change sign dlscontlnuously. At the corresponding time, t - s'/c, z^ 

 or Zj(( - s'/c) has the value stated In [13^]. 



To prove [135]. Equation [116] is first integrated with respect to 

 the time from t^ to tfi 



if D 



M[iSts) - iSh^ = J (2F, + *)d( - p{[zit, - f) - z,(t, - f)] 77(s)c 

 Multiplying by M,(M+Mj) and applying [138] to all terms except 



D 



there results 



D 



+ M^\(M + Myzj^tf- f) -Mz,(t,) - J{2Fi + 0)d( 



- p/2c(«l- t) '?(«■) '^S'][ '?(s)rfs =0 



and by reasoning as before and then using [127], Equation [135] is obtained. 

 To convert this equation into [136], note that, since ri is positive and z^[t) 

 is continuous, there exists a value s" between and D such that 



^/[^^(*i ~ f ) ~ M«i)]'7Wd« = [z\t, - ^) - Mti)]p/'7(s)ds 



by [127]. The terms containing t, in [135] can thus be written 



This expression lies between z^{t^) and zjt^ -s"/c); it is, therefore, the 

 value of z^[t^ - s/c) at some other value s' between and s", or the value 

 of Zj(t) at some time t/ between t^ - D/c and (j. 



