350 OF THE POSITIONS OF EQUILIBRIUM. 



is entirely above the surface of the fluid), and FII the line of 

 floatation. 



Bisect FH in w, and through n draw nm parallel to DC the axis 

 of the parabola, and meeting the curve in the point m ; then is mn 

 when produced to r a diameter of the curve, whose vertex is in the 

 point m. 



Through the point m, draw mt parallel to AB the base of the 

 parabola, and meeting the axis DC in the point K ; produce CD to E, 

 making BE equal to DK, and join EW, then by the property of the 

 parabola, Em is a tangent to the curve in the point m, and it is 

 parallel to FH the line of floatation, or the double ordinate to the 

 diameter mr. 



Let P be the place of the focus; join ?WP, and through H the 

 extreme point of the line of floatation, draw uv parallel to AB the 

 base of the figure, and meeting the diameter mr perpendicularly in 

 the point v; then are the triangles KETW and vnii similar to one 

 another. 



Take DG equal to three fifths of DC, and mg equal to three fifths 

 of mn; then are G and g respectively the centres of gravity of the 

 whole parabola ADB and of the part FDH; join og, then by the 

 principles of floatation, the straight lines Gg and FH are perpendicular 

 to one another, and consequently, the triangles KEW and wng are 

 similar. 



Put a m DC, the axis of the parabola or section of the floating 



body, 

 26zz AB, the base or double ordinate corresponding to the 



axis DC, 



a' zz the area of the whole parabola ADB, 

 a"z= the area of the immersed part FDH, 

 x ~ mn, an abscissa of the diameter mr, 

 y zz: H n, the corresponding ordinate, 



z zz Km, the ordinate passing through m the point of contact, 

 p zz the parameter or latus rectum to the axis, 

 s zz the specific gravity of the floating solid, and 

 s' zz the specific gravity of the fluid on which it floats. 



Now, supposing that all the sections which are perpendicular to the 

 axis of motion, are equal to one another; then, according to the 

 principles of floatation, we have 



a'szza'Y. (276). 



