34 

 and 



PROFESSOR HORACE LAMB ON THE PROPAGATION OF 



. (H6). 



The additions to (143) whicli are required in order to annul the stresses on the 

 plane z = are accordingly found by writing 



a 



in (139), and then integrating with respect to 

 obtain, finally, 



from to oo . In this way we 



...... 



In a similar manner, with the help of Art. 11, we might calculate the effect of a 

 periodic vertical forqe, acting at an internal point. 



13. For the sake of comparison with our previous two-dimensional formulae, it is 

 convenient to write, from (2) and (6), 



7T Jo 



TT 



cosh 



. . . (148). 



The formulae (141) are thus equivalent to 



R r 



<ln = jT cosl1 



/!7r/Ll Jo 

 *T ) * x 



^ _l\J ' 7 



'"'n == ~o~ "' ( 



'27T/U. - l) 



These results are closely comparable with (52), and our previous methods of treat- 

 ment will apply. It is, however, unnecessary to go through all the details of the 

 work, since the definite integrals with respect to which appear in (149) can be 

 derived from those in (52) by performing the operation ?'3/3.x upon the latter, and 

 then replacing x by 57 cosh u. 



Thus, from (67) and (70) we derive 



~ 

 (s) 



= 27r/cH sin (m, cosh u) 



. (150), 



