REGENERATION THEORY 147 



which taken with (67) gives 



iviiy) = AJ{iy). (Bill) 



(Bill) is, therefore, a necessary consequence of (All). (74') taken 

 with (68) shows that 



J{iy) is continuous. (BII) 



It may be shown ^^ that (BI) is a consequence of (AI). Conse- 

 quently all the B conditions are deducible from the A conditions. 



Conversely, it may be inquired whether the A conditions are 

 deducible from the B conditions. This is of interest if AJ{i(S) is given 

 and is known to satisfy the B conditions, whereas nothing is known 

 about G. 



Condition All is a consequence of Bill as may be seen from (67) 

 and (74). On the other hand AI and All I cannot be inferred from 

 the B conditions. It can be shown by examining (5), however, that 

 if the slightly more severe condition 



lim y-'Jiiy) exists, (7 > 1), (Bio) 



is satisfied then 



G{t) exists, - 00 < / < CO , (Ala) 



which, together with All, insures the validity of the reasoning. 



It remains to show that the measured value of J{iw) is equal to that 

 defined by (6). The measurement consists essentially in applying a 

 sinusoidal wave and determining the response after a long period. Let 

 the impressed wave be 



E = real part of g'"', / > 0. (75) 



£ = 0, / < 0. (75') 



The response is 



real part of j AG{\)e^'^^'-^^d\ 

 Jo 



= real part of ^c*"' | G{\)e-^"^d\. (76) 

 For large values of / this approaches 



real part of Ae^'^^J{iw). (77) 



Consequently, the measurements yield the value ^/(/co). 



" See Hobson, loc. cit., vol. II, 2d edition, § 335. It will be apparent that A' de- 

 pends on the total variation but is independent of the limits of integration. 



