90 16 



where a subscript / means that values are to be taken at the end of the In- 

 terval, while d^Zg/dt^ is the acceleration at some unknown instant during the 

 Interval. Alternatively, d^z^/dt^ may merely change discontinuously from a 

 value on one side of that stated to a value on the other side. 



2. Within any time interval of length D/c , at least at one instant t 



1 r ^ V 1 



z^t) = M',) + M+Jtf, (^/[^-(^■~ f) - ^Mi)]ri(s)ds +J(2F, + *)(lf| [135] 



or 



1 ^ 



ic(«) = M«.)+]inriw^ J^2F, + 0)dt [136] 



where ty is the time at the end of the interval and tj is any chosen time not 

 later than its beginning, while t,' Is some unknown instant lying between 

 t, - D/c and tj. Thus 



tj - — < t, < t, < t < (y 



D^ 



c 



and tj and tj are arbitrary except that 



J 1 c 



To prove the first of these statements, multiply [^^6] through by 



D 



(Af + Af,)M,Afz\(t) = M;(M + Af,)[2F^ + *- pizlt -^)*i{s)d^ [137] 

 Now, if Q is any quantity independent of s, by [127] 



D 



M,Q = p[Qri(s)ds [138] 



By applying this transformation to z^{t), F- and 0, it is easily seen that 

 [137] can be written 



p rJAf [(M + M^Yi^U) - 2F. - *] + Af,[(M + M,)z^(t - f) - 2F^ - <t>'f>n(s)ds = 

 ^ 



Now if the second expression in brackets does not vanish for any value of s 

 in the range of integration, and nowhere Jumps from positive to negative or 

 vice versa, then it has everywhere the same sign, and the same sign as the 

 first bracket, which is its own value for 8=0; the entire integrand has. 



