26 PEOFESSOE HOEACE LAMB ON THE PEOPAGATION OF 



about the critical epoch ex. We may proceed, instead, by generalizing the expres- 

 sions (77). This introduces, in addition to the given function Q(t), whose Fourier 

 expression is 



Q(\)cosp(t-\)dX ..... (99), 



TT JO 



the related function 



(100); 



TT Jo J -oo 



viz., we have 



u = - H Q (t - ex) + &c., v = Q x (* - ex) + &c. . . (101). 



It does not appear that the connection between the functions Q (t) and Q v (t) has 

 been specially studied, although it presents itself in more than one department of 

 mathematical physics. The following cases may be noted as of interest from our 

 present point of view : 





(104). 



It is evident, generally, that if Q be an odd function, Q v will be an even function, 

 and vice versd. 



The values of U Q and v , as given by (101), are represented graphically in fig. 6, 

 for the case where Q (t) and Q v (t) have the forms given in (102).* Moreover, writing 



HQ/2T7/AT = /, KQ/27T/AT = g, t ex = T tan x, 

 we have 



2 x) /. v o = sin 2 X 9 ..... ( 105 ) 5 



the orbit of a surface-particle is therefore an ellipse with horizontal and vertical semi- 

 axes f and g. And if from the equilibrium position we project any other position 

 P of the particle on to a vertical straight line, the law of Fs motion is that the 

 projection (E) describes this line with constant velocity. See fig. 7, where the 

 positive direction of y is supposed to be downwards. 



' The relation between the scales of the ordinates in the graphs of w and v depends upon the ratio of 

 the elastic constants A, /*. The figures are constructed on the hypothesis of X = /*. 



