OF FLUIDS ON THE MOTION OF PENDULUMS. [41] 



(95) will only contain the ratio of C to D, which is given by (93), and if F 3 (a) be expressed 

 by (88) C will disappear, inasmuch as D'= 0. 



31. Let us now form the expression for the resultant of the forces which the fluid exerts 

 on the cylinder. Let F be the resultant of the pressures acting on a length dl of the cylinder, 

 which will evidently be a force acting in the direction of the axis of w ; then we get in the 

 same way as the expression (47) was obtained 



F=adl J™ (- P r cos 9 + T e sin 9) a d9, (96) 



and P r , T e are given in terms of R and 9 by the same formulae (46) as before. When the 

 right-hand members of these equations are expressed in terms of v, there will be only one 

 term in which the differentiation with respect to r rises to the second order, and we get from 

 (70), (75), and (76) 



d*X l <*X 1 ^X x d X°- 

 d~? = '" "r~dr~ ' r* d0* 4 'p'~dt' 



We get from this equation and the equations of condition (79) 



,dR\ m i(*X\ _ 1 (J?x_\ m 

 \dr) a a\de)a a?\drde) a ' 



fdR\ 1 fd\\ sin 9 d'£ _ 



U<W/" # " & UflV. a~~di~~a~' 



(dQ\ I _ (d?x\ _l_(<hc\ 1 (d\\ _l(dj(A _ I (dte\ 

 \dr) a WJa a \dr) a a* W6>V a ,/W*/„ n'\dt) m ' 



Hence 



F=adl p*{-p a cos6 + p(-^ sin9}d9. .... (97) 

 We get by integration by parts 



r],Q = n sin ft - f\ 



[d9) 



fp a cos 9 dd = p a sin 9 - f(-^\ sin9d9. 

 » d9 ) a 



The first term vanishes at both limits; and putting for -f- its value given by (77), and 



• d9 



substituting in (97), we get 



F =P adl d i i fl a { d i)^(x^}^ed9, 



or 



F = irpadl.n\/~^\ {aF^a) + F 3 '(a)} e^ 71 "'- 



Observing that F 3 '(a) or F s (a) = ac - F x (a) from (83), and that /\(a) = Aa~\ where 

 A is given by (95), and putting M' for vpcfdl, the mass of the fluid displaced, we get 



F = M'cnV—U .^l^m\^^ 

 [ aF 3 (a) + F 3 (a) J 



Vol. IX. Part II. 30 



