626 Prof. T. H. Havelock on an Integral Equation 



In the same way, the solving function has a limiting form 

 which follows directly from (15) and (19), namely 



air J- x ot 2 + 4-p 2 v/o-' 2 



Using this value as before, we obtain the same result (24). 



6. It is clear that the same procedure is sufficient when 

 the applied force is any assigned function of the time. 

 For example, if the accelerative force is a cospt and the 

 motion starts from rest, we have 



dV {* 



-—■=aco$pt—\ acospr , ^Ar^'^^T, . (25) 



dt Jo ' 



where the summation extends over the roots of the same 

 equation (15), and the coefficients are given by (19) The 

 solution follows on completing the integrations ; it consists 

 of a periodic motion in different phase from the applied 

 force, together with the disturbance due to taking into 

 account the initial conditions. 



7. A final example may be taken from cylindrical motion 

 when there is no limiting steady velocity. Suppose the 

 motion to be symmetrical round an axis ; then if r is dis- 

 tance from the axis and v is the fluid velocity, supposed 

 perpendicular to the radius vector, we have 



|* (|L» + 1|!!_£\. . . . (26) 



^t \$r 2 r dr r 2 J v J 



Consider the motion of a hollow cylinder, of radius a, filled 

 with the liquid. Suppose the motion to start from rest and 

 let the velocity of the cylinder be Q,(t). Then it may be 

 shown that the angular velocity of the fluid at any time is 

 given by 



( T ) (i + 22^4^^ w - T)/a2 }^ ( 27 ) 



I prJi(p) J 



where the summation extends over the positive roots of 

 Ji(p)=0. 



Let the cylindrical shell start from rest under the action 

 of a constant couple N, and let I be its moment of inertia, 

 both quantities being for unit length along the axis. The 

 retarding couple due to fluid friction is the value of 

 27T/Ar 3 ^&)/Br when r — a. Hence the equation of motion 

 of the cylinder is 



n'(0 + (47^/1) \ n / (T)S^-"^- T > / "VT=N/i, (28) 



=j> 



