OF THE MOTION OF FLUIDS. 201 



instant from its centre. Therefore / = 0. Also if the radius of the 



ball be supposed very small, the equation -f- = ^-t^> obtained at the 



end of the preceding Article, will be approximately applicable to the 

 motion of the fluid in contact with the ball. Hence the velocity which 

 is impressed at any point of the spherical surface may be considered 

 to be transmitted instantaneously in the direction of the radius through 

 that point, and to decrease according to the law of the inverse square 

 of the distance. The problem, with the limitations above made is 

 solved in the same manner for air as for water. 



Let now the origin of co-ordinates be A, (Fig. 6.), the position 

 of the centre of the ball when it hangs at rest. I^et its centre perform 

 oscillations of very small extent in nAN, which we will consider to 

 be rectilinear. Suppose N to be the position of the centre at the 

 time t reckoned from a given epoch, and call AN, a. Take P any 

 point of the surface, join NP and produce it to R, and let NPR make 

 an angle Q with ANQ, and the plane RNQ an angle /3 with the 



plane SAQ. The velocity of the centre = ^; and the velocity of 



da 



the air at P — -rrCosO. Hence if NP=h, and NR = r, the velocity 



at ^ = -„ — . -^ . Now if AN be the axis of x, AS of a, and a 



r- at 



line through A perpendicular to the plane SAN, the axis of y, and 



the co-ordinates of R be x, y, %, then r^ = {x — aY + y^ + %^. Consequently 



the velocity (^) at R=, r^ ^ — 2 • ;77- Hence differentiating V^ 



With respect to the time only, 



dr _ d'a b^cos9 2b^cose{x-a) d^ h^ da d.cosO 

 dt ~ W-' r' "^ 7 • dt^ '^ r"' df dt ' 



^ ^ x — a d.cosO 1 da cos^6 da sin^O da 



But as cos9 = , r: — = --77 + 



Therefore 



dt . r ' dt r ' dt r ' dt 



dV_d^ ¥cos9 ^b'cos'e da' b'sin'0 d^ 

 dt ~ df r^ ^ r' ' dt' r' ' dt^ 



