4:90 Lord Kelvin on the Production of 



where ££. 2 &c. are given by (40) with % 2 for % 1 ; % x and X 2 

 being given by (36). 



§ 15. The character of the motion throughout the solid, 

 which is fully specified by (19), will be perfectly understood 

 after a careful study of the details for the equatorial, conal, 

 and axial places, shown clearly by (37) . . . (43) for each 

 constituent, the equivoluminal and the irrotational, separately. 

 The curve <^in the diagram of § 12 shows the history of the 

 motion that must be given to any point of the surface S, for 

 either constituent alone, and therefore for the two together, 

 in any case in which q is exceedingly small in comparison 

 with the smaller of the two quantities ur } vr, which for 



brevity we shall call the wave-lengths. The curve $ shows 

 the history of the motion produced by either wave when it is 

 passing any point at a distance from the centre very great 

 in comparison with its own wave-length. But the three 



algebraic functions <^, $ 9 $ all enter into the expression of 

 the motion due to either wave when the faster has advanced 

 so far that its rear is clear of the front of the slower, but not so 

 far as to make its wave-length (which is the constant thickness 

 of the spherical shell containing it) great in comparison with 

 its inner radius. Look at the diagram, and notice that in 

 the origin at S, a mere motion of each point in one direction 

 and back, represented by $f 3 causes in very distant places, a 



motion (<^) to a certain displacement d, back through the 

 zero to a displacement 1'36 X d in the opposite direction, 

 thence back through zero to d in the first direction and thence 

 back to rest at zero. Remark that the direction of d is radial 

 in the irrotational wave and perpendicular to the radius in the 

 equivoluminal wave. Remark also that the d for every radial 

 line varies inversely as distance from the centre. 



§ 16. Draw any line P K in any fixed direction through 

 0, the centre of the spherical surface S at which the forcive 

 originating the whole motion is applied. In the particular 



case of §§ 12 ... 15, and in any case in which FA t— — 1 and 



t — - J are each assumed to be, from t = to t = t, of the form 



i 3 (^r — t)' B A i f, where i denotes an integer, the time-history of 

 the motion of P is B + IV + •• • + 1>6 f it 6+ % and its space-history 

 (£ constant and r variable) is C_ 3 ? ,-3 + C_ 2 r" 2 + . . . +C 5+ ;r 5+/ ; 

 the complete formula in terms of t and r being given ex- 

 plicitly by (19). The elementary algebraic character of the 

 formula : and the exact nullity of the displacement for 

 every point of the solid for which r>q + vt; and between 



