SBCT . x .] OF STEAM NAVIGATION. 291 



604. If the form be a cubic parabola, then n = 3, and 



8 = b s ' m - l (b- - 3-6 *), and A = *JLL. 



605. If the form be a parabola where p x = y*, as in Fig. 3. Plate xvi. then, 



- *' 



(b* - 4-3 a 2 ), and A = 



5 a b 



606. The stability and capacity both increase as the ordinate of the parabola 

 becomes of a higher power, but a greater breadth is necessary in proportion to the 

 depth. 



607. When the ordinates are parallel to the breadth, 



Jy d x ( y - cos. i + (a x) sin. i\ Jy dx [~- cos. * (a x) sin. i J 



r 2 n" a- y sin. i 



= 2 am. ,Jyd* (a -*)-= (ll + 1)2 , +1)' 



and, 



_ b 3 sin, f 2 2 a 2 b sin, i _ bjiin.i f Js _ 12 n 2 o* \ 



12 ' ( + l)(2n + 1) ~~12 \ ~ ( + 1)(2 + !)/' 



The capacity is ~^ as before ; 



and in the case of the triangle we have the same results. 



608. But if the form be a common parabola, or n = 2, then 



S = *^l (b'- - 3-2 a 2 ), and A = ^f* 

 \~ o 



609. If the form of the parabola be p x = y 5 , as in Fig. 4. Plate xvi. then, 



S = ill (J 2 - 4-5 a*), and A = ill. 



610. This species of figure may be easily traced through all the varieties of 

 form, and it is so easy to compute its capacity and to describe it by ordinates, 

 that it is much to be preferred to the elliptical figures which foreign writers have 

 chosen for calculation. * The breadth should be every where in the same ratio 

 to the depth, to render the stability equal throughout the length, or so that the 

 vessel may undergo no strain from change of position. * 



i For a mode of describing curves of this kind, see my ' Principles of Carpentry,' Sect. i. 

 art. 58. 



4 For other methods see Bossut's Hydrodynamique, torn. i. chap. xiii. et xiv ; or Poisson's 

 ait de Mecanique, torn. ii. p. 389. 



