OF THE MOTION OF FLUIDS. 193 



weight of fluid displaced. Also the shape of the floating body above 

 the part immersed is of no importance to the problem. The form of 

 the whole body may be such as is described in Fig. 4, ABCDEF 

 being a half cylinder of which the axis is GH, and ALC, FKD, the 

 extreme portions of the body, bounded by spherical surfaces which have 

 their centres at M and N. Now in general ^ jjsin^d co^^O co& wdQdw, 



commencing at = sin~'— , and ending at any other value of 9, will 



be found to be 



cosefssin^e + l-^') V sin^e-^ -\(\-—i 



And if we put cos 9 = — ?= , we shall have 



COS0 



a- 



V 



W = w + 



As the second term is necessarily positive, the floating body will be 

 higher than it would be in a state of rest, and consequently the 

 surface against which the resistance acts becomes less by an increase 

 of velocity. 



To obtain a numerical result respecting the rise of the body 

 corresponding to a given velocity, we will suppose for the sake of 

 simplicity of calculation that when the vessel is at rest, the centres 

 of the spherical ends and consequently the axis of the cylindrical part, 

 are in the plane of the horizontal surface of the water. This circum- 

 stance may be produced by loading the upper part of the body 

 . without altering its specific gravity. Let / = the length of the axis 

 of the cylindrical portion. Then the area of the horizontal section of 



the vessel at the level of the water surface is ID H ■— — , its 



4 2 



breadth being D. Now W—w must be equal to the difference of the 



