OF THE STABILITY OF STEAM SHIPS. 



401 



of steam navigation, because the form is altogether unsuitable for 

 vessels of that description, and our only object forgiving it here is to 

 show the method of reducing the equation ; this being the more 

 necessary for the sake of system, as it forms a particular case of the 

 general problem, and is deducible from it by merely assuming a 

 particular value for the exponent of the parabolic ordinate. 



By the rule, we have (b* 2e? 2 ) =r 784 - 380.25X2 = 23.5 ; 

 therefore, by multiplication and division, we obtain 



b sin. rf> 



^(^2rf a )zz28x0.25882X23.5-rl2z=5 14.19 very nearly. 



492. Returning to the general 

 equation, if we suppose wzr2, 

 then the section is in the form of 

 the common or Apollonian para- 

 bola, as represented in the an- 

 nexed diagram, wherein AB is the 

 base or double ordinate of the 

 parabolic section, DC its axis, FII 

 the water-line, or double ordinate 

 of the immersed portion FDH, DE the corresponding abscissa, and IK 

 the horizontal surface of the fluid. Then, with ram 2, the expression 

 for the stability becomes 



(292) . 



The form of the vessel of which the stability is expressed by the 

 above equation, is much better adapted for the purposes of steam 

 navigation, than the triangular form already discussed; but it is obvious 

 from the relation of the parenthetical terms, that it requires a much 

 greater breadth at the water line under the same depth and inclina- 

 tion, to give an equal degree of stability ; and the breadth necessary 

 for this purpose maybe determined by reversing the expression, which 

 will then assume the form of a cubic equation, wanting the second 

 term, and whose reduction will give the necessary breadth. 



493. Now, by the preceding calculation we have found the stability 

 to be 14.19 very nearly, while the depth is 19| feet, and the inclina- 

 tion from the upright position, 15 degrees, of which the natural sine is 

 0.25882; consequently, by substitution we obtain 



0.021576 3 24.66=14.19, 



and if this equation be reduced, we shall find the value of b or the 

 breadth of the vessel at the water line, to be a very small quantity in 



VOL. i. 2 D 



