58 Dr. G. Green on Ship- Waves, and on Waves in 



a being the radius of the sphere, and the coordinates being 

 referred to the instantaneous position of the centre of th© 

 sphere as origin. The corresponding stream function for 

 the fluid motion relative to the sphere is 



2 . 2 a va % sin 2 6 /Qn . 



^i = — vr- snr 6 -\ . . (39) 



We have first to determine from this the stream surface con- 

 taining the fluid particles which, when at rest, lie in the 

 plane at vertical distance z=li above the centre of the 

 moving sphere. On transferring to polar coordinates (#, -zzr),. 

 x in the line of motion, and ot { = ^ / y 2 \ z 2 ) perpendicular to 

 the line of mo f ion, we obtain the stream function in the 

 form 



f^-^+^J 2 (40) 



On any stream line at infinite distance from the sphere, 

 -57 has its value the same for any fluid particle as when the 

 particle is at rest. Putting «r = w at r=oo, we have 

 ^ = — vsr 2 , an equation which enables us to write (40)- 

 in the form 



«*-W="£r, («) 



If we assume, as in the case of the cylinder, that the depth 

 of the centre of the sphere h is large compared with the 

 radius, on this standard we may replace «r by ot- on the 

 right-hand side of (41) and on the left-hand side we may 

 put -or + -btq = 2-3r . This leads to 



3 



a <G7 



,»=a* + t/* + h; . . (42) 



at the plane z = h. The change in the z coordinate of any 

 point on this surface, being («r— otq)^/^, is now easily 

 obtained in the form 



-r=*-* =g : r»=*»+y» + tf. . . (43) 



An equal elevation of surface at each point is produced by 

 the image sphere. This enables us now to obtain the principal 

 term of the pressure system which must be kept applied to the 

 upper surface to maintain the motion represented by <j>i — <pi 

 in the fluid beneath — 



?= _i^/ r 3. r*=x?+f + h 2 . . . . (44) 



