24 



PROFESSOR HORACE LAMB ON THE PROPAGATION OF 



The interpretation of the expression (93) for the vertical displacement r is not 

 quite so simple. For a given value of x, the most important part is that corre- 

 sponding to t = ex, or = c, nearly, when the integrand in the second term changes 

 sign by passing through infinity. This is the epoch of the main shock ; the minor 

 disturbance which sets in when t = ax leads up continuously to this, and only dies 

 out gradually after it. 



As a first step we may tabulate the function V (6} defined by 



V (0) - - - 



* 



-- 



(20* - #>)* + 160* (0* - a 2 ) (6 2 -0 2 ) 



(20* - b-) 3 - 



for a < < b 



, for > b . 



(97). 



The function has a minimum value '45120 when 0/a = 1 '01 170, and a zero 

 maximum when 0/a = 1 '22474 ; it changes from oo to + oo when 0/b = 1 '08767, 

 or 0/a = 1-88389.* Its graph is shown in the lower part of fig. 4, and also (on a 

 smaller scale, so as to bring in a greater range of 0) in fig. 5. 



It is postulated that the function Q (t) is sensible only for values of t lying within 

 a short range on each side of ; the function Q (t - 0x) will therefore be sensible 

 only for values of in the neighbourhood of t/x. We will suppose that for given 

 values of x and t its graph (as a function of 0) has some such form as that of the 



* As in the case of U (6), the calculations are due chiefly to Mr. WOODALL. 



