100 



G. H. KNIBBS. 



(53)... y m = sin m (x + a); (ii) y' m = sin m (x + /?), 



then since the values of each are for the value x = - a 

 and a?= — /3 respectively, we may say that the "lag" of 

 the second behind the first is given by 



(54)... X = a - (|8 + 2n7r) 



In general we may assume that n = 0, in the term 2mr y 

 that is to say, in the majority of cases practically arising, 

 the lag will be less than a whole period. 



Hence if we have reasonable grounds, apart from mathe- 

 matical considerations, to look on them as causally related, 

 then the second may be regarded as an effect of the first 

 brought about after the lapse of time A. Thus the lag may 

 be determined term by term when "a priori" we know the 

 length of the period, and we should thus have for the lag 

 of the "troth" term 



(55)... Ki = a m — <*'m 



in which the accented letter is the epochal angle corres- 

 ponding to the non-accented letter. 



But the resolving of a periodic oscillation under equation 

 (1) is purely arbitrary, the periods being merely assumed 

 instead of being known " a priori." In such a case we may 

 with some propriety regard the whole group of the second 

 series of periodic terms as having what may be called a 

 "group lag" in respect of the first series. It is obvious in 

 the first place that the "group lag" is a function of the 

 epochal angles. Also, since the importance of any term 

 varies with its amplitude, the influence of any epochal 

 angle on the group lag must further be regarded as a func- 

 tion of the amplitude factor. If then we suppose that the 

 weight of each epochal angle enters into the result directly 

 as the amplitude of the term, we shall have approximately 

 for what may be called the mean epochal angle, 



(do;... «o — t" ~r v 



