TREMORS OVER THE SURFACE OF AN ELASTIC SOLID. 5 



Since the mean value of a function <j> which satisfies (14), taken over the surface of 

 a sphere of radius r not enclosing any singularities, is equal to 



sin hr , 



hr ' 



where < is the value at the centre,* the argument already borrowed from THOMSON 

 and TAIT enables us to assert that 



where- 



Finally, we shall require FOURIER'S Theorem in the form 



I 



/(X) *<-*> c?X ....... (19)4 



M _ 



and the analogous formula 



As particular cases, if in (19) we have f(x) = 1 for x* < a~, and = for x- > a", then 



//^.\ 1 I* sin ft* ,-j, 7 > -2 fsina ^ ,,> 

 / ( x ) = g e^dfss- --i30B&dg .... (21); 



7T J - ff 77 J g" 



and, if in (20) /(CT) = 1 for CT < , and = for CT > o, then 



Jo^Ji^df ........ (22). 



These are of course well-known results. || 



* H. WEBER, 'Crelle,' vol. 69 (1868). 



t If in (18) we put z = 0, and then equate separately the real and imaginary parts, we deduce 



I J ( cosh ) cosh du = ~ , 

 Jo C 



J (i sin w) sin it du = --,-- . 



These are known results. Of. RAYLEIGH, ' Scientific Papers,' vol. 3, pp. 46, 98 (1888) ; HOBSON, ' Proc. 

 Lond. Math. Soc.,' vol. 25, p. 71 (1893) ; and SONINE, ' Math. Ann.,' vol. 16). 



f H. WEBER, 'Part. Diff.-Gl. etc.,' vol. 2, p. 190. Since A. occurs here and in (20) only as an inter- 

 mediate variable, no confusion is 'likely to be caused by its subsequent use to denote an elastic constant. 



H. WEBER, 'Part. Diff.-Gl. etc.,' vol. 1, p. 193. 



|| It may be noticed that if in (20) we put / (^) = - i/ " r /tir, we reproduce formulae given in the foot-note t 

 above. 



