36Z Astronomical and Nautical Collections. 



The first three lines may begin from any given points ; the fourth 

 must be so placed that when B 28 and H 28 are brought together, 

 L 30 may stand exactly against C 80. 



Mode of computing the content of a Cask from the wake. 



The wake is the drop of the bung below the cone touching the cask 

 at the head, or half the difference between the bung diameter and 

 that of the base of a cone in which the half cask is inscribed, so as 

 to touch it exactly at the head. This element may be measured 

 without much difficulty by means of a straight rod, with two fixed 

 nails, of equal length, projecting from it near one end, and a third 

 nail sliding along it, so as to stand over the bung when the former 

 two are pressed down upon the stave between the hoops at the 

 head, while the distance of its point from the bung is measured by 

 a scale, or by a pair of compasses. 



The direction of the surface of the staves being given in three 

 given points through which it passes, we shall only have to as- 

 sume that the curvature varies in a uniform manner, from its 

 greatest magnitude at the bung, to its least magnitude at the head, 

 in order to obtain a form which must very nearly coincide with the 

 whole outline of the staves. The most convenient supposition re- 

 specting the curve is that it is of the nature of a parabola, either 

 of an order inferior to the common parabola, and beginning at the 

 bung, or of a higher order , beginning at the head, and meeting its 

 companion at the bung in a direction parallel to the axis ; and the 

 latter form will be found, on examination, to be the most applica- 

 ble to practical cases, the former approaching too much to a cone. 



Now in all parabolas, when the ordinate is ax H , the distance of 

 the tangent from the curve, on the axis, or, in other words, the 

 wake of the cask, is (?i — 1) ax" ; since the fluxion of the ordinate 

 is wax" -1 dx, and, as dx is to this, so is the absciss x to nax", the 

 sum of the ordinate, and of the distance in question ; and making 



this distance = It, the ordinate bein^ here we have ax n — : 



° 2 2 



2k 



while (» — 1) ax* — k; consequently »i- 1 r . ; or, if b —k 



b'l 



