TREMORS OVER THE SURFACE OF AN ELASTIC SOLID. 3 



By a known theorem we have also 



2 f" 

 J () = sin ( cosh u) du (2), 



TT JQ 



provided be real and positive. For our present purpose it is convenient to follow 

 H. WEBER* in adopting as the standard function "of the second kind" 



2 f" 

 KO() = cos ( cosh w) du (3). 



It is further necessary to have a special symbol for that combination of the two 

 functions (2) and (3) which is appropriate to the representation of a diverging wave- 

 system ; we write, after Lord HAYIYEIGH,! 



7T J() 



so that 



We shall also write, in accordance with the usual conventions, 



For large values of we have the asymptotic expansion 



In the two-dimensional problems of this paper we shall have to deal with a 

 number of solutions of the equation 



ff + ft+/^ = ......... (8), 



ox- cy~ 



constructed from the type 



' 



where is real, and 



a = /- or = 



* 'Part. Diff.-Gleichungen d. math. Physik,' Brunswick, 1899-1901, vol. 1, p. 175. HEINE (' Kugel- 

 functionen,' Berlin, 1878-1881, vol. 1, p. 185) omits the factor 2/w. In terms of the more usual 

 notation, 



T = 2 {- Y + (log2- y )J }, 



7T 



where y is EULER'S constant. The function J^Ko has been tabulated (see J. H. MICHELL, ' Phil. Mag.,' 

 Jan., 1898). 



t 'Phil, Mag.,' vol. 43, p. 259 (1897); 'Scientific Papers,' vol. 4, p. 283. I have introduced the 

 factor 2/7T, and reversed the sign. 



B 2 



