MATHEMATICS.-MENSURATION. 



OF A CASK. 



16. To find Ihf Volume of a portion 



by a plane Perpendicular to the Axis of Revolution. 



ID figure 50, suppose we wish to find the volume of 

 the portion of the spheroid corresponding to the portion 

 A P N of the generating ellipse ; then, as in Art. 16, we 

 hall have 



Portion of Spheroid : portion of sphere : : 6* : a* 

 Now (Mensuration of Solids, Art 14), if A N = fc the 



volume of the portion of the sphere is equal to - (3a - h.) 



9 

 .'. If V t is the volume required, 



Similarly, if we wUh to find the volume of the portion 

 corresponding to N P B, and if V 2 is this volume, and 

 A-ON, 



as is evident from CoroL 2, Art. 14, Mensuration of 

 Solids. 



17. To find the Volume of a Cask. 



We may consider a cask to be either the middle portion 

 of a spheroid, or two frustums of equal cones joined 

 together at their bases, though it will not coincide 

 with either of these forms exactly. It is to be observed 

 that excise officers generally consider casks to be of the 

 first form. 



The measurements that are most easily made in prac- 

 tice are the diameters of the end, and of the middle 

 section, and the distance between the ends ; wo will call 

 these il, D, and k, respectively, and investigate the rule 

 in each of the above cases. 



(a). Suppose the cask to bo a portion of a spheroid. 

 Its volume V will be double that of V a in the last article 



Also by a property of the ellipse, (See Fig. 50), 



_ , 

 OA"" 

 .'.PN* A 



TT ? "T 



/. 1 PN* , 1 A J 1 



r ~w + 3 ? = s * 



.'.2 1 PN 1 



3 + 3'~6~ = 1 3 



~ 



2. 



=.fc. 24 -D 



Also ,n ~ -2018 very nearly, hence the rule. " To 



twice the square of the middle diameter add the square 

 of the end diameter, and multiply the sum by the length 

 of the cask ; this product multiplied by .2G18 gives the 

 contents of the cask." 



(6). Suppose the cask to have the form of a double 

 frustum of a cone. 



If v be the volume required, v is clearly double of the 



volume A BCD, Fig. 48. 

 Mensuration of Solids, 



And hence by Article 13, 



NowD = 2r,, d = 2r and 



2Ji 



-jg. k (D+Dd+<P). 



Hence the rule, "To the product of the diameters add 

 the sum of their squares ; multiply this by the length of 

 the cask, then the whole product multiplied by .2018 

 gives the contents of the cask." 



N. B. If the measurements are made in inches, the 

 above rules give the required contents in cubic inches ; 

 to obtain the contents in gallons wo must divide by 

 277.274, since 277-274 cubic inches go to one gallon. 



COR. It is evident that v, the value given by the 

 second rule, is less than the true contents of the ciiak : 

 it is to be observed, also, that V, the value given by the 

 former rule, is generally greater than the true value ; so 

 that the true value will lie somewhere between these two 

 results. Hence we can easily estimate the amount of 

 accuracy in each of the above determinations. Thus, 

 V v is clearly greater than the difference between the 

 true result, and either of those given by rule, and t> is 

 less than the true value of the contents ; hence the error 

 committed by either way of making the calculation 



cannot be so great as the [ ] of the whole. Now 

 V * I 





D(D < 



2 Dn + n". 



= D 2 Dn, andcP=D J 



:; l) 



.'. V p Dn ^ 



3D 2 Sbn+n**". n , n" ' 



D^SD* 



which is an expression for a limit of the part of the whole, 

 by the approximate, which differs from the true value. 

 e.g., Suppose the diameters to be 18 and 20 inches 

 respectively, 



then n =- 2 inches and = , and there the error com- 

 mitted by calculating according to either rule cannot be 



so much as 



30 



of the whole, or so much as 



1 A_u_ 

 10^300 



* or" 1 (very nearly) of the whole. If the cask in 

 271 *7 



uestion had an interior length of two feet, then, by the 

 first rule, its contents are 25 '47 gallons, and, by the 

 second, its contents are 24. 50 gallons. So that the error 

 committed by either way of gauging must be less than 

 one gallon. 



