OF FLUIDS ON THE MOTION OF PENDULUMS. [29] 



of equations (39), (40), (41) under the form U + \/ '- 1 V, and reject the imaginary part. 

 Putting for shortness 



Vr>"^ < 42 ) 



we have m = v (l + y/- l), and we obtain 



f = - sin nt, (48) 



n 



\l/ = ia 8 csin 2 ^| |( 1 + ) cosnt + (1 + — ] sinnH - 



r 2 \L\ 2vaJ 2i>a \ vaj J r 



e~ v ( r ~ a) \ cos (nt - vr + va)+ (l + —\ sin (nt - vr + va)\\, . (44) 



P = - A pacn\[\ + )sinn< (l + — 1 cos«4 cosfl. — (45) 



r ^ r \\ ival 2va \ vaj J r 2 v ' 



The reader will remark that the £, \J/, p of the present article are not the same as the 

 £, ^ , p of the preceding. The latter are the imaginary expressions, of which the real parts 

 constitute the former. It did not appear necessary to change the notation. 



When (i' = 0, v = ■ , and -ty reduces itself to 



a 3 c . a 3 . , „ dp 



— sin 2 9 cos nt, or — sin 2 9 — . 

 2r 2r dt 



In this case we get from (26) 



dt r * dt r 



and Rdr + QrdO is an exact differential d<p where 



d%cos0 



which agrees with the result deduced directly from the ordinary equations of hydrodynamics *. 



18. Let us now form the expression for the resultant of the pressures of the fluid on the 

 several elements of the surface of the sphere. Let P r be the normal, and T e the tangential, 

 component of the pressure at any point in the direction of a plane drawn perpendicular to 

 the radius vector. The formulae (4), (5) are general, and therefore we may replace x, y in 

 these formulae by x', y , where x', y are measured in any two rectangular directions we please. 

 Let the plane of x' y pass through the axis of x and the radius vector, and let the axis of x' 

 be inclined to that of x at an angle $, which after differentiation is made equal to 9. Then 

 P lt T 3 will become P r , T e , respectively. We have 



u = R cos (9 - 9) - 8 sin (9 - 9), v = R sin (0 - $) + 9 cos (9 - $), 



See Camb. Phil. Trans. Vol. VIII. p. 119. 



