CERTAIS l-ACiOKS .IIILlTIXG TLLLCR.ll'll SPEED J41 

 It follow;; at oiiii' tluit, |)r(i\ idrd 



i 



and provided the duration T of the signal is sufficienth' short, the 

 arrival dot is given approximately by the leading term 



.4'(/) f /•(.vV/.v 



.md that this approximation becomes increasingly close as the speed 

 of signaling is increased, i.e., as the duration 7' of the dot is decreased. 

 The conclusions from the foregoing may be stated in the following 

 propositions: 



I. If the speed of signaling is sufficiently high the arri\al signal 

 representing the elementary dot is independent in shape of the form 

 of the impressed signal, and is proportional in amplitude to the time 

 integral or "area" f)f the impressed signal. 



It will be evident, however, that if no restrictions arc imposed on 

 A' (t) and/(0. the foregoing proposition requires, in general, that 

 the duration 7" of the dot shall be so small as to make the series expan- 

 sion rapidly convergent from the start. This, however, requires a 

 speed of signaling \en,- considerably greater than that actually neces- 

 sary in practise in order that the foregoing proposition shall hold 

 to a g(X)d degree of approximation, at least for the types of impressed 

 dot signals specially considered in the present paper. To show this, 

 it is necessary' to establish two less general propositions, valid for 

 the types of impressed signals under consideration. 



II. If the impressed signal / (/) is e\er\'where of the same sign, 

 then a value r exists, such that 0</t/ <T/2, and such that 



S{l + T/2) =A'{1 + t) rf{x)dx (3) 



This proposition follows from the mean \alue theorem. 



III. If/ (/) is everywhere of the same sign, and if further it satisfies 

 the conditions of symmetry, 



/(.t)=/(r-x),(.r<r/2) 



then a value t exists, such that 0<r<7" 2 and such that 



5(/-|-r/2) = l;'2[A'{t + r)+A'{t-T)]£j{x)dx (4) 



This last equation also follows from the mean value theorem. Fur- 

 thermore, the conditions stated in proposition III are satisfied by 



