374 Lord Payleigh on Forced Harmonic 



ment of frequency of vibration is accompanied by a corre- 

 sponding increment of resistance. 



A - . '■., . , n (ipau + by)' 2 



Again, the imaginary part 01 -4 -—. — 



%p a 22 + o 22 



_ . b 12 {2a l2 b 22 — a 22 b 12 ) + p> 2 a 22 an 

 lP ° 22 2 +p 2 a 22 2 



= ip*l - {p M>22-a 22 b, 2 r 



1 a 22 r a 22 (b 22 +^a 22 2 y 



so 



that 



L'=a n -t^ \$ (^-^fa )!. . . .(io) 



a 22 a 22 (b 22 +p 2 a 22 ') v 



In the latter series each term is positive, and continually 

 diminishes as p 1 increases. Hence every increase of frequency 

 is attended by a diminution of the moment of inertia, which 

 tends ultimately to the minimum corresponding to disappear- 

 ance of the dissipative terms. 



Certain very particular cases in which B/ and 1/ remain 

 constant do not require more than a passing allusion. If 

 T and F are of the same form, every such quantity as 

 (a 12 b 22 —a 22 b l2 y vanishes. 



As examples of the general theorem may be mentioned the 

 problems considered by Prof. Stokes in his well-known paper 

 upon " The Effect of the Internal Friction of Fluids on the 

 Motion of Pendulums "*. Consider, for instance, the result 

 for a sphere of radius a, vibrating (according to e ipt ) in a 

 fluid for which the kinematic coefficient of viscosity is yJ , 

 W denoting the mass of the fluid displaced by the sphere, 

 Prof. Stokes's results may be written 





When p is zero, which represents uniform motion of the sphere, 

 ™ VM' T , 



As^> increases, the expressions show that, in agreement with 

 the theorem, B/ continually increases and 1/ continually 

 diminishes. In fact P/ tends to become infinite, and 1/ to 



* Canib. Trans, vol. ix., 1850. 



t That the energy of the motion is infinite in this case does not appear 

 to have been noticed. 



