Astronomical and Nautical Collections. 363 



= d, n — 1 — — , and n sz 1 + — = On the other 



d d d 



hand, for a parabola having its vertex at the head, the wake be- 

 comes the ordinate, and — assumes its place, so that w, or rather m, 



•ii u i i d d + 2k 

 will be 1 + — =r —L 



2k 2k 



For the inferior parabola, the diameter of the cask, at the dis- 

 tance x from the middle, is always b — ax" ; its square 6 3 — 2abx n 

 + a s x Zn , and the content, considering the section as a square, 



2 1 I 



bx — abx" +l + a 9 x 9 " +1 ; but when x as — , ax n 



n+L 2n + 1 2 



= a f — ] = d, and we have, for the whole content, 2 (b* — — 



_JL bd L + — \ d L \ - I (6* - JL W+_i — d«) 



w+ 1 2 2n + I 2 / v w+ 1 2» + 1 



= Z (6« - -A- 6d + — ^ — d»). 

 rf+A 4£ + 3 ' 



For the superior parabola, the diameter, at the distance x from 

 the head, is h + ex — ex m ; c being such, that c — may be equal to 



d + 2k, and el — j being = 2k. Then the square of the dia- 

 meter will be A 2 + 2chx + c 2 j: 2 - 2ehx m - 2cex m+x + eV, and 



12 *> 



the content A 2 x + chx 2 + — cV — ehx m+i — 



3 m + 1 m + 2 



+ 



c m+2 ^. eV"'+' ; which, for the whole length, becomes 



l(h> + —chl + L C H> - 2 eh ( L V" _ ? C e 



2 12 m + 1 \ 2 / »» + 2 



f _L Y' + ' + \ e * ( L Y" = I ( h 2 + ( d + 2k) h + i- 



\2/ 2« + l \2/ J 3 



$ + 2ky - _L_^ M _ *_. (d + 2A) A + jrrirj-tys 

 w+1 m+2 2?«-fl 



