OF FLUIDS ON THE MOTION OF PENDULUMS. [31] 



Substituting in (47), and writing ft p for n, we get 



F = 2-n-a rl-ap a cos6 + p [~j) } sin 9d9. 

 With respect to the first term in this expression, we get by integration by parts 



fp cos 9 sin 9 d9 = £ sin 2 9 . p - \ /sin 2 9 -^ d9. 



du 



The first term vanishes at the limits. Substituting in the second term for — the 



du 



expression got from (29), and putting r = a, we get 



fp, cos 9 sin 9d0 - - $p %- f ( d -p) sin 9 d9. 

 *o dt J \ dr ) a 



Substituting in the expression for F, we get 



F~*f>a- H«^hj +8(^»),J«Qerf0 (49) 



19- The above expression for F, being derived from the general equations (27), (28), 

 combined with the equations of condition (30), holds good, not merely when the fluid is con- 

 fined by a spherical envelope, but whenever the motion is symmetrical about an axis, and that, 

 whether the motion of the sphere be or be not expressed by a single circular function of the 

 time. It might be employed, for instance, in the case of a sphere oscillating in a direction 

 perpendicular to a fixed rigid plane. 



When the fluid is either unconfined, or confined by a spherical envelope concentric with 

 the sphere in its position of equilibrium, the functions xj^, -v^ 2 consist, as we have seen, of 

 sin 2 9 multiplied by two factors independent of 9. If we continue to employ the symbolical 

 expressions, which will be more convenient to work with than the real expressions which 

 might be derived from them, we shall have 



e 



^"7.00, e v - lre 7c00> 



for these factors respectively. Substituting in (49), and performing the integration with 

 respect to 9, we get 



F-^irpans/^Tl {af,'(a) + 2f 2 (a)} e^~ lnt (50) 



20. Consider for the present only the case in which the fluid is unlimited. The arbitrary 

 constants which appear in equations (38) were determined for this case in Art. 16. Substi- 

 tuting in (50) we get 



F= -Zirpa?cn\/- 1 ( 1 + + ^- 9 ) e v - ln '- 



•* r V, ma m 2 a 2 J 



