19* Mr CHALLIS's RESEARCHES IN THE THEORY 



quantities of fluid displaced in the states of rest and motion, and is 

 therefore equal to yg \ID+'^ — j , 7 being small. Therefore 



neglecting powers of — above the first, 



Let ^ = 3. It will then be found that F' = 696** x 7. And if 7 = one 



inch, or ^, this equation gives ^=519 miles per hour; consequently 

 if ^=10*4 miles per hour, 7 = 4 inches. 



2 



In general, neglecting "—, &c. 



TV-w== 



r'a' 



sin e cos e (2 sin'0 + ^ ) ~ |) ' 



also W — w = yg llD + ^---iAO - sin 9 cos6)\ nearly ; 



therefore, as I> = Zasm9, it will be found that 



F- sin2 0(2sin'0 + l)-0 , . ^ / 



y = -r- •'-4 '--TT, — • r.n r,n y w? bemg put for -y\. 



' 4!g 4!msm'9-sm26 + 29 " * D 



If 9 be assumed equal to 15°, and 711 = 3, this equation gives ^"=7-35 

 miles per hour when 7 = 4 inches. 



These results, which probably are but very rough approximations 

 to matters of fact, may yet suffice to shew that when vessels and boats 

 of the usual forms sail in the open sea, they may be expected to rise 

 in some degree upon an increase of their velocity, and so much the 

 more as they are less adapted to cleave the water. Our theory shews 

 that the rise is the same for bodies of the same shape and proportions, 

 moving with the same velocity, whatever be their absolute magnitudes; 

 also that this effect is equally due to the pressures on the front and 



