4 PROFESSOE HORACE LAMB ON THE PROPAGATION OF 



according as 3 5 /r, the radicals being taken positively. In particular, we shall 

 meet with the solution 



. _ 1 f" e-^d( _ 2 f" 



TT J- a TT Jo 



and it is important to recognize that this is identical with D (hr), where 

 r = ^/(x? + i/ 3 ). To see this, we remark that <f>, as given by (11), is an even 

 function of x, and that for x = it assumes the form 



2re-"d 2 F e-^ 



= s = -7775 



TT Jo a TT Jo v /(/i- 



\ ...... 



+ I? 2 ) 



by the method of contour-integration.* This is obviously equal to D (hy}. Again, 

 the mean value of any function <j> which satisfies (8). taken round the circumference 

 of a circle of radius r which does not enclose any singularities, is known to be equal 

 to J () (&r).< , where <^ is the value at the centre. t We can therefore adapt an 

 argument of THOMSON and TAIT} to show that a solution of (8) which has no 

 singularities in the region y > 0, and is symmetrical with respect to the axis of y, is 

 determined by its values at points of this axis. We have, accordingly, 



_! (= 



7T J on (Z 



Again, in some three-dimensional problems where there is symmetry about the 

 axis of z, we have to do with solutions of 



based on the type 



......... (15), 



where TS = \/(x~ -\-y~), and a is defined as in (10). In particular, we have the 

 solution 



which (again) reduces to a known function. At points on the axis of symmetry 

 (CT = 0) it takes the form 



* If we equate severally the real and imaginary parts in the second and third members of (12), we 

 reproduce known results. 



t H. WEBER, ' Math. Ann.,' vol. 1 (1868). 

 J ' Natural Philosophy,' 498. 



