482 Lord Kelvin on tlie Production of 



present communication, regarding which Stokes gives some 

 important indications in §§ 27-29 of his paper. 



§ 4. Poisson in 1819 gave a complete solution of the 

 equation 



5=^V% (8) 



in terms of arbitrary functions of x, y, z representing the 

 initial values of id and -7- ; and showed that for every case 



in which w depends only on distance (r) from a fixed point, 

 it takes the form 



-=H F KWK)}- • • ( 9 >> 



where F and /denote arbitrary functions. In my Baltimore 

 Lectures of 1884 I pointed out that solutions expressing 

 spherical waves, whether equivoluminal (in which there is 

 essentially different range of displacement in different parts 

 of the spherical surface) orirrotational (for which the displace- 

 ment may or may not be different in different parts of the 

 spherical surface), can be very conveniently derived from (9) 

 by differentiations with respect to x, y, z. It may indeed be 

 proved, although I do not know that a formal proof has been 

 anywhere published, that an absolutely general solution of (8) 

 is expressed by the formula 



K/J(0(i/{W'-:>/K-)] 



r= sl\_(x-x'f + (y-y'Y+{z-z'y\ . (10), 



where 2 denotes sums for different integral values of h,i,j, 

 and for any different values of x', y f , z'. 



§5.1 propose at present to consider only the simplest of 

 all the cases in which motion at every point (x, 0, 0) and 

 (0, y, z) is parallel to X' X ; and for all values of y and z, £ is 

 the same for equal positive and negative values of x. For 

 this purpose we of course take a/=y'=z'=0; and we shall 

 find that no values of 7a, i, j greater than 2 can appear in our 

 expressions for f, rj, f, because we confine ourselves to the 

 simplest case fulfilling the specified conditions. Our special 

 subject, under the title of this paper, excludes waves travelling 

 inwards from distant sources, and therefore annuls f(t + r/v). 



§ 6. In §§ 5-8 of his paper, Stokes showed that any motion 

 whatever of a homogeneous elastic solid may, throughout 

 every part of it experiencing no applied force, be analysed 



