486 Lord Kelvin on the Production of 



We find by aid of (18) 



(axial) ^J5 + ^ + J) + J 3 (F 1 + F 2 ); V = ; ?=0 (27 ). 



Next take # = and we find 

 (equatorial) ?=-- ?l_^(J + ^)_^ ( F 1 + F 2 ) ; „=0; ?=0 (28 

 Hence for very small values of r we have 



(axial) f = |(Fi + F 2 ) 



(axial) f — - 5 



(30). 



1 > (29); 



(equatorial) f = — -^ (F x + F 2 ) | 



j 

 and for very large values of r. 



F 2 



1 F , 



(equatorial) £=■ — J 



vi ' * • r u 2 ) 



Thus we see that for very distant places, the motion in the 

 axis is approximately that due to the irrotational wave alone ; 

 and the motion at the equatorial plane is that due to the 

 equivoluminal wave alone : also that with equal values of F 1 

 and F 2 the equivoluminal and the irrotational constituents 

 contribute to these displacements inversely as the squares of 

 the propagational velocities of the two waves. On the other 

 hand, for places very near the centre, (29) shows that 

 both in the axis and in the equatorial plane the irrotational 

 and the equivoluminal constituents contribute equally to the 

 displacements. 



§ 11. Equations (25) and (26) give us full specification of 

 the forcive which must be applied to the boundary of our 

 spherical hollow to cause the motion to be precisely through 

 all time that specified by (19), with F 1 and F 2 any arbitrary 

 functions. Thus we may suppose F x (t — q/u) and F 2 (t—q/v) 

 to be each zero for all negative values of t, and to be zero 

 again for all values of t exceeding a certain limit r. At any 

 distance r from the centre, the disturbance will last during 

 the time 



from r — q , . r— a 



t = to t — — -* + r, 



and from r — q r — q 



t = i to t = % + t. 



y • . . (3i) 



