of Attractive Forces. 325 



vdv 

 is — f2, c being the radius of the sphere. But this force is also 



K (I do 



equal to — -, which by the foregoing equation is equal to 



sin 6 cos 6. 



'^ .-. vdv=^N^ &m 6 C0& 6(16; 



and by iutegratiouj 



r;2=W2sin2^. 



Hence t;=W when ^ = 90". The velocity thus acquired is lost 

 as the particle moves from B to A', by reason of the continuous 

 increase of density between these two points. And as the den- 

 sity at A' is the same as that at A, and the path from B to A' is 

 of the same form and length as that from A to B, and is de- 

 scribed in the same time, it is clear that the circumstances of the 

 motion will be satisfied if the gradations of density are the same 

 in the two cases, and the velocity of the ^'article in its whole 

 course from A to A' is W sin 6. On this supposition the state 

 of the fluid as to density and velocity will be symmetrical with 

 respect to a plane through the centre of the sphere perpendicular 

 to the diameter AA'. That this is the case will appear more 

 clearly from the solutions of the following problems. 



Problem III. The sphere performs small oscillations in the 

 fluid at rest, its centre moving in a given manner in a straight 

 line : it is required to find the pressure on its surface. 



For my present purpose it is only necessary to consider the 

 case of a sphere whose radius (c) is extremely small compared 

 to the breadth (X) of any undulation which it excites in the fluid. 

 In this case it is sufficiently approximate to assume the velocity 

 communicated by the sphere to the fluid to be the same as if the 

 fluid were incompressible, as by this supposition the part of the 



velocity neglected is of the order of- x the part retained. Hence 



by the law of rectilinear transmission, if W be the velocity of the 

 sphere, and 6 be the angle which any radius makes with the 

 line of motion of its centre, the velocity on the prolongation of 



the radius at any distance r from the centre is — ^ ^^^ ^• 



From this expression it is next required to obtain the value of 

 the partial diff'crential coefficient of V with res])ect to t for a given 

 position in space. Now the point of space where the velocity is 

 V being given, r and 6 vary with the varying position of the 

 sphere. Hence 



-Tt=-di'7^''''^--^''''^Tt—l^''''^Tt' 



