232 



where 



Lord Kelvin on the Production of 

 ^(t) = -3^- (K sin cot + L cos at), 



K = 



L = 



4^ 4 (^ + 2^)4-^(^ + 3^-1^ + 1 



'oT 



u — v/3uv .A 

 yw \^ 2 &) 2 / 



[-^- 2 (« 2 + 2^)-lT + ~(u + 2t;) : 

 l_g 2 &) 2 v J q'oy 1 



(65). 



; 



In terms of this notation, (64) gives for great distances 

 from the centre 



ho f ?' 2 —" 3? 3? "*) 



£ = — " •( 5 — (K sin <ot x + L cos a>^) + -5- [ (3 — 2K) sin cot? — 2L cos g>£ 2 ] f 



* • r t *• ?' J 



v = ^i -% (K sin w/x + L cos 0)^)+ *f [(3-2K) sin <y/ 2 -2L cos wf 2 ] j j> (66). 



^ = -iJ —^-"2 (Ksinco^i + LcosG)^) + -j [(3 — 2K) sinci)^ — 2Lcosa)i 2 ] f 



These equations represent two sets of simple harmonic 

 waves, equivolnminal and irrotational, for which the wave- 

 lengths are respectively 27ra/a), "2irvj(o. The maximum dis- 

 placements in the two sets at points of the cone of semi- vertical 

 angle 6 and axis X, are respectively, 



(equivoluminal) sin 6 -&■ V (K z + L 2 ) ; 

 (irrotational) cos 0^- V [(3 - 2K) 2 -f 4L 2 ] 



V 



(67). 



§ 27. The rate of transmission of energy outwards by a 

 single set of waves of either species is equal, per period, to 

 the sum of the kinetic and potential energies, or, which is the 

 same, twice the whole kinetic energy, of the medium between 

 two concentric spherical surfaces, of radii differing by a wave- 

 length. Now the average kinetic energy throughout the 

 wave-length in any part of the spherical shell is half the 

 kinetic energy at the instant of maximum velocity. Hence 

 the total energy transmitted per period is equal to the wave- 

 length multiplied into the surface-integral, over the whole 

 spherical surface, of the maximum kinetic energy at any point 



